dolfinx::fem Namespace Reference

Finite element method functionality. More...

Namespaces
namespace	petsc
	Helper functions for assembly into PETSc data structures.
namespace	sparsitybuild
	Support for building sparsity patterns from degree-of-freedom maps.

Classes
class	Constant
	Constant value which can be attached to a Form. More...
class	CoordinateElement
class	DirichletBC
class	DofMap
	Degree-of-freedom map. More...
class	ElementDofLayout
class	Expression
	Represents a mathematical expression evaluated at a pre-defined set of points on the reference cell. More...
class	FiniteElement
	Model of a finite element. More...
class	Form
	A representation of finite element variational forms. More...
class	Function
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. More...
struct	integral_data
	Represents integral data, containing the integral ID, the kernel, and a list of entities to integrate over. More...

Concepts
concept	MDSpan
concept	DofTransformKernel
	DOF transform kernel concept.
concept	FEkernel
	Finite element cell kernel concept.

Enumerations
enum class	doftransform { standard = 0 , transpose = 1 , inverse = 2 , inverse_transpose = 3 }
	DOF transformation type. More...
enum class	IntegralType : std::int8_t { cell = 0 , exterior_facet = 1 , interior_facet = 2 , vertex = 3 }
	Type of integral. More...

Functions
template<dolfinx::scalar T>
std::map< std::pair< IntegralType, int >, std::pair< std::span< const T >, int > >	make_coefficients_span (const std::map< std::pair< IntegralType, int >, std::pair< std::vector< T >, int > > &coeffs)
	Create a map of std::spans from a map of std::vectors.
template<dolfinx::scalar T, std::floating_point U>
T	assemble_scalar (const Form< T, U > &M, std::span< const T > constants, const std::map< std::pair< IntegralType, int >, std::pair< std::span< const T >, int > > &coefficients)
	Assemble functional into scalar.
template<dolfinx::scalar T, std::floating_point U>
T	assemble_scalar (const Form< T, U > &M)
template<dolfinx::scalar T, std::floating_point U>
void	assemble_vector (std::span< T > b, const Form< T, U > &L, std::span< const T > constants, const std::map< std::pair< IntegralType, int >, std::pair< std::span< const T >, int > > &coefficients)
	Assemble linear form into a vector.
template<dolfinx::scalar T, std::floating_point U>
void	assemble_vector (std::span< T > b, const Form< T, U > &L)
	Assemble linear form into a vector.
template<dolfinx::scalar T, std::floating_point U>
void	apply_lifting (std::span< T > b, const std::vector< std::shared_ptr< const Form< T, U > > > &a, const std::vector< std::span< const T > > &constants, const std::vector< std::map< std::pair< IntegralType, int >, std::pair< std::span< const T >, int > > > &coeffs, const std::vector< std::vector< std::shared_ptr< const DirichletBC< T, U > > > > &bcs1, const std::vector< std::span< const T > > &x0, T alpha)
template<dolfinx::scalar T, std::floating_point U>
void	apply_lifting (std::span< T > b, const std::vector< std::shared_ptr< const Form< T, U > > > &a, const std::vector< std::vector< std::shared_ptr< const DirichletBC< T, U > > > > &bcs1, const std::vector< std::span< const T > > &x0, T alpha)
template<dolfinx::scalar T, std::floating_point U>
void	assemble_matrix (la::MatSet< T > auto mat_add, const Form< T, U > &a, std::span< const T > constants, const std::map< std::pair< IntegralType, int >, std::pair< std::span< const T >, int > > &coefficients, std::span< const std::int8_t > dof_marker0, std::span< const std::int8_t > dof_marker1)
	Assemble bilinear form into a matrix. Matrix must already be initialised. Does not zero or finalise the matrix.
template<dolfinx::scalar T, std::floating_point U>
void	assemble_matrix (auto mat_add, const Form< T, U > &a, std::span< const T > constants, const std::map< std::pair< IntegralType, int >, std::pair< std::span< const T >, int > > &coefficients, const std::vector< std::shared_ptr< const DirichletBC< T, U > > > &bcs)
template<dolfinx::scalar T, std::floating_point U>
void	assemble_matrix (auto mat_add, const Form< T, U > &a, const std::vector< std::shared_ptr< const DirichletBC< T, U > > > &bcs)
template<dolfinx::scalar T, std::floating_point U>
void	assemble_matrix (auto mat_add, const Form< T, U > &a, std::span< const std::int8_t > dof_marker0, std::span< const std::int8_t > dof_marker1)
	Assemble bilinear form into a matrix. Matrix must already be initialised. Does not zero or finalise the matrix.
template<dolfinx::scalar T>
void	set_diagonal (auto set_fn, std::span< const std::int32_t > rows, T diagonal=1.0)
	Sets a value to the diagonal of a matrix for specified rows.
template<dolfinx::scalar T, std::floating_point U>
void	set_diagonal (auto set_fn, const FunctionSpace< U > &V, const std::vector< std::shared_ptr< const DirichletBC< T, U > > > &bcs, T diagonal=1.0)
	Sets a value to the diagonal of the matrix for rows with a Dirichlet boundary conditions applied.
std::vector< std::int32_t >	locate_dofs_topological (const mesh::Topology &topology, const DofMap &dofmap, int dim, std::span< const std::int32_t > entities, bool remote=true)
	Find degrees-of-freedom which belong to the provided mesh entities (topological).
std::array< std::vector< std::int32_t >, 2 >	locate_dofs_topological (const mesh::Topology &topology, std::array< std::reference_wrapper< const DofMap >, 2 > dofmaps, int dim, std::span< const std::int32_t > entities, bool remote=true)
	Find degrees-of-freedom which belong to the provided mesh entities (topological).
template<std::floating_point T, typename U >
std::vector< std::int32_t >	locate_dofs_geometrical (const FunctionSpace< T > &V, U marker_fn)
	Find degrees of freedom whose geometric coordinate is true for the provided marking function.
template<std::floating_point T, typename U >
std::array< std::vector< std::int32_t >, 2 >	locate_dofs_geometrical (const std::array< std::reference_wrapper< const FunctionSpace< T > >, 2 > &V, U marker_fn)
template<dolfinx::scalar T, std::floating_point U = dolfinx::scalar_value_type_t<T>>
void	discrete_gradient (mesh::Topology &topology, std::pair< std::reference_wrapper< const FiniteElement< U > >, std::reference_wrapper< const DofMap > > V0, std::pair< std::reference_wrapper< const FiniteElement< U > >, std::reference_wrapper< const DofMap > > V1, auto &&mat_set)
	Assemble a discrete gradient operator.
template<dolfinx::scalar T, std::floating_point U>
void	interpolation_matrix (const FunctionSpace< U > &V0, const FunctionSpace< U > &V1, auto &&mat_set)
	Assemble an interpolation operator matrix.
graph::AdjacencyList< std::int32_t >	transpose_dofmap (MDSPAN_IMPL_STANDARD_NAMESPACE::mdspan< const std::int32_t, MDSPAN_IMPL_STANDARD_NAMESPACE::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.
std::tuple< common::IndexMap, int, std::vector< std::vector< std::int32_t > > >	build_dofmap_data (MPI_Comm comm, const mesh::Topology &topology, const std::vector< ElementDofLayout > &element_dof_layouts, const std::function< std::vector< int >(const graph::AdjacencyList< std::int32_t > &)> &reorder_fn)
template<dolfinx::scalar T>
std::array< std::vector< std::shared_ptr< const FunctionSpace< T > > >, 2 >	common_function_spaces (const std::vector< std::vector< std::array< std::shared_ptr< const FunctionSpace< T > >, 2 > > > &V)
template<std::floating_point T>
std::vector< std::size_t >	compute_value_shape (std::shared_ptr< const dolfinx::fem::FiniteElement< T > > element, std::size_t tdim, std::size_t gdim)
	Compute the physical value shape of an element for a mesh.
template<typename U , typename V , typename W , typename X >
	FunctionSpace (U mesh, V element, W dofmap, X value_shape) -> FunctionSpace< typename std::remove_cvref< typename U::element_type >::type::geometry_type::value_type >
	Type deduction.
template<std::floating_point T>
std::vector< T >	interpolation_coords (const fem::FiniteElement< T > &element, const mesh::Geometry< T > &geometry, std::span< const std::int32_t > cells)
