# Copyright (C) 2023-2024 Matthew Scroggs # # This file is part of Basix (https://www.fenicsproject.org) # # SPDX-License-Identifier: MIT """Functions for creating finite elements.""" import typing from warnings import warn import numpy as np import numpy.typing as npt from basix._basixcpp import DPCVariant, ElementFamily, LagrangeVariant from basix._basixcpp import FiniteElement_float32 as _FiniteElement_float32 from basix._basixcpp import FiniteElement_float64 as _FiniteElement_float64 from basix._basixcpp import ( create_custom_element_float32 as _create_custom_element_float32, ) from basix._basixcpp import ( create_custom_element_float64 as _create_custom_element_float64, ) from basix._basixcpp import create_element as _create_element from basix._basixcpp import create_tp_element as _create_tp_element from basix._basixcpp import tp_dof_ordering as _tp_dof_ordering from basix._basixcpp import lex_dof_ordering as _lex_dof_ordering from basix._basixcpp import tp_factors as _tp_factors from basix.cell import CellType, geometry, topology, facet_outward_normals from basix import MapType from basix.polynomials import PolysetType from basix.sobolev_spaces import SobolevSpace __all__ = [ "FiniteElement", "create_element", "create_custom_element", "create_tp_element", "lex_dof_ordering", "string_to_family", "tp_factors", "tp_dof_ordering", ] class FiniteElement: """Finite element class.""" _e: typing.Union[_FiniteElement_float32, _FiniteElement_float64] def __init__(self, e: typing.Union[_FiniteElement_float32, _FiniteElement_float64]): """Initialise a finite element wrapper. Note: This initialiser is intended for internal library use. """ self._e = e def tabulate(self, n: int, x: npt.NDArray) -> npt.ArrayLike: """Compute basis values and derivatives at set of points. Note: The version of ``FiniteElement::tabulate`` with the basis data as an out argument should be preferred for repeated call where performance is critical. Args: n: The order of derivatives, up to and including, to compute. Use 0 for the basis functions only. x: The points at which to compute the basis functions. The shape of x is (number of points, geometric dimension). Returns: The basis functions (and derivatives). The shape is ``(derivative, point, basis fn index, value index)``. * The first index is the derivative, with higher derivatives are stored in triangular (2D) or tetrahedral (3D) ordering, i.e. for the ``(x,y)`` derivatives in 2D: ``(0,0), (1,0), (0,1), (2,0), (1,1), (0,2), (3,0)...``. The function basix::indexing::idx can be used to find the appropriate derivative. * The second index is the point index * The fourth index is the basis function component. Its has size * The third index is the basis function index one for scalar basis functions. """ return self._e.tabulate(n, x) def __eq__(self, other) -> bool: """Test element for equality.""" try: return self._e == other._e except TypeError: return False def hash(self) -> int: """Hash.""" return self._e.hash() def __hash__(self) -> int: """Hash.""" return self.hash() def push_forward(self, U, J, detJ, K) -> npt.ArrayLike: """Map function values from the reference to a physical cell. This function can perform the mapping for multiple points, grouped by points that share a common Jacobian. Args: U: The function values on the reference cell. The indices are ``(Jacobian index, point index, components)``. J: The Jacobian of the mapping. The indices are ``(Jacobian index, J_i, J_j)``. detJ: The determinant of the Jacobian of the mapping. It has length ``J.shape(0)``. K: The inverse of the Jacobian of the mapping. The indices are ``(Jacobian index, K_i, K_j)``. Returns: The function values on the cell. The indices are ``(Jacobian index, point index, components)``. """ return self._e.push_forward(U, J, detJ, K) def pull_back( self, u: npt.NDArray, J: npt.NDArray, detJ: npt.NDArray, K: npt.NDArray ) -> npt.ArrayLike: """Map function values from a physical cell to the reference. Args: u: The function values on the cell. J: The Jacobian of the mapping. detJ: