It is often convenient to treat a matrix or vector as a collection of individual blocks. For example, in step-20 (and other tutorial programs), we want to consider the global linear system in the form
-
+ \end{eqnarray*}" src="form_92.png"/>
-
where are the values of velocity and pressure degrees of freedom, respectively, is the mass matrix on the velocity space, corresponds to the negative divergence operator, and is its transpose and corresponds to the negative gradient.
+
where are the values of velocity and pressure degrees of freedom, respectively, is the mass matrix on the velocity space, corresponds to the negative divergence operator, and is its transpose and corresponds to the negative gradient.
Using such a decomposition into blocks, one can then define preconditioners that are based on the individual operators that are present in a system of equations (for example the Schur complement, in the case of step-20), rather than the entire matrix. In essence, blocks are used to reflect the structure of a PDE system in linear algebra, in particular allowing for modular solvers for problems with multiple solution components. On the other hand, the matrix and right hand side vector can also treated as a unit, which is convenient for example during assembly of the linear system when one may not want to make a distinction between the individual components, or for an outer Krylov space solver that doesn't care about the block structure (e.g. if only the preconditioner needs the block structure).
Splitting matrices and vectors into blocks is supported by the BlockSparseMatrix, BlockVector, and related classes. See the overview of the various linear algebra classes in the Linear algebra classes module. The objects present two interfaces: one that makes the object look like a matrix or vector with global indexing operations, and one that makes the object look like a collection of sub-blocks that can be individually addressed. Depending on context, one may wish to use one or the other interface.
Typically, one defines the sub-structure of a matrix or vector by grouping the degrees of freedom that make up groups of physical quantities (for example all velocities) into individual blocks of the linear system. This is defined in more detail below in the glossary entry on Block (finite element).
With the exception of the number of blocks, the two objects are the same for all practical purposes, however.
Global degrees of freedom: While we have defined blocks above in terms of the vector components of a vector-valued solution function (or, equivalently, in terms of the vector-valued finite element space), every shape function of a finite element is part of one block or another. Consequently, we can partition all degrees of freedom defined on a DoFHandler into individual blocks. Since by default the DoFHandler class enumerates degrees of freedom in a more or less random way, you will first want to call the DoFRenumbering::component_wise function to make sure that all degrees of freedom that correspond to a single block are enumerated consecutively.
-
If you do this, you naturally partition matrices and vectors into blocks as well (see block (linear algebra)). In most cases, when you subdivide a matrix or vector into blocks, you do so by creating one block for each block defined by the finite element (i.e. in most practical cases the FESystem object). However, this needs not be so: the DoFRenumbering::component_wise function allows to group several vector components or finite element blocks into the same logical block (see, for example, the step-22 or step-31 tutorial programs, as opposed to step-20). As a consequence, using this feature, we can achieve the same result, i.e. subdividing matrices into blocks and vectors into 2 blocks, for the second way of creating a Stokes element outlined above using an extra argument as we would have using the first way of creating the Stokes element with two blocks right away.
+
If you do this, you naturally partition matrices and vectors into blocks as well (see block (linear algebra)). In most cases, when you subdivide a matrix or vector into blocks, you do so by creating one block for each block defined by the finite element (i.e. in most practical cases the FESystem object). However, this needs not be so: the DoFRenumbering::component_wise function allows to group several vector components or finite element blocks into the same logical block (see, for example, the step-22 or step-31 tutorial programs, as opposed to step-20). As a consequence, using this feature, we can achieve the same result, i.e. subdividing matrices into blocks and vectors into 2 blocks, for the second way of creating a Stokes element outlined above using an extra argument as we would have using the first way of creating the Stokes element with two blocks right away.
More information on this topic can be found in the documentation of FESystem, the Handling vector valued problems module and the tutorial programs referenced therein.
Selecting blocks: Many functions allow you to restrict their operation to certain vector components or blocks. For example, this is the case for the functions that interpolate boundary values: one may want to only interpolate the boundary values for the velocity block of a finite element field but not the pressure block. The way to do this is by passing a BlockMask argument to such functions, see the block mask entry of this glossary.
@@ -164,14 +164,14 @@
Boundary form
For a dim-dimensional triangulation in dim-dimensional space, the boundary form is a vector defined on faces. It is the vector product of the image of coordinate vectors on the surface of the unit cell. It is a vector normal to the surface, pointing outwards and having the length of the surface element.
-
A more general definition would be that (at least up to the length of this vector) it is exactly that vector that is necessary when considering integration by parts, i.e. equalities of the form . Using this definition then also explains what this vector should be in the case of domains (and corresponding triangulations) of dimension dim that are embedded in a space spacedim: in that case, the boundary form is still a vector defined on the faces of the triangulation; it is orthogonal to all tangent directions of the boundary and within the tangent plane of the domain. Note that this is compatible with case dim==spacedim since there the tangent plane is the entire space .
+
A more general definition would be that (at least up to the length of this vector) it is exactly that vector that is necessary when considering integration by parts, i.e. equalities of the form . Using this definition then also explains what this vector should be in the case of domains (and corresponding triangulations) of dimension dim that are embedded in a space spacedim: in that case, the boundary form is still a vector defined on the faces of the triangulation; it is orthogonal to all tangent directions of the boundary and within the tangent plane of the domain. Note that this is compatible with case dim==spacedim since there the tangent plane is the entire space .
In either case, the length of the vector equals the determinant of the transformation of reference face to the face of the current cell.
Boundary indicator
In a Triangulation object, every part of the boundary may be associated with a unique number (of type types::boundary_id) that is used to determine what kinds of boundary conditions are to be applied to a particular part of a boundary. The boundary is composed of the faces of the cells and, in 3d, the edges of these faces.
-
By default, all boundary indicators of a mesh are zero, unless you are reading from a mesh file that specifically sets them to something different, or unless you use one of the mesh generation functions in namespace GridGenerator that have a colorize option. A typical piece of code that sets the boundary indicator on part of the boundary to something else would look like this, here setting the boundary indicator to 42 for all faces located at :
for (auto &face : triangulation.active_face_iterators())
+
By default, all boundary indicators of a mesh are zero, unless you are reading from a mesh file that specifically sets them to something different, or unless you use one of the mesh generation functions in namespace GridGenerator that have a colorize option. A typical piece of code that sets the boundary indicator on part of the boundary to something else would look like this, here setting the boundary indicator to 42 for all faces located at :
for (auto &face : triangulation.active_face_iterators())
When considering systems of equations in which the solution is not just a single scalar function, we say that we have a vector system with a vector-valued solution. For example, the vector solution in the elasticity equation considered in step-8 is consisting of the displacements in each of the three coordinate directions. The solution then has three elements. Similarly, the 3d Stokes equation considered in step-22 has four elements: . We call the elements of the vector-valued solution components in deal.II. To be well-posed, for the solution to have components, there need to be partial differential equations to describe them. This concept is discussed in great detail in the Handling vector valued problems module.
+
When considering systems of equations in which the solution is not just a single scalar function, we say that we have a vector system with a vector-valued solution. For example, the vector solution in the elasticity equation considered in step-8 is consisting of the displacements in each of the three coordinate directions. The solution then has three elements. Similarly, the 3d Stokes equation considered in step-22 has four elements: . We call the elements of the vector-valued solution components in deal.II. To be well-posed, for the solution to have components, there need to be partial differential equations to describe them. This concept is discussed in great detail in the Handling vector valued problems module.
In finite element programs, one frequently wants to address individual elements (components) of this vector-valued solution, or sets of components. For example, we do this extensively in step-8, and a lot of documentation is also provided in the module on Handling vector valued problems. If you are thinking only in terms of the partial differential equation (not in terms of its discretization), then the concept of components is the natural one.
On the other hand, when talking about finite elements and degrees of freedom, components are not always the correct concept because components are not always individually addressable. In particular, this is the case for non-primitive finite elements. Similarly, one may not always want to address individual components but rather sets of components — e.g. all velocity components together, and separate from the pressure in the Stokes system, without further splitting the velocities into their individual components. In either case, the correct concept to think in is that of a block. Since each component, if individually addressable, is also a block, thinking in terms of blocks is most frequently the better strategy.
would result in a mask [true, true, false] in 2d. Of course, in 3d, the result would be [true, true, true, false].
Note
Just as one can think of composed elements as being made up of components or blocks, there are component masks (represented by the ComponentMask class) and block masks (represented by the BlockMask class). The FiniteElement class has functions that convert between the two kinds of objects.
-Not all component masks actually make sense. For example, if you have a FE_RaviartThomas object in 2d, then it doesn't make any sense to have a component mask of the form [true, false] because you try to select individual vector components of a finite element where each shape function has both and velocities. In essence, while you can of course create such a component mask, there is nothing you can do with it.
+Not all component masks actually make sense. For example, if you have a FE_RaviartThomas object in 2d, then it doesn't make any sense to have a component mask of the form [true, false] because you try to select individual vector components of a finite element where each shape function has both and velocities. In essence, while you can of course create such a component mask, there is nothing you can do with it.
Compressing distributed vectors and matrices
For parallel computations, deal.II uses the vector and matrix classes defined in the PETScWrappers and TrilinosWrappers namespaces. When running programs in parallel using MPI, these classes only store a certain number of rows or elements on the current processor, whereas the rest of the vector or matrix is stored on the other processors that belong to our MPI universe. This presents a certain problem when you assemble linear systems: we add elements to the matrix and right hand side vectors that may or may not be stored locally. Sometimes, we may also want to just set an element, not add to it.
@@ -304,9 +304,9 @@
Degree of freedom
-
The term "degree of freedom" (often abbreviated as "DoF") is commonly used in the finite element community to indicate two slightly different, but related things. The first is that we'd like to represent the finite element solution as a linear combination of shape functions, in the form . Here, is a vector of expansion coefficients. Because we don't know their values yet (we will compute them as the solution of a linear or nonlinear system), they are called "unknowns" or "degrees of freedom". The second meaning of the term can be explained as follows: A mathematical description of finite element problem is often to say that we are looking for a finite dimensional function that satisfies some set of equations (e.g. for all test functions ). In other words, all we say here that the solution needs to lie in some space . However, to actually solve this problem on a computer we need to choose a basis of this space; this is the set of shape functions we have used above in the expansion of with coefficients . There are of course many bases of the space , but we will specifically choose the one that is described by the finite element functions that are traditionally defined locally on the cells of the mesh. Describing "degrees of freedom" in this context requires us to simply enumerate the basis functions of the space . For elements this means simply enumerating the vertices of the mesh in some way, but for higher elements one also has to enumerate the shape functions that are associated with edges, faces, or cell interiors of the mesh. The class that provides this enumeration of the basis functions of is called DoFHandler. The process of enumerating degrees of freedom is referred to as "distributing DoFs" in deal.II.
+
The term "degree of freedom" (often abbreviated as "DoF") is commonly used in the finite element community to indicate two slightly different, but related things. The first is that we'd like to represent the finite element solution as a linear combination of shape functions, in the form . Here, is a vector of expansion coefficients. Because we don't know their values yet (we will compute them as the solution of a linear or nonlinear system), they are called "unknowns" or "degrees of freedom". The second meaning of the term can be explained as follows: A mathematical description of finite element problem is often to say that we are looking for a finite dimensional function that satisfies some set of equations (e.g. for all test functions ). In other words, all we say here that the solution needs to lie in some space . However, to actually solve this problem on a computer we need to choose a basis of this space; this is the set of shape functions we have used above in the expansion of with coefficients . There are of course many bases of the space , but we will specifically choose the one that is described by the finite element functions that are traditionally defined locally on the cells of the mesh. Describing "degrees of freedom" in this context requires us to simply enumerate the basis functions of the space . For elements this means simply enumerating the vertices of the mesh in some way, but for higher elements one also has to enumerate the shape functions that are associated with edges, faces, or cell interiors of the mesh. The class that provides this enumeration of the basis functions of is called DoFHandler. The process of enumerating degrees of freedom is referred to as "distributing DoFs" in deal.II.
Direction flags
@@ -327,7 +327,7 @@
Distorted cells
A distorted cell is a cell for which the mapping from the reference cell to real cell has a Jacobian whose determinant is non-positive somewhere in the cell. Typically, we only check the sign of this determinant at the vertices of the cell. The function GeometryInfo::alternating_form_at_vertices computes these determinants at the vertices.
-
By way of example, if all of the determinants are of roughly equal value and on the order of then the cell is well-shaped. For example, a square cell or face has determinants equal to whereas a strongly sheared parallelogram has a determinant much smaller. Similarly, a cell with very unequal edge lengths will have widely varying determinants. Conversely, a pinched cell in which the location of two or more vertices is collapsed to a single point has a zero determinant at this location. Finally, an inverted or twisted cell in which the location of two vertices is out of order will have negative determinants.
+
By way of example, if all of the determinants are of roughly equal value and on the order of then the cell is well-shaped. For example, a square cell or face has determinants equal to whereas a strongly sheared parallelogram has a determinant much smaller. Similarly, a cell with very unequal edge lengths will have widely varying determinants. Conversely, a pinched cell in which the location of two or more vertices is collapsed to a single point has a zero determinant at this location. Finally, an inverted or twisted cell in which the location of two vertices is out of order will have negative determinants.
The following two images show a well-formed, a pinched, and a twisted cell for both 2d and 3d:
@@ -366,19 +366,19 @@
Generalized support points
-
"Generalized support points" are, as the name suggests, a generalization of support points. The latter are used to describe that a finite element simply interpolates values at individual points (the "support points"). If we call these points (where the hat indicates that these points are defined on the reference cell ), then one typically defines shape functions in such a way that the nodal functionals simply evaluate the function at the support point, i.e., that , and the basis is chosen so that where is the Kronecker delta function. This leads to the common Lagrange elements.
-
(In the vector valued case, the only other piece of information besides the support points that one needs to provide is the vector component the th node functional corresponds, so that .)
-
On the other hand, there are other kinds of elements that are not defined this way. For example, for the lowest order Raviart-Thomas element (see the FE_RaviartThomas class), the node functional evaluates not individual components of a vector-valued finite element function with dim components, but the normal component of this vector: "Generalized support points" are, as the name suggests, a generalization of support points. The latter are used to describe that a finite element simply interpolates values at individual points (the "support points"). If we call these points (where the hat indicates that these points are defined on the reference cell ), then one typically defines shape functions in such a way that the nodal functionals simply evaluate the function at the support point, i.e., that , and the basis is chosen so that where is the Kronecker delta function. This leads to the common Lagrange elements.
+
(In the vector valued case, the only other piece of information besides the support points that one needs to provide is the vector component the th node functional corresponds, so that .)
+
On the other hand, there are other kinds of elements that are not defined this way. For example, for the lowest order Raviart-Thomas element (see the FE_RaviartThomas class), the node functional evaluates not individual components of a vector-valued finite element function with dim components, but the normal component of this vector: , where the are the normal vectors to the face of the cell on which is located. In other words, the node functional is a linear combination of the components of when evaluated at . Similar things happen for the BDM, ABF, and Nedelec elements (see the FE_BDM, FE_ABF, FE_Nedelec classes).
-
In these cases, the element does not have support points because it is not purely interpolatory; however, some kind of interpolation is still involved when defining shape functions as the node functionals still require point evaluations at special points . In these cases, we call the points generalized support points.
-
Finally, there are elements that still do not fit into this scheme. For example, some hierarchical basis functions (see, for example the FE_Q_Hierarchical element) are defined so that the node functionals are moments of finite element functions, , where the are the normal vectors to the face of the cell on which is located. In other words, the node functional is a linear combination of the components of when evaluated at . Similar things happen for the BDM, ABF, and Nedelec elements (see the FE_BDM, FE_ABF, FE_Nedelec classes).
+
In these cases, the element does not have support points because it is not purely interpolatory; however, some kind of interpolation is still involved when defining shape functions as the node functionals still require point evaluations at special points . In these cases, we call the points generalized support points.
+
Finally, there are elements that still do not fit into this scheme. For example, some hierarchical basis functions (see, for example the FE_Q_Hierarchical element) are defined so that the node functionals are moments of finite element functions, in 2d, and similarly for 3d, where the are the order of the moment described by shape function . Some other elements use moments over edges or faces. In all of these cases, node functionals are not defined through interpolation at all, and these elements then have neither support points, nor generalized support points.
+ $" src="form_124.png"/> in 2d, and similarly for 3d, where the are the order of the moment described by shape function . Some other elements use moments over edges or faces. In all of these cases, node functionals are not defined through interpolation at all, and these elements then have neither support points, nor generalized support points.
It frequently appears in the solution of time dependent problems where, if one uses an explicit time stepping method, it then leads to the need to solve problems of the form
-
+ \end{align*}" src="form_127.png"/>
-
in time step , where is the solution to be computed, is the known solution from the first time step, and is a matrix related to the differential operator in the PDE. is the size of the time step. A similar linear system of equations also arises out of the discretization of second-order differential equations.
-
The presence of the matrix on the left side is a nuisance because, even though we have used an explicit time stepping method, we still have to solve a linear system in each time step. It would be much preferable if the matrix were diagonal. "Lumping" the mass matrix is a strategy to replace by a matrix that actually is diagonal, yet does not destroy the accuracy of the resulting solution.
-
Historically, mass lumping was performed by adding the elements of a row together and setting the diagonal entries of to that sum. This works for and elements, for example, and can be understood mechanically by replacing the continuous medium we are discretizating by one where the continuous mass distribution is replaced by one where (finite amounts of) mass are located only at the nodes. That is, we are "lumping together" the mass of an element at its vertices, thus giving rise to the name "lumped mass matrix". A more mathematical perspective is to compute the integral above for via special quadrature rules; in particular, we replace the computation of
-, where is the solution to be computed, is the known solution from the first time step, and is a matrix related to the differential operator in the PDE. is the size of the time step. A similar linear system of equations also arises out of the discretization of second-order differential equations.
+
The presence of the matrix on the left side is a nuisance because, even though we have used an explicit time stepping method, we still have to solve a linear system in each time step. It would be much preferable if the matrix were diagonal. "Lumping" the mass matrix is a strategy to replace by a matrix that actually is diagonal, yet does not destroy the accuracy of the resulting solution.
+
Historically, mass lumping was performed by adding the elements of a row together and setting the diagonal entries of to that sum. This works for and elements, for example, and can be understood mechanically by replacing the continuous medium we are discretizating by one where the continuous mass distribution is replaced by one where (finite amounts of) mass are located only at the nodes. That is, we are "lumping together" the mass of an element at its vertices, thus giving rise to the name "lumped mass matrix". A more mathematical perspective is to compute the integral above for via special quadrature rules; in particular, we replace the computation of
+
+ \end{align*}" src="form_134.png"/>
by quadrature
-
+ \end{align*}" src="form_135.png"/>
where we choose the quadrature points as the nodes at which the shape functions are defined. If we order the quadrature points in the same way as the shape functions, then
-
+ \end{align*}" src="form_136.png"/>
and consequently
-
+ \end{align*}" src="form_137.png"/>
-
where the sum extends over those cells on which is nonzero. The so-computed mass matrix is therefore diagonal.
-
Whether or not this particular choice of quadrature formula is sufficient to retain the convergence rate of the discretization is a separate question. For the usual finite elements (implemented by FE_Q and FE_DGQ), the appropriate quadrature formulas are of QGaussLobatto type. Mass lumping can also be done with FE_SimplexP_Bubbles, for example, if appropriate quadrature rules are chosen.
+
where the sum extends over those cells on which is nonzero. The so-computed mass matrix is therefore diagonal.
+
Whether or not this particular choice of quadrature formula is sufficient to retain the convergence rate of the discretization is a separate question. For the usual finite elements (implemented by FE_Q and FE_DGQ), the appropriate quadrature formulas are of QGaussLobatto type. Mass lumping can also be done with FE_SimplexP_Bubbles, for example, if appropriate quadrature rules are chosen.
For an example of where lumped mass matrices play a role, see step-69.
Manifold indicator
Every object that makes up a Triangulation (cells, faces, edges, etc.), is associated with a unique number (of type types::manifold_id) that is used to identify which manifold object is responsible to generate new points when the mesh is refined.
-
By default, all manifold indicators of a mesh are set to numbers::flat_manifold_id. A typical piece of code that sets the manifold indicator on a object to something else would look like this, here setting the manifold indicator to 42 for all cells whose center has an component less than zero:
+
By default, all manifold indicators of a mesh are set to numbers::flat_manifold_id. A typical piece of code that sets the manifold indicator on a object to something else would look like this, here setting the manifold indicator to 42 for all cells whose center has an component less than zero:
for (auto &cell : triangulation.active_cell_iterators())
possibly with a coefficient inside the integral, and where are the shape functions of a finite element. The origin of the term refers to the fact that in structural mechanics (where the finite element method originated), one often starts from the elastodynamics (wave) equation
- are the shape functions of a finite element. The origin of the term refers to the fact that in structural mechanics (where the finite element method originated), one often starts from the elastodynamics (wave) equation
+
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@@ -340,7 +340,7 @@
Improved: The FEValuesViews objects that one gets when writing things like fe_values[velocities] (see Handling vector valued problems) have become a lot smarter. They now compute a significant amount of data at creation time, rather than on the fly. This means that creating such objects becomes more expensive but using them is cheaper. To offset this cost, FEValuesBase objects now create all possible FEValuesViews objects at creation time, rather than whenever you do things like fe_values[velocities], and simply return a reference to a pre-generated object. This turns an effort into an effort, where is the number of cells.
+
Improved: The FEValuesViews objects that one gets when writing things like fe_values[velocities] (see Handling vector valued problems) have become a lot smarter. They now compute a significant amount of data at creation time, rather than on the fly. This means that creating such objects becomes more expensive but using them is cheaper. To offset this cost, FEValuesBase objects now create all possible FEValuesViews objects at creation time, rather than whenever you do things like fe_values[velocities], and simply return a reference to a pre-generated object. This turns an effort into an effort, where is the number of cells.
(WB 2008/12/10)
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@@ -501,7 +501,7 @@
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@@ -839,7 +839,7 @@
-
New: There is now a new class Functions::InterpolatedTensorProductGridData that can be used to (bi-/tri-)linearly interpolate data given on a tensor product mesh of (and and ) values, for example to evaluate experimentally determined coefficients, or to assess the accuracy of a solution by comparing with a solution generated by a different code and written in gridded data. There is also a new class Functions::InterpolatedUniformGridData that can perform the same task more efficiently if the data is stored on meshes that are uniform in each coordinate direction.
+
New: There is now a new class Functions::InterpolatedTensorProductGridData that can be used to (bi-/tri-)linearly interpolate data given on a tensor product mesh of (and and ) values, for example to evaluate experimentally determined coefficients, or to assess the accuracy of a solution by comparing with a solution generated by a different code and written in gridded data. There is also a new class Functions::InterpolatedUniformGridData that can perform the same task more efficiently if the data is stored on meshes that are uniform in each coordinate direction.
(Wolfgang Bangerth, 2013/12/20)
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@@ -518,7 +518,7 @@
-
Fixed: The FE_ABF class reported the maximal polynomial degree (via FiniteElement::degree) for elements of order as , but this is wrong. It should be (see Section 5 of the original paper of Arnold, Boffi, and Falk). This is now fixed.
+
Fixed: The FE_ABF class reported the maximal polynomial degree (via FiniteElement::degree) for elements of order as , but this is wrong. It should be (see Section 5 of the original paper of Arnold, Boffi, and Falk). This is now fixed.
(Wolfgang Bangerth, 2017/01/13)
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@@ -608,7 +608,7 @@
-
Improved: GridGenerator::hyper_shell() in 3d now supports more n_cells options. While previously only 6, 12, or 96 cells were possible, the function now supports any number of the kind with a non-negative integer. The new cases and correspond to refinement in the azimuthal direction of the 6 or 12 cell case with a single mesh layer in radial direction, and are intended for shells that are thin and should be given more resolution in azimuthal direction.
+
Improved: GridGenerator::hyper_shell() in 3d now supports more n_cells options. While previously only 6, 12, or 96 cells were possible, the function now supports any number of the kind with a non-negative integer. The new cases and correspond to refinement in the azimuthal direction of the 6 or 12 cell case with a single mesh layer in radial direction, and are intended for shells that are thin and should be given more resolution in azimuthal direction.
(Martin Kronbichler, 2020/04/07)
@@ -1562,7 +1562,7 @@
-
Improved: The additional roots of the HermiteLikeInterpolation with degree greater than four have been switched to the roots of the Jacobi polynomial , making the interior bubble functions orthogonal and improving the conditioning of interpolation slightly.
+
Improved: The additional roots of the HermiteLikeInterpolation with degree greater than four have been switched to the roots of the Jacobi polynomial , making the interior bubble functions orthogonal and improving the conditioning of interpolation slightly.
(Martin Kronbichler, 2019/07/12)
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@@ -358,9 +358,9 @@
The algorithms used in the implementation of this class are described in some detail in the hp-paper. There is also a significant amount of documentation on how to use this class in the Constraints on degrees of freedom module.
Description of constraints
Each "line" in objects of this class corresponds to one constrained degree of freedom, with the number of the line being i, entered by using add_line() or add_lines(). The entries in this line are pairs of the form (j,aij), which are added by add_entry() or add_entries(). The organization is essentially a SparsityPattern, but with only a few lines containing nonzero elements, and therefore no data wasted on the others. For each line, which has been added by the mechanism above, an elimination of the constrained degree of freedom of the form
-
+\]" src="form_1577.png"/>
is performed, where bi is optional and set by set_inhomogeneity(). Thus, if a constraint is formulated for instance as a zero mean value of several degrees of freedom, one of the degrees has to be chosen to be eliminated.
Note that the constraints are linear in the xi, and that there might be a constant (non-homogeneous) term in the constraint. This is exactly the form we need for hanging node constraints, where we need to constrain one degree of freedom in terms of others. There are other conditions of this form possible, for example for implementing mean value conditions as is done in the step-11 tutorial program. The name of the class stems from the fact that these constraints can be represented in matrix form as Xx = b, and this object then describes the matrix X and the vector b. The most frequent way to create/fill objects of this type is using the DoFTools::make_hanging_node_constraints() function. The use of these objects is first explained in step-6.
@@ -914,13 +914,13 @@
-
Add an entry to a given line. In other words, this function adds a term to the constraints for the th degree of freedom.
+
Add an entry to a given line. In other words, this function adds a term to the constraints for the th degree of freedom.
If an entry with the same indices as the one this function call denotes already exists, then this function simply returns provided that the value of the entry is the same. Thus, it does no harm to enter a constraint twice.
Parameters
-
[in]
constrained_dof_index
The index of the degree of freedom that is being constrained.
-
[in]
column
The index of the degree of freedom being entered into the constraint for degree of freedom .
-
[in]
weight
The factor that multiplies .
+
[in]
constrained_dof_index
The index of the degree of freedom that is being constrained.
+
[in]
column
The index of the degree of freedom being entered into the constraint for degree of freedom .
+
[in]
weight
The factor that multiplies .
@@ -981,11 +981,11 @@
-
Set an inhomogeneity to the constraint for a degree of freedom. In other words, it adds a constant to the constraint for degree of freedom . For this to work, you need to call add_line() first for the given degree of freedom.
+
Set an inhomogeneity to the constraint for a degree of freedom. In other words, it adds a constant to the constraint for degree of freedom . For this to work, you need to call add_line() first for the given degree of freedom.
Parameters
-
[in]
constrained_dof_index
The index of the degree of freedom that is being constrained.
-
[in]
value
The right hand side value for the constraint on the degree of freedom .
+
[in]
constrained_dof_index
The index of the degree of freedom that is being constrained.
+
[in]
value
The right hand side value for the constraint on the degree of freedom .
@@ -1013,9 +1013,9 @@
Close the filling of entries. Since the lines of a matrix of this type are usually filled in an arbitrary order and since we do not want to use associative constrainers to store the lines, we need to sort the lines and within the lines the columns before usage of the matrix. This is done through this function.
Also, zero entries are discarded, since they are not needed.
After closing, no more entries are accepted. If the object was already closed, then this function returns immediately.
-
This function also resolves chains of constraints. For example, degree of freedom 13 may be constrained to while degree of freedom 7 is itself constrained as . Then, the resolution will be that . Note, however, that cycles in this graph of constraints are not allowed, i.e., for example may not itself be constrained, directly or indirectly, to again.
