Finite element (`dolfinx::fem`)#

Finite elements#

template<std::floating_point T> class FiniteElement#

Model of a finite element.

Provides the dof layout on a reference element, and various methods for evaluating and transforming the basis.

Value shapes

An element has two value shapes, and they are not interchangeable.

A basis is tabulated on the reference cell and pushed forward to a physical cell. FiniteElement::reference_value_shape is the shape Basix tabulates in, and is the shape of the data returned by FiniteElement::tabulate. FiniteElement::value_shape is the shape of the field after the push-forward. It is the shape a user of the space sees, and the one UFL reports for a Coefficient or Argument on the space, so it is the one to use for anything user-facing, for checking that two spaces are compatible, and for sizing buffers that hold physical values.

The two differ for three independent reasons:

Blocking. A blocked element repeats a scalar base element at each dof point, e.g. a vector Lagrange space. Its value shape is whatever the caller asked for, while its reference value shape is that of the scalar base element, {}.
Quadrature elements, whose reference value shape is {}.
The map. A Piola-mapped basis is pushed forward with the Jacobian J, which has shape (gdim, tdim). The value axes that J contracts therefore have extent gdim in physical space and tdim on the reference cell. These are equal unless gdim != tdim, i.e. on a manifold. See ::compute_value_shape.

Blocking

FiniteElement::block_size is the number of dofs collocated at a dof point. For a blocked element it equals FiniteElement::value_size, except for a symmetric rank-2 tensor, which stores only its independent components: a {3, 3} symmetric element has value size 9 but block size 6.

Code that needs the number of physical components in one block must therefore call FiniteElement::physical_base_value_size rather than compute value_size() / block_size(), which is wrong for symmetric elements.

Examples

All on a mesh of triangles, so tdim == 2. RT is Raviart-Thomas and sym marks a symmetric element. The columns are FiniteElement::value_shape, FiniteElement::block_size, FiniteElement::reference_value_shape and FiniteElement::physical_base_value_size. FiniteElement::value_size and FiniteElement::reference_value_size are the products of the respective shapes and are not tabulated.

element	gdim	vshape	bs	rvshape	bvsize
P1	2	{}	1	{}	1
P1	3	{}	1	{}	1
P1, shape {2}	2	{2}	2	{}	1
P1, shape {2}	3	{2}	2	{}	1
P1, shape {5}	2	{5}	5	{}	1
P1, sym {3, 3}	2	{3, 3}	6	{}	1
RT 1	2	{2}	1	{2}	2
RT 1	3	{3}	1	{2}	3
Regge 0	2	{2, 2}	1	{2, 2}	4
Regge 0	3	{3, 3}	1	{2, 2}	9
mixed	any	throws	1	throws	throws

Note first that gdim changes nothing for P1 or P1, shape {2}: both are identity mapped, so their physical and reference value shapes agree on a manifold exactly as they do on a 2D mesh, and the blocked shape {2} stays {2} on a gdim == 3 mesh. Contrast the RT 1 and Regge 0 pairs, where the Piola push-forward does introduce gdim.

Some rows deserve further comment:

P1, shape {5}. A blocked element’s value shape is chosen by the caller and has nothing to do with gdim. A 5-vector, or a {3, 3} tensor field, on a mesh of any geometric dimension is legal, and gdim must never be substituted into such a shape.
P1, sym {3, 3}. The only case in which the block size is not the value size: the value size is 9, but only the six independent components are stored, one dof each, and the other three are recovered by symmetry.
RT 1 with gdim == 3, i.e. a triangle embedded in 3D. The contravariant Piola push-forward multiplies by a (3, 2) Jacobian, so the field has three components in physical space while Basix still tabulates two. This is the case that separates the two value shapes for a non-blocked element.
mixed. A mixed element has no value shape of its own, so every shape and size accessor throws; only FiniteElement::block_size is defined, and is 1. Extract a sub-element with ::extract_sub_element to get at its shapes. FiniteElement::is_mixed
reports this case, and is implemented as “has no reference value

shape”.

Public Types

using geometry_type = T #: Geometry type of the Mesh that the FunctionSpace is defined on.

Public Functions

FiniteElement(const basix::FiniteElement<geometry_type> &element, std::size_t gdim, const std::optional<std::vector<std::size_t>> &value_shape = std::nullopt, bool symmetric = false)#

Create a finite element from a Basix finite element.

Parameters:

element – [in] Basix finite element.
gdim – [in] Geometric dimension of the mesh the element will be used on.
value_shape – [in] Value shape for blocked element, e.g. {3} for a vector in 3D or {2, 2} for a rank-2 tensor in 2D. Can only be set for blocked scalar element. For other elements and scalar elements it should be std::nullopt.
symmetric – [in] Is the element a symmetric tensor? Should only set for 2nd-order tensor blocked elements.

FiniteElement(std::vector<BasixElementData<geometry_type>> elements, std::size_t gdim)#

Create a mixed finite element from Basix finite elements.

See FiniteElement(const std::vector<std::shared_ptr<constFiniteElement<geometry_type>>>&) for a discussion of mixed elements.

Parameters:

elements – [in] List of (Basix finite element, block size, symmetric) tuples, one for each element in the mixed element.
gdim – [in] Geometric dimension of the mesh the element will be used on, applied to every sub-element.

FiniteElement(const std::vector<std::shared_ptr<const FiniteElement<geometry_type>>> &elements)#

Create a mixed finite element from a list of finite elements.

This constructs a mixed element \(E_0 \times E_1 \times \ldots \times E_{n-1}\). The *i*th sub-element \(E_i\) can be accessed by ::extract_sub_element. Functions defined on mixed element spaces cannot be interpolated into directly. It is necessary to first extract a sub-Function (view), which can then be interpolated into.

A mixed element can be constructed from one element. In this case the FiniteElement behaves like a mixed element and cannot be interpolated into. The underlying element can be accessed using ::extract_sub_element.

Parameters:: elements – [in] Finite elements to compose the mixed element from.

FiniteElement(mesh::CellType cell_type, std::span<const geometry_type> points, std::array<std::size_t, 2> pshape, std::vector<std::size_t> value_shape = {}, bool symmetric = false)#

Create a quadrature element.

Note

A quadrature element is identity mapped, so value_shape is both its reference and its physical value shape and no geometric dimension is required.

Parameters:

cell_type – [in] Cell type.
points – [in] Quadrature points.
pshape – [in] Shape of points array.
value_shape – [in] Value shape for the element.
symmetric – [in] Is the element a symmetric tensor?

FiniteElement(const FiniteElement &element) = delete#: Copy constructor.

FiniteElement(FiniteElement &&element) = default#: Move constructor.

~FiniteElement() = default#: Destructor.

FiniteElement &operator=(const FiniteElement &element) = delete#: Copy assignment.

FiniteElement &operator=(FiniteElement &&element) = default#: Move assignment.

bool operator==(const FiniteElement &e) const#

Check if two elements are equivalent.

Note

Equality can be checked only for non-mixed elements. For a mixed element, this function will throw an exception.

Returns:: True is the two elements are the same.

bool operator!=(const FiniteElement &e) const#

Check if two elements are not equivalent.

Note

Equality can be checked only for non-mixed elements. For a mixed element, this function will raise an exception.

Returns:: True is the two elements are not the same.

mesh::CellType cell_type() const noexcept#: Cell shape that the element is defined on.

const std::string &signature() const noexcept#

String identifying the finite element.

Warning

The function is provided for convenience, but it should not be relied upon for determining the element type. Use other functions, commonly returning enums, to determine element properties.

Returns:: Element signature

int space_dimension() const noexcept#

Dimension of the finite element function space (the number of degrees-of-freedom for the element).

For ‘blocked’ elements, this function returns the dimension of the full element rather than the dimension of the base element.

Returns:: Dimension of the finite element space.

int block_size() const noexcept#

Block size of the finite element function space.

For non-blocked elements, this is always 1. For blocked elements, this is the number of DOFs collocated at each DOF point, which equals FiniteElement::value_size except for a symmetric rank-2 tensor. A symmetric rank-2 tensor stores only its independent components, so a {2, 2} symmetric element has value size 4 and block size 3, and a {3, 3} one has value size 9 and block size 6.

Returns:: Block size of the finite element space.

int value_size() const#

Value size of the finite element field in physical space.

The value size is the number of components of the finite element field once the basis has been pushed forward to a physical cell. It is the product of FiniteElement::value_shape, e.g. 1 for a scalar function, 2 for a 2D vector, 9 for a second-order tensor in 3D, etc. For blocked elements this is the value size of the full ‘blocked’ element.