	Compute the evaluation points in the physical space at which an expression should be computed to interpolate it in a finite element space.
template<dolfinx::scalar T, std::floating_point U>
void	interpolate (Function< T, U > &u, std::span< const T > f, std::array< std::size_t, 2 > fshape, std::span< const std::int32_t > cells)
	Interpolate an evaluated expression f(x) in a finite element space.
template<std::floating_point T>
geometry::PointOwnershipData< T >	create_interpolation_data (const mesh::Geometry< T > &geometry0, const FiniteElement< T > &element0, const mesh::Mesh< T > &mesh1, std::span< const std::int32_t > cells, T padding)
	Generate data needed to interpolate finite element Functions across different meshes.
template<dolfinx::scalar T, std::floating_point U>
void	interpolate (Function< T, U > &u, const Function< T, U > &v, std::span< const std::int32_t > cells, const geometry::PointOwnershipData< U > &interpolation_data)
	Interpolate a finite element Function defined on a mesh to a finite element Function defined on different (non-matching) mesh.
template<dolfinx::scalar T, std::floating_point U>
void	interpolate (Function< T, U > &u1, std::span< const std::int32_t > cells1, const Function< T, U > &u0, std::span< const std::int32_t > cells0)
	Interpolate from one finite element Function to another Function on the same (sub)mesh.
std::vector< std::int32_t >	compute_integration_domains (IntegralType integral_type, const mesh::Topology &topology, std::span< const std::int32_t > entities, int dim)
	Given an integral type and a set of entities, compute the entities that should be integrated over.
template<dolfinx::scalar T, std::floating_point U>
std::vector< std::vector< std::array< std::shared_ptr< const FunctionSpace< U > >, 2 > > >	extract_function_spaces (const std::vector< std::vector< const Form< T, U > * > > &a)
	Extract test (0) and trial (1) function spaces pairs for each bilinear form for a rectangular array of forms.
template<dolfinx::scalar T, std::floating_point U>
la::SparsityPattern	create_sparsity_pattern (const Form< T, U > &a)
	Create a sparsity pattern for a given form.
template<std::floating_point T>
ElementDofLayout	create_element_dof_layout (const fem::FiniteElement< T > &element, const std::vector< int > &parent_map={})
	Create an ElementDofLayout from a FiniteElement.
DofMap	create_dofmap (MPI_Comm comm, const ElementDofLayout &layout, mesh::Topology &topology, std::function< void(std::span< std::int32_t >, std::uint32_t)> permute_inv, std::function< std::vector< int >(const graph::AdjacencyList< std::int32_t > &)> reorder_fn)
	Create a dof map on mesh.
std::vector< DofMap >	create_dofmaps (MPI_Comm comm, const std::vector< ElementDofLayout > &layouts, mesh::Topology &topology, std::function< void(std::span< std::int32_t >, std::uint32_t)> permute_inv, std::function< std::vector< int >(const graph::AdjacencyList< std::int32_t > &)> reorder_fn)
	Create a set of dofmaps on a given topology.
std::vector< std::string >	get_coefficient_names (const ufcx_form &ufcx_form)
std::vector< std::string >	get_constant_names (const ufcx_form &ufcx_form)
	Get the name of each constant in a UFC form.
template<dolfinx::scalar T, std::floating_point U = scalar_value_type_t<T>>
Form< T, U >	create_form_factory (const ufcx_form &ufcx_form, const std::vector< std::shared_ptr< const FunctionSpace< U > > > &spaces, const std::vector< std::shared_ptr< const Function< T, U > > > &coefficients, const std::vector< std::shared_ptr< const Constant< T > > > &constants, const std::map< IntegralType, std::vector< std::pair< std::int32_t, std::span< const std::int32_t > > > > &subdomains, const std::map< std::shared_ptr< const mesh::Mesh< U > >, std::span< const std::int32_t > > &entity_maps, std::shared_ptr< const mesh::Mesh< U > > mesh=nullptr)
	Create a Form from UFCx input with coefficients and constants passed in the required order.
template<dolfinx::scalar T, std::floating_point U = scalar_value_type_t<T>>
Form< T, U >	create_form (const ufcx_form &ufcx_form, const std::vector< std::shared_ptr< const FunctionSpace< U > > > &spaces, const std::map< std::string, std::shared_ptr< const Function< T, U > > > &coefficients, const std::map< std::string, std::shared_ptr< const Constant< T > > > &constants, const std::map< IntegralType, std::vector< std::pair< std::int32_t, std::span< const std::int32_t > > > > &subdomains, const std::map< std::shared_ptr< const mesh::Mesh< U > >, std::span< const std::int32_t > > &entity_maps, std::shared_ptr< const mesh::Mesh< U > > mesh=nullptr)
	Create a Form from UFC input with coefficients and constants resolved by name.
template<dolfinx::scalar T, std::floating_point U = scalar_value_type_t<T>>
Form< T, U >	create_form (ufcx_form *(*fptr)(), const std::vector< std::shared_ptr< const FunctionSpace< U > > > &spaces, const std::map< std::string, std::shared_ptr< const Function< T, U > > > &coefficients, const std::map< std::string, std::shared_ptr< const Constant< T > > > &constants, const std::map< IntegralType, std::vector< std::pair< std::int32_t, std::span< const std::int32_t > > > > &subdomains, const std::map< std::shared_ptr< const mesh::Mesh< U > >, std::span< const std::int32_t > > &entity_maps, std::shared_ptr< const mesh::Mesh< U > > mesh=nullptr)
	Create a Form using a factory function that returns a pointer to a ufcx_form.
template<std::floating_point T>
FunctionSpace< T >	create_functionspace (std::shared_ptr< mesh::Mesh< T > > mesh, const basix::FiniteElement< T > &e, const std::vector< std::size_t > &value_shape={}, std::function< std::vector< int >(const graph::AdjacencyList< std::int32_t > &)> reorder_fn=nullptr)
	Create a function space from a Basix element.
template<dolfinx::scalar T, std::floating_point U>
std::pair< std::vector< T >, int >	allocate_coefficient_storage (const Form< T, U > &form, IntegralType integral_type, int id)
	Allocate storage for coefficients of a pair (integral_type, / id) from a Form.
template<dolfinx::scalar T, std::floating_point U>
std::map< std::pair< IntegralType, int >, std::pair< std::vector< T >, int > >	allocate_coefficient_storage (const Form< T, U > &form)
	Allocate memory for packed coefficients of a Form.
template<dolfinx::scalar T, std::floating_point U>
void	pack_coefficients (const Form< T, U > &form, IntegralType integral_type, int id, std::span< T > c, int cstride)
	Pack coefficients of a Form for a given integral type and domain id.
template<dolfinx::scalar T, std::floating_point U = scalar_value_type_t<T>>
Expression< T, U >	create_expression (const ufcx_expression &e, const std::vector< std::shared_ptr< const Function< T, U > > > &coefficients, const std::vector< std::shared_ptr< const Constant< T > > > &constants, std::shared_ptr< const FunctionSpace< U > > argument_function_space=nullptr)
	Create Expression from UFC.
template<dolfinx::scalar T, std::floating_point U = scalar_value_type_t<T>>
Expression< T, U >	create_expression (const ufcx_expression &e, const std::map< std::string, std::shared_ptr< const Function< T, U > > > &coefficients, const std::map< std::string, std::shared_ptr< const Constant< T > > > &constants, std::shared_ptr< const FunctionSpace< U > > argument_function_space=nullptr)
	Create Expression from UFC input (with named coefficients and constants).
template<dolfinx::scalar T, std::floating_point U>
void	pack_coefficients (const Form< T, U > &form, std::map< std::pair< IntegralType, int >, std::pair< std::vector< T >, int > > &coeffs)
	Pack coefficients of a Form.
template<dolfinx::scalar T, std::floating_point U>
std::pair< std::vector< T >, int >	pack_coefficients (const Expression< T, U > &e, std::span< const std::int32_t > entities, std::size_t estride)
	Pack coefficients of a Expression u for a give list of active entities.
template<typename U >
std::vector< typename U::scalar_type >	pack_constants (const U &u)
	Pack constants of u into a single array ready for assembly.