+
This function also resolves chains of constraints. For example, degree of freedom 13 may be constrained to while degree of freedom 7 is itself constrained as . Then, the resolution will be that . Note, however, that cycles in this graph of constraints are not allowed, i.e., for example may not itself be constrained, directly or indirectly, to again.
@@ -1445,9 +1445,9 @@
Print the constraints represented by the current object to the given stream.
For each constraint of the form
-
+\]" src="form_1586.png"/>
this function will write a sequence of lines that look like this:
42 2 : 0.5
42 14 : 0.25
@@ -2025,7 +2025,7 @@
This function takes a matrix of local contributions (local_matrix) corresponding to the degrees of freedom indices given in local_dof_indices and distributes them to the global matrix. In other words, this function implements a scatter operation. In most cases, these local contributions will be the result of an integration over a cell or face of a cell. However, as long as local_matrix and local_dof_indices have the same number of elements, this function is happy with whatever it is given.
In contrast to the similar function in the DoFAccessor class, this function also takes care of constraints, i.e. if one of the elements of local_dof_indices belongs to a constrained node, then rather than writing the corresponding element of local_matrix into global_matrix, the element is distributed to the entries in the global matrix to which this particular degree of freedom is constrained.
-
With this scheme, we never write into rows or columns of constrained degrees of freedom. In order to make sure that the resulting matrix can still be inverted, we need to do something with the diagonal elements corresponding to constrained nodes. Thus, if a degree of freedom in local_dof_indices is constrained, we distribute the corresponding entries in the matrix, but also add the absolute value of the diagonal entry of the local matrix to the corresponding entry in the global matrix. Assuming the discretized operator is positive definite, this guarantees that the diagonal entry is always non-zero, positive, and of the same order of magnitude as the other entries of the matrix. On the other hand, when solving a source problem the exact value of the diagonal element is not important, since the value of the respective degree of freedom will be overwritten by the distribute() call later on anyway.
+
With this scheme, we never write into rows or columns of constrained degrees of freedom. In order to make sure that the resulting matrix can still be inverted, we need to do something with the diagonal elements corresponding to constrained nodes. Thus, if a degree of freedom in local_dof_indices is constrained, we distribute the corresponding entries in the matrix, but also add the absolute value of the diagonal entry of the local matrix to the corresponding entry in the global matrix. Assuming the discretized operator is positive definite, this guarantees that the diagonal entry is always non-zero, positive, and of the same order of magnitude as the other entries of the matrix. On the other hand, when solving a source problem the exact value of the diagonal element is not important, since the value of the respective degree of freedom will be overwritten by the distribute() call later on anyway.
Note
The procedure described above adds an unforeseeable number of artificial eigenvalues to the spectrum of the matrix. Therefore, it is recommended to use the equivalent function with two local index vectors in such a case.
By using this function to distribute local contributions to the global object, one saves the call to the condense function after the vectors and matrices are fully assembled.
Note
This function in itself is thread-safe, i.e., it works properly also when several threads call it simultaneously. However, the function call is only thread-safe if the underlying global matrix allows for simultaneous access and the access is not to rows with the same global index at the same time. This needs to be made sure from the caller's site. There is no locking mechanism inside this method to prevent data races.
@@ -2067,7 +2067,7 @@
This function does almost the same as the function above but can treat general rectangular matrices. The main difference to achieve this is that the diagonal entries in constrained rows are left untouched instead of being filled with arbitrary values.
-
Since the diagonal entries corresponding to eliminated degrees of freedom are not set, the result may have a zero eigenvalue, if applied to a square matrix. This has to be considered when solving the resulting problems. For solving a source problem , it is possible to set the diagonal entry after building the matrix by a piece of code of the form
+
Since the diagonal entries corresponding to eliminated degrees of freedom are not set, the result may have a zero eigenvalue, if applied to a square matrix. This has to be considered when solving the resulting problems. For solving a source problem , it is possible to set the diagonal entry after building the matrix by a piece of code of the form
for (unsignedint i=0;i<matrix.m();++i)
if (constraints.is_constrained(i))
matrix.diag_element(i) = 1.;
@@ -2356,7 +2356,7 @@
-
Given a vector, set all constrained degrees of freedom to values so that the constraints are satisfied. For example, if the current object stores the constraint , then this function will read the values of and from the given vector and set the element according to this constraints. Similarly, if the current object stores the constraint , then this function will set the 42nd element of the given vector to 208.
+
Given a vector, set all constrained degrees of freedom to values so that the constraints are satisfied. For example, if the current object stores the constraint , then this function will read the values of and from the given vector and set the element according to this constraints. Similarly, if the current object stores the constraint , then this function will set the 42nd element of the given vector to 208.
Note
If this function is called with a parallel vector vec, then the vector must not contain ghost elements.
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For fixed theta, the Crank-Nicolson scheme is the only second order scheme. Nevertheless, further stability may be achieved by choosing theta larger than ½, thereby introducing a first order error term. In order to avoid a loss of convergence order, the adaptive theta scheme can be used, where theta=½+c dt.
Assume that we want to solve the equation u' + F(u) = 0 with a step size k. A step of the theta scheme can be written as
-
+\]" src="form_351.png"/>
Here, M is the mass matrix. We see, that the right hand side amounts to an explicit Euler step with modified step size in weak form (up to inversion of M). The left hand side corresponds to an implicit Euler step with modified step size (right hand side given). Thus, the implementation of the theta scheme will use two Operator objects, one for the explicit, one for the implicit part. Each of these will use its own TimestepData to account for the modified step sizes (and different times if the problem is not autonomous). Note that once the explicit part has been computed, the left hand side actually constitutes a linear or nonlinear system which has to be solved.
Now we need to study the application of the implicit and explicit operator. We assume that the pointer matrix points to the matrix created in the main program (the constructor did this for us). Here, we first get the time step size from the AnyData object that was provided as input. Then, if we are in the first step or if the timestep has changed, we fill the local matrix , such that with the given matrix , it becomes
-
+
Now we need to study the application of the implicit and explicit operator. We assume that the pointer matrix points to the matrix created in the main program (the constructor did this for us). Here, we first get the time step size from the AnyData object that was provided as input. Then, if we are in the first step or if the timestep has changed, we fill the local matrix , such that with the given matrix , it becomes
+
After we have worked off the notifications, we clear them, such that the matrix is only generated when necessary.
The operator computing the explicit part of the scheme. This will receive in its input data the value at the current time with name "Current time solution". It should obtain the current time and time step size from explicit_data().
-
Its return value is , where is the current state vector, the mass matrix, the operator in space and is the adjusted time step size .
+
Its return value is , where is the current state vector, the mass matrix, the operator in space and is the adjusted time step size .
The operator solving the implicit part of the scheme. It will receive in its input data the vector "Previous time". Information on the timestep should be obtained from implicit_data().
-
Its return value is the solution u of Mu-cF(u)=f, where f is the dual space vector found in the "Previous time" entry of the input data, M the mass matrix, F the operator in space and c is the adjusted time step size
+
Its return value is the solution u of Mu-cF(u)=f, where f is the dual space vector found in the "Previous time" entry of the input data, M the mass matrix, F the operator in space and c is the adjusted time step size
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@@ -154,10 +154,10 @@
Detailed Description
template<int dim>
class AnisotropicPolynomials< dim >
Anisotropic tensor product of given polynomials.
-
Given one-dimensional polynomials in -direction, in -direction, and so on, this class generates polynomials of the form . (With obvious generalization if dim is in fact only 2. If dim is in fact only 1, then the result is simply the same set of one-dimensional polynomials passed to the constructor.)
-
If the elements of each set of base polynomials are mutually orthogonal on the interval or , then the tensor product polynomials are orthogonal on or , respectively.
-
The resulting dim-dimensional tensor product polynomials are ordered as follows: We iterate over the coordinates running fastest, then the coordinate, etc. For example, for dim==2, the first few polynomials are thus , , , ..., , , , etc.
+
Given one-dimensional polynomials in -direction, in -direction, and so on, this class generates polynomials of the form . (With obvious generalization if dim is in fact only 2. If dim is in fact only 1, then the result is simply the same set of one-dimensional polynomials passed to the constructor.)
+
If the elements of each set of base polynomials are mutually orthogonal on the interval or , then the tensor product polynomials are orthogonal on or , respectively.
+
The resulting dim-dimensional tensor product polynomials are ordered as follows: We iterate over the coordinates running fastest, then the coordinate, etc. For example, for dim==2, the first few polynomials are thus , , , ..., , , , etc.
Each tensor product polynomial is a product of one-dimensional polynomials in each space direction. Compute the indices of these one- dimensional polynomials for each space direction, given the index i.
+
Each tensor product polynomial is a product of one-dimensional polynomials in each space direction. Compute the indices of these one- dimensional polynomials for each space direction, given the index i.
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@@ -230,14 +230,14 @@
Detailed Description
Interface for using ARPACK. ARPACK is a collection of Fortran77 subroutines designed to solve large scale eigenvalue problems. Here we interface to the routines dnaupd and dneupd of ARPACK. If the operator is specified to be symmetric we use the symmetric interface dsaupd and dseupd of ARPACK instead. The package is designed to compute a few eigenvalues and corresponding eigenvectors of a general n by n matrix A. It is most appropriate for large sparse matrices A.
-
In this class we make use of the method applied to the generalized eigenspectrum problem , for ; where is a system matrix, is a mass matrix, and are a set of eigenvalues and eigenvectors respectively.
+
In this class we make use of the method applied to the generalized eigenspectrum problem , for ; where is a system matrix, is a mass matrix, and are a set of eigenvalues and eigenvectors respectively.
The ArpackSolver can be used in application codes with serial objects in the following way:
for the generalized eigenvalue problem , where the variable size_of_spectrum tells ARPACK the number of eigenvector/eigenvalue pairs to solve for. Here, lambda is a vector that will contain the eigenvalues computed, x a vector that will contain the eigenvectors computed, and OP is an inverse operation for the matrix A. Shift and invert transformation around zero is applied.
+
for the generalized eigenvalue problem , where the variable size_of_spectrum tells ARPACK the number of eigenvector/eigenvalue pairs to solve for. Here, lambda is a vector that will contain the eigenvalues computed, x a vector that will contain the eigenvectors computed, and OP is an inverse operation for the matrix A. Shift and invert transformation around zero is applied.
Through the AdditionalData the user can specify some of the parameters to be set.
For further information on how the ARPACK routines dsaupd, dseupd, dnaupd and dneupd work and also how to set the parameters appropriately please take a look into the ARPACK manual.
Note
Whenever you eliminate degrees of freedom using AffineConstraints, you generate spurious eigenvalues and eigenvectors. If you make sure that the diagonals of eliminated matrix rows are all equal to one, you get a single additional eigenvalue. But beware that some functions in deal.II set these diagonals to rather arbitrary (from the point of view of eigenvalue problems) values. See also step-36 for an example.
@@ -510,7 +510,7 @@
-
Solve the generalized eigensprectrum problem by calling the dsaupd and dseupd or dnaupd and dneupd functions of ARPACK.
+
Solve the generalized eigensprectrum problem by calling the dsaupd and dseupd or dnaupd and dneupd functions of ARPACK.
The function returns a vector of eigenvalues of length n and a vector of eigenvectors of length n in the symmetric case and of length n+1 in the non-symmetric case. In the symmetric case all eigenvectors are real. In the non-symmetric case complex eigenvalues always occur as complex conjugate pairs. Therefore the eigenvector for an eigenvalue with nonzero complex part is stored by putting the real and the imaginary parts in consecutive real-valued vectors. The eigenvector of the complex conjugate eigenvalue does not need to be stored, since it is just the complex conjugate of the stored eigenvector. Thus, if the last n-th eigenvalue has a nonzero imaginary part, Arpack needs in total n+1 real-valued vectors to store real and imaginary parts of the eigenvectors.
Parameters
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@@ -1025,7 +1025,7 @@
-
Return a reference to the th element of the range represented by the current object.
+
Return a reference to the th element of the range represented by the current object.
This function is marked as const because it does not change the view object. It may however return a reference to a non-const memory location depending on whether the template type of the class is const or not.
This function is only allowed to be called if the underlying data is indeed stored in CPU memory.
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@@ -353,27 +353,27 @@
where each value is the relative weight of each vertex (so the centroid is, in 2d, where each ). Since we only consider convex combinations we can rewrite this equation as
+
where each value is the relative weight of each vertex (so the centroid is, in 2d, where each ). Since we only consider convex combinations we can rewrite this equation as
Detailed Description
template<typename VectorType>
class BaseQR< VectorType >
This class and classes derived from it are meant to build and matrices one row/column at a time, i.e., by growing matrix from an empty matrix to , where is the number of added column vectors.
-
As a consequence, matrices which have the same number of rows as each vector (i.e. matrix) is stored as a collection of vectors of VectorType.
+
This class and classes derived from it are meant to build and matrices one row/column at a time, i.e., by growing matrix from an empty matrix to , where is the number of added column vectors.
+
As a consequence, matrices which have the same number of rows as each vector (i.e. matrix) is stored as a collection of vectors of VectorType.
BlockIndices represents a range of indices (such as the range of valid indices for elements of a vector) and how this one range is broken down into smaller but contiguous "blocks" (such as the velocity and pressure parts of a solution vector). In particular, it provides the ability to translate between global indices and the indices within a block. This class is used, for example, in the BlockVector, BlockSparsityPattern, and BlockMatrixBase classes.
+
BlockIndices represents a range of indices (such as the range of valid indices for elements of a vector) and how this one range is broken down into smaller but contiguous "blocks" (such as the velocity and pressure parts of a solution vector). In particular, it provides the ability to translate between global indices and the indices within a block. This class is used, for example, in the BlockVector, BlockSparsityPattern, and BlockMatrixBase classes.
The information that can be obtained from this class falls into two groups. First, it is possible to query the global size of the index space (through the total_size() member function), and the number of blocks and their sizes (via size() and the block_size() functions).
Secondly, this class manages the conversion of global indices to the local indices within this block, and the other way around. This is required, for example, when you address a global element in a block vector and want to know within which block this is, and which index within this block it corresponds to. It is also useful if a matrix is composed of several blocks, where you have to translate global row and column indices to local ones.
Return a LinearOperator that performs the operations associated with the Schur complement. There are two additional helper functions, condense_schur_rhs() and postprocess_schur_solution(), that are likely necessary to be used in order to perform any useful tasks in linear algebra with this operator.
We construct the definition of the Schur complement in the following way:
Consider a general system of linear equations that can be decomposed into two major sets of equations:
-
+\end{eqnarray*}" src="form_1852.png"/>
-
where represent general subblocks of the matrix and, similarly, general subvectors of are given by .
+
where represent general subblocks of the matrix and, similarly, general subvectors of are given by .
This is equivalent to the following two statements:
-
+\end{eqnarray*}" src="form_1857.png"/>
-
Assuming that are both square and invertible, we could then perform one of two possible substitutions,
- are both square and invertible, we could then perform one of two possible substitutions,
+
+\end{eqnarray*}" src="form_1859.png"/>
which amount to performing block Gaussian elimination on this system of equations.
For the purpose of the current implementation, we choose to substitute (3) into (2)
-
+\end{eqnarray*}" src="form_1860.png"/>
This leads to the result
-
+\]" src="form_1861.png"/>
-
with being the Schur complement and the modified right-hand side vector arising from the condensation step. Note that for this choice of , submatrix need not be invertible and may thus be the null matrix. Ideally should be well-conditioned.
-
So for any arbitrary vector , the Schur complement performs the following operation:
- being the Schur complement and the modified right-hand side vector arising from the condensation step. Note that for this choice of , submatrix need not be invertible and may thus be the null matrix. Ideally should be well-conditioned.
+
So for any arbitrary vector , the Schur complement performs the following operation:
+
+\]" src="form_1868.png"/>
A typical set of steps needed the solve a linear system (1),(2) would be:
Define iterative inverse matrix such that (6) holds. It is necessary to use a solver with a preconditioner to compute the approximate inverse operation of since we never compute directly, but rather the result of its operation. To achieve this, one may again use the inverse_operator() in conjunction with the Schur complement that we've just constructed. Observe that the both and its preconditioner operate over the same space as .
Define iterative inverse matrix such that (6) holds. It is necessary to use a solver with a preconditioner to compute the approximate inverse operation of since we never compute directly, but rather the result of its operation. To achieve this, one may again use the inverse_operator() in conjunction with the Schur complement that we've just constructed. Observe that the both and its preconditioner operate over the same space as .
In the above example, the preconditioner for was defined as the preconditioner for , which is valid since they operate on the same space. However, if and are too dissimilar, then this may lead to a large number of solver iterations as is not a good approximation for .
-
A better preconditioner in such a case would be one that provides a more representative approximation for . One approach is shown in step-22, where is the null matrix and the preconditioner for is derived from the mass matrix over this space.
-
From another viewpoint, a similar result can be achieved by first constructing an object that represents an approximation for wherein expensive operation, namely , is approximated. Thereafter we construct the approximate inverse operator which is then used as the preconditioner for computing .
// Construction of approximate inverse of Schur complement
+
In the above example, the preconditioner for was defined as the preconditioner for , which is valid since they operate on the same space. However, if and are too dissimilar, then this may lead to a large number of solver iterations as is not a good approximation for .
+
A better preconditioner in such a case would be one that provides a more representative approximation for . One approach is shown in step-22, where is the null matrix and the preconditioner for is derived from the mass matrix over this space.
+
From another viewpoint, a similar result can be achieved by first constructing an object that represents an approximation for wherein expensive operation, namely , is approximated. Thereafter we construct the approximate inverse operator which is then used as the preconditioner for computing .
// Construction of approximate inverse of Schur complement
Note that due to the construction of S_inv_approx and subsequently S_inv, there are a pair of nested iterative solvers which could collectively consume a lot of resources. Therefore care should be taken in the choices leading to the construction of the iterative inverse_operators. One might consider the use of a IterationNumberControl (or a similar mechanism) to limit the number of inner solver iterations. This controls the accuracy of the approximate inverse operation which acts only as the preconditioner for . Furthermore, the preconditioner to , which in this example is , should ideally be computationally inexpensive.
+
Note that due to the construction of S_inv_approx and subsequently S_inv, there are a pair of nested iterative solvers which could collectively consume a lot of resources. Therefore care should be taken in the choices leading to the construction of the iterative inverse_operators. One might consider the use of a IterationNumberControl (or a similar mechanism) to limit the number of inner solver iterations. This controls the accuracy of the approximate inverse operation which acts only as the preconditioner for . Furthermore, the preconditioner to , which in this example is , should ideally be computationally inexpensive.
However, if an iterative solver based on IterationNumberControl is used as a preconditioner then the preconditioning operation is not a linear operation. Here a flexible solver like SolverFGMRES (flexible GMRES) is best employed as an outer solver in order to deal with the variable behavior of the preconditioner. Otherwise the iterative solver can stagnate somewhere near the tolerance of the preconditioner or generally behave erratically. Alternatively, using a ReductionControl would ensure that the preconditioner always solves to the same tolerance, thereby rendering its behavior constant.
Further examples of this functionality can be found in the test-suite, such as tests/lac/schur_complement_01.cc . The solution of a multi- component problem (namely step-22) using the schur_complement can be found in tests/lac/schur_complement_03.cc .
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@@ -1296,7 +1296,7 @@
const BlockVectorType &
src&#href_anchor"memdoc">
-
Adding Matrix-vector multiplication. Add on with being this matrix.
+
Adding Matrix-vector multiplication. Add on with being this matrix.
@@ -1385,7 +1385,7 @@
const BlockVectorType &
v&#href_anchor"memdoc">
-
Compute the matrix scalar product .
+
Compute the matrix scalar product .
@@ -1744,7 +1744,7 @@
-
Matrix-vector multiplication: let with being this matrix.
+
Matrix-vector multiplication: let with being this matrix.
Due to problems with deriving template arguments between the block and non-block versions of the vmult/Tvmult functions, the actual functions are implemented in derived classes, with implementations forwarding the calls to the implementations provided here under a unique name for which template arguments can be derived by the compiler.
@@ -1868,7 +1868,7 @@
-
Matrix-vector multiplication: let with being this matrix. This function does the same as vmult() but takes the transposed matrix.
+
Matrix-vector multiplication: let with being this matrix. This function does the same as vmult() but takes the transposed matrix.
Due to problems with deriving template arguments between the block and non-block versions of the vmult/Tvmult functions, the actual functions are implemented in derived classes, with implementations forwarding the calls to the implementations provided here under a unique name for which template arguments can be derived by the compiler.
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+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classBlockSparseMatrix.html 2024-04-12 04:45:49.679562168 +0000
@@ -941,7 +941,7 @@
-
Matrix-vector multiplication: let with being this matrix.
+
Matrix-vector multiplication: let with being this matrix.
Adding Matrix-vector multiplication. Add on with being this matrix.
+
Adding Matrix-vector multiplication. Add on with being this matrix.
@@ -2166,7 +2166,7 @@
-
Compute the matrix scalar product .
+
Compute the matrix scalar product .
@@ -2609,7 +2609,7 @@
-
Matrix-vector multiplication: let with being this matrix.
+
Matrix-vector multiplication: let with being this matrix.
Due to problems with deriving template arguments between the block and non-block versions of the vmult/Tvmult functions, the actual functions are implemented in derived classes, with implementations forwarding the calls to the implementations provided here under a unique name for which template arguments can be derived by the compiler.
@@ -2717,7 +2717,7 @@
-
Matrix-vector multiplication: let with being this matrix. This function does the same as vmult() but takes the transposed matrix.
+
Matrix-vector multiplication: let with being this matrix. This function does the same as vmult() but takes the transposed matrix.
Due to problems with deriving template arguments between the block and non-block versions of the vmult/Tvmult functions, the actual functions are implemented in derived classes, with implementations forwarding the calls to the implementations provided here under a unique name for which template arguments can be derived by the compiler.
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classBlockSparseMatrixEZ.html 2024-04-12 04:45:49.727562501 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classBlockSparseMatrixEZ.html 2024-04-12 04:45:49.731562528 +0000
@@ -754,7 +754,7 @@
/usr/share/doc/packages/dealii/doxygen/deal.II/classBlockVector.html differs (HTML document, ASCII text, with very long lines)
--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classBlockVector.html 2024-04-12 04:45:49.795562972 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classBlockVector.html 2024-04-12 04:45:49.803563027 +0000
@@ -1768,7 +1768,7 @@
-
Return the square of the -norm.
+
Return the square of the -norm.
@@ -1820,7 +1820,7 @@
-
Return the -norm of the vector, i.e. the sum of the absolute values.
+
Return the -norm of the vector, i.e. the sum of the absolute values.
@@ -1846,7 +1846,7 @@
-
Return the -norm of the vector, i.e. the square root of the sum of the squares of the elements.
+
Return the -norm of the vector, i.e. the square root of the sum of the squares of the elements.
@@ -1872,7 +1872,7 @@
-
Return the maximum absolute value of the elements of this vector, which is the -norm of a vector.
+
Return the maximum absolute value of the elements of this vector, which is the -norm of a vector.
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classBlockVectorBase.html 2024-04-12 04:45:49.859563415 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classBlockVectorBase.html 2024-04-12 04:45:49.859563415 +0000
@@ -1218,7 +1218,7 @@
-
Return the square of the -norm.
+
Return the square of the -norm.
@@ -1258,7 +1258,7 @@
-
Return the -norm of the vector, i.e. the sum of the absolute values.
+
Return the -norm of the vector, i.e. the sum of the absolute values.
@@ -1278,7 +1278,7 @@
-
Return the -norm of the vector, i.e. the square root of the sum of the squares of the elements.
+
Return the -norm of the vector, i.e. the square root of the sum of the squares of the elements.
@@ -1298,7 +1298,7 @@
-
Return the maximum absolute value of the elements of this vector, which is the -norm of a vector.
+
Return the maximum absolute value of the elements of this vector, which is the -norm of a vector.
/usr/share/doc/packages/dealii/doxygen/deal.II/classBoundingBox.html differs (HTML document, ASCII text, with very long lines)
--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classBoundingBox.html 2024-04-12 04:45:49.899563692 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classBoundingBox.html 2024-04-12 04:45:49.903563720 +0000
@@ -166,11 +166,11 @@
&#href_anchor"details" id="details">
Detailed Description
template<int spacedim, typename Number = double>
class BoundingBox< spacedim, Number >
A class that represents a box of arbitrary dimension spacedim and with sides parallel to the coordinate axes, that is, a region
-
+\]" src="form_362.png"/>
-
where and denote the two vertices (bottom left and top right) which are used to represent the box. The quantities and denote the "lower" and "upper" bounds of values that are within the box for each coordinate direction .
+
where and denote the two vertices (bottom left and top right) which are used to represent the box. The quantities and denote the "lower" and "upper" bounds of values that are within the box for each coordinate direction .
Geometrically, a bounding box is thus:
1d: a segment (represented by its vertices in the proper order)
2d: a rectangle (represented by the vertices V at bottom left, top right)
.--------V
@@ -186,7 +186,7 @@
Bounding boxes are, for example, useful in parallel distributed meshes to give a general description of the owners of each portion of the mesh. More generally, bounding boxes are often used to roughly describe a region of space in which an object is contained; if a candidate point is not within the bounding box (a test that is cheap to execute), then it is not necessary to perform an expensive test whether the candidate point is in fact inside the object itself. Bounding boxes are therefore often used as a first, cheap rejection test before more detailed checks. As such, bounding boxes serve many of the same purposes as the convex hull, for which it is also relatively straightforward to compute whether a point is inside or outside, though not quite as cheap as for the bounding box.
-
Taking the cross section of a BoundingBox<spacedim> orthogonal to a given direction gives a box in one dimension lower: BoundingBox<spacedim - 1>. In 3d, the 2 coordinates of the cross section of BoundingBox<3> can be ordered in 2 different ways. That is, if we take the cross section orthogonal to the y direction we could either order a 3d-coordinate into a 2d-coordinate as or as . This class uses the second convention, corresponding to the coordinates being ordered cyclicly To be precise, if we take a cross section:
+
Taking the cross section of a BoundingBox<spacedim> orthogonal to a given direction gives a box in one dimension lower: BoundingBox<spacedim - 1>. In 3d, the 2 coordinates of the cross section of BoundingBox<3> can be ordered in 2 different ways. That is, if we take the cross section orthogonal to the y direction we could either order a 3d-coordinate into a 2d-coordinate as or as . This class uses the second convention, corresponding to the coordinates being ordered cyclicly To be precise, if we take a cross section:
Orthogonal to
Cross section coordinates ordered as
@@ -731,7 +731,7 @@
-
Returns the indexth vertex of the box. Vertex is meant in the same way as for a cell, so that index.
+
Returns the indexth vertex of the box. Vertex is meant in the same way as for a cell, so that index.
Return the square of the norm of the vector with respect to the norm induced by this matrix, i.e., . This is useful, e.g., in the finite context, where the norm of a function equals the matrix norm with respect to the mass matrix of the vector representing the nodal values of the finite element function.
+
Return the square of the norm of the vector with respect to the norm induced by this matrix, i.e., . This is useful, e.g., in the finite context, where the norm of a function equals the matrix norm with respect to the mass matrix of the vector representing the nodal values of the finite element function.
Obviously, the matrix needs to be quadratic for this operation.
Return the -norm of the matrix, that is , (max. sum of rows). This is the natural norm that is compatible to the -norm of vectors, i.e., -norm of the matrix, that is , (max. sum of rows). This is the natural norm that is compatible to the -norm of vectors, i.e., .
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classCellAccessor.html 2024-04-12 04:45:50.155565466 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classCellAccessor.html 2024-04-12 04:45:50.151565437 +0000
@@ -4150,7 +4150,7 @@
This function computes a fast approximate transformation from the real to the unit cell by inversion of an affine approximation of the -linear function from the reference -dimensional cell.
-
The affine approximation of the unit to real cell mapping is found by a least squares fit of an affine function to the vertices of the present object. For any valid mesh cell whose geometry is not degenerate, this operation results in a unique affine mapping. Thus, this function will return a finite result for all given input points, even in cases where the actual transformation by an actual bi-/trilinear or higher order mapping might be singular. Besides only approximating the mapping from the vertex points, this function also ignores the attached manifold descriptions. The result is only exact in case the transformation from the unit to the real cell is indeed affine, such as in one dimension or for Cartesian and affine (parallelogram) meshes in 2d/3d.