Throws:: Exception – is thrown for a mixed element as mixed elements do not have a value shape.
Returns:: The value size.

std::span<const std::size_t> value_shape() const#

Value shape of the finite element field in physical space.

The value shape describes the shape of the finite element field once the basis has been pushed forward to a physical cell, e.g. {} for a scalar, {2} for a vector in 2D, {3, 3} for a rank-2 tensor in 3D, etc.

It differs from FiniteElement::reference_value_shape for blocked and quadrature elements, and for a Piola-mapped element on a manifold: Raviart-Thomas on a triangle embedded in 3D has reference value shape {2} and value shape {3}, because the push-forward contracts the reference value axis with a Jacobian of shape (gdim, tdim). See ::compute_value_shape.

Throws:: Exception – is thrown for a mixed element as mixed elements do not have a value shape.
Returns:: The value shape.

int physical_base_value_size() const#

Number of physical components in one block of the finite element field.

A blocked element repeats a scalar base element FiniteElement::block_size times, so one block of its field is a single scalar and this is 1. A non-blocked element has a single block, so this is FiniteElement::value_size.

This is the size of the push-forward of one (non-blocked) basis function, and hence the extent a buffer needs when it holds physical values one block at a time. It is the physical counterpart of FiniteElement::reference_value_size, and equals it unless the element is Piola mapped on a manifold, where it is gdim rather than tdim.

Note

This is not value_size() / block_size(): for a symmetric rank-2 tensor element that expression gives 4/3 or 9/6 rather than the correct value of 1.

Throws:: Exception – is thrown for a mixed element as mixed elements do not have a value shape.
Returns:: Number of physical components per block.

int reference_value_size() const#

Value size of the base (non-blocked) finite element field on the reference cell.

The reference value size is the product of the reference value shape, e.g. it is 1 for a scalar element, 2 for a 2D (non-blocked) vector, 9 for a (non-blocked) second-order tensor in 3D, etc. It is the number of components produced by FiniteElement::tabulate.

For blocked elements, this function returns the value size for the ‘base’ element from which the blocked element is composed.

Throws:: Exception – is thrown for a mixed element as mixed elements do not have a value shape.
Returns:: The value size.

std::span<const std::size_t> reference_value_shape() const#

Value shape of the base (non-blocked) finite element field on the reference cell.

This is the shape Basix tabulates in. For blocked and quadrature elements the returned shape will be {}. For other elements it is the same as FiniteElement::value_shape except on a manifold, where a Piola-mapped element has tdim reference components and gdim physical ones.

Mixed elements do not have a reference value shape.

Throws:: Exception – is thrown for a mixed element as mixed elements do not have a value shape.
Returns:: The value shape.

const std::vector<std::vector<std::vector<int>>> &entity_dofs() const noexcept#: Local DOFs associated with each sub-entity of the cell.

const std::vector<std::vector<std::vector<int>>> &entity_closure_dofs() const noexcept#: Local DOFs associated with the closure of each sub-entity of the cell.

bool symmetric() const#

Does the element represent a symmetric 2-tensor?

A symmetric element has a square rank-2 FiniteElement::value_shape, but stores only the independent components, so its FiniteElement::block_size is d * (d + 1) / 2 rather than d * d. See the examples in the class documentation.

void tabulate(std::span<geometry_type> values, std::span<const geometry_type> X, std::array<std::size_t, 2> shape, int order) const#

Evaluate derivatives of the basis functions up to given order at points in the reference cell.

Parameters:

values – [inout] Array that will be filled with the tabulated basis values. Must have shape (num_derivatives, num_points, num_dofs, reference_value_size) (row-major storage)
X – [in] The reference coordinates at which to evaluate the basis functions. Shape is (num_points, topological dimension) (row-major storage).
shape – [in] Shape of X.
order – [in] Number of derivatives (up to and including this order) to tabulate for.

std::pair<std::vector<geometry_type>, std::array<std::size_t, 4>> tabulate(std::span<const geometry_type> X, std::array<std::size_t, 2> shape, int order) const#

Evaluate all derivatives of the basis functions up to given order at given points in reference cell.

Parameters:

X – [in] The reference coordinates at which to evaluate the basis functions. Shape is (num_points, topological dimension) (row-major storage).
shape – [in] Shape of X.
order – [in] Number of derivatives (up to and including this order) to tabulate for.

Returns:

Basis function values and array shape (row-major storage).

int num_sub_elements() const noexcept#

Number of sub elements (for a mixed or blocked element).

Returns:: Number of sub elements.

bool is_mixed() const noexcept#

Check if element is a mixed element.

A mixed element is composed of two or more elements of different types. A blocked element, e.g. a Lagrange element with block size >= 1 is not considered mixed.

Returns:: True if element is mixed.

const std::vector<std::shared_ptr<const FiniteElement<geometry_type>>> &sub_elements() const noexcept#: Get subelements (if any).

std::shared_ptr<const FiniteElement<geometry_type>> extract_sub_element(const std::vector<int> &component) const#: Extract sub finite element for component.

const basix::FiniteElement<geometry_type> &basix_element() const#

Return underlying Basix element (if it exists).

Throws:: Throws – a std::runtime_error is there no Basix element.

basix::maps::type map_type() const#: Get the map type used by the element.

bool interpolation_ident() const noexcept#

Check if interpolation into the finite element space is an identity operation given the evaluation on an expression at specific points, i.e. the degree-of-freedom are equal to point evaluations. The function will return true for Lagrange elements.

Returns:: True if interpolation is an identity operation

bool map_ident() const noexcept#

Check if the push forward/pull back map from the values on reference to the values on a physical cell for this element is the identity map.

Returns:: True if the map is the identity

std::pair<std::vector<geometry_type>, std::array<std::size_t, 2>> interpolation_points() const#

Points on the reference cell at which an expression needs to be evaluated in order to interpolate the expression in the finite element space.

For Lagrange elements the points will just be the nodal positions. For other elements the points will typically be the quadrature points used to evaluate moment degrees of freedom.

Returns:: Interpolation point coordinates on the reference cell, returning the (0) coordinates data (row-major) storage and (1) the shape (num_points, tdim).

std::pair<std::vector<geometry_type>, std::array<std::size_t, 2>> interpolation_operator() const#

Interpolation operator (matrix) Pi that maps a function evaluated at the points provided by FiniteElement::interpolation_points to the element degrees of freedom, i.e. dofs = Pi f_x. See the Basix documentation for basix::FiniteElement::interpolation_matrix for how the data in f_x should be ordered.

Returns:: The interpolation operator Pi, returning the data for Pi (row-major storage) and the shape (num_dofs, num_points * value_size)

std::pair<std::vector<geometry_type>, std::array<std::size_t, 2>> create_interpolation_operator(const FiniteElement &from) const#

Create a matrix that maps degrees of freedom from one element to this element (interpolation).

Note

Does not support mixed elements.

Parameters:: from – [in] The element to interpolate from.
Returns:: Matrix operator that maps the from degrees-of-freedom to the degrees-of-freedom of this element. The (0) matrix data (row-major storage) and (1) the shape (num_dofs of this element, num_dofs of from) are returned.
Pre:: The two elements must use the same mapping between the reference and physical cells.

bool needs_dof_transformations() const noexcept#

Check if DOF transformations are needed for this element.

DOF transformations will be needed for elements which might not be continuous when two neighbouring cells disagree on the orientation of a shared sub-entity, and when this cannot be corrected for by permuting the DOF numbering in the dofmap.

For example, Raviart-Thomas elements will need DOF transformations, as the neighbouring cells may disagree on the orientation of a basis function, and this orientation cannot be corrected for by permuting the DOF numbers on each cell.

Returns:: True if DOF transformations are required.

bool needs_dof_permutations() const noexcept#

Check if DOF permutations are needed for this element.

DOF permutations will be needed for elements which might not be continuous when two neighbouring cells disagree on the orientation of a shared subentity, and when this can be corrected for by permuting the DOF numbering in the dofmap.

For example, higher order Lagrange elements will need DOF permutations, as the arrangement of DOFs on a shared sub-entity may be different from the point of view of neighbouring cells, and this can be corrected for by permuting the DOF numbers on each cell.

Returns:: True if DOF transformations are required.

template<typename U> inline std::function<void(std::span, std::span<const std::uint32_t>, std::int32_t, int)> dof_transformation_fn(doftransform ttype, bool scalar_element = false) const#

Return a function that applies a DOF transformation operator to some data (see T_apply()).

