`symfem.finite_element`¶

Abstract finite element classes and functions.

Module Contents¶

Classes¶

`FiniteElement`	Abstract finite element.
`CiarletElement`	Finite element defined using the Ciarlet definition.
`DirectElement`	Finite element defined directly.
`EnrichedElement`	Finite element defined directly.
`ElementBasisFunction`	A basis function of a finite element.

Attributes¶

TabulatedBasis

symfem.finite_element.TabulatedBasis¶

exception symfem.finite_element.NoTensorProduct¶

Bases: Exception

Error for element without a tensor representation.

class symfem.finite_element.FiniteElement(reference: symfem.references.Reference, order: int, space_dim: int, domain_dim: int, range_dim: int, range_shape: Tuple[int, Ellipsis] | None = None)¶

Bases: abc.ABC

Abstract finite element.

abstract property maximum_degree: int¶: Get the maximum degree of this polynomial set for the element.

property name: str¶

Get the name of the element.

Returns:: The name of the element’s family

_float_basis_functions: None | List[symfem.functions.AnyFunction]¶

_value_scale: None | sympy.core.expr.Expr¶

names: List[str] = []¶

references: List[str] = []¶

last_updated¶

cache = True¶

_max_continuity_test_order = 4¶

abstract dof_plot_positions() → List[symfem.geometry.PointType]¶

Get the points to plot each DOF at on a DOF diagram.

Returns:: The DOF positions

abstract dof_directions() → List[symfem.geometry.PointType | None]¶

Get the direction associated with each DOF.

Returns:: The DOF directions

abstract dof_entities() → List[Tuple[int, int]]¶

Get the entities that each DOF is associated with.

Returns:: The entities

plot_dof_diagram(filename: str | List[str], plot_options: Dict[str, Any] = {}, **kwargs: Any)¶

Plot a diagram showing the DOFs of the element.

Parameters:

filename – The file name
plot_options – Options for the plot
kwargs – Keyword arguments

abstract entity_dofs(entity_dim: int, entity_number: int) → List[int]¶

Get the numbers of the DOFs associated with the given entity.

Parameters:

entity_dim – The dimension of the entity
entity_number – The number of the entity

Returns:

The numbers of the DOFs associated with the entity

abstract get_basis_functions(use_tensor_factorisation: bool = False) → List[symfem.functions.AnyFunction]¶

Get the basis functions of the element.

Parameters:: use_tensor_factorisation – Should a tensor factorisation be used?
Returns:: The basis functions

get_basis_function(n: int) → symfem.basis_functions.BasisFunction¶

Get a single basis function of the element.

Parameters:: n – The number of the basis function
Returns:: The basis function

tabulate_basis(points_in: symfem.geometry.SetOfPointsInput, order: str = 'xyzxyz') → TabulatedBasis¶

Evaluate the basis functions of the element at the given points.

Parameters:

points_in – The points
order – The order to return the values

Returns:

The tabulated basis functions

tabulate_basis_float(points_in: symfem.geometry.SetOfPointsInput) → TabulatedBasis¶

Evaluate the basis functions of the element at the given points in xyz,xyz order.

Parameters:: points_in – The points
Returns:: The tabulated basis functions

plot_basis_function(n: int, filename: str | List[str], cell: symfem.references.Reference | None = None, **kwargs: Any)¶

Plot a diagram showing a basis function.

Parameters:

n – The basis function number
filename – The file name
cell – The cell to push the basis function to and plot on
kwargs – Keyword arguments

abstract map_to_cell(vertices_in: symfem.geometry.SetOfPointsInput, basis: List[symfem.functions.AnyFunction] | None = None, forward_map: symfem.geometry.PointType | None = None, inverse_map: symfem.geometry.PointType | None = None) → List[symfem.functions.AnyFunction]¶

Map the basis onto a cell using the appropriate mapping for the element.

Parameters:

vertices_in – The vertices of the cell
basis – The basis functions
forward_map – The map from the reference to the cell
inverse_map – The map to the reference from the cell

Returns:

The basis functions mapped to the cell

abstract get_polynomial_basis() → List[symfem.functions.AnyFunction]¶

Get the symbolic polynomial basis for the element.