Detailed Description

Finite element method functionality.

Classes and algorithms for finite element method spaces and operations.

Enumeration Type Documentation

doftransform

enum class doftransform

strong

DOF transformation type.

Enumerator
standard	Standard.
transpose	Transpose.
inverse	Inverse.
inverse_transpose	Transpose inverse.

IntegralType

enum class IntegralType : std::int8_t

strong

Type of integral.

Enumerator
cell	Cell.
exterior_facet	Exterior facet.
interior_facet	Interior facet.
vertex	Vertex.

Function Documentation

allocate_coefficient_storage() [1/2]

template<dolfinx::scalar T, std::floating_point U>

std::map< std::pair< IntegralType, int >, std::pair< std::vector< T >, int > > allocate_coefficient_storage

(

const Form< T, U > &

form

)

Allocate memory for packed coefficients of a Form.

Parameters

[in]

form

The Form

Returns

Map from a form (integral_type, domain_id) pair to a (coeffs, cstride) pair

allocate_coefficient_storage() [2/2]

template<dolfinx::scalar T, std::floating_point U>

std::pair< std::vector< T >, int > allocate_coefficient_storage	(	const Form< T, U > &	form,
		IntegralType	integral_type,
		int	id )

Allocate storage for coefficients of a pair (integral_type, / id) from a Form.