The transformation is applied from the left-hand side, i.e.

\[ u \leftarrow T u. \]

If the transformation for the (sub)element is a permutation only, the returned function will do change the ordering for the (sub)element as it is assumed that permutations are incorporated into the degree-of-freedom map.

See the documentation for T_apply() for a description of the transformation for a single element type. This function generates a function that can apply the transformation to a mixed element.

The signature of the returned function has four arguments:

[in,out] data The data to be transformed. This data is flattened with row-major layout, shape=(num_dofs, block_size)
[in] cell_info Permutation data for the cell. The size of this is num_cells. For elements where no transformations are required, an empty span can be passed in.
[in] cell The cell number.
[in] n The block_size of the input data.

Parameters:

ttype – [in] The transformation type. Typical usage is:
- doftransform::standard Transforms basis function data from the reference element to the conforming ‘physical’ element, e.g. \(\phi = T \tilde{\phi}\).
- doftransform::transpose Transforms degree-of-freedom data from the conforming (physical) ordering to the reference ordering, e.g. \(\tilde{u} = T^{T} u\).
- doftransform::inverse: Transforms basis function data from the the conforming (physical) ordering to the reference ordering, e.g. \(\tilde{\phi} = T^{-1} \phi\).
- doftransform::inverse_transpose: Transforms degree-of-freedom data from the reference element to the conforming (physical) ordering, e.g. \(u = T^{-t} \tilde{u}\).
scalar_element – [in] Indicates whether the scalar transformations should be returned for a vector element.

template<typename U> inline std::function<void(std::span, std::span<const std::uint32_t>, std::int32_t, int)> dof_transformation_right_fn(doftransform ttype, bool scalar_element = false) const#

Return a function that applies DOF transformation to some transposed data (see T_apply_right()).

The transformation is applied from the right-hand side, i.e.

\[ u^{t} \leftarrow u^{t} T. \]

If the transformation for the (sub)element is a permutation only, the returned function will do change the ordering for the (sub)element as it is assumed that permutations are incorporated into the degree-of-freedom map.

The signature of the returned function has four arguments:

[in,out] data The data to be transformed. This data is flattened with row-major layout, shape=(num_dofs, block_size)
[in] cell_info Permutation data for the cell. The size of this is num_cells. For elements where no transformations are required, an empty span can be passed in.
[in] cell The cell number
[in] block_size The block_size of the input data

Parameters:

ttype – [in] Transformation type. See dof_transformation_fn().
scalar_element – [in] Indicate if the scalar transformations should be returned for a vector element.

template<typename U> inline void T_apply(std::span data, std::uint32_t cell_permutation, int n) const#

Transform basis functions from the reference element ordering and orientation to the globally consistent physical element ordering and orientation.

Consider that the value of a finite element function \(f_{h}\) at a point is given by

\[ f_{h} = \phi^{T} c, \]

where \(f_{h}\) has shape \(r \times 1\), \(\phi\) has shape \(d \times r\) and holds the finite element basis functions, and \(c\) has shape \(d \times 1\) and holds the degrees-of-freedom. The basis functions and degree-of-freedom are with respect to the physical element orientation. If the degrees-of-freedom on the physical element orientation are given by

\[\phi = T \tilde{\phi}, \]

where \(T\) is a \(d \times d\) matrix, it follows from \(f_{h} = \phi^{T} c = \tilde{\phi}^{T} T^{T} c\) that

\[ \tilde{c} = T^{T} c. \]

This function applies \(T\) to data. The transformation is performed in-place. The operator \(T\) is orthogonal for many elements, but not all.

This function calls the corresponding Basix function.

Parameters:

data – [inout] Data to transform. The shape is (m, n), where m is the number of dgerees-of-freedom and the storage is row-major.
cell_permutation – [in] Permutation data for the cell
n – [in] Number of columns in data.

template<typename U> inline void Tt_inv_apply(std::span data, std::uint32_t cell_permutation, int n) const#

Apply the inverse transpose of the operator applied by T_apply().

The transformation

\[ v = T^{-T} u \]

is performed in-place.

Parameters:

data – [inout] The data to be transformed. This data is flattened with row-major layout, shape=(num_dofs, block_size).
cell_permutation – [in] Permutation data for the cell.
n – [in] Block_size of the input data.

template<typename U> inline void Tt_apply(std::span data, std::uint32_t cell_permutation, int n) const#

Apply the transpose of the operator applied by T_apply().

The transformation

\[ u \leftarrow T^{T} u \]

is performed in-place.

Parameters:

data – [inout] The data to be transformed. This data is flattened with row-major layout, shape=(num_dofs, block_size).
cell_permutation – [in] Permutation data for the cell.
n – [in] The block size of the input data.

template<typename U> inline void Tinv_apply(std::span data, std::uint32_t cell_permutation, int n) const#

Apply the inverse of the operator applied by T_apply().

The transformation

\[ v = T^{-1} u \]

is performed in-place.

Parameters:

data – [inout] The data to be transformed. This data is flattened with row-major layout, shape=(num_dofs, block_size).
cell_permutation – [in] Permutation data for the cell.
n – [in] Block size of the input data.

template<typename U> inline void T_apply_right(std::span data, std::uint32_t cell_permutation, int n) const#

Right(post)-apply the operator applied by T_apply().

Computes

\[ v^{T} = u^{T} T \]

in-place.

Parameters:

data – [inout] The data to be transformed. This data is flattened with row-major layout, shape=(num_dofs, block_size).
cell_permutation – [in] Permutation data for the cell
n – [in] Block size of the input data

template<typename U> inline void Tinv_apply_right(std::span data, std::uint32_t cell_permutation, int n) const#

Right(post)-apply the inverse of the operator applied by T_apply().

Computes

\[ v^{T} = u^{T} T^{-1} \]

in-place.

Parameters:

data – [inout] Data to be transformed. This data is flattened with row-major layout, shape=(num_dofs, block_size).
cell_permutation – [in] Permutation data for the cell
n – [in] Block size of the input data

template<typename U> inline void Tt_apply_right(std::span data, std::uint32_t cell_permutation, int n) const#

Right(post)-apply the transpose of the operator applied by T_apply().

Computes

\[ v^{T} = u^{T} T^{T} \]

in-place.

Parameters:

data – [inout] Data to be transformed. The data is flattened with row-major layout, shape=(num_dofs, block_size).
cell_permutation – [in] Permutation data for the cell
n – [in] Block size of the input data.

template<typename U> inline void Tt_inv_apply_right(std::span data, std::uint32_t cell_permutation, int n) const#

Right(post)-apply the transpose inverse of the operator applied by T_apply().

Computes

\[ v^{T} = u^{T} T^{-T} \]

in-place.

Parameters:

data – [inout] Data to be transformed. This data is flattened with row-major layout, shape=(num_dofs, block_size).
cell_permutation – [in] Permutation data for the cell.
n – [in] Block size of the input data.

void permute(std::span<std::int32_t> doflist, std::uint32_t cell_permutation) const#

Permute indices associated with degree-of-freedoms on the reference element ordering to the globally consistent physical element degree-of-freedom ordering.

Given an array \(\tilde{d}\) that holds an integer associated with each degree-of-freedom and following the reference element degree-of-freedom ordering, this function computes

\[ d = P \tilde{d},\]

where \(P\) is a permutation matrix and \(d\) holds the integers in \(\tilde{d}\) but permuted to follow the globally consistent physical element degree-of-freedom ordering. The permutation is computed in-place.

Parameters:

doflist – [inout] Indices associated with the degrees-of-freedom. Size=num_dofs.
cell_permutation – [in] Permutation data for the cell.

void permute_inv(std::span<std::int32_t> doflist, std::uint32_t cell_permutation) const#

Perform the inverse of the operation applied by permute.

Given an array \(d\) that holds an integer associated with each degree-of-freedom and following the globally consistent physical element degree-of-freedom ordering, this function computes

\[ \tilde{d} = P^{T} d, \]

where \(P^{T}\) is a permutation matrix and \(\tilde{d}\) holds the integers in \(d\) but permuted to follow the reference element degree-of-freedom ordering. The permutation is computed in-place.

Parameters:

doflist – [inout] Indices associated with the degrees-of-freedom. Size=num_dofs.
cell_permutation – [in] Permutation data for the cell.

std::function<void(std::span<std::int32_t>, std::uint32_t)> dof_permutation_fn(bool inverse = false, bool scalar_element = false) const#

Return a function that applies a degree-of-freedom permutation to some data.

The returned function can apply permute to mixed-elements.