The determinant of the Jacobian of the mapping. K: The inverse of the Jacobian of the mapping. Returns: The function values on the reference. The indices are ``(Jacobian index, point index, components``). """ return self._e.pull_back(u, J, detJ, K) def T_apply(self, data, block_size, cell_info) -> None: """Apply DOF transformations to some data in-place. Note: This function is designed to be called at runtime, so its performance is critical. Args: data: The data block_size: The number of data points per DOF cell_info: The permutation info for the cell """ self._e.T_apply(data, block_size, cell_info) def Tt_apply_right(self, data, block_size, cell_info) -> None: """Post-apply DOF transformations to some transposed data in-place. Note: This function is designed to be called at runtime, so its performance is critical. Args: data: The data. block_size: The number of data points per DOF. cell_info: The permutation info for the cell. """ self._e.Tt_apply_right(data, block_size, cell_info) def Tt_inv_apply(self, data, block_size, cell_info) -> None: """Pre-apply inverse transpose DOF transformations to some data. Note: This function is designed to be called at runtime, so its performance is critical. Args: data: The data. block_size: The number of data points per DOF. cell_info: The permutation info for the cell. """ self._e.Tt_inv_apply(data, block_size, cell_info) def base_transformations(self) -> npt.ArrayLike: r"""Get the base transformations. The base transformations represent the effect of rotating or reflecting a subentity of the cell on the numbering and orientation of the DOFs. This returns a list of matrices with one matrix for each subentity permutation in the following order: Reversing edge 0, reversing edge 1, ... Rotate face 0, reflect face 0, rotate face 1, reflect face 1, ... *Example: Order 3 Lagrange on a triangle* This space has 10 dofs arranged like: .. code-block:: 2 |\ 6 4 | \ 5 9 3 | \ 0-7-8-1 For this element, the base transformations are: [Matrix swapping 3 and 4, Matrix swapping 5 and 6, Matrix swapping 7 and 8] The first row shows the effect of reversing the diagonal edge. The second row shows the effect of reversing the vertical edge. The third row shows the effect of reversing the horizontal edge. *Example: Order 1 Raviart-Thomas on a triangle* This space has 3 dofs arranged like: .. code-block:: |\ | \ | \ <-1 0 | / \ | L ^ \ | | \ ---2--- These DOFs are integrals of normal components over the edges: DOFs 0 and 2 are oriented inward, DOF 1 is oriented outwards. For this element, the base transformation matrices are: .. code-block:: 0: [[-1, 0, 0], [ 0, 1, 0], [ 0, 0, 1]] 1: [[1, 0, 0], [0, -1, 0], [0, 0, 1]] 2: [[1, 0, 0], [0, 1, 0], [0, 0, -1]] The first matrix reverses DOF 0 (as this is on the first edge). The second matrix reverses DOF 1 (as this is on the second edge). The third matrix reverses DOF 2 (as this is on the third edge). *Example: DOFs on the face of Order 2 Nedelec first kind on a tetrahedron* On a face of this tetrahedron, this space has two face tangent DOFs: .. code-block:: |\ |\ | \ | \ | \ | ^\ | \ | | \ | 0->\ | 1 \ | \ | \ ------ ------ For these DOFs, the subblocks of the base transformation matrices are: .. code-block:: rotation: [[-1, 1], [ 1, 0]] reflection: [[0, 1], [1, 0]] Returns: The base transformations for this element. The shape is ``(ntranformations, ndofs, ndofs)``. """ return self._e.base_transformations() def entity_transformations(self) -> dict: """Entity dof transformation matrices. Returns: The base transformations for this element. The shape is ``(ntranformations, ndofs, ndofs)``. """ return self._e.entity_transformations() def get_tensor_product_representation(self) -> list[list["FiniteElement"]]: """Get the tensor product representation of this element. Raises an exception if no such factorisation exists. The tensor product representation will be a vector of tuples. Each tuple contains a vector of finite elements, and a vector of integers. The vector of finite elements gives the elements on an