+
The affine approximation of the unit to real cell mapping is found by a least squares fit of an affine function to the vertices of the present object. For any valid mesh cell whose geometry is not degenerate, this operation results in a unique affine mapping. Thus, this function will return a finite result for all given input points, even in cases where the actual transformation by an actual bi-/trilinear or higher order mapping might be singular. Besides only approximating the mapping from the vertex points, this function also ignores the attached manifold descriptions. The result is only exact in case the transformation from the unit to the real cell is indeed affine, such as in one dimension or for Cartesian and affine (parallelogram) meshes in 2d/3d.
If dim<spacedim we first project p onto the plane.
@@ -4213,15 +4213,15 @@
-
Return the barycenter (also called centroid) of the object. The barycenter for an object of dimension in space dimensions is given by the -dimensional vector defined by
- of dimension in space dimensions is given by the -dimensional vector defined by
+
+\]" src="form_1482.png"/>
where the measure of the object is given by
-
+\]" src="form_1483.png"/>
This function assumes that is mapped by a -linear function from the reference -dimensional cell. Then the integrals above can be pulled back to the reference cell and evaluated exactly (if through lengthy and, compared to the center() function, expensive computations).
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classChartManifold.html 2024-04-12 04:45:50.203565798 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classChartManifold.html 2024-04-12 04:45:50.211565854 +0000
@@ -206,37 +206,37 @@
Detailed Description
template<int dim, int spacedim = dim, int chartdim = dim>
class ChartManifold< dim, spacedim, chartdim >
This class describes mappings that can be expressed in terms of charts. Specifically, this class with its template arguments describes a chart of dimension chartdim, which is part of a Manifold<dim,spacedim> and is used in an object of type Triangulation<dim,spacedim>: It specializes a Manifold of dimension chartdim embedded in a manifold of dimension spacedim, for which you have explicit pull_back() and push_forward() transformations. Its use is explained in great detail in step-53.
-
This is a helper class which is useful when you have an explicit map from an Euclidean space of dimension chartdim to an Euclidean space of dimension spacedim which represents your manifold, i.e., when your manifold can be represented by a map
-
+
This is a helper class which is useful when you have an explicit map from an Euclidean space of dimension chartdim to an Euclidean space of dimension spacedim which represents your manifold, i.e., when your manifold can be represented by a map
+
(the push_forward() function) and that admits the inverse transformation
The get_new_point() function of the ChartManifold class is implemented by calling the pull_back() method for all surrounding_points, computing their weighted average in the chartdim Euclidean space, and calling the push_forward() method with the resulting point, i.e.,
-
+
Derived classes are required to implement the push_forward() and the pull_back() methods. All other functions (with the exception of the push_forward_gradient() function, see below) that are required by mappings will then be provided by this class.
Providing function gradients
-
In order to compute vectors that are tangent to the manifold (for example, tangent to a surface embedded in higher dimensional space, or simply the three unit vectors of ), one needs to also have access to the gradient of the push-forward function . The gradient is the matrix , where we take the derivative with regard to the chartdim reference coordinates on the flat Euclidean space in which is located. In other words, at a point , is a matrix of size spacedim times chartdim.
+
In order to compute vectors that are tangent to the manifold (for example, tangent to a surface embedded in higher dimensional space, or simply the three unit vectors of ), one needs to also have access to the gradient of the push-forward function . The gradient is the matrix , where we take the derivative with regard to the chartdim reference coordinates on the flat Euclidean space in which is located. In other words, at a point , is a matrix of size spacedim times chartdim.
Only the ChartManifold::get_tangent_vector() function uses the gradient of the push-forward, but only a subset of all finite element codes actually require the computation of tangent vectors. Consequently, while derived classes need to implement the abstract virtual push_forward() and pull_back() functions of this class, they do not need to implement the virtual push_forward_gradient() function. Rather, that function has a default implementation (and consequently is not abstract, therefore not forcing derived classes to overload it), but the default implementation clearly can not compute anything useful and therefore simply triggers and exception.
A note on the template arguments
The dimension arguments chartdim, dim and spacedim must satisfy the following relationships:
dim <= spacedim
chartdim <= spacedim
However, there is no a priori relationship between dim and chartdim. For example, if you want to describe a mapping for an edge (a 1d object) in a 2d triangulation embedded in 3d space, you could do so by parameterizing it via a line
-
+ \]" src="form_1426.png"/>
in which case chartdim is 1. On the other hand, there is no reason why one can't describe this as a mapping
-
+ \]" src="form_1427.png"/>
-
in such a way that the line happens to be mapped onto the edge in question. Here, chartdim is 3. This may seem cumbersome but satisfies the requirements of an invertible function just fine as long as it is possible to get from the edge to the pull-back space and then back again. Finally, given that we are dealing with a 2d triangulation in 3d, one will often have a mapping from, say, the 2d unit square or unit disk to the domain in 3d space, and the edge in question may simply be the mapped edge of the unit domain in 2d space. In this case, chartdim is 2.
+
in such a way that the line happens to be mapped onto the edge in question. Here, chartdim is 3. This may seem cumbersome but satisfies the requirements of an invertible function just fine as long as it is possible to get from the edge to the pull-back space and then back again. Finally, given that we are dealing with a 2d triangulation in 3d, one will often have a mapping from, say, the 2d unit square or unit disk to the domain in 3d space, and the edge in question may simply be the mapped edge of the unit domain in 2d space. In this case, chartdim is 2.
Given a point in the chartdim dimensional Euclidean space, this method returns the derivatives of the function that maps from the chartdim-dimensional to the spacedim-dimensional space. In other words, it is a matrix of size .
+
Given a point in the chartdim dimensional Euclidean space, this method returns the derivatives of the function that maps from the chartdim-dimensional to the spacedim-dimensional space. In other words, it is a matrix of size .
This function is used in the computations required by the get_tangent_vector() function. Since not all users of the Manifold class interface will require calling that function, the current function is implemented but will trigger an exception whenever called. This allows derived classes to avoid implementing the push_forward_gradient function if this functionality is not needed in the user program.
Refer to the general documentation of this class for more information.
@@ -600,24 +600,24 @@
-
Return a vector that, at , is tangential to the geodesic that connects two points . See the documentation of the Manifold class and of Manifold::get_tangent_vector() for a more detailed description.
-
For the current class, we assume that this geodesic is the image under the push_forward() operation of a straight line of the pre-images of x1 and x2 (where pre-images are computed by pulling back the locations x1 and x2). In other words, if these preimages are , then the geodesic in preimage (the chartdim-dimensional Euclidean) space is
-, is tangential to the geodesic that connects two points . See the documentation of the Manifold class and of Manifold::get_tangent_vector() for a more detailed description.
+
For the current class, we assume that this geodesic is the image under the push_forward() operation of a straight line of the pre-images of x1 and x2 (where pre-images are computed by pulling back the locations x1 and x2). In other words, if these preimages are , then the geodesic in preimage (the chartdim-dimensional Euclidean) space is
+
+\end{align*}" src="form_1440.png"/>
In image space, i.e., in the space in which we operate, this leads to the curve
-
+\end{align*}" src="form_1441.png"/>
-
What the current function is supposed to return is . By the chain rule, this is equal to
-. By the chain rule, this is equal to
+
+\end{align*}" src="form_1442.png"/>
This formula may then have to be slightly modified by considering any periodicity that was assumed in the call to the constructor.
-
Thus, the computation of tangent vectors also requires the implementation of derivatives of the push-forward mapping. Here, is a chartdim-dimensional vector, and is a spacedim-times-chartdim-dimensional matrix. Consequently, and as desired, the operation results in a spacedim-dimensional vector.
+
Thus, the computation of tangent vectors also requires the implementation of derivatives of the push-forward mapping. Here, is a chartdim-dimensional vector, and is a spacedim-times-chartdim-dimensional matrix. Consequently, and as desired, the operation results in a spacedim-dimensional vector.
Parameters
x1
The first point that describes the geodesic, and the one at which the "direction" is to be evaluated.
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classChunkSparseMatrix.html 2024-04-12 04:45:50.279566324 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classChunkSparseMatrix.html 2024-04-12 04:45:50.283566352 +0000
@@ -1036,7 +1036,7 @@
-
Symmetrize the matrix by forming the mean value between the existing matrix and its transpose, .
+
Symmetrize the matrix by forming the mean value between the existing matrix and its transpose, .
This operation assumes that the underlying sparsity pattern represents a symmetric object. If this is not the case, then the result of this operation will not be a symmetric matrix, since it only explicitly symmetrizes by looping over the lower left triangular part for efficiency reasons; if there are entries in the upper right triangle, then these elements are missed in the symmetrization. Symmetrization of the sparsity pattern can be obtain by ChunkSparsityPattern::symmetrize().
@@ -1367,7 +1367,7 @@
-
Return the square of the norm of the vector with respect to the norm induced by this matrix, i.e. . This is useful, e.g. in the finite element context, where the norm of a function equals the matrix norm with respect to the mass matrix of the vector representing the nodal values of the finite element function.
+
Return the square of the norm of the vector with respect to the norm induced by this matrix, i.e. . This is useful, e.g. in the finite element context, where the norm of a function equals the matrix norm with respect to the mass matrix of the vector representing the nodal values of the finite element function.
Obviously, the matrix needs to be quadratic for this operation, and for the result to actually be a norm it also needs to be either real symmetric or complex hermitian.
The underlying template types of both this matrix and the given vector should either both be real or complex-valued, but not mixed, for this function to make sense.
Return the l1-norm of the matrix, that is , (max. sum of columns). This is the natural matrix norm that is compatible to the l1-norm for vectors, i.e. . (cf. Haemmerlin-Hoffmann : Numerische Mathematik)
+
Return the l1-norm of the matrix, that is , (max. sum of columns). This is the natural matrix norm that is compatible to the l1-norm for vectors, i.e. . (cf. Haemmerlin-Hoffmann : Numerische Mathematik)
@@ -1462,8 +1462,8 @@
-
Return the linfty-norm of the matrix, that is , (max. sum of rows). This is the natural matrix norm that is compatible to the linfty-norm of vectors, i.e. . (cf. Haemmerlin-Hoffmann : Numerische Mathematik)
+
Return the linfty-norm of the matrix, that is , (max. sum of rows). This is the natural matrix norm that is compatible to the linfty-norm of vectors, i.e. . (cf. Haemmerlin-Hoffmann : Numerische Mathematik)
@@ -2157,7 +2157,7 @@
-
Return the location of entry within the val array.
+
Return the location of entry within the val array.
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classChunkSparsityPattern.html 2024-04-12 04:45:50.335566712 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classChunkSparsityPattern.html 2024-04-12 04:45:50.343566767 +0000
@@ -1123,7 +1123,7 @@
-
Compute the bandwidth of the matrix represented by this structure. The bandwidth is the maximum of for which the index pair represents a nonzero entry of the matrix. Consequently, the maximum bandwidth a matrix can have is .
+
Compute the bandwidth of the matrix represented by this structure. The bandwidth is the maximum of for which the index pair represents a nonzero entry of the matrix. Consequently, the maximum bandwidth a matrix can have is .
/usr/share/doc/packages/dealii/doxygen/deal.II/classCompositionManifold.html differs (HTML document, ASCII text, with very long lines)
--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classCompositionManifold.html 2024-04-12 04:45:50.399567155 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classCompositionManifold.html 2024-04-12 04:45:50.403567183 +0000
@@ -594,24 +594,24 @@
-
Return a vector that, at , is tangential to the geodesic that connects two points . See the documentation of the Manifold class and of Manifold::get_tangent_vector() for a more detailed description.
-
For the current class, we assume that this geodesic is the image under the push_forward() operation of a straight line of the pre-images of x1 and x2 (where pre-images are computed by pulling back the locations x1 and x2). In other words, if these preimages are , then the geodesic in preimage (the chartdim-dimensional Euclidean) space is
-, is tangential to the geodesic that connects two points . See the documentation of the Manifold class and of Manifold::get_tangent_vector() for a more detailed description.
+
For the current class, we assume that this geodesic is the image under the push_forward() operation of a straight line of the pre-images of x1 and x2 (where pre-images are computed by pulling back the locations x1 and x2). In other words, if these preimages are , then the geodesic in preimage (the chartdim-dimensional Euclidean) space is
+
+\end{align*}" src="form_1440.png"/>
In image space, i.e., in the space in which we operate, this leads to the curve
-
+\end{align*}" src="form_1441.png"/>
-
What the current function is supposed to return is . By the chain rule, this is equal to
-. By the chain rule, this is equal to
+
+\end{align*}" src="form_1442.png"/>
This formula may then have to be slightly modified by considering any periodicity that was assumed in the call to the constructor.
-
Thus, the computation of tangent vectors also requires the implementation of derivatives of the push-forward mapping. Here, is a chartdim-dimensional vector, and is a spacedim-times-chartdim-dimensional matrix. Consequently, and as desired, the operation results in a spacedim-dimensional vector.
+
Thus, the computation of tangent vectors also requires the implementation of derivatives of the push-forward mapping. Here, is a chartdim-dimensional vector, and is a spacedim-times-chartdim-dimensional matrix. Consequently, and as desired, the operation results in a spacedim-dimensional vector.
Parameters
x1
The first point that describes the geodesic, and the one at which the "direction" is to be evaluated.
/usr/share/doc/packages/dealii/doxygen/deal.II/classConvergenceTable.html differs (HTML document, ASCII text, with very long lines)
--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classConvergenceTable.html 2024-04-12 04:45:50.443567460 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classConvergenceTable.html 2024-04-12 04:45:50.443567460 +0000
@@ -362,14 +362,14 @@
Evaluate the convergence rates of the data column data_column_key due to the RateMode in relation to the reference column reference_column_key. Be sure that the value types of the table entries of the data column and the reference data column is a number, i.e. double, float, (unsigned) int, and so on.
-
As this class has no information on the space dimension upon which the reference column vs. the value column is based upon, it needs to be passed as last argument to this method. The default dimension for the reference column is 2, which is appropriate for the number of cells in 2d. If you work in 3d, set the number to 3. If the reference column is , remember to set the dimension to 1 also when working in 3d to get correct rates.
+
As this class has no information on the space dimension upon which the reference column vs. the value column is based upon, it needs to be passed as last argument to this method. The default dimension for the reference column is 2, which is appropriate for the number of cells in 2d. If you work in 3d, set the number to 3. If the reference column is , remember to set the dimension to 1 also when working in 3d to get correct rates.
The new rate column and the data column will be merged to a supercolumn. The tex caption of the supercolumn will be (by default) the same as the one of the data column. This may be changed by using the set_tex_supercaption (...) function of the base class TableHandler.
This method behaves in the following way:
-
If RateMode is reduction_rate, then the computed output is where is the reference column (no dimension dependence!).
-
If RateMode is reduction_rate_log2, then the computed output is .
-
This is useful, for example, if we use as reference key the number of degrees of freedom or better, the number of cells. Assuming that the error is proportional to in 2d, then this method will produce the rate as a result. For general dimension, as described by the last parameter of this function, the formula needs to be .
+
If RateMode is reduction_rate, then the computed output is where is the reference column (no dimension dependence!).
+
If RateMode is reduction_rate_log2, then the computed output is .
+
This is useful, for example, if we use as reference key the number of degrees of freedom or better, the number of cells. Assuming that the error is proportional to in 2d, then this method will produce the rate as a result. For general dimension, as described by the last parameter of this function, the formula needs to be .
Note
Since this function adds columns to the table after several rows have already been filled, it switches off the auto fill mode of the TableHandler base class. If you intend to add further data with auto fill, you will have to re-enable it after calling this function.
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classCylindricalManifold.html 2024-04-12 04:45:50.503567876 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classCylindricalManifold.html 2024-04-12 04:45:50.511567930 +0000
@@ -413,7 +413,7 @@
-
Compute the cylindrical coordinates for the given space point where denotes the distance from the axis, the angle between the given point and the computed normal direction, and the axial position.
+
Compute the cylindrical coordinates for the given space point where denotes the distance from the axis, the angle between the given point and the computed normal direction, and the axial position.
Compute the Cartesian coordinates for a chart point given in cylindrical coordinates , where denotes the distance from the axis, the angle between the given point and the computed normal direction, and the axial position.
+
Compute the Cartesian coordinates for a chart point given in cylindrical coordinates , where denotes the distance from the axis, the angle between the given point and the computed normal direction, and the axial position.
Compute the derivatives of the mapping from cylindrical coordinates to cartesian coordinates where denotes the distance from the axis, the angle between the given point and the computed normal direction, and the axial position.
+
Compute the derivatives of the mapping from cylindrical coordinates to cartesian coordinates where denotes the distance from the axis, the angle between the given point and the computed normal direction, and the axial position.
Given a point in the chartdim dimensional Euclidean space, this method returns the derivatives of the function that maps from the chartdim-dimensional to the spacedim-dimensional space. In other words, it is a matrix of size .
+
Given a point in the chartdim dimensional Euclidean space, this method returns the derivatives of the function that maps from the chartdim-dimensional to the spacedim-dimensional space. In other words, it is a matrix of size .
This function is used in the computations required by the get_tangent_vector() function. Since not all users of the Manifold class interface will require calling that function, the current function is implemented but will trigger an exception whenever called. This allows derived classes to avoid implementing the push_forward_gradient function if this functionality is not needed in the user program.
Refer to the general documentation of this class for more information.
@@ -678,24 +678,24 @@
-
Return a vector that, at , is tangential to the geodesic that connects two points . See the documentation of the Manifold class and of Manifold::get_tangent_vector() for a more detailed description.
-
For the current class, we assume that this geodesic is the image under the push_forward() operation of a straight line of the pre-images of x1 and x2 (where pre-images are computed by pulling back the locations x1 and x2). In other words, if these preimages are , then the geodesic in preimage (the chartdim-dimensional Euclidean) space is
-, is tangential to the geodesic that connects two points . See the documentation of the Manifold class and of Manifold::get_tangent_vector() for a more detailed description.
+
For the current class, we assume that this geodesic is the image under the push_forward() operation of a straight line of the pre-images of x1 and x2 (where pre-images are computed by pulling back the locations x1 and x2). In other words, if these preimages are , then the geodesic in preimage (the chartdim-dimensional Euclidean) space is
+
+\end{align*}" src="form_1440.png"/>
In image space, i.e., in the space in which we operate, this leads to the curve
-
+\end{align*}" src="form_1441.png"/>
-
What the current function is supposed to return is . By the chain rule, this is equal to
-. By the chain rule, this is equal to
+
+\end{align*}" src="form_1442.png"/>
This formula may then have to be slightly modified by considering any periodicity that was assumed in the call to the constructor.
-
Thus, the computation of tangent vectors also requires the implementation of derivatives of the push-forward mapping. Here, is a chartdim-dimensional vector, and is a spacedim-times-chartdim-dimensional matrix. Consequently, and as desired, the operation results in a spacedim-dimensional vector.
+
Thus, the computation of tangent vectors also requires the implementation of derivatives of the push-forward mapping. Here, is a chartdim-dimensional vector, and is a spacedim-times-chartdim-dimensional matrix. Consequently, and as desired, the operation results in a spacedim-dimensional vector.
Parameters
x1
The first point that describes the geodesic, and the one at which the "direction" is to be evaluated.
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classDataPostprocessor.html 2024-04-12 04:45:50.539568125 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classDataPostprocessor.html 2024-04-12 04:45:50.547568180 +0000
@@ -183,7 +183,7 @@
As a consequence, DataOut is forced to take things apart into their real and imaginary parts, and both are output as separate quantities. This is the case for data that is written directly to a file by DataOut, but it is also the case for data that is first routed through DataPostprocessor objects (or objects of their derived classes): All these objects see is a collection of real values, even if the underlying solution vector was complex-valued.
Implementations of the DataPostprocessor::evaluate_vector_field() in derived classes must understand how the solution values are arranged in the DataPostprocessorInputs::Vector objects they receive as input. The rule here is: If the finite element has vector components (including the case , i.e., a scalar element), then the inputs for complex-valued solution vectors will have components. These first contain the values (or gradients, or Hessians) of the real parts of all solution components, and then the values (or gradients, or Hessians) of the imaginary parts of all solution components.
+
Implementations of the DataPostprocessor::evaluate_vector_field() in derived classes must understand how the solution values are arranged in the DataPostprocessorInputs::Vector objects they receive as input. The rule here is: If the finite element has vector components (including the case , i.e., a scalar element), then the inputs for complex-valued solution vectors will have components. These first contain the values (or gradients, or Hessians) of the real parts of all solution components, and then the values (or gradients, or Hessians) of the imaginary parts of all solution components.
step-58 provides an example of how this class (or, rather, the derived DataPostprocessorScalar class) is used in a complex-valued situation.
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classDataPostprocessorTensor.html 2024-04-12 04:45:50.579568401 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classDataPostprocessorTensor.html 2024-04-12 04:45:50.587568457 +0000
@@ -255,7 +255,7 @@
These pictures show an ellipse representing the gradient tensor at, on average, every tenth mesh point. You may want to read through the documentation of the VisIt visualization program (see https://wci.llnl.gov/simulation/computer-codes/visit/) for an interpretation of how exactly tensors are visualizated.
-
In elasticity, one is often interested not in the gradient of the displacement, but in the "strain", i.e., the symmetrized version of the gradient . This is easily facilitated with the following minor modification:
template <int dim>
+
In elasticity, one is often interested not in the gradient of the displacement, but in the "strain", i.e., the symmetrized version of the gradient . This is easily facilitated with the following minor modification:
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classDataPostprocessorVector.html 2024-04-12 04:45:50.631568762 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classDataPostprocessorVector.html 2024-04-12 04:45:50.627568735 +0000
@@ -246,7 +246,7 @@
In the second image, the background color corresponds to the magnitude of the gradient vector and the vector glyphs to the gradient itself. It may be surprising at first to see that from each vertex, multiple vectors originate, going in different directions. But that is because the solution is only continuous: in general, the gradient is discontinuous across edges, and so the multiple vectors originating from each vertex simply represent the differing gradients of the solution at each adjacent cell.
-
The output above – namely, the gradient of the solution – corresponds to the temperature gradient if one interpreted step-6 as solving a steady-state heat transfer problem. It is very small in the central part of the domain because in step-6 we are solving an equation that has a coefficient that is large in the central part and small on the outside. This can be thought as a material that conducts heat well, and consequently the temperature gradient is small. On the other hand, the "heat flux" corresponds to the quantity . For the solution of that equation, the flux should be continuous across the interface. This is easily verified by the following modification of the postprocessor:
template <int dim>
+
The output above – namely, the gradient of the solution – corresponds to the temperature gradient if one interpreted step-6 as solving a steady-state heat transfer problem. It is very small in the central part of the domain because in step-6 we are solving an equation that has a coefficient that is large in the central part and small on the outside. This can be thought as a material that conducts heat well, and consequently the temperature gradient is small. On the other hand, the "heat flux" corresponds to the quantity . For the solution of that equation, the flux should be continuous across the interface. This is easily verified by the following modification of the postprocessor:
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+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classDerivativeApproximation_1_1internal_1_1SecondDerivative.html 2024-04-12 04:45:50.659568955 +0000
@@ -235,7 +235,7 @@
-
Return the norm of the derivative object. Here, for the (symmetric) tensor of second derivatives, we choose the absolute value of the largest eigenvalue, which is the matrix norm associated to the norm of vectors. It is also the largest value of the curvature of the solution.
+
Return the norm of the derivative object. Here, for the (symmetric) tensor of second derivatives, we choose the absolute value of the largest eigenvalue, which is the matrix norm associated to the norm of vectors. It is also the largest value of the curvature of the solution.
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classDerivativeApproximation_1_1internal_1_1ThirdDerivative.html 2024-04-12 04:45:50.687569150 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classDerivativeApproximation_1_1internal_1_1ThirdDerivative.html 2024-04-12 04:45:50.691569177 +0000
@@ -230,7 +230,7 @@
-
Return the norm of the derivative object. Here, for the (symmetric) tensor of second derivatives, we choose the absolute value of the largest eigenvalue, which is the matrix norm associated to the norm of vectors. It is also the largest value of the curvature of the solution.
+
Return the norm of the derivative object. Here, for the (symmetric) tensor of second derivatives, we choose the absolute value of the largest eigenvalue, which is the matrix norm associated to the norm of vectors. It is also the largest value of the curvature of the solution.
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classDerivativeForm.html 2024-04-12 04:45:50.723569399 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classDerivativeForm.html 2024-04-12 04:45:50.731569455 +0000
@@ -164,24 +164,24 @@
DerivativeForm< 1, spacedim, dim, Number >&#href_anchor"memTemplItemRight" valign="bottom">transpose (const DerivativeForm< 1, dim, spacedim, Number > &DF)
&#href_anchor"details" id="details">
Detailed Description
template<int order, int dim, int spacedim, typename Number = double>
-class DerivativeForm< order, dim, spacedim, Number >
This class represents the (tangential) derivatives of a function . Such functions are always used to map the reference dim-dimensional cell into spacedim-dimensional space. For such objects, the first derivative of the function is a linear map from to , i.e., it can be represented as a matrix in . This makes sense since one would represent the first derivative, with . Such functions are always used to map the reference dim-dimensional cell into spacedim-dimensional space. For such objects, the first derivative of the function is a linear map from to , i.e., it can be represented as a matrix in . This makes sense since one would represent the first derivative, with , in such a way that the directional derivative in direction so that
-, in such a way that the directional derivative in direction so that
+
+\end{align*}" src="form_387.png"/>
-
i.e., one needs to be able to multiply the matrix by a vector in , and the result is a difference of function values, which are in . Consequently, the matrix must be of size .
-
Similarly, the second derivative is a bilinear map from to , which one can think of a rank-3 object of size .
+
i.e., one needs to be able to multiply the matrix by a vector in , and the result is a difference of function values, which are in . Consequently, the matrix must be of size .
+
Similarly, the second derivative is a bilinear map from to , which one can think of a rank-3 object of size .
In deal.II we represent these derivatives using objects of type DerivativeForm<1,dim,spacedim,Number>, DerivativeForm<2,dim,spacedim,Number> and so on.
Converts a DerivativeForm <order, dim, dim, Number> to Tensor<order+1, dim, Number>. In particular, if order == 1 and the derivative is the Jacobian of , then Tensor[i] = .
+
Converts a DerivativeForm <order, dim, dim, Number> to Tensor<order+1, dim, Number>. In particular, if order == 1 and the derivative is the Jacobian of , then Tensor[i] = .
@@ -453,8 +453,8 @@
-
Compute the Frobenius norm of this form, i.e., the expression .
+
Compute the Frobenius norm of this form, i.e., the expression .
@@ -474,7 +474,7 @@
-
Compute the volume element associated with the jacobian of the transformation . That is to say if is square, it computes , in case DF is not square returns .
+
Compute the volume element associated with the jacobian of the transformation . That is to say if is square, it computes , in case DF is not square returns .
@@ -494,9 +494,9 @@
-
Assuming that the current object stores the Jacobian of a mapping , then the current function computes the covariant form of the derivative, namely , where . If is a square matrix (i.e., ), then this function simplifies to computing .
+
Assuming that the current object stores the Jacobian of a mapping , then the current function computes the covariant form of the derivative, namely , where . If is a square matrix (i.e., ), then this function simplifies to computing .
@@ -552,7 +552,7 @@
-
Auxiliary function that computes where A represents the current object.
+
Auxiliary function that computes where A represents the current object.
@@ -581,21 +581,21 @@
-
One of the uses of DerivativeForm is to apply it as a linear transformation. This function returns , which approximates the change in when is changed by the amount
-DerivativeForm is to apply it as a linear transformation. This function returns , which approximates the change in when is changed by the amount
+
+\]" src="form_396.png"/>
The transformation corresponds to
-
+\]" src="form_397.png"/>
-
in index notation and corresponds to in matrix notation.
+
in index notation and corresponds to in matrix notation.
Similar to the previous apply_transformation(). Each row of the result corresponds to one of the rows of D_X transformed by grad_F, equivalent to in matrix notation.