The signature of the returned function has three arguments:

[in,out] doflist The numbers of the DOFs, a span of length num_dofs
[in] cell_permutation Permutation data for the cell
[in] block_size The block_size of the input data

Parameters:

inverse – [in] Indicates if the inverse transformation should be returned.
scalar_element – [in] Indicates is the scalar transformations should be returned for a vector element.

Function spaces and functions#

Finite element functions, expressions and constants

Function spaces#

template<std::floating_point T> class FunctionSpace#

This class represents a finite element function space defined by a mesh, a finite element, and a local-to-global map of the degrees-of-freedom.

Template Parameters:: T – The floating point (real) type of the mesh geometry and the finite element basis.

Public Types

using geometry_type = T #: Geometry type of the Mesh that the FunctionSpace is defined on.

Public Functions

inline FunctionSpace(std::shared_ptr<const mesh::Mesh<geometry_type>> mesh, std::shared_ptr<const FiniteElement<geometry_type>> element, std::shared_ptr<const DofMap> dofmap)#

Create function space for given mesh, element and degree-of-freedom map.

Parameters:

mesh – [in] Mesh that the space is defined on.
element – [in] Finite element for the space.
dofmap – [in] Degree-of-freedom map for the space.

inline FunctionSpace(std::shared_ptr<const mesh::Mesh<geometry_type>> mesh, std::vector<std::shared_ptr<const FiniteElement<geometry_type>>> elements, std::vector<std::shared_ptr<const DofMap>> dofmaps)#

Create function space for given mesh, elements and degree-of-freedom maps.

Parameters:

mesh – [in] Mesh that the space is defined on.
elements – [in] Finite elements for the space, one for each cell type. The elements must be ordered to be consistent with mesh::topology::cell_types.
dofmaps – [in] Degree-of-freedom maps for the space, one for each element. The dofmaps must be ordered in the same way as the elements.

FunctionSpace(FunctionSpace &&V) = default#: Move constructor.

~FunctionSpace() = default#: Destructor.

FunctionSpace &operator=(FunctionSpace &&V) = default#: Move assignment operator.

inline FunctionSpace sub(const std::vector<int> &component) const#

Create a subspace (view) for a specific component.

Note

If the subspace is re-used, for performance reasons the returned subspace should be stored by the caller to avoid repeated re-computation of the subspace.

Parameters:: component – [in] Subspace component.
Returns:: A subspace.

inline bool contains(const FunctionSpace &V) const#

Check whether V is subspace of this, or this itself.

Parameters:: V – [in] The space to be tested for inclusion
Returns:: True if V is contained in or is equal to this FunctionSpace

inline std::pair<FunctionSpace, std::vector<std::vector<std::int32_t>>> collapse() const#

Collapse a subspace and return a new function space and a map from new to old dofs

Returns:: The new function space and a map from new to old dofs

inline std::vector<int> component() const#

Get the component with respect to the root superspace.

Returns:: The component with respect to the root superspace, i.e. W.sub(1).sub(0) == [1, 0].

inline bool symmetric() const#: Indicate whether this function space represents a symmetric 2-tensor.

inline std::vector<geometry_type> tabulate_dof_coordinates(bool transpose) const#

Tabulate the physical coordinates of all dofs on this process.

Todo:: Remove - see function in interpolate.h

Parameters:: transpose – [in] If false the returned data has shape (num_points, 3), otherwise it is transposed and has shape (3, num_points).
Returns:: The dof coordinates [([x0, y0, z0], [x1, y1, z1], ...) if transpose is false, and otherwise the returned data is transposed. Storage is row-major.

inline std::shared_ptr<const mesh::Mesh<geometry_type>> mesh() const#: The mesh.

inline std::shared_ptr<const FiniteElement<geometry_type>> element() const#: The finite element.

inline std::shared_ptr<const FiniteElement<geometry_type>> elements(int cell_type_idx) const#: The finite elements.

inline std::shared_ptr<const DofMap> dofmap() const#: The dofmap.

inline const std::vector<std::shared_ptr<const DofMap>> &dofmaps() const#: The dofmaps.

Functions#

template<dolfinx::scalar T, std::floating_point U = dolfinx::scalar_value_t<T>> class Function#

This class represents a function \( u_h \) in a finite element function space \( V_h \), given by

\[ u_h = \sum_{i=1}^{n} U_i \phi_i, \]

where \( \{\phi_i\}_{i=1}^{n} \) is a basis for \( V_h \), and \( U \) is a vector of expansion coefficients for \( u_h \).

Template Parameters:

T – The function scalar type.
U – The mesh geometry scalar type.

Public Types

using value_type = T #: Field type for the Function, e.g. double, std::complex<float>, etc.

using geometry_type = U #: Geometry type of the Mesh that the Function is defined on.

Public Functions

inline explicit Function(std::shared_ptr<const FunctionSpace<geometry_type>> V)#

Create function on given function space.

Parameters:: V – [in] The function space

inline Function(std::shared_ptr<const FunctionSpace<geometry_type>> V, std::shared_ptr<la::Vector<value_type>> x)#

Create function on given function space with a given vector.

Warning

This constructor is intended for internal library use only.

Parameters:

V – [in] The function space.
x – [in] The vector.

Function(Function &&v) = default#: Move constructor.

~Function() = default#: Destructor.

Function &operator=(Function &&v) = default#: Move assignment.

inline Function sub(int i) const#

Extract a sub-function (a view into the Function).

Parameters:: i – [in] Index of subfunction
Returns:: The sub-function

inline Function collapse() const#

Collapse a subfunction (view into a Function) to a stand-alone Function.

Returns:: New collapsed Function.

inline std::shared_ptr<const FunctionSpace<geometry_type>> function_space() const#

Access the function space.

Returns:: The function space.

inline std::shared_ptr<const la::Vector<value_type>> x() const#: Underlying vector (const version).

inline std::shared_ptr<la::Vector<value_type>> x()#: Underlying vector.

inline void interpolate(const std::function<std::pair<std::vector<value_type>, std::vector<std::size_t>>(md::mdspan<const geometry_type, md::extents<std::size_t, 3, md::dynamic_extent>>)> &f, mesh::CellRange auto &&cells)#

Interpolate an expression f(x) over a set of cells.

Parameters:

f – [in] Expression function to be interpolated.
cells – [in] Cells to interpolate on.

inline void interpolate(const std::function<std::pair<std::vector<value_type>, std::vector<std::size_t>>(md::mdspan<const geometry_type, md::extents<std::size_t, 3, md::dynamic_extent>>)> &f)#

Interpolate an expression f(x) on the whole domain.

Parameters:: f – [in] Expression to be interpolated.

inline void interpolate(const Function<value_type, geometry_type> &u0, mesh::CellRange auto &&cells0, mesh::CellRange auto &&cells1)#

Interpolate a Function over a subset of cells.

The Function being interpolated from and the Function being interpolated into can be defined on different sub-meshes, i.e. views into a subset a cells.

Parameters:

u0 – [in] Function to be interpolated.
cells0 – [in] Cells to interpolate from. These are the indices of the cells in the mesh associated with u0.
cells1 – [in] Cell indices associated with the mesh of this that will be interpolated to. If cells0[i] is the index of a cell in the mesh associated with u0, then cells1[i] is the index of the same cell but in the mesh associated with this.

Pre:

cells0 and cells1 must have the same length.

inline void interpolate(const Function<value_type, geometry_type> &u, mesh::CellRange auto &&cells)#

Interpolate a Function over a subset of cells.

The Functions must be defined on the same mesh.

Parameters:

u – [in] Function to be interpolated.
cells – [in] Cells to interpolate from.

inline void interpolate(const Function<value_type, geometry_type> &u)#

Interpolate a Function over all cells.

The Functions must be defined on the same mesh.

Parameters:: u – [in] Function to be interpolated.

inline void interpolate(const Expression<value_type, geometry_type> &e0, mesh::CellRange auto &&cells0, mesh::CellRange auto &&cells1)#

Interpolate an Expression over a subset of cells.

Parameters:

e0 – [in] Expression to be interpolated. The Expression must have been created using the reference coordinates created by FiniteElement::interpolation_points for the element associated with this.
cells0 – [in] Cells in the mesh associated with e0 to interpolate from if e0 has Function coefficients. If no mesh can be associated with e0 then the mesh associated with this is used.
cells1 – [in] Cell indices associated with the mesh of this that will be interpolated to. If cells0[i] is the index of a cell in the mesh associated with u0, then cells1[i] is the index of the same cell but in the mesh associated with this.