Returns:: The polynomial basis

test()¶: Run tests for this element.

test_continuity()¶: Test that this element has the correct continuity.

get_tensor_factorisation() → List[Tuple[str, List[FiniteElement], List[int]]]¶

Get the representation of the element as a tensor product.

Returns:: The tensor factorisation

_get_basis_functions_tensor() → List[symfem.functions.AnyFunction]¶

Compute the basis functions using the space’s tensor product factorisation.

Returns:: The basis functions

init_kwargs() → Dict[str, Any]¶

Return the keyword arguments used to create this element.

Returns:: Keyword arguments dictionary

class symfem.finite_element.CiarletElement(reference: symfem.references.Reference, order: int, basis: List[symfem.functions.FunctionInput], dofs: symfem.functionals.ListOfFunctionals, domain_dim: int, range_dim: int, range_shape: Tuple[int, Ellipsis] | None = None)¶

Bases: FiniteElement

Finite element defined using the Ciarlet definition.

property maximum_degree: int¶: Get the maximum degree of this polynomial set for the element.

entity_dofs(entity_dim: int, entity_number: int) → List[int]¶

Get the numbers of the DOFs associated with the given entity.

Parameters:

entity_dim – The dimension of the entity
entity_number – The number of the entity

Returns:

The numbers of the DOFs associated with the entity

dof_plot_positions() → List[symfem.geometry.PointType]¶

Get the points to plot each DOF at on a DOF diagram.

Returns:: The DOF positions

dof_directions() → List[symfem.geometry.PointType | None]¶

Get the direction associated with each DOF.

Returns:: The DOF directions

dof_entities() → List[Tuple[int, int]]¶

Get the entities that each DOF is associated with.

Returns:: The entities

get_polynomial_basis() → List[symfem.functions.AnyFunction]¶

Get the symbolic polynomial basis for the element.

Returns:: The polynomial basis

get_dual_matrix(inverse=False, caching=True) → sympy.matrices.dense.MutableDenseMatrix¶

Get the dual matrix.

Parameters:

inverse – Should the dual matrix be inverted?
caching – Should the result be cached

Returns:

The dual matrix

get_basis_functions(use_tensor_factorisation: bool = False) → List[symfem.functions.AnyFunction]¶

Get the basis functions of the element.

Parameters:: use_tensor_factorisation – Should a tensor factorisation be used?
Returns:: The basis functions

plot_basis_function(n: int, filename: str | List[str], cell: symfem.references.Reference | None = None, **kwargs: Any)¶

Plot a diagram showing a basis function.

Parameters:

n – The basis function number
filename – The file name
cell – The cell to push the basis function to and plot on
kwargs – Keyword arguments

map_to_cell(vertices_in: symfem.geometry.SetOfPointsInput, basis: List[symfem.functions.AnyFunction] | None = None, forward_map: symfem.geometry.PointType | None = None, inverse_map: symfem.geometry.PointType | None = None) → List[symfem.functions.AnyFunction]¶

Map the basis onto a cell using the appropriate mapping for the element.

Parameters:

vertices_in – The vertices of the cell
basis – The basis functions
forward_map – The map from the reference to the cell
inverse_map – The map to the reference from the cell

Returns:

The basis functions mapped to the cell

test()¶: Run tests for this element.

test_dof_points()¶: Test that DOF points are valid.

test_functional_entities()¶: Test that the dof entities are valid and match the references of integrals.

test_functionals()¶: Test that the functionals are satisfied by the basis functions.

class symfem.finite_element.DirectElement(reference: symfem.references.Reference, order: int, basis_functions: List[symfem.functions.FunctionInput], basis_entities: List[Tuple[int, int]], domain_dim: int, range_dim: int, range_shape: Tuple[int, Ellipsis] | None = None)¶

Bases: FiniteElement

Finite element defined directly.

abstract property maximum_degree: int¶: Get the maximum degree of this polynomial set for the element.