+
Similar to the previous apply_transformation(). Each row of the result corresponds to one of the rows of D_X transformed by grad_F, equivalent to in matrix notation.
Similar to the previous apply_transformation(), specialized for the case dim == spacedim where we can return a rank-2 tensor instead of the more general DerivativeForm. Each row of the result corresponds to one of the rows of D_X transformed by grad_F, equivalent to in matrix notation.
+
Similar to the previous apply_transformation(), specialized for the case dim == spacedim where we can return a rank-2 tensor instead of the more general DerivativeForm. Each row of the result corresponds to one of the rows of D_X transformed by grad_F, equivalent to in matrix notation.
Similar to the previous apply_transformation(). In matrix notation, it computes . Moreover, the result of this operation can be interpreted as a metric tensor in which corresponds to the Euclidean metric tensor in . For every pair of vectors , we have:
-apply_transformation(). In matrix notation, it computes . Moreover, the result of this operation can be interpreted as a metric tensor in which corresponds to the Euclidean metric tensor in . For every pair of vectors , we have:
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+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classDifferentiation_1_1AD_1_1CellLevelBase.html 2024-04-12 04:45:50.791569870 +0000
@@ -514,7 +514,7 @@
-
Compute the value of the residual vector field .
+
Compute the value of the residual vector field .
Parameters
[out]
residual
A Vector object with the value for each component of the vector field evaluated at the point defined by the independent variable values.
@@ -552,9 +552,9 @@
Compute the gradient (first derivative) of the residual vector field with respect to all independent variables, i.e.
-
+\]" src="form_904.png"/>
Parameters
@@ -1295,7 +1295,7 @@
-
Set the actual value of the independent variable .
+
Set the actual value of the independent variable .
Parameters
[in]
index
The index in the vector of independent variables.
@@ -1336,7 +1336,7 @@
-
Initialize an independent variable such that subsequent operations performed with it are tracked.
+
Initialize an independent variable such that subsequent operations performed with it are tracked.
Note
Care must be taken to mark each independent variable only once.
The order in which the independent variables are marked defines the order of all future internal operations. They must be manipulated in the same order as that in which they are first marked. If not then, for example, ADOL-C won't throw an error, but rather it might complain nonsensically during later computations or produce garbage results.
@@ -1411,7 +1411,7 @@
-
Initialize an independent variable .
+
Initialize an independent variable .
Parameters
[out]
out
An auto-differentiable number that is ready for use in standard computations. The operations that are performed with it are not recorded on the tape, and so should only be used when not in recording mode.
@@ -1542,7 +1542,7 @@
-
Register the definition of the index'th dependent variable .
+
Register the definition of the index'th dependent variable .
Parameters
[in]
index
The index of the entry in the global list of dependent variables that this function belongs to.
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classDifferentiation_1_1AD_1_1EnergyFunctional.html 2024-04-12 04:45:50.847570258 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classDifferentiation_1_1AD_1_1EnergyFunctional.html 2024-04-12 04:45:50.851570286 +0000
@@ -440,11 +440,11 @@
The constructor for the class.
Parameters
-
[in]
n_independent_variables
The number of independent variables that will be used in the definition of the functions that it is desired to compute the sensitivities of. In the computation of , this will be the number of inputs , i.e., the dimension of the domain space.
+
[in]
n_independent_variables
The number of independent variables that will be used in the definition of the functions that it is desired to compute the sensitivities of. In the computation of , this will be the number of inputs , i.e., the dimension of the domain space.
-
Note
There is only one dependent variable associated with the total energy attributed to the local finite element. That is to say, this class assumes that the (local) right hand side and matrix contribution is computed from the first and second derivatives of a scalar function .
+
Note
There is only one dependent variable associated with the total energy attributed to the local finite element. That is to say, this class assumes that the (local) right hand side and matrix contribution is computed from the first and second derivatives of a scalar function .
Register the definition of the total cell energy .
+
Register the definition of the total cell energy .
Parameters
[in]
energy
A recorded function that defines the total cell energy. This represents the single dependent variable from which both the residual and its linearization are to be computed.
@@ -527,9 +527,9 @@
Evaluation of the total scalar energy functional for a chosen set of degree of freedom values, i.e.
The value of the energy functional at the evaluation point corresponding to a chosen set of local degree of freedom values.
@@ -562,12 +562,12 @@
-
Evaluation of the residual for a chosen set of degree of freedom values. Underlying this is the computation of the gradient (first derivative) of the scalar function with respect to all independent variables, i.e.
Compute the linearization of the residual vector around a chosen set of degree of freedom values. Underlying this is the computation of the Hessian (second derivative) of the scalar function with respect to all independent variables, i.e.
Set the actual value of the independent variable .
+
Set the actual value of the independent variable .
Parameters
[in]
index
The index in the vector of independent variables.
@@ -1592,7 +1592,7 @@
-
Initialize an independent variable such that subsequent operations performed with it are tracked.
+
Initialize an independent variable such that subsequent operations performed with it are tracked.
Note
Care must be taken to mark each independent variable only once.
The order in which the independent variables are marked defines the order of all future internal operations. They must be manipulated in the same order as that in which they are first marked. If not then, for example, ADOL-C won't throw an error, but rather it might complain nonsensically during later computations or produce garbage results.
@@ -1667,7 +1667,7 @@
-
Initialize an independent variable .
+
Initialize an independent variable .
Parameters
[out]
out
An auto-differentiable number that is ready for use in standard computations. The operations that are performed with it are not recorded on the tape, and so should only be used when not in recording mode.
@@ -1798,7 +1798,7 @@
-
Register the definition of the index'th dependent variable .
+
Register the definition of the index'th dependent variable .
Parameters
[in]
index
The index of the entry in the global list of dependent variables that this function belongs to.
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classDifferentiation_1_1AD_1_1HelperBase.html 2024-04-12 04:45:50.899570618 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classDifferentiation_1_1AD_1_1HelperBase.html 2024-04-12 04:45:50.907570674 +0000
@@ -991,7 +991,7 @@
-
Set the actual value of the independent variable .
+
Set the actual value of the independent variable .
Parameters
[in]
index
The index in the vector of independent variables.
@@ -1032,7 +1032,7 @@
-
Initialize an independent variable such that subsequent operations performed with it are tracked.
+
Initialize an independent variable such that subsequent operations performed with it are tracked.
Note
Care must be taken to mark each independent variable only once.
The order in which the independent variables are marked defines the order of all future internal operations. They must be manipulated in the same order as that in which they are first marked. If not then, for example, ADOL-C won't throw an error, but rather it might complain nonsensically during later computations or produce garbage results.
@@ -1107,7 +1107,7 @@
-
Initialize an independent variable .
+
Initialize an independent variable .
Parameters
[out]
out
An auto-differentiable number that is ready for use in standard computations. The operations that are performed with it are not recorded on the tape, and so should only be used when not in recording mode.
@@ -1238,7 +1238,7 @@
-
Register the definition of the index'th dependent variable .
+
Register the definition of the index'th dependent variable .
Parameters
[in]
index
The index of the entry in the global list of dependent variables that this function belongs to.
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classDifferentiation_1_1AD_1_1PointLevelFunctionsBase.html 2024-04-12 04:45:50.975571145 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classDifferentiation_1_1AD_1_1PointLevelFunctionsBase.html 2024-04-12 04:45:50.975571145 +0000
@@ -437,7 +437,7 @@
const ExtractorType &
extractor&#href_anchor"memdoc">
-
Register the subset of independent variables .
+
Register the subset of independent variables .
Parameters
[in]
value
A field that defines a number of independent variables. When considering taped AD numbers with branching functions, to avoid potential issues with branch switching it may be a good idea to choose these values close or equal to those that will be later evaluated and differentiated around.
@@ -551,7 +551,7 @@
const ExtractorType &
extractor&#href_anchor"memdoc">
-
Set the values for a subset of independent variables .
+
Set the values for a subset of independent variables .
Parameters
[in]
value
A field that defines the values of a number of independent variables.
@@ -600,7 +600,7 @@
-
Set the actual value of the independent variable .
+
Set the actual value of the independent variable .
Parameters
[in]
index
The index in the vector of independent variables.
@@ -1353,7 +1353,7 @@
-
Set the actual value of the independent variable .
+
Set the actual value of the independent variable .
Parameters
[in]
index
The index in the vector of independent variables.
@@ -1394,7 +1394,7 @@
-
Initialize an independent variable such that subsequent operations performed with it are tracked.
+
Initialize an independent variable such that subsequent operations performed with it are tracked.
Note
Care must be taken to mark each independent variable only once.
The order in which the independent variables are marked defines the order of all future internal operations. They must be manipulated in the same order as that in which they are first marked. If not then, for example, ADOL-C won't throw an error, but rather it might complain nonsensically during later computations or produce garbage results.
@@ -1469,7 +1469,7 @@
-
Initialize an independent variable .
+
Initialize an independent variable .
Parameters
[out]
out
An auto-differentiable number that is ready for use in standard computations. The operations that are performed with it are not recorded on the tape, and so should only be used when not in recording mode.
@@ -1600,7 +1600,7 @@
-
Register the definition of the index'th dependent variable .
+
Register the definition of the index'th dependent variable .
Parameters
[in]
index
The index of the entry in the global list of dependent variables that this function belongs to.
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classDifferentiation_1_1AD_1_1ResidualLinearization.html 2024-04-12 04:45:51.031571533 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classDifferentiation_1_1AD_1_1ResidualLinearization.html 2024-04-12 04:45:51.035571560 +0000
@@ -454,8 +454,8 @@
The constructor for the class.
Parameters
-
[in]
n_independent_variables
The number of independent variables that will be used in the definition of the functions that it is desired to compute the sensitivities of. In the computation of , this will be the number of inputs , i.e., the dimension of the domain space.
-
[in]
n_dependent_variables
The number of scalar functions to be defined that will have a sensitivity to the given independent variables. In the computation of , this will be the number of outputs , i.e., the dimension of the image space.
+
[in]
n_independent_variables
The number of independent variables that will be used in the definition of the functions that it is desired to compute the sensitivities of. In the computation of , this will be the number of inputs , i.e., the dimension of the domain space.
+
[in]
n_dependent_variables
The number of scalar functions to be defined that will have a sensitivity to the given independent variables. In the computation of , this will be the number of outputs , i.e., the dimension of the image space.
@@ -509,7 +509,7 @@
-
Register the definition of the cell residual vector .
+
Register the definition of the cell residual vector .
Parameters
[in]
residual
A vector of recorded functions that defines the residual. The components of this vector represents the dependent variables.
@@ -549,9 +549,9 @@
Evaluation of the residual for a chosen set of degree of freedom values. This corresponds to the computation of the residual vector, i.e.
Compute the linearization of the residual vector around a chosen set of degree of freedom values. Underlying this is the computation of the gradient (first derivative) of the residual vector with respect to all independent variables, i.e.
Set the actual value of the independent variable .
+
Set the actual value of the independent variable .
Parameters
[in]
index
The index in the vector of independent variables.
@@ -1573,7 +1573,7 @@
-
Initialize an independent variable such that subsequent operations performed with it are tracked.
+
Initialize an independent variable such that subsequent operations performed with it are tracked.
Note
Care must be taken to mark each independent variable only once.
The order in which the independent variables are marked defines the order of all future internal operations. They must be manipulated in the same order as that in which they are first marked. If not then, for example, ADOL-C won't throw an error, but rather it might complain nonsensically during later computations or produce garbage results.
@@ -1648,7 +1648,7 @@
-
Initialize an independent variable .
+
Initialize an independent variable .
Parameters
[out]
out
An auto-differentiable number that is ready for use in standard computations. The operations that are performed with it are not recorded on the tape, and so should only be used when not in recording mode.
@@ -1779,7 +1779,7 @@
-
Register the definition of the index'th dependent variable .
+
Register the definition of the index'th dependent variable .
Parameters
[in]
index
The index of the entry in the global list of dependent variables that this function belongs to.
/usr/share/doc/packages/dealii/doxygen/deal.II/classDifferentiation_1_1AD_1_1ScalarFunction.html differs (HTML document, ASCII text, with very long lines)
--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classDifferentiation_1_1AD_1_1ScalarFunction.html 2024-04-12 04:45:51.103572031 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classDifferentiation_1_1AD_1_1ScalarFunction.html 2024-04-12 04:45:51.111572087 +0000
@@ -520,7 +520,7 @@
-
Register the definition of the scalar field .
+
Register the definition of the scalar field .
Parameters
[in]
func
The recorded function that defines a dependent variable.
@@ -551,7 +551,7 @@
-
Compute the value of the scalar field using the tape as opposed to executing the source code.
+
Compute the value of the scalar field using the tape as opposed to executing the source code.
Returns
A scalar object with the value for the scalar field evaluated at the point defined by the independent variable values.
Compute the gradient (first derivative) of the scalar field with respect to all independent variables, i.e.
-
+\]" src="form_912.png"/>
Parameters
@@ -607,10 +607,10 @@
Compute the Hessian (second derivative) of the scalar field with respect to all independent variables, i.e.
-
+\]" src="form_913.png"/>
Parameters
@@ -653,10 +653,10 @@
-
Extract the function gradient for a subset of independent variables , i.e.
-, i.e.
+
+\]" src="form_914.png"/>
Parameters
@@ -704,13 +704,13 @@
-
Extract the function Hessian for a subset of independent variables , i.e.
-, i.e.
+
+\]" src="form_916.png"/>
Parameters
@@ -753,11 +753,11 @@
-
Extract the function Hessian for a subset of independent variables , i.e.
-, i.e.
+
+\]" src="form_917.png"/>
This function is a specialization of the above for rank-0 tensors (scalars). This corresponds to extracting a single entry of the Hessian matrix because both extractors imply selection of just a single row or column of the matrix.
@@ -794,11 +794,11 @@
-
Extract the function Hessian for a subset of independent variables , i.e.
-, i.e.
+
+\]" src="form_917.png"/>
This function is a specialization of the above for rank-4 symmetric tensors.
@@ -919,7 +919,7 @@
-
Register the subset of independent variables .
+
Register the subset of independent variables .
Parameters
[in]
value
A field that defines a number of independent variables. When considering taped AD numbers with branching functions, to avoid potential issues with branch switching it may be a good idea to choose these values close or equal to those that will be later evaluated and differentiated around.
@@ -1064,7 +1064,7 @@
-
Set the values for a subset of independent variables .
+
Set the values for a subset of independent variables .
Parameters
[in]
value
A field that defines the values of a number of independent variables.
@@ -1113,7 +1113,7 @@
-
Set the actual value of the independent variable .
+
Set the actual value of the independent variable .
Parameters
[in]
index
The index in the vector of independent variables.
@@ -1155,7 +1155,7 @@
-
Set the actual value of the independent variable .
+
Set the actual value of the independent variable .
Parameters
[in]
index
The index in the vector of independent variables.
@@ -1907,7 +1907,7 @@
-
Initialize an independent variable such that subsequent operations performed with it are tracked.
+
Initialize an independent variable such that subsequent operations performed with it are tracked.
Note
Care must be taken to mark each independent variable only once.
The order in which the independent variables are marked defines the order of all future internal operations. They must be manipulated in the same order as that in which they are first marked. If not then, for example, ADOL-C won't throw an error, but rather it might complain nonsensically during later computations or produce garbage results.
@@ -1982,7 +1982,7 @@
-
Initialize an independent variable .
+
Initialize an independent variable .
Parameters
[out]
out
An auto-differentiable number that is ready for use in standard computations. The operations that are performed with it are not recorded on the tape, and so should only be used when not in recording mode.
@@ -2113,7 +2113,7 @@
-
Register the definition of the index'th dependent variable .
+
Register the definition of the index'th dependent variable .
Parameters
[in]
index
The index of the entry in the global list of dependent variables that this function belongs to.
/usr/share/doc/packages/dealii/doxygen/deal.II/classDifferentiation_1_1AD_1_1VectorFunction.html differs (HTML document, ASCII text, with very long lines)
--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classDifferentiation_1_1AD_1_1VectorFunction.html 2024-04-12 04:45:51.175572530 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classDifferentiation_1_1AD_1_1VectorFunction.html 2024-04-12 04:45:51.183572585 +0000
@@ -524,7 +524,7 @@
-
Register the definition of the vector field .
+
Register the definition of the vector field .
Parameters
[in]
funcs
A vector of recorded functions that defines the dependent variables.
@@ -558,7 +558,7 @@
const ExtractorType &
extractor&#href_anchor"memdoc">
-
Register the definition of the vector field that may represent a subset of the dependent variables.
+
Register the definition of the vector field that may represent a subset of the dependent variables.
Parameters
[in]
funcs
The recorded functions that define a set of dependent variables.
@@ -588,7 +588,7 @@
-
Compute the value of the vector field .
+
Compute the value of the vector field .
Parameters
[out]
values
A Vector object with the value for each component of the vector field evaluated at the point defined by the independent variable values. The output values vector has a length corresponding to n_dependent_variables.
@@ -617,10 +617,10 @@
Compute the Jacobian (first derivative) of the vector field with respect to all independent variables, i.e.
-
+\]" src="form_920.png"/>
Parameters
@@ -663,7 +663,7 @@
-
Extract the set of functions' values for a subset of dependent variables .
+
Extract the set of functions' values for a subset of dependent variables .
Parameters
[in]
values
A Vector object with the value for each component of the vector field evaluated at the point defined by the independent variable values.
@@ -709,13 +709,13 @@
-
Extract the Jacobian of the subset of dependent functions for a subset of independent variables , i.e.
- for a subset of independent variables , i.e.
+
+\]" src="form_922.png"/>
-
The first index of the Jacobian matrix relates to the dependent variables, while the second index relates to the independent variables.
+
The first index of the Jacobian matrix relates to the dependent variables, while the second index relates to the independent variables.
Parameters
[in]
jacobian
The Jacobian of the vector function with respect to all independent variables, i.e., that returned by compute_jacobian().
@@ -757,11 +757,11 @@
-
Extract the Jacobian of the subset of dependent functions for a subset of independent variables , i.e.
- for a subset of independent variables , i.e.
+
+\]" src="form_922.png"/>
This function is a specialization of the above for rank-0 tensors (scalars). This corresponds to extracting a single entry of the Jacobian matrix because both extractors imply selection of just a single row or column of the matrix.
@@ -798,11 +798,11 @@
-
Extract the Jacobian of the subset of dependent functions for a subset of independent variables , i.e.
- for a subset of independent variables , i.e.
+
+\]" src="form_922.png"/>
This function is a specialization of the above for rank-4 symmetric tensors.
@@ -923,7 +923,7 @@
-
Register the subset of independent variables .
+
Register the subset of independent variables .
Parameters
[in]
value
A field that defines a number of independent variables. When considering taped AD numbers with branching functions, to avoid potential issues with branch switching it may be a good idea to choose these values close or equal to those that will be later evaluated and differentiated around.
@@ -1068,7 +1068,7 @@
-
Set the values for a subset of independent variables .
+
Set the values for a subset of independent variables .
Parameters
[in]
value
A field that defines the values of a number of independent variables.
@@ -1117,7 +1117,7 @@
-
Set the actual value of the independent variable .
+
Set the actual value of the independent variable .
Parameters
[in]
index
The index in the vector of independent variables.
@@ -1159,7 +1159,7 @@
-
Set the actual value of the independent variable .
+
Set the actual value of the independent variable .
Parameters
[in]
index
The index in the vector of independent variables.
@@ -1911,7 +1911,7 @@
-
Initialize an independent variable such that subsequent operations performed with it are tracked.
+
Initialize an independent variable such that subsequent operations performed with it are tracked.
Note
Care must be taken to mark each independent variable only once.
The order in which the independent variables are marked defines the order of all future internal operations. They must be manipulated in the same order as that in which they are first marked. If not then, for example, ADOL-C won't throw an error, but rather it might complain nonsensically during later computations or produce garbage results.
@@ -1986,7 +1986,7 @@
-
Initialize an independent variable .
+
Initialize an independent variable .
Parameters
[out]
out
An auto-differentiable number that is ready for use in standard computations. The operations that are performed with it are not recorded on the tape, and so should only be used when not in recording mode.
@@ -2117,7 +2117,7 @@
-
Register the definition of the index'th dependent variable .
+
Register the definition of the index'th dependent variable .
Parameters
[in]
index
The index of the entry in the global list of dependent variables that this function belongs to.
/usr/share/doc/packages/dealii/doxygen/deal.II/classDiscreteTime.html differs (HTML document, ASCII text, with very long lines)
--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classDiscreteTime.html 2024-04-12 04:45:51.215572807 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classDiscreteTime.html 2024-04-12 04:45:51.219572834 +0000
@@ -185,7 +185,7 @@
Since time is marched forward in a discrete manner in our simulations, we need to discuss how we increment time. During time stepping we enter two separate alternating regimes in every step.
The snapshot stage (the current stage, the consistent stage): In this part of the algorithm, we are at and all quantities of the simulation (displacements, strains, temperatures, etc.) are up-to-date for . In this stage, current time refers to , next time refers to , previous time refers to . The other useful notation quantities are the next time step size and previous time step size . In this stage, it is a perfect occasion to generate text output using print commands within the user's code. Additionally, post-processed outputs can be prepared here, which can then later be viewed by visualization programs such as Tecplot, Paraview, and VisIt. Additionally, during the snapshot stage, the code can assess the quality of the previous step and decide whether it wants to increase or decrease the time step size. The step size for the next time step can be modified here, by calling set_desired_next_step_size().
-
The update stage (the transition stage, the inconsistent stage): In this section of the program, the internal state of the simulation is getting updated from to . All of the variables need to be updated one by one, the step number is incremented, the time is incremented by , and time-integration algorithms are used to update the other simulation quantities. In the middle of this stage, some variables have been updated to but other variables still represent their value at . Thus, we call this the inconsistent stage, requiring that no post-processing output related to the state variables take place within it. The state variables, namely those related to time, the solution field and any internal variables, are not synchronized and then get updated one by one. In general, the order of updating variables is arbitrary, but some care should be taken if there are interdependencies between them. For example, if some variable such as depends on the calculation of another variable such as , then must be updated before can be updated.
+
The update stage (the transition stage, the inconsistent stage): In this section of the program, the internal state of the simulation is getting updated from to . All of the variables need to be updated one by one, the step number is incremented, the time is incremented by , and time-integration algorithms are used to update the other simulation quantities. In the middle of this stage, some variables have been updated to but other variables still represent their value at . Thus, we call this the inconsistent stage, requiring that no post-processing output related to the state variables take place within it. The state variables, namely those related to time, the solution field and any internal variables, are not synchronized and then get updated one by one. In general, the order of updating variables is arbitrary, but some care should be taken if there are interdependencies between them. For example, if some variable such as depends on the calculation of another variable such as , then must be updated before can be updated.
The question arises whether time should be incremented before updating state quantities. Multiple possibilities exist, depending on program and formulation requirements, and possibly the programmer's preferences:
Time is incremented before the rest of the updates. In this case, even though time is incremented to , not all variables are updated yet. During this update phase, equals the previous time step size. Previous means that it is referring to the of the advance_time() command that was performed previously. In the following example code, we are assuming that a and b are two state variables that need to be updated in this time step.
Given a triangulation and a description of a finite element, this class enumerates degrees of freedom on all vertices, edges, faces, and cells of the triangulation. As a result, it also provides a basis for a discrete space whose elements are finite element functions defined on each cell by a FiniteElement object. This class satisfies the MeshType concept requirements.
+class DoFHandler< dim, spacedim >
Given a triangulation and a description of a finite element, this class enumerates degrees of freedom on all vertices, edges, faces, and cells of the triangulation. As a result, it also provides a basis for a discrete space whose elements are finite element functions defined on each cell by a FiniteElement object. This class satisfies the MeshType concept requirements.
For each 0d, 1d, 2d, and 3d subobject, this class stores a list of the indices of degrees of freedom defined on this DoFHandler. These indices refer to the unconstrained degrees of freedom, i.e. constrained degrees of freedom are numbered in the same way as unconstrained ones, and are only later eliminated. This leads to the fact that indices in global vectors and matrices also refer to all degrees of freedom and some kind of condensation is needed to restrict the systems of equations to the unconstrained degrees of freedom only. The actual layout of storage of the indices is described in the internal::DoFHandlerImplementation::DoFLevel class documentation.
The class offers iterators to traverse all cells, in much the same way as the Triangulation class does. Using the begin() and end() functions (and companions, like begin_active()), one can obtain iterators to walk over cells, and query the degree of freedom structures as well as the triangulation data. These iterators are built on top of those of the Triangulation class, but offer the additional information on degrees of freedom functionality compared to pure triangulation iterators. The order in which dof iterators are presented by the ++ and -- operators is the same as that for the corresponding iterators traversing the triangulation on which this DoFHandler is constructed.
@@ -434,7 +434,7 @@
Like many other classes in deal.II, the DoFHandler class can stream its contents to an archive using BOOST's serialization facilities. The data so stored can later be retrieved again from the archive to restore the contents of this object. This facility is frequently used to save the state of a program to disk for possible later resurrection, often in the context of checkpoint/restart strategies for long running computations or on computers that aren't very reliable (e.g. on very large clusters where individual nodes occasionally fail and then bring down an entire MPI job).
The model for doing so is similar for the DoFHandler class as it is for the Triangulation class (see the section in the general documentation of that class). In particular, the load() function does not exactly restore the same state as was stored previously using the save() function. Rather, the function assumes that you load data into a DoFHandler object that is already associated with a triangulation that has a content that matches the one that was used when the data was saved. Likewise, the load() function assumes that the current object is already associated with a finite element object that matches the one that was associated with it when data was saved; the latter can be achieved by calling DoFHandler::distribute_dofs() using the same kind of finite element before re-loading data from the serialization archive.
hp-adaptive finite element methods
-
Instead of only using one particular FiniteElement on all cells, this class also allows for an enumeration of degrees of freedom on different finite elements on every cells. To this end, one assigns an active_fe_index to every cell that indicates which element within a collection of finite elements (represented by an object of type hp::FECollection) is the one that lives on this cell. The class then enumerates the degree of freedom associated with these finite elements on each cell of a triangulation and, if possible, identifies degrees of freedom at the interfaces of cells if they match. If neighboring cells have degrees of freedom along the common interface that do not immediate match (for example, if you have and elements meeting at a common face), then one needs to compute constraints to ensure that the resulting finite element space on the mesh remains conforming.
+
Instead of only using one particular FiniteElement on all cells, this class also allows for an enumeration of degrees of freedom on different finite elements on every cells. To this end, one assigns an active_fe_index to every cell that indicates which element within a collection of finite elements (represented by an object of type hp::FECollection) is the one that lives on this cell. The class then enumerates the degree of freedom associated with these finite elements on each cell of a triangulation and, if possible, identifies degrees of freedom at the interfaces of cells if they match. If neighboring cells have degrees of freedom along the common interface that do not immediate match (for example, if you have and elements meeting at a common face), then one needs to compute constraints to ensure that the resulting finite element space on the mesh remains conforming.
The whole process of working with objects of this type is explained in step-27. Many of the algorithms this class implements are described in the hp-paper.
Active FE indices and their behavior under mesh refinement
The typical workflow for using this class is to create a mesh, assign an active FE index to every active cell, call DoFHandler::distribute_dofs(), and then assemble a linear system and solve a problem on this finite element space.
@@ -983,7 +983,7 @@
-
Go through the triangulation and "distribute" the degrees of freedom needed for the given finite element. "Distributing" degrees of freedom involves allocating memory to store the indices on all entities on which degrees of freedom can be located (e.g., vertices, edges, faces, etc.) and to then enumerate all degrees of freedom. In other words, while the mesh and the finite element object by themselves simply define a finite element space , the process of distributing degrees of freedom makes sure that there is a basis for this space and that the shape functions of this basis are enumerated in an indexable, predictable way.
+
Go through the triangulation and "distribute" the degrees of freedom needed for the given finite element. "Distributing" degrees of freedom involves allocating memory to store the indices on all entities on which degrees of freedom can be located (e.g., vertices, edges, faces, etc.) and to then enumerate all degrees of freedom. In other words, while the mesh and the finite element object by themselves simply define a finite element space , the process of distributing degrees of freedom makes sure that there is a basis for this space and that the shape functions of this basis are enumerated in an indexable, predictable way.