Pre:

cells0 cells1 must have the same length.

inline void interpolate(const Expression<value_type, geometry_type> &e0, mesh::CellRange auto &&cells)#

Interpolate an Expression over a subset of cells.

Parameters:

e0 – [in] Expression to be interpolated. The Expression must have been created using the reference coordinates created by FiniteElement::interpolation_points for the element associated with this.
cells – [in] Cells in the mesh associated with e0 to interpolate from if e0 has Function coefficients. If no mesh can be associated with e0 then the mesh associated with this is used.

inline void interpolate(const Expression<value_type, geometry_type> &e)#

Interpolate an Expression on all cells.

Parameters:: e – [in] Expression to be interpolated.
Pre:: If a mesh is associated with Function coefficients of e, it must be the same as the mesh::Mesh associated with this.

inline void interpolate(const Function<value_type, geometry_type> &u, mesh::CellRange auto &&cells, double tol, int maxit, const geometry::PointOwnershipData &interpolation_data)#

Interpolate a Function defined on a different mesh.

Parameters:

u – [in] Function to be interpolated.
cells – [in] Cells in the mesh associated with this to interpolate into.
tol – [in] Tolerance for convergence in Newton method for non-affine pullbacks. If the mesh geometry is affine this argument is ignored.
maxit – [in] Maximum number of Newton iterations in non-affine pull-back. If the mesh geometry is affine this argument is ignored.
interpolation_data – [in] Data required for associating the interpolation points of this with cells in u. Can be computed with fem::create_interpolation_data.

inline void eval(std::span<const geometry_type> x, std::array<std::size_t, 2> xshape, mesh::CellRange auto &&cells, std::span<value_type> u, std::array<std::size_t, 2> ushape, double tol, int maxit) const#

Evaluate the Function at points.

Parameters:

x – [in] The coordinates of the points. It has shape (num_points, 3) and storage is row-major.
xshape – [in] Shape of x.
cells – [in] Cell indices such that cells[i] is the index of the cell that contains the point x(i). Negative cell indices can be passed, in which case the corresponding point is ignored.
u – [out] Values at the points. Values are not computed for points with a negative cell index. This argument must be passed with the correct size. Storage is row-major.
ushape – [in] Shape of u.
tol – [in] Tolerance for convergence in Newton method for non-affine pullbacks. If the mesh geometry is affine this argument is ignored.
maxit – [in] Maximum number of Newton iterations in non-affine pull-back. If the mesh geometry is affine this argument is ignored.

Public Members

std::string name = "u"#: Name.

Constants#

template<dolfinx::scalar T> class Constant#

Constant (in space) value which can be attached to a Form.

Constants may be scalar (rank 0), vector (rank 1), or tensor-valued.

Template Parameters:: T – Scalar type of the Constant.

Public Types

using value_type = T #: Field type.

Public Functions

inline explicit Constant(value_type c)#

Create a rank-0 (scalar-valued) constant.

Parameters:: c – [in] Value of the constant.

inline explicit Constant(std::span<const value_type> c)#

Create a rank-1 (vector-valued) constant.

Parameters:: c – [in] Value of the constant.

inline Constant(std::span<const value_type> c, std::span<const std::size_t> shape)#

Create a rank-d constant.

Parameters:

c – [in] Value of the Constant (row-majors storage)
shape – [in] Shape of the Constant

Public Members

std::vector<value_type> value#: Values, stored as a row-major flattened array.

std::vector<std::size_t> shape#: Shape.

Forms#

template<dolfinx::scalar T, std::floating_point U = dolfinx::scalar_value_t<T>> class Form#

A representation of finite element variational forms.

A note on the order of trial and test spaces: FEniCS numbers argument spaces starting with the leading dimension of the corresponding tensor (matrix). In other words, the test space is numbered 0 and the trial space is numbered 1. However, in order to have a notation that agrees with most existing finite element literature, in particular

\[ a = a(u, v) \]

the spaces are numbered from right to left

\[ a: V_1 \times V_0 \rightarrow \mathbb{R} \]

This is reflected in the ordering of the spaces that should be supplied to generated subclasses. In particular, when a bilinear form is initialized, it should be initialized as a(V_1, V_0) = ..., where V_1 is the trial space and V_0 is the test space. However, when a form is initialized by a list of argument spaces (the variable function_spaces in the constructors below), the list of spaces should start with space number 0 (the test space) and then space number 1 (the trial space).

Template Parameters:

T – Scalar type in the form.
U – Float (real) type used for the finite element and geometry.

Public Types

using scalar_type = T #: Scalar type.

using geometry_type = U #: Geometry type.

Public Functions

template<typename X> inline Form(const std::vector<std::shared_ptr<const FunctionSpace<geometry_type>>> &V, X &&integrals, std::shared_ptr<const mesh::Mesh<geometry_type>> mesh, const std::vector<std::shared_ptr<const Function<scalar_type, geometry_type>>> &coefficients, const std::vector<std::shared_ptr<const Constant<scalar_type>>> &constants, bool needs_facet_permutations, const std::vector<std::reference_wrapper<const mesh::EntityMap>> &entity_maps)#

Create a finite element form.

Note

User applications will normally call a factory function rather using this interface directly.

Note

For the single domain case, pass an empty entity_maps.

Parameters:

V – [in] Function spaces for the form arguments, e.g. test and trial function spaces.
integrals – [in] Integrals in the form, where integrals[{type, i, kernel_index}] gives the integral_data of type type at position i in the flattened, sorted-by-subdomain-id list of integrals, for kernel kernel_index. The subdomain ids can contain duplicate entries referring to different kernels over the same subdomain.
coefficients – [in] Coefficients in the form.
constants – [in] Constants in the form.
mesh – [in] Mesh of the domain to integrate over (the ‘integration domain’).
needs_facet_permutations – [in] Set to true if any of the integration kernels require cell permutation data.
entity_maps – [in] A list of EntityMaps. For every mesh other than mesh on which a trial function, test function, or coefficient is defined, entity_maps must contain an EntityMap relating that mesh and mesh.

Form(Form &&form) = default#: Move constructor

Note

Valid because ::_integrals is a std::map, whose elements keep a stable address across a move, so the std::spans cached in ::_edata and ::_cdata remain valid after the move.

~Form() = default#: Destructor.

Form &operator=(Form &&form) = default#: Move assignment

Note

Valid for the same reason as the move constructor: move assigning a std::map transfers its nodes, so the entity vectors aliased by the std::spans cached in ::_edata and ::_cdata keep their addresses.

inline int rank() const#

Rank of the form.

bilinear form = 2, linear form = 1, functional = 0, etc.

Returns:: The rank of the form.

inline std::shared_ptr<const mesh::Mesh<geometry_type>> mesh() const#

Common mesh for the form (the ‘integration domain’).

Returns:: The integration domain mesh.

inline const std::vector<std::shared_ptr<const FunctionSpace<geometry_type>>> &function_spaces() const#

Function spaces for all arguments.

Returns:: Function spaces.

inline std::function<void(scalar_type*, const scalar_type*, const scalar_type*, const geometry_type*, const int*, const uint8_t*, void*)> kernel(IntegralType type, int idx, int kernel_idx) const#

Get the kernel function for an integral.

Parameters:

type – [in] Integral type.
idx – [in] Integral index in the flattened list of integral kernels (see ::domain).
kernel_idx – [in] Index of the kernel (we may have multiple kernels for a given idx in mixed-topology meshes).

Returns:

Function to call for tabulate_tensor.

inline std::set<IntegralType> integral_types() const#

Get types of integrals in the form.

Returns:: Integrals types.

inline std::vector<int> active_coeffs(IntegralType type, int idx) const#

Indices of coefficients that are active for a given integral (kernel).

A form is split into multiple integrals (kernels) and each integral might contain only a subset of all coefficients in the form. This function returns an indicator array for a given integral kernel that signifies which coefficients are present.

Parameters:

type – [in] Integral type.
idx – [in] Integral index in the flattened list of integral kernels (see ::domain).

Returns:

Indices of the coefficients that are active (present) in the given integral.

inline int num_integrals(IntegralType type, int kernel_idx) const#

Get number of integrals (kernels) for a given integral type and kernel index.

For a form containing two integrals integral_a and integral_b with subdomain ids (1, 4) and (3, 4, 5) respectively, the integrals are stored as a flattened list, sorted by subdomain id:

auto form_integrals = {integral_a, integral_b, integral_a,
                       integral_b, integral_b};
auto form_integral_ids = {1, 3, 4, 4, 5};

Parameters:

type – [in] Integral type.
kernel_idx – [in] Index of the kernel (we may have multiple kernels for a integral type in mixed-topology meshes).