Parameters

[in]	form	The Form
[in]	integral_type	Type of integral
[in]	id	The id of the integration domain

Returns

A storage container and the column stride

apply_lifting() [1/2]

template<dolfinx::scalar T, std::floating_point U>

void apply_lifting	(	std::span< T >	b,
		const std::vector< std::shared_ptr< const Form< T, U > > > &	a,
		const std::vector< std::span< const T > > &	constants,
		const std::vector< std::map< std::pair< IntegralType, int >, std::pair< std::span< const T >, int > > > &	coeffs,
		const std::vector< std::vector< std::shared_ptr< const DirichletBC< T, U > > > > &	bcs1,
		const std::vector< std::span< const T > > &	x0,
		T	alpha )

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.

apply_lifting() [2/2]

template<dolfinx::scalar T, std::floating_point U>

void apply_lifting	(	std::span< T >	b,
		const std::vector< std::shared_ptr< const Form< T, U > > > &	a,
		const std::vector< std::vector< std::shared_ptr< const DirichletBC< T, U > > > > &	bcs1,
		const std::vector< std::span< const T > > &	x0,
		T	alpha )

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.

assemble_matrix() [1/4]

template<dolfinx::scalar T, std::floating_point U>

void assemble_matrix	(	auto	mat_add,
		const Form< T, U > &	a,
		const std::vector< std::shared_ptr< const DirichletBC< T, U > > > &	bcs )

Assemble bilinear form into a matrix

Parameters

[in]	mat_add	The function for adding values into the matrix
[in]	a	The bilinear from to assemble
[in]	bcs	Boundary conditions to apply. For boundary condition dofs the row and column are zeroed. The diagonal entry is not set.

assemble_matrix() [2/4]

template<dolfinx::scalar T, std::floating_point U>

void assemble_matrix	(	auto	mat_add,
		const Form< T, U > &	a,
		std::span< const std::int8_t >	dof_marker0,
		std::span< const std::int8_t >	dof_marker1 )

Assemble bilinear form into a matrix. Matrix must already be initialised. Does not zero or finalise the matrix.

Parameters

[in]	mat_add	The function for adding values into the matrix
[in]	a	The bilinear form to assemble
[in]	dof_marker0	Boundary condition markers for the rows. If bc[i] is true then rows i in A will be zeroed. The index i is a local index.
[in]	dof_marker1	Boundary condition markers for the columns. If bc[i] is true then rows i in A will be zeroed. The index i is a local index.

assemble_matrix() [3/4]

template<dolfinx::scalar T, std::floating_point U>

void assemble_matrix	(	auto	mat_add,
		const Form< T, U > &	a,
		std::span< const T >	constants,
		const std::map< std::pair< IntegralType, int >, std::pair< std::span< const T >, int > > &	coefficients,
		const std::vector< std::shared_ptr< const DirichletBC< T, U > > > &	bcs )

Assemble bilinear form into a matrix

Parameters

[in]	mat_add	The function for adding values into the matrix
[in]	a	The bilinear from to assemble
[in]	constants	Constants that appear in a
[in]	coefficients	Coefficients that appear in a
[in]	bcs	Boundary conditions to apply. For boundary condition dofs the row and column are zeroed. The diagonal entry is not set.

assemble_matrix() [4/4]

template<dolfinx::scalar T, std::floating_point U>

void assemble_matrix	(	la::MatSet< T > auto	mat_add,
		const Form< T, U > &	a,
		std::span< const T >	constants,
		const std::map< std::pair< IntegralType, int >, std::pair< std::span< const T >, int > > &	coefficients,
		std::span< const std::int8_t >	dof_marker0,
		std::span< const std::int8_t >	dof_marker1 )

Assemble bilinear form into a matrix. Matrix must already be initialised. Does not zero or finalise the matrix.

Parameters

[in]	mat_add	The function for adding values into the matrix
[in]	a	The bilinear form to assemble
[in]	constants	Constants that appear in a
[in]	coefficients	Coefficients that appear in a
[in]	dof_marker0	Boundary condition markers for the rows. If bc[i] is true then rows i in A will be zeroed. The index i is a local index.
[in]	dof_marker1	Boundary condition markers for the columns. If bc[i] is true then rows i in A will be zeroed. The index i is a local index.

assemble_scalar() [1/2]

template<dolfinx::scalar T, std::floating_point U>

T assemble_scalar

(

const Form< T, U > &

M

)

Assemble functional into scalar

Note

Caller is responsible for accumulation across processes.

Parameters

[in]

M

The form (functional) to assemble

Returns

The contribution to the form (functional) from the local process

assemble_scalar() [2/2]

template<dolfinx::scalar T, std::floating_point U>

T assemble_scalar	(	const Form< T, U > &	M,
		std::span< const T >	constants,
		const std::map< std::pair< IntegralType, int >, std::pair< std::span< const T >, int > > &	coefficients )

Assemble functional into scalar.

The caller supplies the form constants and coefficients for this version, which has efficiency benefits if the data can be re-used for multiple calls.

Note

Caller is responsible for accumulation across processes.

Parameters

[in]	M	The form (functional) to assemble
[in]	constants	The constants that appear in M
[in]	coefficients	The coefficients that appear in M

Returns

The contribution to the form (functional) from the local process

assemble_vector() [1/2]

template<dolfinx::scalar T, std::floating_point U>

void assemble_vector	(	std::span< T >	b,
		const Form< T, U > &	L )

Assemble linear form into a vector.

Parameters

[in,out]	b	The vector to be assembled. It will not be zeroed before assembly.
[in]	L	The linear forms to assemble into b

assemble_vector() [2/2]

template<dolfinx::scalar T, std::floating_point U>

void assemble_vector	(	std::span< T >	b,
		const Form< T, U > &	L,
		std::span< const T >	constants,
		const std::map< std::pair< IntegralType, int >, std::pair< std::span< const T >, int > > &	coefficients )

Assemble linear form into a vector.

The caller supplies the form constants and coefficients for this version, which has efficiency benefits if the data can be re-used for multiple calls.

Parameters

[in,out]	b	The vector to be assembled. It will not be zeroed before assembly.
[in]	L	The linear forms to assemble into b
[in]	constants	The constants that appear in L
[in]	coefficients	The coefficients that appear in L

build_dofmap_data()

std::tuple< common::IndexMap, int, std::vector< std::vector< std::int32_t > > > build_dofmap_data	(	MPI_Comm	comm,
		const mesh::Topology &	topology,
		const std::vector< ElementDofLayout > &	element_dof_layouts,
		const std::function< std::vector< int >(const graph::AdjacencyList< std::int32_t > &)> &	reorder_fn )

Build dofmap data for elements on a mesh topology

Parameters

[in]	comm	MPI communicator
[in]	topology	The mesh topology
[in]	element_dof_layouts	The element dof layouts for each cell type in topology
[in]	reorder_fn	Graph reordering function that is applied to the dofmaps

Returns

The index map, block size, and dofmaps for each element type

common_function_spaces()

template<dolfinx::scalar T>

std::array< std::vector< std::shared_ptr< const FunctionSpace< T > > >, 2 > common_function_spaces

(

const std::vector< std::vector< std::array< std::shared_ptr< const FunctionSpace< T > >, 2 > > > &

V

)

Extract FunctionSpaces for (0) rows blocks and (1) columns blocks from a rectangular array of (test, trial) space pairs. The test space must be the same for each row and the trial spaces must be the same for each column. Raises an exception if there is an inconsistency. e.g. if each form in row i does not have the same test space then an exception is raised.