_basis_functions: List[symfem.functions.AnyFunction]¶

entity_dofs(entity_dim: int, entity_number: int) → List[int]¶

Get the numbers of the DOFs associated with the given entity.

Parameters:

entity_dim – The dimension of the entity
entity_number – The number of the entity

Returns:

The numbers of the DOFs associated with the entity

dof_plot_positions() → List[symfem.geometry.PointType]¶

Get the points to plot each DOF at on a DOF diagram.

Returns:: The DOF positions

dof_directions() → List[symfem.geometry.PointType | None]¶

Get the direction associated with each DOF.

Returns:: The DOF directions

dof_entities() → List[Tuple[int, int]]¶

Get the entities that each DOF is associated with.

Returns:: The entities

get_basis_functions(use_tensor_factorisation: bool = False) → List[symfem.functions.AnyFunction]¶

Get the basis functions of the element.

Parameters:: use_tensor_factorisation – Should a tensor factorisation be used?
Returns:: The basis functions

map_to_cell(vertices_in: symfem.geometry.SetOfPointsInput, basis: List[symfem.functions.AnyFunction] | None = None, forward_map: symfem.geometry.PointType | None = None, inverse_map: symfem.geometry.PointType | None = None) → List[symfem.functions.AnyFunction]¶

Map the basis onto a cell using the appropriate mapping for the element.

Parameters:

vertices_in – The vertices of the cell
basis – The basis functions
forward_map – The map from the reference to the cell
inverse_map – The map to the reference from the cell

Returns:

The basis functions mapped to the cell

abstract get_polynomial_basis() → List[symfem.functions.AnyFunction]¶

Get the symbolic polynomial basis for the element.

Returns:: The polynomial basis

test()¶: Run tests for this element.

test_independence()¶: Test that the basis functions of this element are linearly independent.

class symfem.finite_element.EnrichedElement(subelements: List[FiniteElement])¶

Bases: FiniteElement

Finite element defined directly.

abstract property maximum_degree: int¶: Get the maximum degree of this polynomial set for the element.

_basis_functions: List[symfem.functions.AnyFunction] | None¶

entity_dofs(entity_dim: int, entity_number: int) → List[int]¶

Get the numbers of the DOFs associated with the given entity.

Parameters:

entity_dim – The dimension of the entity
entity_number – The number of the entity

Returns:

The numbers of the DOFs associated with the entity

dof_plot_positions() → List[symfem.geometry.PointType]¶

Get the points to plot each DOF at on a DOF diagram.

Returns:: The DOF positions

dof_directions() → List[symfem.geometry.PointType | None]¶

Get the direction associated with each DOF.

Returns:: The DOF directions

dof_entities() → List[Tuple[int, int]]¶

Get the entities that each DOF is associated with.

Returns:: The entities

get_basis_functions(use_tensor_factorisation: bool = False) → List[symfem.functions.AnyFunction]¶

Get the basis functions of the element.

Parameters:: use_tensor_factorisation – Should a tensor factorisation be used?
Returns:: The basis functions

map_to_cell(vertices_in: symfem.geometry.SetOfPointsInput, basis: List[symfem.functions.AnyFunction] | None = None, forward_map: symfem.geometry.PointType | None = None, inverse_map: symfem.geometry.PointType | None = None) → List[symfem.functions.AnyFunction]¶

Map the basis onto a cell using the appropriate mapping for the element.

Parameters:

vertices_in – The vertices of the cell
basis – The basis functions
forward_map – The map from the reference to the cell
inverse_map – The map to the reference from the cell

Returns:

The basis functions mapped to the cell

abstract get_polynomial_basis() → List[symfem.functions.AnyFunction]¶

Get the symbolic polynomial basis for the element.

Returns:: The polynomial basis

test()¶: Run tests for this element.

class symfem.finite_element.ElementBasisFunction(element: FiniteElement, n: int)¶

Bases: symfem.basis_functions.BasisFunction

A basis function of a finite element.

get_function() → symfem.functions.AnyFunction¶

Get the actual basis function.

Returns:: The basis function

`symfem.finite_element`¶

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