Returns:

Number of integrals (kernels) of the given type and kernel index.

inline std::span<const std::int32_t> domain(IntegralType type, int idx, int kernel_idx) const#

Mesh entity indices to integrate over for a given integral (kernel).

These are the entities in the mesh returned by ::mesh that are integrated over by a given integral (kernel).

For IntegralType::cell, returns a list of cell indices.
For IntegralType::exterior_facet, returns a list with shape (num_facets, 2) (row-major storage), where for row i, [i, 0] is the cell index and [i, 1] is the local facet index relative to the cell.
For IntegralType::interior_facet, returns a list with shape (num_facets, 4) (row-major storage), where for row i, [i, 0] is the index of one attached cell, [i, 1] is the local facet index relative to that cell, [i, 2] is the index of the other attached cell, and [i, 3] is the local facet index relative to that cell.

Parameters:

type – [in] Integral type.
idx – [in] Integral index in the flattened list of integral kernels. For a form containing two integrals integral_a and integral_b with subdomain ids (1, 4) and (3, 4, 5) respectively, the integrals are stored as a flattened list, sorted by subdomain id:
```
auto form_integrals = {integral_a, integral_b, integral_a,
                       integral_b, integral_b};
auto form_integral_ids = {1, 3, 4, 4, 5};
```
kernel_idx – [in] Index of the kernel within the domain (we may have multiple kernels for a given id in mixed-topology meshes).

Returns:

Entity indices, with respect to the mesh::Mesh returned by ::mesh, to integrate over.

inline std::span<const std::int32_t> domain_arg(IntegralType type, int rank, int idx, int kernel_idx) const#

Argument function mesh integration entity indices.

Integration can be performed over cells/facets involving functions that are defined on different meshes but which share common cells, i.e. meshes can be ‘views’ into a common mesh. Meshes can share some cells but a common cell will have a different index in each mesh::Mesh. Consider:

auto mesh = this->mesh();
auto entities = this->domain(type, idx, kernel_idx);
auto entities0 = this->domain_arg(type, rank, idx, kernel_idx);

Assembly is performed over entities, where entities[i] is an entity index (e.g., cell index) in mesh. entities0 holds the corresponding entity indices but in the mesh associated with the argument function (test/trial function) space. entities[i] and entities0[i] point to the same mesh entity, but with respect to different mesh views. In some cases, such as when integrating over the interface between two domains that do not overlap, an entity may exist in one domain but not another. In this case, the entity is marked with -1.

Parameters:

type – [in] Integral type.
rank – [in] Argument index, e.g. 0 for the test function space, 1 for the trial function space.
idx – [in] Integral identifier.
kernel_idx – [in] Index of the kernel (we may have multiple kernels for a given id in mixed-topology meshes).

Returns:

Entity indices in the argument function space mesh that is integrated over.

For cell integrals it has shape (num_cells,).
For exterior/interior facet integrals, it has shape (num_facets, 2) (row-major storage), where [i, 0] is the index of a cell and [i, 1] is the local index of the facet relative to the cell.

inline std::span<const std::int32_t> domain_coeff(IntegralType type, int idx, int c) const#

Coefficient function mesh integration entity indices.

This method is equivalent to ::domain_arg, but returns mesh entity indices for coefficient Functions.

Parameters:

type – [in] Integral type.
idx – [in] Integral identifier.
c – [in] Coefficient index.

Returns:

Entity indices in the coefficient function space mesh that is integrated over.

For cell integrals it has shape (num_cells,).
For exterior/interior facet integrals, it has shape (num_facets, 2) (row-major storage), where [i, 0] is the index of a cell and [i, 1] is the local index of the facet relative to the cell.

inline const std::vector<std::shared_ptr<const Function<scalar_type, geometry_type>>> &coefficients() const#

Access coefficients.

Returns:: Coefficients in the form.

inline bool needs_facet_permutations() const#

Get bool indicating whether permutation data needs to be passed into these integrals.

Returns:: True if cell permutation data is required

inline std::vector<int> coefficient_offsets() const#

Offset for each coefficient expansion array on a cell.

Used to pack data for multiple coefficients in a flat array. The last entry is the size required to store all coefficients.

Returns:: Coefficient offsets.

inline const std::vector<std::shared_ptr<const Constant<scalar_type>>> &constants() const#

Access constants.

Returns:: Constants in the form.

Dirichlet boundary conditions#

template<dolfinx::scalar T, std::floating_point U = dolfinx::scalar_value_t<T>> class DirichletBC#

Object for setting (strong) Dirichlet boundary conditions

\[u = g \ \text{on} \ G,\]

where \(u\) is the solution to be computed, \(g\) is a function and \(G\) is a sub domain of the mesh.

A DirichletBC is specified by the function \(g\), the function space (trial space) and degrees of freedom to which the boundary condition applies.

Public Functions

template<typename S, typename X> inline DirichletBC(const S &g, X &&dofs, std::shared_ptr<const FunctionSpace> V)#

Create a representation of a Dirichlet boundary condition constrained by a scalar- or vector-valued constant.

Note

Can be used only with point-evaluation elements.

Note

The indices in dofs are for blocks, e.g. a block index corresponds to 3 degrees-of-freedom if the dofmap associated with g has block size 3.

Note

The size of g must be equal to the block size if V. Use the Function version if this is not the case, e.g. for some mixed spaces.

Parameters:

g – [in] The boundary condition value (T or convertible to std::span<const T>)
dofs – [in] Degree-of-freedom block indices to be constrained. The indices must be sorted.
V – [in] The function space to be constrained

Pre:

dofs must be sorted.

template<typename X> inline DirichletBC(std::shared_ptr<const Constant<T>> g, X &&dofs, std::shared_ptr<const FunctionSpace> V)#

Create a representation of a Dirichlet boundary condition constrained by a fem::Constant.

Note

Can be used only with point-evaluation elements.

Note

The indices in dofs are for blocks, e.g. a block index corresponds to 3 degrees-of-freedom if the dofmap associated with g has block size 3.

Note

The size of g must be equal to the block size if V. Use the Function version if this is not the case, e.g. for some mixed spaces.

Parameters:

g – [in] The boundary condition value.
dofs – [in] Degree-of-freedom block indices to be constrained.
V – [in] The function space to be constrained

Pre:

dofs must be sorted.

template<typename X> inline DirichletBC(std::shared_ptr<const Function<T, U>> g, X &&dofs)#

Create a representation of a Dirichlet boundary condition where the space being constrained is the same as the function that defines the constraint Function, i.e. share the same fem::FunctionSpace.

Note

The indices in dofs are for blocks, e.g. a block index corresponds to 3 degrees-of-freedom if the dofmap associated with g has block size 3.

Parameters:

g – [in] The boundary condition value.
dofs – [in] Degree-of-freedom block indices to be constrained.

Pre:

dofs must be sorted.

template<typename X> inline DirichletBC(std::shared_ptr<const Function<T, U>> g, X &&V_g_dofs, std::shared_ptr<const FunctionSpace> V)#

Create a representation of a Dirichlet boundary condition where the space being constrained and the function that defines the constraint values do not share the same fem::FunctionSpace.

A typical example is when applying a constraint on a subspace. The (sub)space and the constrain function must have the same finite element.

Note

The indices in dofs are unrolled and not for blocks.

Parameters:

g – [in] The boundary condition value
V_g_dofs – [in] Two arrays of degree-of-freedom indices (std::array<std::vector<std::int32_t>, 2>). First array are indices in the space where boundary condition is applied (V), second array are indices in the space of the boundary condition value function g. The dof indices are unrolled, i.e. are not by dof block.
V – [in] The function (sub)space on which the boundary condition is applied

Pre:

The two degree-of-freedom arrays in V_g_dofs must be sorted by the indices in the first array.

DirichletBC(const DirichletBC &bc) = default#

Copy constructor

Parameters:: bc – [in] The object to be copied

DirichletBC(DirichletBC &&bc) = default#

Move constructor

Parameters:: bc – [in] The object to be moved

~DirichletBC() = default#: Destructor.