Parameters

[in]

V

Vector function spaces for (0) each row block and (1) each column block

compute_integration_domains()

std::vector< std::int32_t > compute_integration_domains	(	fem::IntegralType	integral_type,
		const mesh::Topology &	topology,
		std::span< const std::int32_t >	entities,
		int	dim )

Given an integral type and a set of entities, compute the entities that should be integrated over.

This function returns a list [(id, entities)]. For cell integrals entities are the cell indices. For exterior facet integrals, entities is a list of (cell_index, local_facet_index) pairs. For interior facet integrals, entities is a list of (cell_index0, local_facet_index0, cell_index1, local_facet_index1). id refers to the subdomain id used in the definition of the integration measures of the variational form.

Note

Owned mesh entities only are returned. Ghost entities are not included.

Parameters

[in]	integral_type	Integral type
[in]	topology	Mesh topology
[in]	entities	List of mesh entities
[in]	dim	Topological dimension of entities

Returns

List of integration entities

Precondition

For facet integrals, the topology facet-to-cell and cell-to-facet connectivity must be computed before calling this function.

compute_value_shape()

template<std::floating_point T>

std::vector< std::size_t > compute_value_shape	(	std::shared_ptr< const dolfinx::fem::FiniteElement< T > >	element,
		std::size_t	tdim,
		std::size_t	gdim )

Compute the physical value shape of an element for a mesh.

Parameters

[in]	element	The element
[in]	tdim	Topological dimension
[in]	gdim	Geometric dimension

Returns

Physical valus shape

create_dofmap()

fem::DofMap create_dofmap	(	MPI_Comm	comm,
		const ElementDofLayout &	layout,
		mesh::Topology &	topology,
		std::function< void(std::span< std::int32_t >, std::uint32_t)>	permute_inv,
		std::function< std::vector< int >(const graph::AdjacencyList< std::int32_t > &)>	reorder_fn )

Create a dof map on mesh.

Parameters

[in]	comm	MPI communicator
[in]	layout	Dof layout on an element
[in]	topology	Mesh topology
[in]	permute_inv	Function to un-permute dofs. nullptr when transformation is not required.
[in]	reorder_fn	Graph reordering function called on the dofmap

Returns

A new dof map

create_dofmaps()

std::vector< fem::DofMap > create_dofmaps	(	MPI_Comm	comm,
		const std::vector< ElementDofLayout > &	layouts,
		mesh::Topology &	topology,
		std::function< void(std::span< std::int32_t >, std::uint32_t)>	permute_inv,
		std::function< std::vector< int >(const graph::AdjacencyList< std::int32_t > &)>	reorder_fn )

Create a set of dofmaps on a given topology.

Parameters

[in]	comm	MPI communicator
[in]	layouts	Dof layout on each element type
[in]	topology	Mesh topology
[in]	permute_inv	Function to un-permute dofs. nullptr when transformation is not required.
[in]	reorder_fn	Graph reordering function called on the dofmaps

Returns

The list of new dof maps

Note

The number of layouts must match the number of cell types in the topology

create_form() [1/2]

template<dolfinx::scalar T, std::floating_point U = scalar_value_type_t<T>>

Form< T, U > create_form	(	const ufcx_form &	ufcx_form,
		const std::vector< std::shared_ptr< const FunctionSpace< U > > > &	spaces,
		const std::map< std::string, std::shared_ptr< const Function< T, U > > > &	coefficients,
		const std::map< std::string, std::shared_ptr< const Constant< T > > > &	constants,
		const std::map< IntegralType, std::vector< std::pair< std::int32_t, std::span< const std::int32_t > > > > &	subdomains,
		const std::map< std::shared_ptr< const mesh::Mesh< U > >, std::span< const std::int32_t > > &	entity_maps,
		std::shared_ptr< const mesh::Mesh< U > >	mesh = nullptr )

Create a Form from UFC input with coefficients and constants resolved by name.

Parameters

[in]	ufcx_form	UFC form
[in]	spaces	Function spaces for the Form arguments.
[in]	coefficients	Coefficient fields in the form (by name).
[in]	constants	Spatial constants in the form (by name).
[in]	subdomains	Subdomain markers.

Precondition

Each value in subdomains must be sorted by domain id.

Parameters

[in]	entity_maps	The entity maps for the form. Empty for single domain problems.
[in]	mesh	Mesh of the domain. This is required if the form has no arguments, e.g. a functional.

Returns

A Form

create_form() [2/2]

template<dolfinx::scalar T, std::floating_point U = scalar_value_type_t<T>>

Form< T, U > create_form	(	ufcx_form *(*	fptr )(),
		const std::vector< std::shared_ptr< const FunctionSpace< U > > > &	spaces,
		const std::map< std::string, std::shared_ptr< const Function< T, U > > > &	coefficients,
		const std::map< std::string, std::shared_ptr< const Constant< T > > > &	constants,
		const std::map< IntegralType, std::vector< std::pair< std::int32_t, std::span< const std::int32_t > > > > &	subdomains,
		const std::map< std::shared_ptr< const mesh::Mesh< U > >, std::span< const std::int32_t > > &	entity_maps,
		std::shared_ptr< const mesh::Mesh< U > >	mesh = nullptr )

Create a Form using a factory function that returns a pointer to a ufcx_form.

Coefficients and constants are resolved by name/string.

Parameters

[in]	fptr	Pointer to a function returning a pointer to ufcx_form.
[in]	spaces	Function spaces for the Form arguments.
[in]	coefficients	Coefficient fields in the form (by name),
[in]	constants	Spatial constants in the form (by name),
[in]	subdomains	Subdomain markers.