The exact order in which degrees of freedom on a mesh are ordered, i.e., the order in which basis functions of the finite element space are enumerated, is something that deal.II treats as an implementation detail. By and large, degrees of freedom are enumerated in the same order in which we traverse cells, but you should not rely on any specific numbering. In contrast, if you want a particular ordering, use the functions in namespace DoFRenumbering.
This function is first discussed in the introduction to the step-2 tutorial program.
Note
This function makes a copy of the finite element given as argument, and stores it as a member variable, similarly to the above function set_fe().
/usr/share/doc/packages/dealii/doxygen/deal.II/classDynamicSparsityPattern.html differs (HTML document, ASCII text, with very long lines)
--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classDynamicSparsityPattern.html 2024-04-12 04:45:51.359573804 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classDynamicSparsityPattern.html 2024-04-12 04:45:51.363573831 +0000
@@ -1106,7 +1106,7 @@
-
Compute the bandwidth of the matrix represented by this structure. The bandwidth is the maximum of for which the index pair represents a nonzero entry of the matrix.
+
Compute the bandwidth of the matrix represented by this structure. The bandwidth is the maximum of for which the index pair represents a nonzero entry of the matrix.
/usr/share/doc/packages/dealii/doxygen/deal.II/classEigenInverse.html differs (HTML document, ASCII text, with very long lines)
--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classEigenInverse.html 2024-04-12 04:45:51.391574025 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classEigenInverse.html 2024-04-12 04:45:51.399574081 +0000
@@ -191,7 +191,7 @@
template<typename VectorType = Vector<double>>
class EigenInverse< VectorType >
Inverse iteration (Wieland) for eigenvalue computations.
This class implements an adaptive version of the inverse iteration by Wieland.
-
There are two choices for the stopping criterion: by default, the norm of the residual is computed. Since this might not converge to zero for non-symmetric matrices with non-trivial Jordan blocks, it can be replaced by checking the difference of successive eigenvalues. Use AdditionalData::use_residual for switching this option.
+
There are two choices for the stopping criterion: by default, the norm of the residual is computed. Since this might not converge to zero for non-symmetric matrices with non-trivial Jordan blocks, it can be replaced by checking the difference of successive eigenvalues. Use AdditionalData::use_residual for switching this option.
Usually, the initial guess entering this method is updated after each step, replacing it with the new approximation of the eigenvalue. Using a parameter AdditionalData::relaxation between 0 and 1, this update can be damped. With relaxation parameter 0, no update is performed. This damping allows for slower adaption of the shift value to make sure that the method converges to the eigenvalue closest to the initial guess. This can be aided by the parameter AdditionalData::start_adaption, which indicates the first iteration step in which the shift value should be adapted.
/usr/share/doc/packages/dealii/doxygen/deal.II/classEigenPower.html differs (HTML document, ASCII text, with very long lines)
--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classEigenPower.html 2024-04-12 04:45:51.435574330 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classEigenPower.html 2024-04-12 04:45:51.439574358 +0000
@@ -190,7 +190,7 @@
Detailed Description
template<typename VectorType = Vector<double>>
class EigenPower< VectorType >
Power method (von Mises) for eigenvalue computations.
-
This method determines the largest eigenvalue of a matrix by applying increasing powers of this matrix to a vector. If there is an eigenvalue with dominant absolute value, the iteration vectors will become aligned to its eigenspace and .
+
This method determines the largest eigenvalue of a matrix by applying increasing powers of this matrix to a vector. If there is an eigenvalue with dominant absolute value, the iteration vectors will become aligned to its eigenspace and .
A shift parameter allows to shift the spectrum, so it is possible to compute the smallest eigenvalue, too.
Convergence of this method is known to be slow.
/usr/share/doc/packages/dealii/doxygen/deal.II/classEllipticalManifold.html differs (HTML document, ASCII text, with very long lines)
--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classEllipticalManifold.html 2024-04-12 04:45:51.491574718 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classEllipticalManifold.html 2024-04-12 04:45:51.499574773 +0000
@@ -223,16 +223,16 @@
Detailed Description
template<int dim, int spacedim = dim>
class EllipticalManifold< dim, spacedim >
Elliptical manifold description derived from ChartManifold. More information on the elliptical coordinate system can be found at Wikipedia .
-
This is based on the definition of elliptic coordinates
-
+
+\]" src="form_1454.png"/>
-
in which are coordinates of the center of the cartesian system.
-
The current implementation uses coordinates , instead of , and fixes according to a given eccentricity. Therefore, this choice of coordinates generates an elliptical manifold characterized by a constant eccentricity: , with .
+
in which are coordinates of the center of the cartesian system.
+
The current implementation uses coordinates , instead of , and fixes according to a given eccentricity. Therefore, this choice of coordinates generates an elliptical manifold characterized by a constant eccentricity: , with .
The constructor of this class will throw an exception if both dim and spacedim are different from two.
This manifold can be used to produce hyper_shells with elliptical curvature. As an example, the test elliptical_manifold_01 produces the following triangulation:
@@ -348,7 +348,7 @@
center
Center of the manifold.
major_axis_direction
Direction of the major axis of the manifold.
-
eccentricity
Eccentricity of the manifold .
+
eccentricity
Eccentricity of the manifold .
@@ -485,7 +485,7 @@
-
Given a point in the chartdim dimensional Euclidean space, this method returns the derivatives of the function that maps from the chartdim-dimensional to the spacedim-dimensional space. In other words, it is a matrix of size .
+
Given a point in the chartdim dimensional Euclidean space, this method returns the derivatives of the function that maps from the chartdim-dimensional to the spacedim-dimensional space. In other words, it is a matrix of size .
This function is used in the computations required by the get_tangent_vector() function. Since not all users of the Manifold class interface will require calling that function, the current function is implemented but will trigger an exception whenever called. This allows derived classes to avoid implementing the push_forward_gradient function if this functionality is not needed in the user program.
Refer to the general documentation of this class for more information.
@@ -520,7 +520,7 @@
Return the periodicity associated with the submanifold.
-
For and , the first coordinate is non-periodic, while the second coordinate has a periodicity of .
+
For and , the first coordinate is non-periodic, while the second coordinate has a periodicity of .
Given a point in the chartdim dimensional Euclidean space, this method returns the derivatives of the function that maps from the chartdim-dimensional to the spacedim-dimensional space. In other words, it is a matrix of size .
+
Given a point in the chartdim dimensional Euclidean space, this method returns the derivatives of the function that maps from the chartdim-dimensional to the spacedim-dimensional space. In other words, it is a matrix of size .
This function is used in the computations required by the get_tangent_vector() function. Since not all users of the Manifold class interface will require calling that function, the current function is implemented but will trigger an exception whenever called. This allows derived classes to avoid implementing the push_forward_gradient function if this functionality is not needed in the user program.
Refer to the general documentation of this class for more information.
@@ -782,24 +782,24 @@
-
Return a vector that, at , is tangential to the geodesic that connects two points . See the documentation of the Manifold class and of Manifold::get_tangent_vector() for a more detailed description.
-
For the current class, we assume that this geodesic is the image under the push_forward() operation of a straight line of the pre-images of x1 and x2 (where pre-images are computed by pulling back the locations x1 and x2). In other words, if these preimages are , then the geodesic in preimage (the chartdim-dimensional Euclidean) space is
-, is tangential to the geodesic that connects two points . See the documentation of the Manifold class and of Manifold::get_tangent_vector() for a more detailed description.
+
For the current class, we assume that this geodesic is the image under the push_forward() operation of a straight line of the pre-images of x1 and x2 (where pre-images are computed by pulling back the locations x1 and x2). In other words, if these preimages are , then the geodesic in preimage (the chartdim-dimensional Euclidean) space is
+
+\end{align*}" src="form_1440.png"/>
In image space, i.e., in the space in which we operate, this leads to the curve
-
+\end{align*}" src="form_1441.png"/>
-
What the current function is supposed to return is . By the chain rule, this is equal to
-. By the chain rule, this is equal to
+
+\end{align*}" src="form_1442.png"/>
This formula may then have to be slightly modified by considering any periodicity that was assumed in the call to the constructor.
-
Thus, the computation of tangent vectors also requires the implementation of derivatives of the push-forward mapping. Here, is a chartdim-dimensional vector, and is a spacedim-times-chartdim-dimensional matrix. Consequently, and as desired, the operation results in a spacedim-dimensional vector.
+
Thus, the computation of tangent vectors also requires the implementation of derivatives of the push-forward mapping. Here, is a chartdim-dimensional vector, and is a spacedim-times-chartdim-dimensional matrix. Consequently, and as desired, the operation results in a spacedim-dimensional vector.
Parameters
x1
The first point that describes the geodesic, and the one at which the "direction" is to be evaluated.
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+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classFEEvaluation.html 2024-04-12 04:45:51.623575632 +0000
@@ -459,7 +459,7 @@
Likewise, a gradient of the finite element solution represented by vector can be interpolated to the quadrature points by fe_eval.get_gradient(q). The combination of read_dof_values(), evaluate() and get_value() is similar to what FEValues::get_function_values or FEValues::get_function_gradients does, but it is in general much faster because it makes use of the tensor product, see the description of the evaluation routines below, and can do this operation for several cells at once through vectorization.
-
The second class of tasks done by FEEvaluation are integration tasks for right hand sides. In finite element computations, these typically consist of multiplying a quantity on quadrature points (a function value, or a field interpolated by the finite element space itself) by a set of test functions and integrating over the cell through summation of the values in each quadrature point, multiplied by the quadrature weight and the Jacobian determinant of the transformation. If a generic Function object is given and we want to compute , this is done by the following cell-wise integration:
+
The second class of tasks done by FEEvaluation are integration tasks for right hand sides. In finite element computations, these typically consist of multiplying a quantity on quadrature points (a function value, or a field interpolated by the finite element space itself) by a set of test functions and integrating over the cell through summation of the values in each quadrature point, multiplied by the quadrature weight and the Jacobian determinant of the transformation. If a generic Function object is given and we want to compute , this is done by the following cell-wise integration:
Return the derivative of a finite element function at quadrature point number q_point after a call to FEEvaluation::evaluate(EvaluationFlags::gradients) the direction normal to the face:
+
Return the derivative of a finite element function at quadrature point number q_point after a call to FEEvaluation::evaluate(EvaluationFlags::gradients) the direction normal to the face:
This call is equivalent to calling get_gradient() * normal_vector() but will use a more efficient internal representation of data.
Note
The derived class FEEvaluationAccess overloads this operation with specializations for the scalar case (n_components == 1) and for the vector-valued case (n_components == dim).
@@ -2209,7 +2209,7 @@
-
Return the curl of the vector field, after a call to evaluate(EvaluationFlags::gradients).
+
Return the curl of the vector field, after a call to evaluate(EvaluationFlags::gradients).
Note
Only available for the vector-valued case (n_components == dim).
@@ -2659,8 +2659,8 @@
-
Return the inverse and transposed version of the Jacobian of the mapping between the unit to the real cell defined as . The entry of the returned tensor contains , i.e., columns refer to reference space coordinates and rows to real cell coordinates. Thus, the returned tensor represents a covariant transformation, which is used in the FEEvaluationBase::get_gradient() function to transform the unit cell gradients to gradients on the real cell by a multiplication .
+
Return the inverse and transposed version of the Jacobian of the mapping between the unit to the real cell defined as . The entry of the returned tensor contains , i.e., columns refer to reference space coordinates and rows to real cell coordinates. Thus, the returned tensor represents a covariant transformation, which is used in the FEEvaluationBase::get_gradient() function to transform the unit cell gradients to gradients on the real cell by a multiplication .
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classFEEvaluationAccess.html 2024-04-12 04:45:51.723576324 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classFEEvaluationAccess.html 2024-04-12 04:45:51.723576324 +0000
@@ -1154,8 +1154,8 @@
-
Return the derivative of a finite element function at quadrature point number q_point after a call to FEEvaluation::evaluate(EvaluationFlags::gradients) the direction normal to the face:
+
Return the derivative of a finite element function at quadrature point number q_point after a call to FEEvaluation::evaluate(EvaluationFlags::gradients) the direction normal to the face:
This call is equivalent to calling get_gradient() * normal_vector() but will use a more efficient internal representation of data.
Note
The derived class FEEvaluationAccess overloads this operation with specializations for the scalar case (n_components == 1) and for the vector-valued case (n_components == dim).
@@ -1419,7 +1419,7 @@
-
Return the curl of the vector field, after a call to evaluate(EvaluationFlags::gradients).
+
Return the curl of the vector field, after a call to evaluate(EvaluationFlags::gradients).
Note
Only available for the vector-valued case (n_components == dim).
@@ -1869,8 +1869,8 @@
-
Return the inverse and transposed version of the Jacobian of the mapping between the unit to the real cell defined as . The entry of the returned tensor contains , i.e., columns refer to reference space coordinates and rows to real cell coordinates. Thus, the returned tensor represents a covariant transformation, which is used in the FEEvaluationBase::get_gradient() function to transform the unit cell gradients to gradients on the real cell by a multiplication .
+
Return the inverse and transposed version of the Jacobian of the mapping between the unit to the real cell defined as . The entry of the returned tensor contains , i.e., columns refer to reference space coordinates and rows to real cell coordinates. Thus, the returned tensor represents a covariant transformation, which is used in the FEEvaluationBase::get_gradient() function to transform the unit cell gradients to gradients on the real cell by a multiplication .
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classFEEvaluationAccess_3_011_00_011_00_01Number_00_01is__face_00_01VectorizedArrayType_01_4.html 2024-04-12 04:45:51.815576961 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classFEEvaluationAccess_3_011_00_011_00_01Number_00_01is__face_00_01VectorizedArrayType_01_4.html 2024-04-12 04:45:51.823577016 +0000
@@ -940,8 +940,8 @@
-
Return the derivative of a finite element function at quadrature point number q_point after a call to FEEvaluation::evaluate(EvaluationFlags::gradients) the direction normal to the face:
+
Return the derivative of a finite element function at quadrature point number q_point after a call to FEEvaluation::evaluate(EvaluationFlags::gradients) the direction normal to the face:
This call is equivalent to calling get_gradient() * normal_vector() but will use a more efficient internal representation of data.
Note
The derived class FEEvaluationAccess overloads this operation with specializations for the scalar case (n_components == 1) and for the vector-valued case (n_components == dim).
@@ -1567,7 +1567,7 @@
-
Return the curl of the vector field, after a call to evaluate(EvaluationFlags::gradients).
+
Return the curl of the vector field, after a call to evaluate(EvaluationFlags::gradients).
Note
Only available for the vector-valued case (n_components == dim).
@@ -1966,8 +1966,8 @@
-
Return the inverse and transposed version of the Jacobian of the mapping between the unit to the real cell defined as . The entry of the returned tensor contains , i.e., columns refer to reference space coordinates and rows to real cell coordinates. Thus, the returned tensor represents a covariant transformation, which is used in the FEEvaluationBase::get_gradient() function to transform the unit cell gradients to gradients on the real cell by a multiplication .
+
Return the inverse and transposed version of the Jacobian of the mapping between the unit to the real cell defined as . The entry of the returned tensor contains , i.e., columns refer to reference space coordinates and rows to real cell coordinates. Thus, the returned tensor represents a covariant transformation, which is used in the FEEvaluationBase::get_gradient() function to transform the unit cell gradients to gradients on the real cell by a multiplication .
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classFEEvaluationAccess_3_01dim_00_011_00_01Number_00_01is__face_00_01VectorizedArrayType_01_4.html 2024-04-12 04:45:51.915577653 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classFEEvaluationAccess_3_01dim_00_011_00_01Number_00_01is__face_00_01VectorizedArrayType_01_4.html 2024-04-12 04:45:51.919577682 +0000
@@ -914,8 +914,8 @@
-
Return the derivative of a finite element function at quadrature point number q_point after a call to FEEvaluation::evaluate(EvaluationFlags::gradients) the direction normal to the face:
+
Return the derivative of a finite element function at quadrature point number q_point after a call to FEEvaluation::evaluate(EvaluationFlags::gradients) the direction normal to the face:
This call is equivalent to calling get_gradient() * normal_vector() but will use a more efficient internal representation of data.
Note
The derived class FEEvaluationAccess overloads this operation with specializations for the scalar case (n_components == 1) and for the vector-valued case (n_components == dim).
@@ -1494,7 +1494,7 @@
-
Return the curl of the vector field, after a call to evaluate(EvaluationFlags::gradients).
+
Return the curl of the vector field, after a call to evaluate(EvaluationFlags::gradients).
Note
Only available for the vector-valued case (n_components == dim).
@@ -1893,8 +1893,8 @@
-
Return the inverse and transposed version of the Jacobian of the mapping between the unit to the real cell defined as . The entry of the returned tensor contains , i.e., columns refer to reference space coordinates and rows to real cell coordinates. Thus, the returned tensor represents a covariant transformation, which is used in the FEEvaluationBase::get_gradient() function to transform the unit cell gradients to gradients on the real cell by a multiplication .
+
Return the inverse and transposed version of the Jacobian of the mapping between the unit to the real cell defined as . The entry of the returned tensor contains , i.e., columns refer to reference space coordinates and rows to real cell coordinates. Thus, the returned tensor represents a covariant transformation, which is used in the FEEvaluationBase::get_gradient() function to transform the unit cell gradients to gradients on the real cell by a multiplication .
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classFEEvaluationAccess_3_01dim_00_01dim_00_01Number_00_01is__face_00_01VectorizedArrayType_01_4.html 2024-04-12 04:45:52.011578319 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classFEEvaluationAccess_3_01dim_00_01dim_00_01Number_00_01is__face_00_01VectorizedArrayType_01_4.html 2024-04-12 04:45:52.011578319 +0000
@@ -860,7 +860,7 @@
-
Return the curl of the vector field, after a call to evaluate(EvaluationFlags::gradients).
+
Return the curl of the vector field, after a call to evaluate(EvaluationFlags::gradients).
@@ -1339,8 +1339,8 @@
-
Return the derivative of a finite element function at quadrature point number q_point after a call to FEEvaluation::evaluate(EvaluationFlags::gradients) the direction normal to the face:
+
Return the derivative of a finite element function at quadrature point number q_point after a call to FEEvaluation::evaluate(EvaluationFlags::gradients) the direction normal to the face:
This call is equivalent to calling get_gradient() * normal_vector() but will use a more efficient internal representation of data.
Note
The derived class FEEvaluationAccess overloads this operation with specializations for the scalar case (n_components == 1) and for the vector-valued case (n_components == dim).
@@ -1790,8 +1790,8 @@
-
Return the inverse and transposed version of the Jacobian of the mapping between the unit to the real cell defined as . The entry of the returned tensor contains , i.e., columns refer to reference space coordinates and rows to real cell coordinates. Thus, the returned tensor represents a covariant transformation, which is used in the FEEvaluationBase::get_gradient() function to transform the unit cell gradients to gradients on the real cell by a multiplication .
+
Return the inverse and transposed version of the Jacobian of the mapping between the unit to the real cell defined as . The entry of the returned tensor contains , i.e., columns refer to reference space coordinates and rows to real cell coordinates. Thus, the returned tensor represents a covariant transformation, which is used in the FEEvaluationBase::get_gradient() function to transform the unit cell gradients to gradients on the real cell by a multiplication .
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@@ -1053,8 +1053,8 @@
-
Return the derivative of a finite element function at quadrature point number q_point after a call to FEEvaluation::evaluate(EvaluationFlags::gradients) the direction normal to the face:
+
Return the derivative of a finite element function at quadrature point number q_point after a call to FEEvaluation::evaluate(EvaluationFlags::gradients) the direction normal to the face:
This call is equivalent to calling get_gradient() * normal_vector() but will use a more efficient internal representation of data.
Note
The derived class FEEvaluationAccess overloads this operation with specializations for the scalar case (n_components == 1) and for the vector-valued case (n_components == dim).
@@ -1249,7 +1249,7 @@
-
Return the curl of the vector field, after a call to evaluate(EvaluationFlags::gradients).
+
Return the curl of the vector field, after a call to evaluate(EvaluationFlags::gradients).
Note
Only available for the vector-valued case (n_components == dim).
@@ -1690,8 +1690,8 @@
-
Return the inverse and transposed version of the Jacobian of the mapping between the unit to the real cell defined as . The entry of the returned tensor contains , i.e., columns refer to reference space coordinates and rows to real cell coordinates. Thus, the returned tensor represents a covariant transformation, which is used in the FEEvaluationBase::get_gradient() function to transform the unit cell gradients to gradients on the real cell by a multiplication .
+
Return the inverse and transposed version of the Jacobian of the mapping between the unit to the real cell defined as . The entry of the returned tensor contains , i.e., columns refer to reference space coordinates and rows to real cell coordinates. Thus, the returned tensor represents a covariant transformation, which is used in the FEEvaluationBase::get_gradient() function to transform the unit cell gradients to gradients on the real cell by a multiplication .
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classFEEvaluationData.html 2024-04-12 04:45:52.175579453 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classFEEvaluationData.html 2024-04-12 04:45:52.179579482 +0000
@@ -768,8 +768,8 @@
-
Return the inverse and transposed version of the Jacobian of the mapping between the unit to the real cell defined as . The entry of the returned tensor contains , i.e., columns refer to reference space coordinates and rows to real cell coordinates. Thus, the returned tensor represents a covariant transformation, which is used in the FEEvaluationBase::get_gradient() function to transform the unit cell gradients to gradients on the real cell by a multiplication .
+
Return the inverse and transposed version of the Jacobian of the mapping between the unit to the real cell defined as . The entry of the returned tensor contains , i.e., columns refer to reference space coordinates and rows to real cell coordinates. Thus, the returned tensor represents a covariant transformation, which is used in the FEEvaluationBase::get_gradient() function to transform the unit cell gradients to gradients on the real cell by a multiplication .
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classFEFaceEvaluation.html 2024-04-12 04:45:52.287580229 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classFEFaceEvaluation.html 2024-04-12 04:45:52.295580285 +0000
@@ -1579,8 +1579,8 @@
-
Return the derivative of a finite element function at quadrature point number q_point after a call to FEEvaluation::evaluate(EvaluationFlags::gradients) the direction normal to the face:
+
Return the derivative of a finite element function at quadrature point number q_point after a call to FEEvaluation::evaluate(EvaluationFlags::gradients) the direction normal to the face:
This call is equivalent to calling get_gradient() * normal_vector() but will use a more efficient internal representation of data.
Note
The derived class FEEvaluationAccess overloads this operation with specializations for the scalar case (n_components == 1) and for the vector-valued case (n_components == dim).
@@ -1844,7 +1844,7 @@
-
Return the curl of the vector field, after a call to evaluate(EvaluationFlags::gradients).
+
Return the curl of the vector field, after a call to evaluate(EvaluationFlags::gradients).
Note
Only available for the vector-valued case (n_components == dim).
@@ -2294,8 +2294,8 @@
-
Return the inverse and transposed version of the Jacobian of the mapping between the unit to the real cell defined as . The entry of the returned tensor contains , i.e., columns refer to reference space coordinates and rows to real cell coordinates. Thus, the returned tensor represents a covariant transformation, which is used in the FEEvaluationBase::get_gradient() function to transform the unit cell gradients to gradients on the real cell by a multiplication .
+
Return the inverse and transposed version of the Jacobian of the mapping between the unit to the real cell defined as . The entry of the returned tensor contains , i.e., columns refer to reference space coordinates and rows to real cell coordinates. Thus, the returned tensor represents a covariant transformation, which is used in the FEEvaluationBase::get_gradient() function to transform the unit cell gradients to gradients on the real cell by a multiplication .
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--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classFEFaceValues.html 2024-04-12 04:45:52.403581032 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classFEFaceValues.html 2024-04-12 04:45:52.407581060 +0000
@@ -931,7 +931,7 @@
If the shape function is vector-valued, then this returns the only non- zero component. If the shape function has more than one non-zero component (i.e. it is not primitive), then throw an exception of type ExcShapeFunctionNotPrimitive. In that case, use the shape_value_component() function.
Parameters
-
i
Number of the shape function to be evaluated. Note that this number runs from zero to dofs_per_cell, even in the case of an FEFaceValues or FESubfaceValues object.
+
i
Number of the shape function to be evaluated. Note that this number runs from zero to dofs_per_cell, even in the case of an FEFaceValues or FESubfaceValues object.
q_point
Number of the quadrature point at which function is to be evaluated
@@ -972,7 +972,7 @@
Compute one vector component of the value of a shape function at a quadrature point. If the finite element is scalar, then only component zero is allowed and the return value equals that of the shape_value() function. If the finite element is vector valued but all shape functions are primitive (i.e. they are non-zero in only one component), then the value returned by shape_value() equals that of this function for exactly one component. This function is therefore only of greater interest if the shape function is not primitive, but then it is necessary since the other function cannot be used.
Parameters
-
i
Number of the shape function to be evaluated.
+
i
Number of the shape function to be evaluated.
q_point
Number of the quadrature point at which function is to be evaluated.
component
vector component to be evaluated.
@@ -1011,7 +1011,7 @@
The same holds for the arguments of this function as for the shape_value() function.
Parameters
-
i
Number of the shape function to be evaluated.
+
i
Number of the shape function to be evaluated.
q_point
Number of the quadrature point at which function is to be evaluated.
@@ -1213,17 +1213,17 @@
-
Return the values of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and the related get_function_gradients() function is also used in step-15 along with numerous other tutorial programs.
+
Return the values of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and the related get_function_gradients() function is also used in step-15 along with numerous other tutorial programs.
If the current cell is not active (i.e., it has children), then the finite element function is, strictly speaking, defined by shape functions that live on these child cells. Rather than evaluating the shape functions on the child cells, with the quadrature points defined on the current cell, this function first interpolates the finite element function to shape functions defined on the current cell, and then evaluates this interpolated function.
This function may only be used if the finite element in use is a scalar one, i.e. has only one vector component. To get values of multi-component elements, there is another get_function_values() below, returning a vector of vectors of results.
Parameters
[in]
fe_function
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
values
The values of the function specified by fe_function at the quadrature points of the current cell. The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the values of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the solution vector.
+
[out]
values
The values of the function specified by fe_function at the quadrature points of the current cell. The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the values of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the solution vector.
-
Postcondition
values[q] will contain the value of the field described by fe_function at the th quadrature point.
+
Postcondition
values[q] will contain the value of the field described by fe_function at the th quadrature point.
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_values(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
values[q] is a vector of values of the field described by fe_function at the th quadrature point. The size of the vector accessed by values[q] equals the number of components of the finite element, i.e. values[q](c) returns the value of the th vector component at the th quadrature point.
+
Postcondition
values[q] is a vector of values of the field described by fe_function at the th quadrature point. The size of the vector accessed by values[q] equals the number of components of the finite element, i.e. values[q](c) returns the value of the th vector component at the th quadrature point.
Return the gradients of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and it is also used in step-15 along with numerous other tutorial programs.
+
Return the gradients of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and it is also used in step-15 along with numerous other tutorial programs.
This function may only be used if the finite element in use is a scalar one, i.e. has only one vector component. There is a corresponding function of the same name for vector-valued finite elements.
Parameters
[in]
fe_function
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
gradients
The gradients of the function specified by fe_function at the quadrature points of the current cell. The gradients are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the gradients of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
[out]
gradients
The gradients of the function specified by fe_function at the quadrature points of the current cell. The gradients are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the gradients of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
-
Postcondition
gradients[q] will contain the gradient of the field described by fe_function at the th quadrature point. gradients[q][d] represents the derivative in coordinate direction at quadrature point .
+
Postcondition
gradients[q] will contain the gradient of the field described by fe_function at the th quadrature point. gradients[q][d] represents the derivative in coordinate direction at quadrature point .
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_gradients(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
gradients[q] is a vector of gradients of the field described by fe_function at the th quadrature point. The size of the vector accessed by gradients[q] equals the number of components of the finite element, i.e. gradients[q][c] returns the gradient of the th vector component at the th quadrature point. Consequently, gradients[q][c][d] is the derivative in coordinate direction of the th vector component of the vector field at quadrature point of the current cell.
+
Postcondition
gradients[q] is a vector of gradients of the field described by fe_function at the th quadrature point. The size of the vector accessed by gradients[q] equals the number of components of the finite element, i.e. gradients[q][c] returns the gradient of the th vector component at the th quadrature point. Consequently, gradients[q][c][d] is the derivative in coordinate direction of the th vector component of the vector field at quadrature point of the current cell.