DirichletBC &operator=(const DirichletBC &bc) = default#

Assignment operator

Parameters:: bc – [in] Another DirichletBC object

DirichletBC &operator=(DirichletBC &&bc) = default#: Move assignment operator.

inline std::shared_ptr<const FunctionSpace> function_space() const#

The function space to which boundary conditions are applied

Returns:: The function space

inline std::variant<std::shared_ptr<const Function<T, U>>, std::shared_ptr<const Constant<T>>> value() const#

Return boundary value function g

Returns:: The boundary values Function

inline std::pair<std::span<const std::int32_t>, std::int32_t> dof_indices() const#

Access dof indices (local indices, unrolled), including ghosts, to which a Dirichlet condition is applied, and the index to the first non-owned (ghost) index. The array of indices is sorted.

Returns:: Sorted array of dof indices (unrolled) and index to the first entry in the dof index array that is not owned. Entries dofs[:pos] are owned and entries dofs[pos:] are ghosts.

inline void set(std::span<T> x, std::optional<std::span<const T>> x0, T alpha = 1) const#

Set entries in an array that are constrained by Dirichlet boundary conditions.

Entries in x that are constrained by a Dirichlet boundary conditions are set to alpha * (x_bc - x0), where x_bc is the (interpolated) boundary condition value.

For elements with point-wise evaluated degrees-of-freedom, e.g. Lagrange elements, x_bc is the value of the boundary condition at the degree-of-freedom. For elements with moment degrees-of-freedom, x_bc is the value of the boundary condition interpolated into the finite element space.

x may hold only owned entries or also include ghosts (entries available on the calling rank but owned by another rank); a constrained degree-of-freedom beyond the end of x is silently skipped, so an owned-only x sets only owned entries.

Note

If x0 is provided, it must be at least as long as x (checked only in Debug builds).

Parameters:

x – [inout] Array to modify for Dirichlet boundary conditions. May include ghost entries.
x0 – [in] Optional array used in computing the value to set. If not provided it is treated as zero.
alpha – [in] Scaling to apply.

inline void mark_dofs(std::span<std::int8_t> markers) const#

Set markers[i] = true if dof i has a boundary condition applied.

Value of markers[i] is not changed otherwise.

Parameters:: markers – [inout] Entry makers[i] is set to true if dof i in V0 had a boundary condition applied, i.e. dofs which are fixed by a boundary condition. Other entries in markers are left unchanged.

Degree-of-freedom maps#

graph::AdjacencyList<std::int32_t> dolfinx::fem::transpose_dofmap(md::mdspan<const std::int32_t, md::dextents<std::size_t, 2>> dofmap, std::int32_t num_cells)#

Create an adjacency list that maps a global index (process-wise) to the ‘unassembled’ cell-wise contributions.

It is built from the usual (cell, local index) -> global index dof map. An ‘unassembled’ vector is the stacked cell contributions, ordered by cell index. If the usual dof map is:

Cell: 0 1 2 3Global index: [ [0, 3, 5], [3, 2, 4], [4, 3, 2], [2, 1, 0]]

the ‘transpose’ dof map will be:

Global index: 0 1 2 3 4 5Unassembled index: [ [0, 11], [10], [4, 8, 9], [1, 3, 7], [5, 6], [2] ]

Parameters:

dofmap – [in] The standard dof map that for each cell (node) gives the global (process-wise) index of each local (cell-wise) index.
num_cells – [in] The number of cells (nodes) in dofmap to consider. The first num_cells are used. This is argument is typically used to exclude ghost cell contributions.

Returns:

Map from global (process-wise) index to positions in an unaassembled array. The links for each node are sorted.

class DofMap#

Degree-of-freedom map.

This class handles the mapping of degrees of freedom. It builds a dof map based on an ElementDofLayout on a specific mesh topology. It will reorder the dofs when running in parallel. Sub-dofmaps, both views and copies, are supported.

Public Functions

template<typename E, typename U> inline DofMap(E &&element, std::shared_ptr<const common::IndexMap> index_map, int index_map_bs, U &&dofmap, int bs)#

Create a DofMap from the layout of dofs on a reference element, an IndexMap defining the distribution of dofs across processes and a vector of indices.

Parameters:

element – [in] The layout of the degrees of freedom on an element
index_map – [in] The map describing the parallel distribution of the degrees of freedom.
index_map_bs – [in] The block size associated with the index_map.
dofmap – [in] Adjacency list with the degrees-of-freedom for each cell.
bs – [in] The block size of the dofmap.

DofMap(DofMap &&dofmap) = default#: Move constructor.

DofMap &operator=(DofMap &&dofmap) = default#: Move assignment.

bool operator==(const DofMap &map) const#

Equality operator.

Returns:: Returns true if the data for the two dofmaps are equal

inline std::span<const std::int32_t> cell_dofs(std::int32_t c) const#

Local-to-global mapping of dofs on a cell.

Parameters:: c – [in] The cell index
Returns:: Local-global dof map for the cell (using process-local indices)

int bs() const noexcept#: Return the block size for the dofmap.

DofMap extract_sub_dofmap(std::span<const int> component) const#

Extract subdofmap component.

Parameters:: component – [in] The component indices
Returns:: The dofmap for the component

std::pair<DofMap, std::vector<std::int32_t>> collapse(MPI_Comm comm, const mesh::Topology &topology, std::function<std::vector<int>(const graph::AdjacencyList<std::int32_t>&)> &&reorder_fn = nullptr) const#

Create a “collapsed” dofmap (collapses a sub-dofmap).

Parameters:

comm – [in] MPI Communicator
topology – [in] Mesh topology that the dofmap is defined on
reorder_fn – [in] Graph re-ordering function to apply to the dof data

Returns:

The collapsed dofmap

md::mdspan<const std::int32_t, md::dextents<std::size_t, 2>> map() const#

Get dofmap data.

Returns:: The adjacency list with dof indices for each cell

inline const ElementDofLayout &element_dof_layout() const#: Layout of dofs on an element.

int index_map_bs() const#: Block size associated with the index_map.

Public Members

std::shared_ptr<const common::IndexMap> index_map#: Index map that describes the parallel distribution of the dofmap.

Assembly#

Warning

doxygenfile: Cannot find file “fem/assembler.h

Interpolation#

template<std::floating_point T> std::vector<T> dolfinx::fem::interpolation_coords(const fem::FiniteElement<T> &element, const mesh::Geometry<T> &geometry, mesh::CellRange auto &&cells)#

Compute the evaluation points in the physical space at which an expression should be computed to interpolate it in a finite element space.

Parameters:

element – [in] Element to be interpolated into.
geometry – [in] Mesh geometry.
cells – [in] Indices of the cells in the mesh to compute interpolation coordinates for.

Returns:

Coordinates in the physical space at which to evaluate an expression. The shape is (3, num_points) and storage is row-major.

Sparsity pattern construction#

template<dolfinx::scalar T, std::floating_point U> la::SparsityPattern dolfinx::fem::create_sparsity_pattern(const Form<T, U> &a)#

Create a sparsity pattern for a given form.

Note

The pattern is not finalised, i.e. the caller is responsible for calling SparsityPattern::assemble.

Parameters:: a – [in] A bilinear form
Returns:: The corresponding sparsity pattern

PETSc helpers#

template<std::floating_point T> Mat dolfinx::fem::petsc::create_matrix(const Form<PetscScalar, T> &a, std::optional<std::string> type = std::nullopt)#

Create a matrix.

Parameters:

a – [in] A bilinear form
type – [in] The PETSc matrix type to create

Returns:

A sparse matrix with a layout and sparsity that matches the bilinear form. The caller is responsible for destroying the Mat object.

template<std::floating_point T> Mat dolfinx::fem::petsc::create_matrix_block(const std::vector<std::vector<const Form<PetscScalar, T>*>> &a, std::optional<std::string> type = std::nullopt)#

Initialise a monolithic matrix for an array of bilinear forms.

Parameters:

a – [in] Rectangular array of bilinear forms. The a(i, j) form will correspond to the (i, j) block in the returned matrix
type – [in] The type of PETSc Mat. If empty the PETSc default is used.

Returns:

A sparse matrix with a layout and sparsity that matches the bilinear forms. The caller is responsible for destroying the Mat object.

template<std::floating_point T> Mat dolfinx::fem::petsc::create_matrix_nest(const std::vector<std::vector<const Form<PetscScalar, T>*>> &a, std::optional<std::vector<std::vector<std::optional<std::string>>>> types)#: Create nested (MatNest) matrix.

Note

The caller is responsible for destroying the Mat object.

Vec dolfinx::fem::petsc::create_vector_block(const std::vector<std::pair<std::reference_wrapper<const common::IndexMap>, int>> &maps)#

Initialise monolithic vector. Vector is not zeroed.