Precondition

Each value in subdomains must be sorted by domain id.

Parameters

[in]	entity_maps	The entity maps for the form. Empty for single domain problems.
[in]	mesh	Mesh of the domain. This is required if the form has no arguments, e.g. a functional.

Returns

A Form

create_form_factory()

template<dolfinx::scalar T, std::floating_point U = scalar_value_type_t<T>>

Form< T, U > create_form_factory	(	const ufcx_form &	ufcx_form,
		const std::vector< std::shared_ptr< const FunctionSpace< U > > > &	spaces,
		const std::vector< std::shared_ptr< const Function< T, U > > > &	coefficients,
		const std::vector< std::shared_ptr< const Constant< T > > > &	constants,
		const std::map< IntegralType, std::vector< std::pair< std::int32_t, std::span< const std::int32_t > > > > &	subdomains,
		const std::map< std::shared_ptr< const mesh::Mesh< U > >, std::span< const std::int32_t > > &	entity_maps,
		std::shared_ptr< const mesh::Mesh< U > >	mesh = nullptr )

Create a Form from UFCx input with coefficients and constants passed in the required order.

Use fem::create_form to create a fem::Form with coefficients and constants associated with the name/string.

Parameters

[in]	ufcx_form	The UFCx form.
[in]	spaces	Vector of function spaces. The number of spaces is equal to the rank of the form.
[in]	coefficients	Coefficient fields in the form.
[in]	constants	Spatial constants in the form.
[in]	subdomains	Subdomain markers.
[in]	entity_maps	The entity maps for the form. Empty for single domain problems.
[in]	mesh	The mesh of the domain.

Precondition

Each value in subdomains must be sorted by domain id.

create_functionspace()

template<std::floating_point T>

FunctionSpace< T > create_functionspace	(	std::shared_ptr< mesh::Mesh< T > >	mesh,
		const basix::FiniteElement< T > &	e,
		const std::vector< std::size_t > &	value_shape = {},
		std::function< std::vector< int >(const graph::AdjacencyList< std::int32_t > &)>	reorder_fn = nullptr )

Create a function space from a Basix element.

Parameters

[in]	mesh	Mesh
[in]	e	Basix finite element.
[in]	value_shape	Value shape for 'blocked' elements, e.g. vector-valued Lagrange elements where each component for the vector field is a Lagrange element. For example, a vector-valued element in 3D will have value_shape equal to {3}, and for a second-order tensor element in 2D value_shape equal to {2, 2}.
[in]	reorder_fn	The graph reordering function to call on the dofmap. If nullptr, the default re-ordering is used.

Returns

The created function space

create_interpolation_data()

template<std::floating_point T>

geometry::PointOwnershipData< T > create_interpolation_data	(	const mesh::Geometry< T > &	geometry0,
		const FiniteElement< T > &	element0,
		const mesh::Mesh< T > &	mesh1,
		std::span< const std::int32_t >	cells,
		T	padding )

Generate data needed to interpolate finite element Functions across different meshes.

Parameters

[in]	geometry0	Mesh geometry of the space to interpolate into
[in]	element0	Element of the space to interpolate into
[in]	mesh1	Mesh of the function to interpolate from
[in]	cells	Indices of the cells in the destination mesh on which to interpolate. Should be the same as the list used when calling interpolation_coords.
[in]	padding	Absolute padding of bounding boxes of all entities on mesh1. This is used avoid floating point issues when an interpolation point from mesh0 is on the surface of a cell in mesh1. This parameter can also be used for extrapolation, i.e. if cells in mesh0 is not overlapped by mesh1.

Note

Setting the padding to a large value will increase the runtime of this function, as one has to determine what entity is closest if there is no intersection.

create_sparsity_pattern()

template<dolfinx::scalar T, std::floating_point U>

la::SparsityPattern 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

[in]

a

A bilinear form

Returns

The corresponding sparsity pattern

discrete_gradient()

template<dolfinx::scalar T, std::floating_point U = dolfinx::scalar_value_type_t<T>>

void discrete_gradient	(	mesh::Topology &	topology,
		std::pair< std::reference_wrapper< const FiniteElement< U > >, std::reference_wrapper< const DofMap > >	V0,
		std::pair< std::reference_wrapper< const FiniteElement< U > >, std::reference_wrapper< const DofMap > >	V1,
		auto &&	mat_set )

Assemble a discrete gradient operator.

The discrete gradient operator \(A\) interpolates the gradient of a Lagrange finite element function in \(V_0 \subset H^1\) into a Nédélec (first kind) space \(V_1 \subset H({\rm curl})\), i.e. \(\nabla V_0 \rightarrow V_1\). If \(u_0\) is the degree-of-freedom vector associated with \(V_0\), the hen \(u_1=Au_0\) where \(u_1\) is the degrees-of-freedom vector for interpolating function in the \(H({\rm curl})\) space. An example of where discrete gradient operators are used is the creation of algebraic multigrid solvers for \(H({\rm curl})\) and \(H({\rm div})\) problems.

Note

The sparsity pattern for a discrete operator can be initialised using sparsitybuild::cells. The space V1 should be used for the rows of the sparsity pattern, V0 for the columns.

Warning

This function relies on the user supplying appropriate input and output spaces. See parameter descriptions.

Parameters

[in]	topology	Mesh topology
[in]	V0	Lagrange element and dofmap for corresponding space to interpolate the gradient from
[in]	V1	Nédélec (first kind) element and and dofmap for corresponding space to interpolate into
[in]	mat_set	A functor that sets values in a matrix

extract_function_spaces()

template<dolfinx::scalar T, std::floating_point U>

std::vector< std::vector< std::array< std::shared_ptr< const FunctionSpace< U > >, 2 > > > extract_function_spaces

(

const std::vector< std::vector< const Form< T, U > * > > &

a

)

Extract test (0) and trial (1) function spaces pairs for each bilinear form for a rectangular array of forms.