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
hessians
The Hessians of the function specified by fe_function at the quadrature points of the current cell. The Hessians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Hessians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
[out]
hessians
The Hessians of the function specified by fe_function at the quadrature points of the current cell. The Hessians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Hessians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
-
Postcondition
hessians[q] will contain the Hessian of the field described by fe_function at the th quadrature point. hessians[q][i][j] represents the th component of the matrix of second derivatives at quadrature point .
+
Postcondition
hessians[q] will contain the Hessian of the field described by fe_function at the th quadrature point. hessians[q][i][j] represents the th component of the matrix of second derivatives at quadrature point .
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_hessians(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
hessians[q] is a vector of Hessians of the field described by fe_function at the th quadrature point. The size of the vector accessed by hessians[q] equals the number of components of the finite element, i.e. hessians[q][c] returns the Hessian of the th vector component at the th quadrature point. Consequently, hessians[q][c][i][j] is the th component of the matrix of second derivatives of the th vector component of the vector field at quadrature point of the current cell.
+
Postcondition
hessians[q] is a vector of Hessians of the field described by fe_function at the th quadrature point. The size of the vector accessed by hessians[q] equals the number of components of the finite element, i.e. hessians[q][c] returns the Hessian of the th vector component at the th quadrature point. Consequently, hessians[q][c][i][j] is the th component of the matrix of second derivatives of the th vector component of the vector field at quadrature point of the current cell.
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
laplacians
The Laplacians of the function specified by fe_function at the quadrature points of the current cell. The Laplacians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Laplacians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the input vector.
+
[out]
laplacians
The Laplacians of the function specified by fe_function at the quadrature points of the current cell. The Laplacians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Laplacians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the input vector.
-
Postcondition
laplacians[q] will contain the Laplacian of the field described by fe_function at the th quadrature point.
+
Postcondition
laplacians[q] will contain the Laplacian of the field described by fe_function at the th quadrature point.
For each component of the output vector, there holds laplacians[q]=trace(hessians[q]), where hessians would be the output of the get_function_hessians() function.
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
@@ -1801,7 +1801,7 @@
This function does the same as the other get_function_laplacians(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
laplacians[q] is a vector of Laplacians of the field described by fe_function at the th quadrature point. The size of the vector accessed by laplacians[q] equals the number of components of the finite element, i.e. laplacians[q][c] returns the Laplacian of the th vector component at the th quadrature point.
+
Postcondition
laplacians[q] is a vector of Laplacians of the field described by fe_function at the th quadrature point. The size of the vector accessed by laplacians[q] equals the number of components of the finite element, i.e. laplacians[q][c] returns the Laplacian of the th vector component at the th quadrature point.
For each component of the output vector, there holds laplacians[q][c]=trace(hessians[q][c]), where hessians would be the output of the get_function_hessians() function.
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
third_derivatives
The third derivatives of the function specified by fe_function at the quadrature points of the current cell. The third derivatives are computed in real space (as opposed to on the unit cell). The object is assumed to already have the correct size. The data type stored by this output vector must be what you get when you multiply the third derivatives of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
[out]
third_derivatives
The third derivatives of the function specified by fe_function at the quadrature points of the current cell. The third derivatives are computed in real space (as opposed to on the unit cell). The object is assumed to already have the correct size. The data type stored by this output vector must be what you get when you multiply the third derivatives of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
-
Postcondition
third_derivatives[q] will contain the third derivatives of the field described by fe_function at the th quadrature point. third_derivatives[q][i][j][k] represents the th component of the 3rd order tensor of third derivatives at quadrature point .
+
Postcondition
third_derivatives[q] will contain the third derivatives of the field described by fe_function at the th quadrature point. third_derivatives[q][i][j][k] represents the th component of the 3rd order tensor of third derivatives at quadrature point .
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_third_derivatives(), but applied to multi-component (vector- valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
third_derivatives[q] is a vector of third derivatives of the field described by fe_function at the th quadrature point. The size of the vector accessed by third_derivatives[q] equals the number of components of the finite element, i.e. third_derivatives[q][c] returns the third derivative of the th vector component at the th quadrature point. Consequently, third_derivatives[q][c][i][j][k] is the th component of the tensor of third derivatives of the th vector component of the vector field at quadrature point of the current cell.
+
Postcondition
third_derivatives[q] is a vector of third derivatives of the field described by fe_function at the th quadrature point. The size of the vector accessed by third_derivatives[q] equals the number of components of the finite element, i.e. third_derivatives[q][c] returns the third derivative of the th vector component at the th quadrature point. Consequently, third_derivatives[q][c][i][j][k] is the th component of the tensor of third derivatives of the th vector component of the vector field at quadrature point of the current cell.
Mapped quadrature weight. If this object refers to a volume evaluation (i.e. the derived class is of type FEValues), then this is the Jacobi determinant times the weight of the q_pointth unit quadrature point.
For surface evaluations (i.e. classes FEFaceValues or FESubfaceValues), it is the mapped surface element times the weight of the quadrature point.
-
You can think of the quantity returned by this function as the volume or surface element in the integral that we implement here by quadrature.
+
You can think of the quantity returned by this function as the volume or surface element in the integral that we implement here by quadrature.
Return the second derivative of the transformation from unit to real cell, i.e. the first derivative of the Jacobian, at the specified quadrature point, i.e. .
+
Return the second derivative of the transformation from unit to real cell, i.e. the first derivative of the Jacobian, at the specified quadrature point, i.e. .
Return the second derivative of the transformation from unit to real cell, i.e. the first derivative of the Jacobian, at the specified quadrature point, pushed forward to the real cell coordinates, i.e. .
+
Return the second derivative of the transformation from unit to real cell, i.e. the first derivative of the Jacobian, at the specified quadrature point, pushed forward to the real cell coordinates, i.e. .
Note
For this function to work properly, the underlying FEValues, FEFaceValues, or FESubfaceValues object on which you call it must have computed the information you are requesting. To do so, the update_jacobian_pushed_forward_grads flag must be an element of the list of UpdateFlags that you passed to the constructor of this object. See The interplay of UpdateFlags, Mapping, and FiniteElement in FEValues for more information.
@@ -2581,7 +2581,7 @@
-
Return the third derivative of the transformation from unit to real cell, i.e. the second derivative of the Jacobian, at the specified quadrature point, i.e. .
+
Return the third derivative of the transformation from unit to real cell, i.e. the second derivative of the Jacobian, at the specified quadrature point, i.e. .
Note
For this function to work properly, the underlying FEValues, FEFaceValues, or FESubfaceValues object on which you call it must have computed the information you are requesting. To do so, the update_jacobian_2nd_derivatives flag must be an element of the list of UpdateFlags that you passed to the constructor of this object. See The interplay of UpdateFlags, Mapping, and FiniteElement in FEValues for more information.
/usr/share/doc/packages/dealii/doxygen/deal.II/classFEFaceValuesBase.html differs (HTML document, ASCII text, with very long lines)
--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classFEFaceValuesBase.html 2024-04-12 04:45:52.511581780 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classFEFaceValuesBase.html 2024-04-12 04:45:52.515581807 +0000
@@ -649,7 +649,7 @@
If the shape function is vector-valued, then this returns the only non- zero component. If the shape function has more than one non-zero component (i.e. it is not primitive), then throw an exception of type ExcShapeFunctionNotPrimitive. In that case, use the shape_value_component() function.
Parameters
-
i
Number of the shape function to be evaluated. Note that this number runs from zero to dofs_per_cell, even in the case of an FEFaceValues or FESubfaceValues object.
+
i
Number of the shape function to be evaluated. Note that this number runs from zero to dofs_per_cell, even in the case of an FEFaceValues or FESubfaceValues object.
q_point
Number of the quadrature point at which function is to be evaluated
@@ -690,7 +690,7 @@
Compute one vector component of the value of a shape function at a quadrature point. If the finite element is scalar, then only component zero is allowed and the return value equals that of the shape_value() function. If the finite element is vector valued but all shape functions are primitive (i.e. they are non-zero in only one component), then the value returned by shape_value() equals that of this function for exactly one component. This function is therefore only of greater interest if the shape function is not primitive, but then it is necessary since the other function cannot be used.
Parameters
-
i
Number of the shape function to be evaluated.
+
i
Number of the shape function to be evaluated.
q_point
Number of the quadrature point at which function is to be evaluated.
component
vector component to be evaluated.
@@ -729,7 +729,7 @@
The same holds for the arguments of this function as for the shape_value() function.
Parameters
-
i
Number of the shape function to be evaluated.
+
i
Number of the shape function to be evaluated.
q_point
Number of the quadrature point at which function is to be evaluated.
@@ -931,17 +931,17 @@
-
Return the values of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and the related get_function_gradients() function is also used in step-15 along with numerous other tutorial programs.
+
Return the values of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and the related get_function_gradients() function is also used in step-15 along with numerous other tutorial programs.
If the current cell is not active (i.e., it has children), then the finite element function is, strictly speaking, defined by shape functions that live on these child cells. Rather than evaluating the shape functions on the child cells, with the quadrature points defined on the current cell, this function first interpolates the finite element function to shape functions defined on the current cell, and then evaluates this interpolated function.
This function may only be used if the finite element in use is a scalar one, i.e. has only one vector component. To get values of multi-component elements, there is another get_function_values() below, returning a vector of vectors of results.
Parameters
[in]
fe_function
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
values
The values of the function specified by fe_function at the quadrature points of the current cell. The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the values of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the solution vector.
+
[out]
values
The values of the function specified by fe_function at the quadrature points of the current cell. The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the values of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the solution vector.
-
Postcondition
values[q] will contain the value of the field described by fe_function at the th quadrature point.
+
Postcondition
values[q] will contain the value of the field described by fe_function at the th quadrature point.
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_values(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
values[q] is a vector of values of the field described by fe_function at the th quadrature point. The size of the vector accessed by values[q] equals the number of components of the finite element, i.e. values[q](c) returns the value of the th vector component at the th quadrature point.
+
Postcondition
values[q] is a vector of values of the field described by fe_function at the th quadrature point. The size of the vector accessed by values[q] equals the number of components of the finite element, i.e. values[q](c) returns the value of the th vector component at the th quadrature point.
Return the gradients of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and it is also used in step-15 along with numerous other tutorial programs.
+
Return the gradients of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and it is also used in step-15 along with numerous other tutorial programs.
This function may only be used if the finite element in use is a scalar one, i.e. has only one vector component. There is a corresponding function of the same name for vector-valued finite elements.
Parameters
[in]
fe_function
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
gradients
The gradients of the function specified by fe_function at the quadrature points of the current cell. The gradients are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the gradients of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
[out]
gradients
The gradients of the function specified by fe_function at the quadrature points of the current cell. The gradients are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the gradients of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
-
Postcondition
gradients[q] will contain the gradient of the field described by fe_function at the th quadrature point. gradients[q][d] represents the derivative in coordinate direction at quadrature point .
+
Postcondition
gradients[q] will contain the gradient of the field described by fe_function at the th quadrature point. gradients[q][d] represents the derivative in coordinate direction at quadrature point .
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_gradients(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
gradients[q] is a vector of gradients of the field described by fe_function at the th quadrature point. The size of the vector accessed by gradients[q] equals the number of components of the finite element, i.e. gradients[q][c] returns the gradient of the th vector component at the th quadrature point. Consequently, gradients[q][c][d] is the derivative in coordinate direction of the th vector component of the vector field at quadrature point of the current cell.
+
Postcondition
gradients[q] is a vector of gradients of the field described by fe_function at the th quadrature point. The size of the vector accessed by gradients[q] equals the number of components of the finite element, i.e. gradients[q][c] returns the gradient of the th vector component at the th quadrature point. Consequently, gradients[q][c][d] is the derivative in coordinate direction of the th vector component of the vector field at quadrature point of the current cell.
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
hessians
The Hessians of the function specified by fe_function at the quadrature points of the current cell. The Hessians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Hessians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
[out]
hessians
The Hessians of the function specified by fe_function at the quadrature points of the current cell. The Hessians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Hessians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
-
Postcondition
hessians[q] will contain the Hessian of the field described by fe_function at the th quadrature point. hessians[q][i][j] represents the th component of the matrix of second derivatives at quadrature point .
+
Postcondition
hessians[q] will contain the Hessian of the field described by fe_function at the th quadrature point. hessians[q][i][j] represents the th component of the matrix of second derivatives at quadrature point .
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_hessians(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
hessians[q] is a vector of Hessians of the field described by fe_function at the th quadrature point. The size of the vector accessed by hessians[q] equals the number of components of the finite element, i.e. hessians[q][c] returns the Hessian of the th vector component at the th quadrature point. Consequently, hessians[q][c][i][j] is the th component of the matrix of second derivatives of the th vector component of the vector field at quadrature point of the current cell.
+
Postcondition
hessians[q] is a vector of Hessians of the field described by fe_function at the th quadrature point. The size of the vector accessed by hessians[q] equals the number of components of the finite element, i.e. hessians[q][c] returns the Hessian of the th vector component at the th quadrature point. Consequently, hessians[q][c][i][j] is the th component of the matrix of second derivatives of the th vector component of the vector field at quadrature point of the current cell.
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
laplacians
The Laplacians of the function specified by fe_function at the quadrature points of the current cell. The Laplacians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Laplacians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the input vector.
+
[out]
laplacians
The Laplacians of the function specified by fe_function at the quadrature points of the current cell. The Laplacians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Laplacians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the input vector.
-
Postcondition
laplacians[q] will contain the Laplacian of the field described by fe_function at the th quadrature point.
+
Postcondition
laplacians[q] will contain the Laplacian of the field described by fe_function at the th quadrature point.
For each component of the output vector, there holds laplacians[q]=trace(hessians[q]), where hessians would be the output of the get_function_hessians() function.
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
@@ -1519,7 +1519,7 @@
This function does the same as the other get_function_laplacians(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
laplacians[q] is a vector of Laplacians of the field described by fe_function at the th quadrature point. The size of the vector accessed by laplacians[q] equals the number of components of the finite element, i.e. laplacians[q][c] returns the Laplacian of the th vector component at the th quadrature point.
+
Postcondition
laplacians[q] is a vector of Laplacians of the field described by fe_function at the th quadrature point. The size of the vector accessed by laplacians[q] equals the number of components of the finite element, i.e. laplacians[q][c] returns the Laplacian of the th vector component at the th quadrature point.
For each component of the output vector, there holds laplacians[q][c]=trace(hessians[q][c]), where hessians would be the output of the get_function_hessians() function.
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
third_derivatives
The third derivatives of the function specified by fe_function at the quadrature points of the current cell. The third derivatives are computed in real space (as opposed to on the unit cell). The object is assumed to already have the correct size. The data type stored by this output vector must be what you get when you multiply the third derivatives of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
[out]
third_derivatives
The third derivatives of the function specified by fe_function at the quadrature points of the current cell. The third derivatives are computed in real space (as opposed to on the unit cell). The object is assumed to already have the correct size. The data type stored by this output vector must be what you get when you multiply the third derivatives of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
-
Postcondition
third_derivatives[q] will contain the third derivatives of the field described by fe_function at the th quadrature point. third_derivatives[q][i][j][k] represents the th component of the 3rd order tensor of third derivatives at quadrature point .
+
Postcondition
third_derivatives[q] will contain the third derivatives of the field described by fe_function at the th quadrature point. third_derivatives[q][i][j][k] represents the th component of the 3rd order tensor of third derivatives at quadrature point .
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_third_derivatives(), but applied to multi-component (vector- valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
third_derivatives[q] is a vector of third derivatives of the field described by fe_function at the th quadrature point. The size of the vector accessed by third_derivatives[q] equals the number of components of the finite element, i.e. third_derivatives[q][c] returns the third derivative of the th vector component at the th quadrature point. Consequently, third_derivatives[q][c][i][j][k] is the th component of the tensor of third derivatives of the th vector component of the vector field at quadrature point of the current cell.
+
Postcondition
third_derivatives[q] is a vector of third derivatives of the field described by fe_function at the th quadrature point. The size of the vector accessed by third_derivatives[q] equals the number of components of the finite element, i.e. third_derivatives[q][c] returns the third derivative of the th vector component at the th quadrature point. Consequently, third_derivatives[q][c][i][j][k] is the th component of the tensor of third derivatives of the th vector component of the vector field at quadrature point of the current cell.
Mapped quadrature weight. If this object refers to a volume evaluation (i.e. the derived class is of type FEValues), then this is the Jacobi determinant times the weight of the q_pointth unit quadrature point.
For surface evaluations (i.e. classes FEFaceValues or FESubfaceValues), it is the mapped surface element times the weight of the quadrature point.
-
You can think of the quantity returned by this function as the volume or surface element in the integral that we implement here by quadrature.
+
You can think of the quantity returned by this function as the volume or surface element in the integral that we implement here by quadrature.
Return the second derivative of the transformation from unit to real cell, i.e. the first derivative of the Jacobian, at the specified quadrature point, i.e. .
+
Return the second derivative of the transformation from unit to real cell, i.e. the first derivative of the Jacobian, at the specified quadrature point, i.e. .
Return the second derivative of the transformation from unit to real cell, i.e. the first derivative of the Jacobian, at the specified quadrature point, pushed forward to the real cell coordinates, i.e. .
+
Return the second derivative of the transformation from unit to real cell, i.e. the first derivative of the Jacobian, at the specified quadrature point, pushed forward to the real cell coordinates, i.e. .
Note
For this function to work properly, the underlying FEValues, FEFaceValues, or FESubfaceValues object on which you call it must have computed the information you are requesting. To do so, the update_jacobian_pushed_forward_grads flag must be an element of the list of UpdateFlags that you passed to the constructor of this object. See The interplay of UpdateFlags, Mapping, and FiniteElement in FEValues for more information.
@@ -2299,7 +2299,7 @@
-
Return the third derivative of the transformation from unit to real cell, i.e. the second derivative of the Jacobian, at the specified quadrature point, i.e. .
+
Return the third derivative of the transformation from unit to real cell, i.e. the second derivative of the Jacobian, at the specified quadrature point, i.e. .
Note
For this function to work properly, the underlying FEValues, FEFaceValues, or FESubfaceValues object on which you call it must have computed the information you are requesting. To do so, the update_jacobian_2nd_derivatives flag must be an element of the list of UpdateFlags that you passed to the constructor of this object. See The interplay of UpdateFlags, Mapping, and FiniteElement in FEValues for more information.
/usr/share/doc/packages/dealii/doxygen/deal.II/classFEInterfaceValues.html differs (HTML document, ASCII text, with very long lines)
--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classFEInterfaceValues.html 2024-04-12 04:45:52.579582251 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classFEInterfaceValues.html 2024-04-12 04:45:52.583582278 +0000
@@ -488,8 +488,8 @@
If the q_index and mapping_index arguments to this function are explicitly specified (rather than leaving them at their default values), then these indices will be used to select which element of the hp::QCollection and hp::MappingCollection passed to the constructor should serve as the quadrature and mapping to be used.
If one of these arguments is left at its default value, then the function will need to choose a quadrature and/or mapping that is appropriate for the two finite element spaces used on the two cells adjacent to the current interface. As the first choice, if the quadrature or mapping collection we are considering has only one element, then that is clearly the one that should be used.
If the quadrature or mapping collection have multiple elements, then we need to dig further. For quadrature objects, we can compare whether the two quadrature objects that correspond to the active_fe_index values of the two adjacent cells are identical (i.e., have quadrature points at the same locations, and have the same weights). If this is so, then it does not matter which one of the two we take, and we choose one or the other.
-
If this has still not helped, we try to find out which of the two finite element spaces on the two adjacent cells is "larger" (say, if you had used and elements on the two adjacent cells, then the element is the larger one); the determination of which space is "larger" is made using the hp::FECollection::find_dominated_fe() function, which is not necessarily intended for this kind of query, but yields a result that serves just fine for our purposes here. We then operate on the assumption that the quadrature object associated with the "larger" of the two spaces is the appropriate one to use for the face that separates these two spaces.
-
If this function returns that one of the two elements in question is dominated by the other, then presumably it is "larger" one and we take the quadrature formula and mapping that corresponds to this "larger" element is. For example, for the element mentioned above, one would generally use a QGauss(3) quadrature formula, whereas for the element, one would use QGauss(5). To integrate jump and average terms on the interface between cells using these two elements, QGauss(5) is appropriate. Because, typically, people will order elements in the hp::FECollection in the same order as the quadrature and mapping objects in hp::QCollection and hp::MappingCollection, this function will use the index of the "larger" element in the hp::FECollection to also index into the hp::QCollection and hp::MappingCollection to retrieve quadrature and mapping objects appropriate for the current face.
+
If this has still not helped, we try to find out which of the two finite element spaces on the two adjacent cells is "larger" (say, if you had used and elements on the two adjacent cells, then the element is the larger one); the determination of which space is "larger" is made using the hp::FECollection::find_dominated_fe() function, which is not necessarily intended for this kind of query, but yields a result that serves just fine for our purposes here. We then operate on the assumption that the quadrature object associated with the "larger" of the two spaces is the appropriate one to use for the face that separates these two spaces.
+
If this function returns that one of the two elements in question is dominated by the other, then presumably it is "larger" one and we take the quadrature formula and mapping that corresponds to this "larger" element is. For example, for the element mentioned above, one would generally use a QGauss(3) quadrature formula, whereas for the element, one would use QGauss(5). To integrate jump and average terms on the interface between cells using these two elements, QGauss(5) is appropriate. Because, typically, people will order elements in the hp::FECollection in the same order as the quadrature and mapping objects in hp::QCollection and hp::MappingCollection, this function will use the index of the "larger" element in the hp::FECollection to also index into the hp::QCollection and hp::MappingCollection to retrieve quadrature and mapping objects appropriate for the current face.
There are cases where neither element dominates the other. For example, if one uses and elements on neighboring cells, neither of the two spaces dominates the other – or, in the context of the current function, neither space is "larger" than the other. In that case, there is no way for the current function to determine quadrature and mapping objects associated with the two elements are the appropriate ones. If that happens, you will get an error – and the only way to avoid the error is to explicitly specify for these interfaces which quadrature and mapping objects you want to use, by providing non-default values for the q_index and mapping_index arguments to this function.
@@ -825,7 +825,7 @@
Mapped quadrature weight. This value equals the mapped surface element times the weight of the quadrature point.
-
You can think of the quantity returned by this function as the surface element in the integral that we implement here by quadrature.
+
You can think of the quantity returned by this function as the surface element in the integral that we implement here by quadrature.
Return the jump on the interface for the shape function interface_dof_index at the quadrature point q_point of component component.
+
Return the jump on the interface for the shape function interface_dof_index at the quadrature point q_point of component component.
Note that one can define the jump in different ways (the value "there" minus the value "here", or the other way around; both are used in the finite element literature). The definition here uses "value here minus value there", as seen from the first cell.
-
If this is a boundary face (at_boundary() returns true), then , that is "the value here (minus zero)".
+
If this is a boundary face (at_boundary() returns true), then , that is "the value here (minus zero)".
Note
The name of the function is supposed to be read as "the jump
(singular) in the values (plural: one or two possible values)
of the shape function (singular)".
Return the jump in the gradient on the interface for the shape function interface_dof_index at the quadrature point q_point of component component.
-
If this is a boundary face (at_boundary() returns true), then .
+
Return the jump in the gradient on the interface for the shape function interface_dof_index at the quadrature point q_point of component component.
+
If this is a boundary face (at_boundary() returns true), then .
Note
The name of the function is supposed to be read as "the jump
(singular) in the gradients (plural: one or two possible gradients)
of the shape function (singular)".
Return the jump in the Hessian on the interface for the shape function interface_dof_index at the quadrature point q_point of component component.
-
If this is a boundary face (at_boundary() returns true), then .
+
Return the jump in the Hessian on the interface for the shape function interface_dof_index at the quadrature point q_point of component component.
+
If this is a boundary face (at_boundary() returns true), then .
Note
The name of the function is supposed to be read as "the jump
(singular) in the Hessians (plural: one or two possible values
for the derivative) of the shape function (singular)".
Return the jump in the third derivative on the interface for the shape function interface_dof_index at the quadrature point q_point of component component.
-
If this is a boundary face (at_boundary() returns true), then .
+
Return the jump in the third derivative on the interface for the shape function interface_dof_index at the quadrature point q_point of component component.
+
If this is a boundary face (at_boundary() returns true), then .
Note
The name of the function is supposed to be read as "the jump
(singular) in the third derivatives (plural: one or two possible values
for the derivative) of the shape function (singular)".
Return the average on the interface for the shape function interface_dof_index at the quadrature point q_point of component component.
-
If this is a boundary face (at_boundary() returns true), then .
+
Return the average on the interface for the shape function interface_dof_index at the quadrature point q_point of component component.
+
If this is a boundary face (at_boundary() returns true), then .
Note
The name of the function is supposed to be read as "the average
(singular) of the values (plural: one or two possible values)
of the shape function (singular)".
Return the average of the gradient on the interface for the shape function interface_dof_index at the quadrature point q_point of component component.
-
If this is a boundary face (at_boundary() returns true), then .
+
Return the average of the gradient on the interface for the shape function interface_dof_index at the quadrature point q_point of component component.
+
If this is a boundary face (at_boundary() returns true), then .
Note
The name of the function is supposed to be read as "the average
(singular) of the gradients (plural: one or two possible values
for the gradient) of the shape function (singular)".
Return the average of the Hessian on the interface for the shape function interface_dof_index at the quadrature point q_point of component component.
-
If this is a boundary face (at_boundary() returns true), then .
+u_{\text{cell1}}$" src="form_1094.png"/> on the interface for the shape function interface_dof_index at the quadrature point q_point of component component.
+
If this is a boundary face (at_boundary() returns true), then .
Note
The name of the function is supposed to be read as "the average
(singular) of the Hessians (plural: one or two possible values
for the second derivatives) of the shape function (singular)".
/usr/share/doc/packages/dealii/doxygen/deal.II/classFEInterfaceViews_1_1Scalar.html differs (HTML document, ASCII text, with very long lines)
--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classFEInterfaceViews_1_1Scalar.html 2024-04-12 04:45:52.647582721 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classFEInterfaceViews_1_1Scalar.html 2024-04-12 04:45:52.655582777 +0000
@@ -454,7 +454,7 @@
Return the jump on the interface for the shape function interface_dof_index in the quadrature point q_point of the component selected by this view.
+
Return the jump on the interface for the shape function interface_dof_index in the quadrature point q_point of the component selected by this view.
Note
The name of the function is supposed to be read as "the jump
(singular) in the values (plural: one or two possible values) of
the shape function (singular)".
Return the jump of the gradient on the interface for the shape function interface_dof_index in the quadrature point q_point of the component selected by this view.
+
Return the jump of the gradient on the interface for the shape function interface_dof_index in the quadrature point q_point of the component selected by this view.
Note
The name of the function is supposed to be read as "the jump
(singular) in the gradients (plural: one or two possible gradients)
of the shape function (singular)".
Return the jump in the gradient on the interface for the shape function interface_dof_index at the quadrature point q_point of the component selected by this view.
+
Return the jump in the gradient on the interface for the shape function interface_dof_index at the quadrature point q_point of the component selected by this view.
Note
The name of the function is supposed to be read as "the jump
(singular) in the Hessians (plural: one or two possible values
for the second derivative) of the shape function (singular)".
Return the jump in the third derivative on the interface for the shape function interface_dof_index at the quadrature point q_point of the component selected by this view.
+
Return the jump in the third derivative on the interface for the shape function interface_dof_index at the quadrature point q_point of the component selected by this view.
Note
The name of the function is supposed to be read as "the jump
(singular) in the third derivatives (plural: one or two possible values
for the third derivative) of the shape function (singular)".
Return the average value on the interface for the shape function interface_dof_index in the quadrature point q_point of the component selected by this view.
+
Return the average value on the interface for the shape function interface_dof_index in the quadrature point q_point of the component selected by this view.
Note
The name of the function is supposed to be read as "the average
(singular) of the values (plural: one or two possible values) of
the shape function (singular)".
Return the average of the gradient on the interface for the shape function interface_dof_index in the quadrature point q_point of the component selected by this view.