The caller is responsible for destroying the Vec object

Vec dolfinx::fem::petsc::create_vector_nest(const std::vector<std::pair<std::reference_wrapper<const common::IndexMap>, int>> &maps)#: Create nested (VecNest) vector. Vector is not zeroed.

template<std::floating_point T> void dolfinx::fem::petsc::assemble_vector(Vec b, const Form<PetscScalar, T> &L, std::span<const PetscScalar> constants, const std::map<std::pair<IntegralType, int>, std::pair<std::span<const PetscScalar>, int>> &coeffs)#

Assemble linear form into an already allocated PETSc vector.

Ghost contributions are not accumulated (not sent to owner). Caller is responsible for calling VecGhostUpdateBegin/End.

Parameters:

b – [inout] The PETsc vector to assemble the form into. The vector must already be initialised with the correct size. The process-local contribution of the form is assembled into this vector. It is not zeroed before assembly.
L – [in] The linear form to assemble
constants – [in] The constants that appear in L
coeffs – [in] The coefficients that appear in L

template<std::floating_point T> void dolfinx::fem::petsc::assemble_vector(Vec b, const Form<PetscScalar, T> &L)#

Assemble linear form into an already allocated PETSc vector.

Ghost contributions are not accumulated (not sent to owner). Caller is responsible for calling VecGhostUpdateBegin/End.

Parameters:

b – [inout] Vector to assemble the form into. The vector must already be initialised with the correct size. The process-local contribution of the form is assembled into this vector. It is not zeroed before assembly.
L – [in] Linear form to assemble.

template<std::floating_point T> void dolfinx::fem::petsc::apply_lifting(Vec b, std::vector<std::optional<std::reference_wrapper<const Form<PetscScalar, T>>>> a, const std::vector<std::span<const PetscScalar>> &constants, const std::vector<std::map<std::pair<IntegralType, int>, std::pair<std::span<const PetscScalar>, int>>> &coeffs, const std::vector<std::vector<std::reference_wrapper<const DirichletBC<PetscScalar, T>>>> &bcs1, const std::vector<Vec> &x0, PetscScalar alpha)#

Modify RHS vector to account for Dirichlet boundary conditions.

Modify b such that:

b <- b - alpha * A_j (g_j - x0_j)

where j is a block (nest) index. For a non-blocked problem j = 0. The boundary conditions bcs1 are on the trial spaces V_j. The forms in [a] must have the same test space as L (from which b was built), but the trial space may differ. If x0 is not supplied, then it is treated as zero.

Ghost contributions are not accumulated (not sent to owner). Caller is responsible for calling VecGhostUpdateBegin/End.

Parameters:

b – [inout] Vector to modify by lifting.
a – [in] Bilinear forms, one per block j. A std::nullopt entry skips that block.
constants – [in] Constants that appear in each form in a, one entry per block j.
coeffs – [in] Coefficients that appear in each form in a, one entry per block j.
bcs1 – [in] Boundary conditions on the trial space V_j for each block j.
x0 – [in] Vectors used in the lifting, one per block j. If empty, x0_j is treated as zero for every block. Otherwise must have the same length as a.
alpha – [in] Scaling to apply.

template<std::floating_point T> void dolfinx::fem::petsc::apply_lifting(Vec b, const std::vector<std::optional<std::reference_wrapper<const Form<PetscScalar, T>>>> &a, const std::vector<std::vector<std::reference_wrapper<const DirichletBC<PetscScalar, T>>>> &bcs1, const std::vector<Vec> &x0, PetscScalar alpha)#

Modify RHS vector to account for Dirichlet boundary conditions.

Modify b such that:

b <- b - alpha * A_j (g_j - x0_j)

where j is a block (nest) index. For a non-blocked problem j = 0. The boundary conditions bcs1 are on the trial spaces V_j. The forms in [a] must have the same test space as L (from which b was built), but the trial space may differ. If x0 is not supplied, then it is treated as zero.

Ghost contributions are not accumulated (not sent to owner). Caller is responsible for calling VecGhostUpdateBegin/End.

Parameters:

b – [inout] Vector to modify by lifting.
a – [in] Bilinear forms, one per block j. A std::nullopt entry skips that block.
bcs1 – [in] Boundary conditions on the trial space V_j for each block j.
x0 – [in] Vectors used in the lifting, one per block j. If empty, x0_j is treated as zero for every block. Otherwise must have the same length as a.
alpha – [in] Scaling to apply.

template<std::floating_point T> void dolfinx::fem::petsc::set_bc(Vec b, const std::vector<std::reference_wrapper<const DirichletBC<PetscScalar, T>>> &bcs, std::optional<const Vec> x0, PetscScalar alpha = 1)#

Entries in b that are constrained by a Dirichlet boundary conditions are set to alpha * (x_bc - x0), where x_bc is the (interpolated) boundary condition value.

Parameters:

b – [in] The vector to apply the boundary condition to. The local (owned) part of this vector is modified. The user is responsible for scattering the changes to the ghost part of the vector if necessary.
bcs – [in] The boundary conditions to apply.
x0 – [in] Optional vector used in computing the value to set. If not provided it is treated as zero. The local (owned) part of this vector is used.
alpha – [in] Scaling to apply.

template<std::floating_point T> void dolfinx::fem::petsc::assemble_residual(const Vec x, Vec b, const Form<PetscScalar, T> &F, const Form<PetscScalar, T> &J, const std::vector<std::reference_wrapper<const DirichletBC<PetscScalar, T>>> &bcs, Function<PetscScalar, T> &u)#

Assemble the residual \(F(x)\) of a nonlinear problem into b, with Dirichlet conditions applied.

Intended as the body of the residual callback of nls::petsc::SNESSolver, which passes the point to evaluate at x and the vector to assemble into b:

solver.set_F([&](const Vec x, Vec b)
             { assemble_residual(x, b, F, J, bcs, u); }, b_layout);

Entries of b constrained by bcs are set to x - g, so that a Newton update drives x to the boundary condition value g.

Parameters:

x – [in] Point at which to evaluate the residual, e.g. a line search trial point. Must be ghosted. Its ghost values are updated before use.
b – [out] Vector to assemble into, which is the one the solver passed to the callback and not necessarily the one registered with set_F. Zeroed first, and its ghost values are updated on return.
F – [in] Residual form.
J – [in] Jacobian form, used to lift bcs.
bcs – [in] Dirichlet boundary conditions.
u – [out] Function that F and J hold as a coefficient. Its degrees-of-freedom are set to x before assembly.

template<std::floating_point T> void dolfinx::fem::petsc::assemble_jacobian(const Vec x, Mat Jmat, Mat Pmat, const Form<PetscScalar, T> &J, const std::vector<std::reference_wrapper<const DirichletBC<PetscScalar, T>>> &bcs, Function<PetscScalar, T> &u, const Form<PetscScalar, T> *P = nullptr)#

Assemble the Jacobian \(dF/dx\) of a nonlinear problem into Jmat, and a preconditioner into Pmat.

Intended as the body of the Jacobian callback of nls::petsc::SNESSolver, which passes the point to evaluate at x and the matrices to assemble into Jmat and Pmat:

solver.set_J([&](const Vec x, Mat Jmat, Mat Pmat)
             { assemble_jacobian(x, Jmat, Pmat, J, bcs, u); },
             A_layout);

Rows and columns constrained by bcs are zeroed, and for a form whose test and trial spaces are the same a unit diagonal is set on the constrained rows, matching the residual assembled by assemble_residual.

Parameters:

x – [in] Point at which to evaluate the Jacobian, e.g. a line search trial point. Must be ghosted. Its ghost values are updated before use.
Jmat – [out] Matrix to assemble the Jacobian into, which is the one the solver passed to the callback and not necessarily the one registered with set_J. Zeroed first.
Pmat – [out] Matrix to assemble the preconditioner into. Zeroed first. Unused, and may be nullptr, if P is not given.
J – [in] Jacobian form.
bcs – [in] Dirichlet boundary conditions.
u – [out] Function that J and P hold as a coefficient. Its degrees-of-freedom are set to x before assembly.
P – [in] Preconditioner form. If not given, Pmat is left alone and PETSc preconditions with the Jacobian.

Misc#

Warning

doxygenfile: Cannot find file “fem/utils.h

Finite element (dolfinx::fem)#

Finite elements#

Function spaces and functions#

Function spaces#

Functions#

Constants#

Forms#

Dirichlet boundary conditions#

Degree-of-freedom maps#

Assembly#

Interpolation#

Sparsity pattern construction#

PETSc helpers#

Misc#

Finite element (`dolfinx::fem`)#