Parameters

[in]

a

A rectangular block on bilinear forms

Returns

Rectangular array of the same shape as a with a pair of function spaces in each array entry. If a form is null, then the returned function space pair is (null, null).

get_coefficient_names()

std::vector< std::string > get_coefficient_names

(

const ufcx_form &

ufcx_form

)

Get the name of each coefficient in a UFC form

Parameters

[in]

ufcx_form

The UFC form

Returns

The name of each coefficient

get_constant_names()

std::vector< std::string > get_constant_names

(

const ufcx_form &

ufcx_form

)

Get the name of each constant in a UFC form.

Parameters

[in]

ufcx_form

The UFC form

Returns

The name of each constant

interpolate() [1/3]

template<dolfinx::scalar T, std::floating_point U>

void interpolate	(	Function< T, U > &	u,
		const Function< T, U > &	v,
		std::span< const std::int32_t >	cells,
		const geometry::PointOwnershipData< U > &	interpolation_data )

Interpolate a finite element Function defined on a mesh to a finite element Function defined on different (non-matching) mesh.

Template Parameters

T	Function scalar type.
U	mesh::Mesh geometry scalar type.

Parameters

u	Function to interpolate into.
v	Function to interpolate from.
cells	Cells indices relative to the mesh associated with u that will be interpolated into.
interpolation_data	Data required for associating the interpolation points of u with cells in v. This is computed by fem::create_interpolation_data.

interpolate() [2/3]

template<dolfinx::scalar T, std::floating_point U>

void interpolate	(	Function< T, U > &	u,
		std::span< const T >	f,
		std::array< std::size_t, 2 >	fshape,
		std::span< const std::int32_t >	cells )

Interpolate an evaluated expression f(x) in a finite element space.

Template Parameters

T	Scalar type
U	Mesh geometry type

Parameters

[out]	u	Function object to interpolate into
[in]	f	Evaluation of the function f(x) at the physical points x given by interpolation_coords. The element used in interpolation_coords should be the same element as associated with u. The shape of f is (value_size, num_points), with row-major storage.
[in]	fshape	Shape of f.
[in]	cells	Indices of the cells in the mesh on which to interpolate. Should be the same as the list of cells used when calling interpolation_coords.

interpolate() [3/3]

template<dolfinx::scalar T, std::floating_point U>

void interpolate	(	Function< T, U > &	u1,
		std::span< const std::int32_t >	cells1,
		const Function< T, U > &	u0,
		std::span< const std::int32_t >	cells0 )

Interpolate from one finite element Function to another Function on the same (sub)mesh.

Interpolation can be performed on a subset of mesh cells and Functions may be defined on 'sub-meshes'.

Parameters

[out]	u1	Function to interpolate into.
[in]	cells1	Cell indices associated with the mesh of u1 that will be interpolated onto.
[in]	u0	Function to b interpolated from.
[in]	cells0	Cell indices associated with the mesh of u0 that will be interpolated from. If cells1[i] is the index of a cell in the mesh associated with u1, then cells0[i] is the index of the same cell but in the mesh associated with u0. cells0 and cells1 must be the same size.

interpolation_coords()

template<std::floating_point T>

std::vector< T > interpolation_coords	(	const fem::FiniteElement< T > &	element,
		const mesh::Geometry< T > &	geometry,
		std::span< const std::int32_t >	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

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

Returns

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

interpolation_matrix()

template<dolfinx::scalar T, std::floating_point U>

void interpolation_matrix	(	const FunctionSpace< U > &	V0,
		const FunctionSpace< U > &	V1,
		auto &&	mat_set )

Assemble an interpolation operator matrix.

The interpolation operator \(A\) interpolates a function in the space \(V_0\) into a space \(V_1\). If \(u_0\) is the degree-of-freedom vector associated with \(V_0\), then the degree-of-freedom vector \(u_1\) for the interpolated function in \(V_1\) is given by \(u_1=Au_0\).

Note

The sparsity pattern for a discrete operator can be initialised using sparsitybuild::cells. The space V1 should be used for the rows of the sparsity pattern, V0 for the columns.

Parameters

[in]	V0	The space to interpolate from
[in]	V1	The space to interpolate to
[in]	mat_set	A functor that sets values in a matrix

locate_dofs_geometrical() [1/2]

template<std::floating_point T, typename U >

std::vector< std::int32_t > locate_dofs_geometrical	(	const FunctionSpace< T > &	V,
		U	marker_fn )

Find degrees of freedom whose geometric coordinate is true for the provided marking function.

Attention

This function is slower than the topological version.

Parameters

[in]	V	The function (sub)space on which degrees of freedom will be located.
[in]	marker_fn	Function marking tabulated degrees of freedom

Returns

Array of DOF index blocks (local to the MPI rank) in the space V. The array uses the block size of the dofmap associated with V.

locate_dofs_geometrical() [2/2]

template<std::floating_point T, typename U >

std::array< std::vector< std::int32_t >, 2 > locate_dofs_geometrical	(	const std::array< std::reference_wrapper< const FunctionSpace< T > >, 2 > &	V,
		U	marker_fn )

Finds degrees of freedom whose geometric coordinate is true for the provided marking function.

Attention

This function is slower than the topological version

Parameters

[in]	V	The function (sub)space(s) on which degrees of freedom will be located. The spaces must share the same mesh and element type.
[in]	marker_fn	Function marking tabulated degrees of freedom

Returns

Array of DOF indices (local to the MPI rank) in the spaces V[0] and V[1]. The array[0](i) entry is the DOF index in the space V[0] and array[1](i) is the corresponding DOF entry in the space V[1]. The returned dofs are 'unrolled', i.e. block size = 1.

locate_dofs_topological() [1/2]

std::vector< std::int32_t > locate_dofs_topological	(	const mesh::Topology &	topology,
		const DofMap &	dofmap,
		int	dim,
		std::span< const std::int32_t >	entities,
		bool	remote = true )

Find degrees-of-freedom which belong to the provided mesh entities (topological).

Note

Degrees-of-freedom for discontinuous elements are associated with the cell even if they may appear to be associated with a facet/edge/vertex.