+
Return the average of the gradient on the interface for the shape function interface_dof_index in the quadrature point q_point of the component selected by this view.
Note
The name of the function is supposed to be read as "the average
(singular) of the gradients (plural: one or two possible values of
the derivative) of the shape function (singular)".
Return the average of the Hessian on the interface for the shape function interface_dof_index at the quadrature point q_point of the component selected by this view.
+u_{\text{cell1}}$" src="form_1094.png"/> on the interface for the shape function interface_dof_index at the quadrature point q_point of the component selected by this view.
Note
The name of the function is supposed to be read as "the average
(singular) in the Hessians (plural: one or two possible values of
the second derivative) of the shape function (singular)".
Return the values of the selected scalar component of the finite element function characterized by fe_function at the quadrature points of the cell interface selected the last time the reinit function of the FEInterfaceValues object was called.
The argument here_or_there selects between the value on cell 0 (here, true) and cell 1 (there, false). You can also interpret it as "upstream" (true) and "downstream" (false) as defined by the direction of the normal vector in this quadrature point. If here_or_there is true, the values from the first cell of the interface is used.
-
The data type stored by the output vector must be what you get when you multiply the values of shape functions (i.e., value_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
The data type stored by the output vector must be what you get when you multiply the values of shape functions (i.e., value_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
Return the jump in the values of the selected scalar component of the finite element function characterized by fe_function at the quadrature points of the cell interface selected the last time the reinit function of the FEInterfaceValues object was called.
-
The data type stored by the output vector must be what you get when you multiply the values of shape functions (i.e., value_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
The data type stored by the output vector must be what you get when you multiply the values of shape functions (i.e., value_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
Return the jump in the gradients of the selected scalar components of the finite element function characterized by fe_function at the quadrature points of the cell interface selected the last time the reinit function of the FEInterfaceValues object was called.
-
The data type stored by the output vector must be what you get when you multiply the gradients of shape functions (i.e., gradient_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
The data type stored by the output vector must be what you get when you multiply the gradients of shape functions (i.e., gradient_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
Return the jump in the Hessians of the selected scalar component of the finite element function characterized by fe_function at the quadrature points of the cell, face or subface selected the last time the reinit function of the FEInterfaceValues object was called.
-
The data type stored by the output vector must be what you get when you multiply the Hessians of shape functions (i.e., hessian_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
The data type stored by the output vector must be what you get when you multiply the Hessians of shape functions (i.e., hessian_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
Return the jump in the third derivatives of the selected scalar component of the finite element function characterized by fe_function at the quadrature points of the cell, face or subface selected the last time the reinit function of the FEInterfaceValues object was called.
-
The data type stored by the output vector must be what you get when you multiply the third derivatives of shape functions (i.e., third_derivative_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
The data type stored by the output vector must be what you get when you multiply the third derivatives of shape functions (i.e., third_derivative_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
Note
For this function to work properly, the underlying FEValues, FEFaceValues, or FESubfaceValues object on which you call it must have computed the information you are requesting. To do so, the update_third_derivatives flag must be an element of the list of UpdateFlags that you passed to the constructor of this object. See The interplay of UpdateFlags, Mapping, and FiniteElement in FEValues for more information.
Return the average of the values of the selected scalar component of the finite element function characterized by fe_function at the quadrature points of the cell interface selected the last time the reinit function of the FEInterfaceValues object was called.
-
The data type stored by the output vector must be what you get when you multiply the values of shape functions (i.e., value_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
The data type stored by the output vector must be what you get when you multiply the values of shape functions (i.e., value_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
Return the average of the gradients of the selected scalar components of the finite element function characterized by fe_function at the quadrature points of the cell interface selected the last time the reinit function of the FEInterfaceValues object was called.
-
The data type stored by the output vector must be what you get when you multiply the gradients of shape functions (i.e., gradient_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
The data type stored by the output vector must be what you get when you multiply the gradients of shape functions (i.e., gradient_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
Return the average of the Hessians of the selected scalar component of the finite element function characterized by fe_function at the quadrature points of the cell, face or subface selected the last time the reinit function of the FEInterfaceValues object was called.
-
The data type stored by the output vector must be what you get when you multiply the Hessians of shape functions (i.e., hessian_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
The data type stored by the output vector must be what you get when you multiply the Hessians of shape functions (i.e., hessian_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
Return the jump vector on the interface for the shape function interface_dof_index in the quadrature point q_point.
+
Return the jump vector on the interface for the shape function interface_dof_index in the quadrature point q_point.
Note
The name of the function is supposed to be read as "the jump
(singular) in the values (plural: one or two possible values) of
the shape function (singular)".
Return the jump of the gradient (a tensor of rank 2) on the interface for the shape function interface_dof_index in the quadrature point q_point.
+
Return the jump of the gradient (a tensor of rank 2) on the interface for the shape function interface_dof_index in the quadrature point q_point.
Note
The name of the function is supposed to be read as "the jump
(singular) in the gradients (plural: one or two possible gradients)
of the shape function (singular)".
Return the jump in the gradient on the interface for the shape function interface_dof_index at the quadrature point q_point of the component selected by this view.
+
Return the jump in the gradient on the interface for the shape function interface_dof_index at the quadrature point q_point of the component selected by this view.
Note
The name of the function is supposed to be read as "the jump
(singular) in the Hessians (plural: one or two possible values
for the second derivative) of the shape function (singular)".
Return the jump in the third derivative on the interface for the shape function interface_dof_index at the quadrature point q_point of the component selected by this view.
+
Return the jump in the third derivative on the interface for the shape function interface_dof_index at the quadrature point q_point of the component selected by this view.
Note
The name of the function is supposed to be read as "the jump
(singular) in the third derivatives (plural: one or two possible values
for the third derivative) of the shape function (singular)".
Return the average vector on the interface for the shape function interface_dof_index in the quadrature point q_point.
+
Return the average vector on the interface for the shape function interface_dof_index in the quadrature point q_point.
Note
The name of the function is supposed to be read as "the average
(singular) of the values (plural: one or two possible values) of
the shape function (singular)".
Return the average of the gradient (a tensor of rank 2) on the interface for the shape function interface_dof_index in the quadrature point q_point.
+
Return the average of the gradient (a tensor of rank 2) on the interface for the shape function interface_dof_index in the quadrature point q_point.
Note
The name of the function is supposed to be read as "the average
(singular) of the gradients (plural: one or two possible values
of the derivative) of the shape function (singular)".
Return the average of the Hessian on the interface for the shape function interface_dof_index at the quadrature point q_point of the component selected by this view.
+u_{\text{cell1}}$" src="form_1094.png"/> on the interface for the shape function interface_dof_index at the quadrature point q_point of the component selected by this view.
Note
The name of the function is supposed to be read as "the average
(singular) in the Hessians (plural: one or two possible values of
the second derivative) of the shape function (singular)".
@@ -796,7 +796,7 @@
Return the values of the selected vector component of the finite element function characterized by fe_function at the quadrature points of the cell interface selected the last time the reinit function of the FEInterfaceValues object was called.
The argument here_or_there selects between the value on cell 0 (here, true) and cell 1 (there, false). You can also interpret it as "upstream" (true) and "downstream" (false) as defined by the direction of the normal vector in this quadrature point. If here_or_there is true, the values from the first cell of the interface is used.
-
The data type stored by the output vector must be what you get when you multiply the values of shape functions (i.e., value_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
The data type stored by the output vector must be what you get when you multiply the values of shape functions (i.e., value_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
Return the jump in the values of the selected vector component of the finite element function characterized by fe_function at the quadrature points of the cell interface selected the last time the reinit function of the FEInterfaceValues object was called.
-
The data type stored by the output vector must be what you get when you multiply the values of shape functions (i.e., value_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
The data type stored by the output vector must be what you get when you multiply the values of shape functions (i.e., value_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
Return the jump in the gradients of the selected vector components of the finite element function characterized by fe_function at the quadrature points of the cell interface selected the last time the reinit function of the FEInterfaceValues object was called.
-
The data type stored by the output vector must be what you get when you multiply the gradients of shape functions (i.e., gradient_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
The data type stored by the output vector must be what you get when you multiply the gradients of shape functions (i.e., gradient_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
Return the jump in the Hessians of the selected vector component of the finite element function characterized by fe_function at the quadrature points of the cell, face or subface selected the last time the reinit function of the FEInterfaceValues object was called.
-
The data type stored by the output vector must be what you get when you multiply the Hessians of shape functions (i.e., hessian_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
The data type stored by the output vector must be what you get when you multiply the Hessians of shape functions (i.e., hessian_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
Return the jump in the third derivatives of the selected vector component of the finite element function characterized by fe_function at the quadrature points of the cell, face or subface selected the last time the reinit function of the FEInterfaceValues object was called.
-
The data type stored by the output vector must be what you get when you multiply the third derivatives of shape functions (i.e., third_derivative_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
The data type stored by the output vector must be what you get when you multiply the third derivatives of shape functions (i.e., third_derivative_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
Note
For this function to work properly, the underlying FEValues, FEFaceValues, or FESubfaceValues object on which you call it must have computed the information you are requesting. To do so, the update_third_derivatives flag must be an element of the list of UpdateFlags that you passed to the constructor of this object. See The interplay of UpdateFlags, Mapping, and FiniteElement in FEValues for more information.
@@ -1085,7 +1085,7 @@
Return the average of the values of the selected vector component of the finite element function characterized by fe_function at the quadrature points of the cell interface selected the last time the reinit function of the FEInterfaceValues object was called.
-
The data type stored by the output vector must be what you get when you multiply the values of shape functions (i.e., value_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
The data type stored by the output vector must be what you get when you multiply the values of shape functions (i.e., value_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
Return the average of the gradients of the selected vector components of the finite element function characterized by fe_function at the quadrature points of the cell interface selected the last time the reinit function of the FEInterfaceValues object was called.
-
The data type stored by the output vector must be what you get when you multiply the gradients of shape functions (i.e., gradient_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
The data type stored by the output vector must be what you get when you multiply the gradients of shape functions (i.e., gradient_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
Return the average of the Hessians of the selected vector component of the finite element function characterized by fe_function at the quadrature points of the cell, face or subface selected the last time the reinit function of the FEInterfaceValues object was called.
-
The data type stored by the output vector must be what you get when you multiply the Hessians of shape functions (i.e., hessian_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
The data type stored by the output vector must be what you get when you multiply the Hessians of shape functions (i.e., hessian_type) times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
A class to calculate expansion of a scalar FE (or a single component of vector-valued FE) field into Fourier series on a reference element. The exponential form of the Fourier series is based on completeness and Hermitian orthogonality of the set of exponential functions . For example in 1d the L2-orthogonality condition reads
-Fourier series on a reference element. The exponential form of the Fourier series is based on completeness and Hermitian orthogonality of the set of exponential functions . For example in 1d the L2-orthogonality condition reads
+
+\]" src="form_1176.png"/>
-
Note that .
+
Note that .
The arbitrary scalar FE field on the reference element can be expanded in the complete orthogonal exponential basis as
-
+\]" src="form_1178.png"/>
From the orthogonality property of the basis, it follows that
-
+\]" src="form_1179.png"/>
-
It is this complex-valued expansion coefficients, that are calculated by this class. Note that , where are real-valued FiniteElement shape functions. Consequently and we only need to compute for positive indices .
+
It is this complex-valued expansion coefficients, that are calculated by this class. Note that , where are real-valued FiniteElement shape functions. Consequently and we only need to compute for positive indices .
/usr/share/doc/packages/dealii/doxygen/deal.II/classFESeries_1_1Legendre.html differs (HTML document, ASCII text, with very long lines)
--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classFESeries_1_1Legendre.html 2024-04-12 04:45:52.799583773 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classFESeries_1_1Legendre.html 2024-04-12 04:45:52.799583773 +0000
@@ -196,39 +196,39 @@
template<int dim, int spacedim = dim>
class FESeries::Legendre< dim, spacedim >
A class to calculate expansion of a scalar FE (or a single component of vector-valued FE) field into series of Legendre functions on a reference element.
Legendre functions are solutions to Legendre's differential equation
-
+\]" src="form_1185.png"/>
and can be expressed using Rodrigues' formula
-
+\]" src="form_1186.png"/>
-
These polynomials are orthogonal with respect to the inner product on the interval
- inner product on the interval
+
+\]" src="form_1189.png"/>
-
and are complete. A family of -orthogonal polynomials on can be constructed via
--orthogonal polynomials on can be constructed via
+
+\]" src="form_1191.png"/>
-
An arbitrary scalar FE field on the reference element can be expanded in the complete orthogonal basis as
- can be expanded in the complete orthogonal basis as
+
+\]" src="form_1192.png"/>
From the orthogonality property of the basis, it follows that
-
+\]" src="form_1193.png"/>
-
This class calculates coefficients using -dimensional Legendre polynomials constructed from using tensor product rule.
+
This class calculates coefficients using -dimensional Legendre polynomials constructed from using tensor product rule.
/usr/share/doc/packages/dealii/doxygen/deal.II/classFESubfaceValues.html differs (HTML document, ASCII text, with very long lines)
--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classFESubfaceValues.html 2024-04-12 04:45:52.907584521 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classFESubfaceValues.html 2024-04-12 04:45:52.911584548 +0000
@@ -959,7 +959,7 @@
If the shape function is vector-valued, then this returns the only non- zero component. If the shape function has more than one non-zero component (i.e. it is not primitive), then throw an exception of type ExcShapeFunctionNotPrimitive. In that case, use the shape_value_component() function.
Parameters
-
i
Number of the shape function to be evaluated. Note that this number runs from zero to dofs_per_cell, even in the case of an FEFaceValues or FESubfaceValues object.
+
i
Number of the shape function to be evaluated. Note that this number runs from zero to dofs_per_cell, even in the case of an FEFaceValues or FESubfaceValues object.
q_point
Number of the quadrature point at which function is to be evaluated
@@ -1000,7 +1000,7 @@
Compute one vector component of the value of a shape function at a quadrature point. If the finite element is scalar, then only component zero is allowed and the return value equals that of the shape_value() function. If the finite element is vector valued but all shape functions are primitive (i.e. they are non-zero in only one component), then the value returned by shape_value() equals that of this function for exactly one component. This function is therefore only of greater interest if the shape function is not primitive, but then it is necessary since the other function cannot be used.
Parameters
-
i
Number of the shape function to be evaluated.
+
i
Number of the shape function to be evaluated.
q_point
Number of the quadrature point at which function is to be evaluated.
component
vector component to be evaluated.
@@ -1039,7 +1039,7 @@
The same holds for the arguments of this function as for the shape_value() function.
Parameters
-
i
Number of the shape function to be evaluated.
+
i
Number of the shape function to be evaluated.
q_point
Number of the quadrature point at which function is to be evaluated.
@@ -1241,17 +1241,17 @@
-
Return the values of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and the related get_function_gradients() function is also used in step-15 along with numerous other tutorial programs.
+
Return the values of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and the related get_function_gradients() function is also used in step-15 along with numerous other tutorial programs.
If the current cell is not active (i.e., it has children), then the finite element function is, strictly speaking, defined by shape functions that live on these child cells. Rather than evaluating the shape functions on the child cells, with the quadrature points defined on the current cell, this function first interpolates the finite element function to shape functions defined on the current cell, and then evaluates this interpolated function.
This function may only be used if the finite element in use is a scalar one, i.e. has only one vector component. To get values of multi-component elements, there is another get_function_values() below, returning a vector of vectors of results.
Parameters
[in]
fe_function
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
values
The values of the function specified by fe_function at the quadrature points of the current cell. The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the values of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the solution vector.
+
[out]
values
The values of the function specified by fe_function at the quadrature points of the current cell. The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the values of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the solution vector.
-
Postcondition
values[q] will contain the value of the field described by fe_function at the th quadrature point.
+
Postcondition
values[q] will contain the value of the field described by fe_function at the th quadrature point.
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_values(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
values[q] is a vector of values of the field described by fe_function at the th quadrature point. The size of the vector accessed by values[q] equals the number of components of the finite element, i.e. values[q](c) returns the value of the th vector component at the th quadrature point.
+
Postcondition
values[q] is a vector of values of the field described by fe_function at the th quadrature point. The size of the vector accessed by values[q] equals the number of components of the finite element, i.e. values[q](c) returns the value of the th vector component at the th quadrature point.
Return the gradients of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and it is also used in step-15 along with numerous other tutorial programs.
+
Return the gradients of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and it is also used in step-15 along with numerous other tutorial programs.
This function may only be used if the finite element in use is a scalar one, i.e. has only one vector component. There is a corresponding function of the same name for vector-valued finite elements.
Parameters
[in]
fe_function
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
gradients
The gradients of the function specified by fe_function at the quadrature points of the current cell. The gradients are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the gradients of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
[out]
gradients
The gradients of the function specified by fe_function at the quadrature points of the current cell. The gradients are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the gradients of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
-
Postcondition
gradients[q] will contain the gradient of the field described by fe_function at the th quadrature point. gradients[q][d] represents the derivative in coordinate direction at quadrature point .
+
Postcondition
gradients[q] will contain the gradient of the field described by fe_function at the th quadrature point. gradients[q][d] represents the derivative in coordinate direction at quadrature point .
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_gradients(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
gradients[q] is a vector of gradients of the field described by fe_function at the th quadrature point. The size of the vector accessed by gradients[q] equals the number of components of the finite element, i.e. gradients[q][c] returns the gradient of the th vector component at the th quadrature point. Consequently, gradients[q][c][d] is the derivative in coordinate direction of the th vector component of the vector field at quadrature point of the current cell.
+
Postcondition
gradients[q] is a vector of gradients of the field described by fe_function at the th quadrature point. The size of the vector accessed by gradients[q] equals the number of components of the finite element, i.e. gradients[q][c] returns the gradient of the th vector component at the th quadrature point. Consequently, gradients[q][c][d] is the derivative in coordinate direction of the th vector component of the vector field at quadrature point of the current cell.
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
hessians
The Hessians of the function specified by fe_function at the quadrature points of the current cell. The Hessians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Hessians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
[out]
hessians
The Hessians of the function specified by fe_function at the quadrature points of the current cell. The Hessians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Hessians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
-
Postcondition
hessians[q] will contain the Hessian of the field described by fe_function at the th quadrature point. hessians[q][i][j] represents the th component of the matrix of second derivatives at quadrature point .
+
Postcondition
hessians[q] will contain the Hessian of the field described by fe_function at the th quadrature point. hessians[q][i][j] represents the th component of the matrix of second derivatives at quadrature point .
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_hessians(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
hessians[q] is a vector of Hessians of the field described by fe_function at the th quadrature point. The size of the vector accessed by hessians[q] equals the number of components of the finite element, i.e. hessians[q][c] returns the Hessian of the th vector component at the th quadrature point. Consequently, hessians[q][c][i][j] is the th component of the matrix of second derivatives of the th vector component of the vector field at quadrature point of the current cell.
+
Postcondition
hessians[q] is a vector of Hessians of the field described by fe_function at the th quadrature point. The size of the vector accessed by hessians[q] equals the number of components of the finite element, i.e. hessians[q][c] returns the Hessian of the th vector component at the th quadrature point. Consequently, hessians[q][c][i][j] is the th component of the matrix of second derivatives of the th vector component of the vector field at quadrature point of the current cell.
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
laplacians
The Laplacians of the function specified by fe_function at the quadrature points of the current cell. The Laplacians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Laplacians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the input vector.
+
[out]
laplacians
The Laplacians of the function specified by fe_function at the quadrature points of the current cell. The Laplacians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Laplacians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the input vector.
-
Postcondition
laplacians[q] will contain the Laplacian of the field described by fe_function at the th quadrature point.
+
Postcondition
laplacians[q] will contain the Laplacian of the field described by fe_function at the th quadrature point.
For each component of the output vector, there holds laplacians[q]=trace(hessians[q]), where hessians would be the output of the get_function_hessians() function.
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
@@ -1829,7 +1829,7 @@
This function does the same as the other get_function_laplacians(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
laplacians[q] is a vector of Laplacians of the field described by fe_function at the th quadrature point. The size of the vector accessed by laplacians[q] equals the number of components of the finite element, i.e. laplacians[q][c] returns the Laplacian of the th vector component at the th quadrature point.
+
Postcondition
laplacians[q] is a vector of Laplacians of the field described by fe_function at the th quadrature point. The size of the vector accessed by laplacians[q] equals the number of components of the finite element, i.e. laplacians[q][c] returns the Laplacian of the th vector component at the th quadrature point.
For each component of the output vector, there holds laplacians[q][c]=trace(hessians[q][c]), where hessians would be the output of the get_function_hessians() function.
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
third_derivatives
The third derivatives of the function specified by fe_function at the quadrature points of the current cell. The third derivatives are computed in real space (as opposed to on the unit cell). The object is assumed to already have the correct size. The data type stored by this output vector must be what you get when you multiply the third derivatives of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
[out]
third_derivatives
The third derivatives of the function specified by fe_function at the quadrature points of the current cell. The third derivatives are computed in real space (as opposed to on the unit cell). The object is assumed to already have the correct size. The data type stored by this output vector must be what you get when you multiply the third derivatives of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
-
Postcondition
third_derivatives[q] will contain the third derivatives of the field described by fe_function at the th quadrature point. third_derivatives[q][i][j][k] represents the th component of the 3rd order tensor of third derivatives at quadrature point .
+
Postcondition
third_derivatives[q] will contain the third derivatives of the field described by fe_function at the th quadrature point. third_derivatives[q][i][j][k] represents the th component of the 3rd order tensor of third derivatives at quadrature point .
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_third_derivatives(), but applied to multi-component (vector- valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
third_derivatives[q] is a vector of third derivatives of the field described by fe_function at the th quadrature point. The size of the vector accessed by third_derivatives[q] equals the number of components of the finite element, i.e. third_derivatives[q][c] returns the third derivative of the th vector component at the th quadrature point. Consequently, third_derivatives[q][c][i][j][k] is the th component of the tensor of third derivatives of the th vector component of the vector field at quadrature point of the current cell.
+
Postcondition
third_derivatives[q] is a vector of third derivatives of the field described by fe_function at the th quadrature point. The size of the vector accessed by third_derivatives[q] equals the number of components of the finite element, i.e. third_derivatives[q][c] returns the third derivative of the th vector component at the th quadrature point. Consequently, third_derivatives[q][c][i][j][k] is the th component of the tensor of third derivatives of the th vector component of the vector field at quadrature point of the current cell.
Mapped quadrature weight. If this object refers to a volume evaluation (i.e. the derived class is of type FEValues), then this is the Jacobi determinant times the weight of the q_pointth unit quadrature point.
For surface evaluations (i.e. classes FEFaceValues or FESubfaceValues), it is the mapped surface element times the weight of the quadrature point.
-
You can think of the quantity returned by this function as the volume or surface element in the integral that we implement here by quadrature.
+
You can think of the quantity returned by this function as the volume or surface element in the integral that we implement here by quadrature.
Return the second derivative of the transformation from unit to real cell, i.e. the first derivative of the Jacobian, at the specified quadrature point, i.e. .
+
Return the second derivative of the transformation from unit to real cell, i.e. the first derivative of the Jacobian, at the specified quadrature point, i.e. .
Return the second derivative of the transformation from unit to real cell, i.e. the first derivative of the Jacobian, at the specified quadrature point, pushed forward to the real cell coordinates, i.e. .
+
Return the second derivative of the transformation from unit to real cell, i.e. the first derivative of the Jacobian, at the specified quadrature point, pushed forward to the real cell coordinates, i.e. .
Note
For this function to work properly, the underlying FEValues, FEFaceValues, or FESubfaceValues object on which you call it must have computed the information you are requesting. To do so, the update_jacobian_pushed_forward_grads flag must be an element of the list of UpdateFlags that you passed to the constructor of this object. See The interplay of UpdateFlags, Mapping, and FiniteElement in FEValues for more information.
@@ -2609,7 +2609,7 @@
-
Return the third derivative of the transformation from unit to real cell, i.e. the second derivative of the Jacobian, at the specified quadrature point, i.e. .
+
Return the third derivative of the transformation from unit to real cell, i.e. the second derivative of the Jacobian, at the specified quadrature point, i.e. .
Note
For this function to work properly, the underlying FEValues, FEFaceValues, or FESubfaceValues object on which you call it must have computed the information you are requesting. To do so, the update_jacobian_2nd_derivatives flag must be an element of the list of UpdateFlags that you passed to the constructor of this object. See The interplay of UpdateFlags, Mapping, and FiniteElement in FEValues for more information.
/usr/share/doc/packages/dealii/doxygen/deal.II/classFESystem.html differs (HTML document, ASCII text, with very long lines)
--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classFESystem.html 2024-04-12 04:45:53.067585628 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classFESystem.html 2024-04-12 04:45:53.075585684 +0000
@@ -500,11 +500,11 @@
This class provides an interface to group several elements together into one, vector-valued element. As example, consider the Taylor-Hood element that is used for the solution of the Stokes and Navier-Stokes equations: There, the velocity (of which there are as many components as the dimension of the domain) is discretized with elements and the pressure with elements. Mathematically, the finite element space for the coupled problem is then often written as where the exponentiation is understood to be the tensor product of spaces – i.e., in 2d, we have – and tensor products lead to vectors where each component of the vector-valued function space corresponds to a scalar function in one of the or spaces. Using the FESystem class, this space is created using
This class provides an interface to group several elements together into one, vector-valued element. As example, consider the Taylor-Hood element that is used for the solution of the Stokes and Navier-Stokes equations: There, the velocity (of which there are as many components as the dimension of the domain) is discretized with elements and the pressure with elements. Mathematically, the finite element space for the coupled problem is then often written as where the exponentiation is understood to be the tensor product of spaces – i.e., in 2d, we have – and tensor products lead to vectors where each component of the vector-valued function space corresponds to a scalar function in one of the or spaces. Using the FESystem class, this space is created using
The creation of this element here corresponds to taking tensor-product powers of the element in the first line of the list of arguments to the FESystem constructor, and then concatenation via another tensor product with the element in the second line. This kind of construction is used, for example, in the step-22 tutorial program.
+
The creation of this element here corresponds to taking tensor-product powers of the element in the first line of the list of arguments to the FESystem constructor, and then concatenation via another tensor product with the element in the second line. This kind of construction is used, for example, in the step-22 tutorial program.
Similarly, step-8 solves an elasticity equation where we need to solve for the displacement of a solid object. The displacement again has components if the domain is -dimensional, and so the combined finite element is created using
where now each (vector) component of the combined element corresponds to a space.
To the outside world, FESystem objects look just like a usual finite element object, they just happen to be composed of several other finite elements that are possibly of different type. These "base elements" can themselves have multiple components and, in particular, could also be vector-valued – for example, if one of the base elements is an FESystem itself (see also below). An example is given in the documentation of namespace FETools::Compositing, when using the "tensor product" strategy.
@@ -3836,7 +3836,7 @@
Return a block mask with as many elements as this object has blocks and of which exactly the one component is true that corresponds to the given argument. See the glossary for more information.
-
Note
This function will only succeed if the scalar referenced by the argument encompasses a complete block. In other words, if, for example, you pass an extractor for the single velocity and this object represents an FE_RaviartThomas object, then the single scalar object you selected is part of a larger block and consequently there is no block mask that would represent it. The function will then produce an exception.
+
Note
This function will only succeed if the scalar referenced by the argument encompasses a complete block. In other words, if, for example, you pass an extractor for the single velocity and this object represents an FE_RaviartThomas object, then the single scalar object you selected is part of a larger block and consequently there is no block mask that would represent it. The function will then produce an exception.
Parameters
scalar
An object that represents a single scalar vector component of this finite element.
@@ -3944,7 +3944,7 @@
Given a component mask (see this glossary entry), produce a block mask (see this glossary entry) that represents the blocks that correspond to the components selected in the input argument. This is essentially a conversion operator from ComponentMask to BlockMask.
-
Note
This function will only succeed if the components referenced by the argument encompasses complete blocks. In other words, if, for example, you pass an component mask for the single velocity and this object represents an FE_RaviartThomas object, then the single component you selected is part of a larger block and consequently there is no block mask that would represent it. The function will then produce an exception.
+
Note
This function will only succeed if the components referenced by the argument encompasses complete blocks. In other words, if, for example, you pass an component mask for the single velocity and this object represents an FE_RaviartThomas object, then the single component you selected is part of a larger block and consequently there is no block mask that would represent it. The function will then produce an exception.
Parameters
component_mask
The mask that selects individual components of the finite element
@@ -4162,9 +4162,9 @@
For a given degree of freedom, return whether it is logically associated with a vertex, line, quad or hex.
-
For instance, for continuous finite elements this coincides with the lowest dimensional object the support point of the degree of freedom lies on. To give an example, for elements in 3d, every degree of freedom is defined by a shape function that we get by interpolating using support points that lie on the vertices of the cell. The support of these points of course extends to all edges connected to this vertex, as well as the adjacent faces and the cell interior, but we say that logically the degree of freedom is associated with the vertex as this is the lowest- dimensional object it is associated with. Likewise, for elements in 3d, the degrees of freedom with support points at edge midpoints would yield a value of GeometryPrimitive::line from this function, whereas those on the centers of faces in 3d would return GeometryPrimitive::quad.
-
To make this more formal, the kind of object returned by this function represents the object so that the support of the shape function corresponding to the degree of freedom, (i.e., that part of the domain where the function "lives") is the union of all of the cells sharing this object. To return to the example above, for in 3d, the shape function with support point at an edge midpoint has support on all cells that share the edge and not only the cells that share the adjacent faces, and consequently the function will return GeometryPrimitive::line.
-
On the other hand, for discontinuous elements of type , a degree of freedom associated with an interpolation polynomial that has its support point physically located at a line bounding a cell, but is nonzero only on one cell. Consequently, it is logically associated with the interior of that cell (i.e., with a GeometryPrimitive::quad in 2d and a GeometryPrimitive::hex in 3d).
+
For instance, for continuous finite elements this coincides with the lowest dimensional object the support point of the degree of freedom lies on. To give an example, for elements in 3d, every degree of freedom is defined by a shape function that we get by interpolating using support points that lie on the vertices of the cell. The support of these points of course extends to all edges connected to this vertex, as well as the adjacent faces and the cell interior, but we say that logically the degree of freedom is associated with the vertex as this is the lowest- dimensional object it is associated with. Likewise, for elements in 3d, the degrees of freedom with support points at edge midpoints would yield a value of GeometryPrimitive::line from this function, whereas those on the centers of faces in 3d would return GeometryPrimitive::quad.
+
To make this more formal, the kind of object returned by this function represents the object so that the support of the shape function corresponding to the degree of freedom, (i.e., that part of the domain where the function "lives") is the union of all of the cells sharing this object. To return to the example above, for in 3d, the shape function with support point at an edge midpoint has support on all cells that share the edge and not only the cells that share the adjacent faces, and consequently the function will return GeometryPrimitive::line.
+
On the other hand, for discontinuous elements of type , a degree of freedom associated with an interpolation polynomial that has its support point physically located at a line bounding a cell, but is nonzero only on one cell. Consequently, it is logically associated with the interior of that cell (i.e., with a GeometryPrimitive::quad in 2d and a GeometryPrimitive::hex in 3d).
Parameters
[in]
cell_dof_index
The index of a shape function or degree of freedom. This index must be in the range [0,dofs_per_cell).
/usr/share/doc/packages/dealii/doxygen/deal.II/classFEValues.html differs (HTML document, ASCII text, with very long lines)
--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classFEValues.html 2024-04-12 04:45:53.187586459 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classFEValues.html 2024-04-12 04:45:53.187586459 +0000
@@ -743,7 +743,7 @@
If the shape function is vector-valued, then this returns the only non- zero component. If the shape function has more than one non-zero component (i.e. it is not primitive), then throw an exception of type ExcShapeFunctionNotPrimitive. In that case, use the shape_value_component() function.
Parameters
-
i
Number of the shape function to be evaluated. Note that this number runs from zero to dofs_per_cell, even in the case of an FEFaceValues or FESubfaceValues object.
+
i
Number of the shape function to be evaluated. Note that this number runs from zero to dofs_per_cell, even in the case of an FEFaceValues or FESubfaceValues object.
q_point
Number of the quadrature point at which function is to be evaluated
@@ -784,7 +784,7 @@
Compute one vector component of the value of a shape function at a quadrature point. If the finite element is scalar, then only component zero is allowed and the return value equals that of the shape_value() function. If the finite element is vector valued but all shape functions are primitive (i.e. they are non-zero in only one component), then the value returned by shape_value() equals that of this function for exactly one component. This function is therefore only of greater interest if the shape function is not primitive, but then it is necessary since the other function cannot be used.
Parameters
-
i
Number of the shape function to be evaluated.
+
i
Number of the shape function to be evaluated.
q_point
Number of the quadrature point at which function is to be evaluated.
component
vector component to be evaluated.
@@ -823,7 +823,7 @@
The same holds for the arguments of this function as for the shape_value() function.
Parameters
-
i
Number of the shape function to be evaluated.
+
i
Number of the shape function to be evaluated.
q_point
Number of the quadrature point at which function is to be evaluated.
@@ -1025,17 +1025,17 @@
-
Return the values of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and the related get_function_gradients() function is also used in step-15 along with numerous other tutorial programs.
+
Return the values of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and the related get_function_gradients() function is also used in step-15 along with numerous other tutorial programs.
If the current cell is not active (i.e., it has children), then the finite element function is, strictly speaking, defined by shape functions that live on these child cells. Rather than evaluating the shape functions on the child cells, with the quadrature points defined on the current cell, this function first interpolates the finite element function to shape functions defined on the current cell, and then evaluates this interpolated function.
This function may only be used if the finite element in use is a scalar one, i.e. has only one vector component. To get values of multi-component elements, there is another get_function_values() below, returning a vector of vectors of results.
Parameters
[in]
fe_function
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
values
The values of the function specified by fe_function at the quadrature points of the current cell. The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the values of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the solution vector.
+
[out]
values
The values of the function specified by fe_function at the quadrature points of the current cell. The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the values of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the solution vector.
-
Postcondition
values[q] will contain the value of the field described by fe_function at the th quadrature point.
+
Postcondition
values[q] will contain the value of the field described by fe_function at the th quadrature point.
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_values(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
values[q] is a vector of values of the field described by fe_function at the th quadrature point. The size of the vector accessed by values[q] equals the number of components of the finite element, i.e. values[q](c) returns the value of the th vector component at the th quadrature point.
+
Postcondition
values[q] is a vector of values of the field described by fe_function at the th quadrature point. The size of the vector accessed by values[q] equals the number of components of the finite element, i.e. values[q](c) returns the value of the th vector component at the th quadrature point.
Return the gradients of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and it is also used in step-15 along with numerous other tutorial programs.
+
Return the gradients of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and it is also used in step-15 along with numerous other tutorial programs.
This function may only be used if the finite element in use is a scalar one, i.e. has only one vector component. There is a corresponding function of the same name for vector-valued finite elements.
Parameters
[in]
fe_function
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
gradients
The gradients of the function specified by fe_function at the quadrature points of the current cell. The gradients are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the gradients of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
[out]
gradients
The gradients of the function specified by fe_function at the quadrature points of the current cell. The gradients are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the gradients of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
-
Postcondition
gradients[q] will contain the gradient of the field described by fe_function at the th quadrature point. gradients[q][d] represents the derivative in coordinate direction at quadrature point .
+
Postcondition
gradients[q] will contain the gradient of the field described by fe_function at the th quadrature point. gradients[q][d] represents the derivative in coordinate direction at quadrature point .
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_gradients(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
gradients[q] is a vector of gradients of the field described by fe_function at the th quadrature point. The size of the vector accessed by gradients[q] equals the number of components of the finite element, i.e. gradients[q][c] returns the gradient of the th vector component at the th quadrature point. Consequently, gradients[q][c][d] is the derivative in coordinate direction of the th vector component of the vector field at quadrature point of the current cell.
+
Postcondition
gradients[q] is a vector of gradients of the field described by fe_function at the th quadrature point. The size of the vector accessed by gradients[q] equals the number of components of the finite element, i.e. gradients[q][c] returns the gradient of the th vector component at the th quadrature point. Consequently, gradients[q][c][d] is the derivative in coordinate direction of the th vector component of the vector field at quadrature point of the current cell.
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
hessians
The Hessians of the function specified by fe_function at the quadrature points of the current cell. The Hessians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Hessians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
[out]
hessians
The Hessians of the function specified by fe_function at the quadrature points of the current cell. The Hessians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Hessians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
-
Postcondition
hessians[q] will contain the Hessian of the field described by fe_function at the th quadrature point. hessians[q][i][j] represents the th component of the matrix of second derivatives at quadrature point .
+
Postcondition
hessians[q] will contain the Hessian of the field described by fe_function at the th quadrature point. hessians[q][i][j] represents the th component of the matrix of second derivatives at quadrature point .
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_hessians(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
hessians[q] is a vector of Hessians of the field described by fe_function at the th quadrature point. The size of the vector accessed by hessians[q] equals the number of components of the finite element, i.e. hessians[q][c] returns the Hessian of the th vector component at the th quadrature point. Consequently, hessians[q][c][i][j] is the th component of the matrix of second derivatives of the th vector component of the vector field at quadrature point of the current cell.
+
Postcondition
hessians[q] is a vector of Hessians of the field described by fe_function at the th quadrature point. The size of the vector accessed by hessians[q] equals the number of components of the finite element, i.e. hessians[q][c] returns the Hessian of the th vector component at the th quadrature point. Consequently, hessians[q][c][i][j] is the th component of the matrix of second derivatives of the th vector component of the vector field at quadrature point of the current cell.
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
laplacians
The Laplacians of the function specified by fe_function at the quadrature points of the current cell. The Laplacians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Laplacians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the input vector.
+
[out]
laplacians
The Laplacians of the function specified by fe_function at the quadrature points of the current cell. The Laplacians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Laplacians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the input vector.
-
Postcondition
laplacians[q] will contain the Laplacian of the field described by fe_function at the th quadrature point.
+
Postcondition
laplacians[q] will contain the Laplacian of the field described by fe_function at the th quadrature point.
For each component of the output vector, there holds laplacians[q]=trace(hessians[q]), where hessians would be the output of the get_function_hessians() function.
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
@@ -1613,7 +1613,7 @@
This function does the same as the other get_function_laplacians(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
laplacians[q] is a vector of Laplacians of the field described by fe_function at the th quadrature point. The size of the vector accessed by laplacians[q] equals the number of components of the finite element, i.e. laplacians[q][c] returns the Laplacian of the th vector component at the th quadrature point.
+
Postcondition
laplacians[q] is a vector of Laplacians of the field described by fe_function at the th quadrature point. The size of the vector accessed by laplacians[q] equals the number of components of the finite element, i.e. laplacians[q][c] returns the Laplacian of the th vector component at the th quadrature point.
For each component of the output vector, there holds laplacians[q][c]=trace(hessians[q][c]), where hessians would be the output of the get_function_hessians() function.
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
third_derivatives
The third derivatives of the function specified by fe_function at the quadrature points of the current cell. The third derivatives are computed in real space (as opposed to on the unit cell). The object is assumed to already have the correct size. The data type stored by this output vector must be what you get when you multiply the third derivatives of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
[out]
third_derivatives
The third derivatives of the function specified by fe_function at the quadrature points of the current cell. The third derivatives are computed in real space (as opposed to on the unit cell). The object is assumed to already have the correct size. The data type stored by this output vector must be what you get when you multiply the third derivatives of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
-
Postcondition
third_derivatives[q] will contain the third derivatives of the field described by fe_function at the th quadrature point. third_derivatives[q][i][j][k] represents the th component of the 3rd order tensor of third derivatives at quadrature point .
+
Postcondition
third_derivatives[q] will contain the third derivatives of the field described by fe_function at the th quadrature point. third_derivatives[q][i][j][k] represents the th component of the 3rd order tensor of third derivatives at quadrature point .
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_third_derivatives(), but applied to multi-component (vector- valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
third_derivatives[q] is a vector of third derivatives of the field described by fe_function at the th quadrature point. The size of the vector accessed by third_derivatives[q] equals the number of components of the finite element, i.e. third_derivatives[q][c] returns the third derivative of the th vector component at the th quadrature point. Consequently, third_derivatives[q][c][i][j][k] is the th component of the tensor of third derivatives of the th vector component of the vector field at quadrature point of the current cell.
+
Postcondition
third_derivatives[q] is a vector of third derivatives of the field described by fe_function at the th quadrature point. The size of the vector accessed by third_derivatives[q] equals the number of components of the finite element, i.e. third_derivatives[q][c] returns the third derivative of the th vector component at the th quadrature point. Consequently, third_derivatives[q][c][i][j][k] is the th component of the tensor of third derivatives of the th vector component of the vector field at quadrature point of the current cell.
Mapped quadrature weight. If this object refers to a volume evaluation (i.e. the derived class is of type FEValues), then this is the Jacobi determinant times the weight of the q_pointth unit quadrature point.
For surface evaluations (i.e. classes FEFaceValues or FESubfaceValues), it is the mapped surface element times the weight of the quadrature point.
-
You can think of the quantity returned by this function as the volume or surface element in the integral that we implement here by quadrature.
+
You can think of the quantity returned by this function as the volume or surface element in the integral that we implement here by quadrature.
Return the second derivative of the transformation from unit to real cell, i.e. the first derivative of the Jacobian, at the specified quadrature point, i.e. .
+
Return the second derivative of the transformation from unit to real cell, i.e. the first derivative of the Jacobian, at the specified quadrature point, i.e. .
Return the second derivative of the transformation from unit to real cell, i.e. the first derivative of the Jacobian, at the specified quadrature point, pushed forward to the real cell coordinates, i.e. .
+
Return the second derivative of the transformation from unit to real cell, i.e. the first derivative of the Jacobian, at the specified quadrature point, pushed forward to the real cell coordinates, i.e. .
Note
For this function to work properly, the underlying FEValues, FEFaceValues, or FESubfaceValues object on which you call it must have computed the information you are requesting. To do so, the update_jacobian_pushed_forward_grads flag must be an element of the list of UpdateFlags that you passed to the constructor of this object. See The interplay of UpdateFlags, Mapping, and FiniteElement in FEValues for more information.
@@ -2393,7 +2393,7 @@
-
Return the third derivative of the transformation from unit to real cell, i.e. the second derivative of the Jacobian, at the specified quadrature point, i.e. .
+
Return the third derivative of the transformation from unit to real cell, i.e. the second derivative of the Jacobian, at the specified quadrature point, i.e. .
Note
For this function to work properly, the underlying FEValues, FEFaceValues, or FESubfaceValues object on which you call it must have computed the information you are requesting. To do so, the update_jacobian_2nd_derivatives flag must be an element of the list of UpdateFlags that you passed to the constructor of this object. See The interplay of UpdateFlags, Mapping, and FiniteElement in FEValues for more information.
/usr/share/doc/packages/dealii/doxygen/deal.II/classFEValuesBase.html differs (HTML document, ASCII text, with very long lines)
--- old//usr/share/doc/packages/dealii/doxygen/deal.II/classFEValuesBase.html 2024-04-12 04:45:53.283587123 +0000
+++ new//usr/share/doc/packages/dealii/doxygen/deal.II/classFEValuesBase.html 2024-04-12 04:45:53.287587152 +0000
@@ -614,7 +614,7 @@
If the shape function is vector-valued, then this returns the only non- zero component. If the shape function has more than one non-zero component (i.e. it is not primitive), then throw an exception of type ExcShapeFunctionNotPrimitive. In that case, use the shape_value_component() function.
Parameters
-
i
Number of the shape function to be evaluated. Note that this number runs from zero to dofs_per_cell, even in the case of an FEFaceValues or FESubfaceValues object.
+
i
Number of the shape function to be evaluated. Note that this number runs from zero to dofs_per_cell, even in the case of an FEFaceValues or FESubfaceValues object.
q_point
Number of the quadrature point at which function is to be evaluated
@@ -648,7 +648,7 @@
Compute one vector component of the value of a shape function at a quadrature point. If the finite element is scalar, then only component zero is allowed and the return value equals that of the shape_value() function. If the finite element is vector valued but all shape functions are primitive (i.e. they are non-zero in only one component), then the value returned by shape_value() equals that of this function for exactly one component. This function is therefore only of greater interest if the shape function is not primitive, but then it is necessary since the other function cannot be used.
Parameters
-
i
Number of the shape function to be evaluated.
+
i
Number of the shape function to be evaluated.
q_point
Number of the quadrature point at which function is to be evaluated.
component
vector component to be evaluated.
@@ -680,7 +680,7 @@
The same holds for the arguments of this function as for the shape_value() function.
Parameters
-
i
Number of the shape function to be evaluated.
+
i
Number of the shape function to be evaluated.
q_point
Number of the quadrature point at which function is to be evaluated.
@@ -840,17 +840,17 @@
std::vector< typename InputVector::value_type > &
values&#href_anchor"memdoc">
-
Return the values of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and the related get_function_gradients() function is also used in step-15 along with numerous other tutorial programs.
+
Return the values of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and the related get_function_gradients() function is also used in step-15 along with numerous other tutorial programs.
If the current cell is not active (i.e., it has children), then the finite element function is, strictly speaking, defined by shape functions that live on these child cells. Rather than evaluating the shape functions on the child cells, with the quadrature points defined on the current cell, this function first interpolates the finite element function to shape functions defined on the current cell, and then evaluates this interpolated function.
This function may only be used if the finite element in use is a scalar one, i.e. has only one vector component. To get values of multi-component elements, there is another get_function_values() below, returning a vector of vectors of results.
Parameters
[in]
fe_function
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
values
The values of the function specified by fe_function at the quadrature points of the current cell. The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the values of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the solution vector.
+
[out]
values
The values of the function specified by fe_function at the quadrature points of the current cell. The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the values of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the solution vector.
-
Postcondition
values[q] will contain the value of the field described by fe_function at the th quadrature point.
+
Postcondition
values[q] will contain the value of the field described by fe_function at the th quadrature point.
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_values(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
values[q] is a vector of values of the field described by fe_function at the th quadrature point. The size of the vector accessed by values[q] equals the number of components of the finite element, i.e. values[q](c) returns the value of the th vector component at the th quadrature point.
+
Postcondition
values[q] is a vector of values of the field described by fe_function at the th quadrature point. The size of the vector accessed by values[q] equals the number of components of the finite element, i.e. values[q](c) returns the value of the th vector component at the th quadrature point.
Return the gradients of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and it is also used in step-15 along with numerous other tutorial programs.
+
Return the gradients of a finite element function at the quadrature points of the current cell, face, or subface (selected the last time the reinit() function was called). That is, if the first argument fe_function is a vector of nodal values of a finite element function defined on a DoFHandler object, then the output vector (the second argument, values) is the vector of values where are the quadrature points on the current cell . This function is first discussed in the Results section of step-4, and it is also used in step-15 along with numerous other tutorial programs.
This function may only be used if the finite element in use is a scalar one, i.e. has only one vector component. There is a corresponding function of the same name for vector-valued finite elements.
Parameters
[in]
fe_function
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
gradients
The gradients of the function specified by fe_function at the quadrature points of the current cell. The gradients are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the gradients of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
[out]
gradients
The gradients of the function specified by fe_function at the quadrature points of the current cell. The gradients are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the gradients of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
-
Postcondition
gradients[q] will contain the gradient of the field described by fe_function at the th quadrature point. gradients[q][d] represents the derivative in coordinate direction at quadrature point .
+
Postcondition
gradients[q] will contain the gradient of the field described by fe_function at the th quadrature point. gradients[q][d] represents the derivative in coordinate direction at quadrature point .
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_gradients(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
gradients[q] is a vector of gradients of the field described by fe_function at the th quadrature point. The size of the vector accessed by gradients[q] equals the number of components of the finite element, i.e. gradients[q][c] returns the gradient of the th vector component at the th quadrature point. Consequently, gradients[q][c][d] is the derivative in coordinate direction of the th vector component of the vector field at quadrature point of the current cell.
+
Postcondition
gradients[q] is a vector of gradients of the field described by fe_function at the th quadrature point. The size of the vector accessed by gradients[q] equals the number of components of the finite element, i.e. gradients[q][c] returns the gradient of the th vector component at the th quadrature point. Consequently, gradients[q][c][d] is the derivative in coordinate direction of the th vector component of the vector field at quadrature point of the current cell.
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
hessians
The Hessians of the function specified by fe_function at the quadrature points of the current cell. The Hessians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Hessians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
+
[out]
hessians
The Hessians of the function specified by fe_function at the quadrature points of the current cell. The Hessians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Hessians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument).
-
Postcondition
hessians[q] will contain the Hessian of the field described by fe_function at the th quadrature point. hessians[q][i][j] represents the th component of the matrix of second derivatives at quadrature point .
+
Postcondition
hessians[q] will contain the Hessian of the field described by fe_function at the th quadrature point. hessians[q][i][j] represents the th component of the matrix of second derivatives at quadrature point .
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_hessians(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
hessians[q] is a vector of Hessians of the field described by fe_function at the th quadrature point. The size of the vector accessed by hessians[q] equals the number of components of the finite element, i.e. hessians[q][c] returns the Hessian of the th vector component at the th quadrature point. Consequently, hessians[q][c][i][j] is the th component of the matrix of second derivatives of the th vector component of the vector field at quadrature point of the current cell.
+
Postcondition
hessians[q] is a vector of Hessians of the field described by fe_function at the th quadrature point. The size of the vector accessed by hessians[q] equals the number of components of the finite element, i.e. hessians[q][c] returns the Hessian of the th vector component at the th quadrature point. Consequently, hessians[q][c][i][j] is the th component of the matrix of second derivatives of the th vector component of the vector field at quadrature point of the current cell.
A vector of values that describes (globally) the finite element function that this function should evaluate at the quadrature points of the current cell.
-
[out]
laplacians
The Laplacians of the function specified by fe_function at the quadrature points of the current cell. The Laplacians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Laplacians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the input vector.
+
[out]
laplacians
The Laplacians of the function specified by fe_function at the quadrature points of the current cell. The Laplacians are computed in real space (as opposed to on the unit cell). The object is assume to already have the correct size. The data type stored by this output vector must be what you get when you multiply the Laplacians of shape function times the type used to store the values of the unknowns of your finite element vector (represented by the fe_function argument). This happens to be equal to the type of the elements of the input vector.
-
Postcondition
laplacians[q] will contain the Laplacian of the field described by fe_function at the th quadrature point.
+
Postcondition
laplacians[q] will contain the Laplacian of the field described by fe_function at the th quadrature point.
For each component of the output vector, there holds laplacians[q]=trace(hessians[q]), where hessians would be the output of the get_function_hessians() function.
Note
The actual data type of the input vector may be either a Vector<T>, BlockVector<T>, or one of the PETSc or Trilinos vector wrapper classes. It represents a global vector of DoF values associated with the DoFHandler object with which this FEValues object was last initialized.
This function does the same as the other get_function_laplacians(), but applied to multi-component (vector-valued) elements. The meaning of the arguments is as explained there.
-
Postcondition
laplacians[q] is a vector of Laplacians of the field described by fe_function at the th quadrature point. The size of the vector accessed by laplacians[q] equals the number of components of the finite element, i.e. laplacians[q][c] returns the Laplacian of the th vector component at the th quadrature point.
@@ -366,19 +366,19 @@
![$\Psi_i[\cdot]$](form_141_dark.png)
![$\Psi_i[\varphi]=\varphi(\hat{\mathbf{x}}_i)$](form_142_dark.png)
![$\Psi_i[\varphi_j]=\delta_{ij}$](form_143_dark.png)
![$\Psi_i[\varphi]=\varphi(\hat{\mathbf{x}}_i)_{c(i)}$](form_146_dark.png)
![$\Psi_i[\varphi]
+<dd><p class=](form_147_dark.png)
"Generalized support points" are, as the name suggests, a generalization of 
![$\Psi_i[\cdot]$](form_115_dark.png)
![$\Psi_i[\varphi]=\varphi(\hat{\mathbf{x}}_i)$](form_116_dark.png)
![$\Psi_i[\varphi_j]=\delta_{ij}$](form_117_dark.png)
![$\Psi_i[\varphi]=\varphi(\hat{\mathbf{x}}_i)_{c(i)}$](form_120_dark.png)
![$\Psi_i[\varphi]
=
\varphi(\hat{\mathbf{x}}_i) \cdot \mathbf{n}_i
- $](form_121_dark.png)
![$\Psi_i[\varphi]
+ $](form_150_dark.png)
![$\Psi_i[\varphi]
=
\int_{\hat{K}} \varphi(\hat{\mathbf{x}})
{\hat{x}_1}^{p_1(i)}
{\hat{x}_2}^{p_2(i)}
- $](form_124_dark.png)
![\[ m = I - \Delta t M. \]](form_368_dark.png)
![\[ m = I - \Delta t M. \]](form_353_dark.png)
![$[-1,1]$](form_582_dark.png)
![$[0,1]$](form_581_dark.png)
![$[-1,1]^d$](form_464_dark.png)
![$[0,1]^d$](form_872_dark.png)
![\[
(6) \quad Sa = (D - C \: A^{-1} \: B)a
-\]](form_1868_dark.png)
![\[
y = S^{-1} g'
- \]](form_1872_dark.png)
![$\in [0, 2^{\text{dim}} - 1]$](form_362_dark.png)
![\[
\mathbf x_K = \frac{1}{|K|} \int_K \mathbf x \; \textrm{d}x
-\]](form_1482_dark.png)
![\[ F: \mathcal{B} \subset
-R^{\text{chartdim}} \mapsto \mathcal{M} \subset R^{\text{spacedim}} \]](form_1401_dark.png)
![\[ F: \mathcal{B} \subset
+R^{\text{chartdim}} \mapsto \mathcal{M} \subset R^{\text{spacedim}} \]](form_1419_dark.png)
![\[ F^{-1}: \mathcal{M} \subset R^{\text{spacedim}} \mapsto \mathcal{B}
-\subset R^{\text{chartdim}} \]](form_1402_dark.png)
![\[ F^{-1}: \mathcal{M} \subset R^{\text{spacedim}} \mapsto \mathcal{B}
+\subset R^{\text{chartdim}} \]](form_1420_dark.png)
![\[
-\mathbf x^{\text{new}} = F(\sum_i w_i F^{-1}(\mathbf x_i)). \]](form_1403_dark.png)
![\[
+\mathbf x^{\text{new}} = F(\sum_i w_i F^{-1}(\mathbf x_i)). \]](form_1421_dark.png)
![$[0,1]\times \{0\}\times \{0\}$](form_1410_dark.png)
![$[0,1]\times \{0\}\times \{0\}$](form_1428_dark.png)
![\begin{align*}
\zeta(t) &= \xi_1 + t (\xi_2-\xi_1)
\\ &= F^{-1}(\mathbf x_1) + t\left[F^{-1}(\mathbf x_2)
-F^{-1}(\mathbf x_1)\right]
-\end{align*}](form_1440_dark.png)
![$ C (1/\sqrt[dim]{k})^r $](form_415_dark.png)
![$ C (1/\sqrt[dim]{k})^r $](form_379_dark.png)
![\[
\nabla \mathbf F(\mathbf x) \; \Delta \mathbf x
\approx
\mathbf F(\mathbf x + \Delta \mathbf x) - \mathbf F(\mathbf x).
-\]](form_396_dark.png)
![$[\Delta \mathbf x] [\nabla \mathbf F(\mathbf x)]^T$](form_398_dark.png)
![\[
\left\lbrace\begin{aligned}
x &= x_0 + c \cosh(u) \cos(v) \\
y &= y_0 + c \sinh(u) \sin(v)
\end{aligned}\right.
-\]](form_1454_dark.png)
![$e\in\left]0,1\right[$](form_1458_dark.png)
@@ -348,7 +348,7 @@
![$[\mathbf{u}]=\mathbf{u_1} - \mathbf{u_2}$](form_1183_dark.png)
![$[\mathbf{u}]=\mathbf{u_1} - \mathbf{u_2}$](form_1095_dark.png)
Fourier series on a reference element. The exponential form of the 
![\[
\int_0^1 \phi_k(x) \phi_l^\ast(x) dx=\delta_{kl}.
-\]](form_1176_dark.png)
![$ [-1;1] $](form_1167_dark.png)
![$ [0;1] $](form_1169_dark.png)