Parameters

[in]	topology	Mesh topology.
[in]	dofmap	Dofmap that associated DOFs with cells.
[in]	dim	Topological dimension of mesh entities on which degrees-of-freedom will be located
[in]	entities	Indices of mesh entities. All DOFs associated with the closure of these indices will be returned
[in]	remote	True to return also "remotely located" degree-of-freedom indices. Remotely located degree-of-freedom indices are local/owned by the current process, but which the current process cannot identify because it does not recognize mesh entity as a marked. For example, a boundary condition dof at a vertex where this process does not have the associated boundary facet. This commonly occurs with partitioned meshes.

Returns

Array of DOF index blocks (local to the MPI rank) in the space V. The array uses the block size of the dofmap associated with V.

Precondition

The topology cell->entity and entity->cell connectivity must have been computed before calling this function.

locate_dofs_topological() [2/2]

std::array< std::vector< std::int32_t >, 2 > locate_dofs_topological	(	const mesh::Topology &	topology,
		std::array< std::reference_wrapper< const DofMap >, 2 >	dofmaps,
		int	dim,
		std::span< const std::int32_t >	entities,
		bool	remote = true )

Find degrees-of-freedom which belong to the provided mesh entities (topological).

Note

Degrees-of-freedom for discontinuous elements are associated with the cell even if they may appear to be associated with a facet/edge/vertex.

Parameters

[in]	topology	Mesh topology.
[in]	dofmaps	The dofmaps.
[in]	dim	Topological dimension of mesh entities on which degrees-of-freedom will be located
[in]	entities	Indices of mesh entities. All DOFs associated with the closure of these indices will be returned
[in]	remote	True to return also "remotely located" degree-of-freedom indices. Remotely located degree-of-freedom indices are local/owned by the current process, but which the current process cannot identify because it does not recognize mesh entity as a marked. For example, a boundary condition dof at a vertex where this process does not have the associated boundary facet. This commonly occurs with partitioned meshes.

Returns

Array of DOF indices (local to the MPI rank) in the spaces V[0] and V[1]. The array[0](i) entry is the DOF index in the space V[0] and array[1](i) is the corresponding DOF entry in the space V[1]. The returned dofs are 'unrolled', i.e. block size = 1.

Precondition

The topology cell->entity and entity->cell connectivity must have been computed before calling this function.

pack_coefficients() [1/3]

template<dolfinx::scalar T, std::floating_point U>

std::pair< std::vector< T >, int > pack_coefficients	(	const Expression< T, U > &	e,
		std::span< const std::int32_t >	entities,
		std::size_t	estride )

Pack coefficients of a Expression u for a give list of active entities.

Parameters

[in]	e	The Expression
[in]	entities	A list of active entities
[in]	estride	Stride for each entity in active entities (1 for cells, 2 for facets)

Returns

A pair of the form (coeffs, cstride)

pack_coefficients() [2/3]

template<dolfinx::scalar T, std::floating_point U>

void pack_coefficients	(	const Form< T, U > &	form,
		IntegralType	integral_type,
		int	id,
		std::span< T >	c,
		int	cstride )

Pack coefficients of a Form for a given integral type and domain id.

Parameters

[in]	form	The Form
[in]	integral_type	Type of integral
[in]	id	The id of the integration domain
[in,out]	c	The coefficient array
[in]	cstride	The coefficient stride

pack_coefficients() [3/3]

template<dolfinx::scalar T, std::floating_point U>

void pack_coefficients	(	const Form< T, U > &	form,
		std::map< std::pair< IntegralType, int >, std::pair< std::vector< T >, int > > &	coeffs )

Pack coefficients of a Form.

Warning

This is subject to change

Parameters

[in]	form	The Form
[in,out]	coeffs	A map from an (integral_type, domain_id) pair to a (coeffs, cstride) pair. coeffs is a storage container representing an array of shape (num_int_entities, cstride) in which to pack the coefficient data, where num_int_entities is the number of entities being integrated over and cstride is the number of coefficient data entries per integration entity. coeffs is flattened into row-major layout.

pack_constants()

template<typename U >

std::vector< typename U::scalar_type > pack_constants

(

const U &

u

)

Pack constants of u into a single array ready for assembly.

Warning

This function is subject to change.

set_diagonal() [1/2]

template<dolfinx::scalar T, std::floating_point U>

void set_diagonal	(	auto	set_fn,
		const FunctionSpace< U > &	V,
		const std::vector< std::shared_ptr< const DirichletBC< T, U > > > &	bcs,
		T	diagonal = 1.0 )

Sets a value to the diagonal of the matrix for rows with a Dirichlet boundary conditions applied.

This function is typically called after assembly. The assembly function zeroes Dirichlet rows and columns. This function adds the value only to rows that are locally owned, and therefore does not create a need for parallel communication. For block matrices, this function should normally be called only on the diagonal blocks, i.e. blocks for which the test and trial spaces are the same.

Parameters

[in]	set_fn	The function for setting values to a matrix
[in]	V	The function space for the rows and columns of the matrix. It is used to extract only the Dirichlet boundary conditions that are define on V or subspaces of V.
[in]	bcs	The Dirichlet boundary conditions
[in]	diagonal	The value to add to the diagonal for rows with a boundary condition applied

set_diagonal() [2/2]

template<dolfinx::scalar T>

void set_diagonal	(	auto	set_fn,
		std::span< const std::int32_t >	rows,
		T	diagonal = 1.0 )

Sets a value to the diagonal of a matrix for specified rows.

This function is typically called after assembly. The assembly function zeroes Dirichlet rows and columns. For block matrices, this function should normally be called only on the diagonal blocks, i.e. blocks for which the test and trial spaces are the same.

Parameters

[in]	set_fn	The function for setting values to a matrix
[in]	rows	The row blocks, in local indices, for which to add a value to the diagonal
[in]	diagonal	The value to add to the diagonal for the specified rows

transpose_dofmap()

graph::AdjacencyList< std::int32_t > transpose_dofmap	(	MDSPAN_IMPL_STANDARD_NAMESPACE::mdspan< const std::int32_t, MDSPAN_IMPL_STANDARD_NAMESPACE::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 3
Global 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 5
Unassembled index: [ [0, 11], [10], [4, 8, 9], [1, 3, 7], [5, 6], [2] ]

Parameters

[in]	dofmap	The standard dof map that for each cell (node) gives the global (process-wise) index of each local (cell-wise) index.
[in]	num_cells	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.