interval that appear in the tensor product representation. The vector of integers gives the permutation between the numbering of the tensor product DOFs and the number of the DOFs of this Basix element. Returns: The tensor product representation """ factors = self._e.get_tensor_product_representation() return [[FiniteElement(e) for e in elements] for elements in factors] def permute_subentity_closure( self, indices: npt.NDArray, cell_or_entity_info: int, entity_type: CellType, entity_index: typing.Optional[int] = None, ) -> npt.NDArray: """Permute DOF indices on the closure of a sub-entity. Args: indices: The indices to permute cell_or_entity_info: Bit packed entity info (if entity_index is None) or cell info (if entity_index is not None) entity_type: The cell type of the entity entity_index: The index of the entity """ if entity_index is None: return np.array( self._e.permute_subentity_closure(indices, cell_or_entity_info, entity_type) ) else: return np.array( self._e.permute_subentity_closure( indices, cell_or_entity_info, entity_type, entity_index ) ) def permute_subentity_closure_inv( self, indices: npt.NDArray, cell_or_entity_info: int, entity_type: CellType, entity_index: typing.Optional[int] = None, ) -> npt.NDArray: """Apply inverse permutation to DOF indices on the closure of a sub-entity. Args: indices: The indices to permute cell_or_entity_info: Bit packed entity info (if entity_index is None) or cell info (if entity_index is not None) entity_type: The cell type of the entity entity_index: The index of the entity """ if entity_index is None: return np.array( self._e.permute_subentity_closure_inv(indices, cell_or_entity_info, entity_type) ) else: return np.array( self._e.permute_subentity_closure_inv( indices, cell_or_entity_info, entity_type, entity_index ) ) @property def degree(self) -> int: """Element polynomial degree.""" return self._e.degree @property def embedded_superdegree(self) -> int: """Embedded polynomial degree. Lowest degree ``n`` such that the highest degree polynomial in this element is contained in a Lagrange (or vector Lagrange) element of degree ``n``. """ return self._e.embedded_superdegree @property def embedded_subdegree(self) -> int: """Embedded polynomial sub-degree. Highest degree ``n`` such that a Lagrange (or vector Lagrange) element of degree ``n`` is a subspace of this element. """ return self._e.embedded_subdegree @property def cell_type(self) -> CellType: """Element cell type.""" return self._e.cell_type @property def polyset_type(self) -> PolysetType: """Element polyset type.""" return getattr(PolysetType, self._e.polyset_type.name) @property def dim(self) -> int: """Dimension of the finite element space. This is the number of degrees-of-freedom for the element. """ return self._e.dim @property def num_entity_dofs(self) -> list[list[int]]: """Number of entity dofs. Warning: This property may be removed. """ return self._e.num_entity_dofs @property def entity_dofs(self) -> list[list[list[int]]]: """Dofs on each topological entity. Data is order ``(vertices, edges, faces, cell)``. For example, Lagrange degree 2 on a triangle has vertices: ``[[0], [1], [2]]``, edges: ``[[3], [4], [5]]``, cell: ``[[]]``. """ return self._e.entity_dofs @property def num_entity_closure_dofs(self) -> list[list[int]]: """Number of entity closure dofs. Warning: This property may be removed. """ return self._e.num_entity_closure_dofs @property def entity_closure_dofs(self) -> list[list[list[int]]]: """Get the dofs on the closure of each topological entity. Data is in the order ``(vertices, edges, faces, cell)``. For example, Lagrange degree 2 on a triangle has vertices: ``[[0], [1], [2]]``, edges: ``[[1, 2, 3], [0, 2, 4], [0, 1, 5]]``, cell: ``[[0, 1, 2, 3, 4, 5]]``. """ return self._e.entity_closure_dofs @property def value_size(self) -> int: """Value size.""" return self._e.value_size @property def value_shape(self) -> list[int]: """Element value tensor shape. E.g., returning ``(,)`` for scalars, ``(3,)`` for vectors in 3D, ``(2, 2)`` for a rank-2 tensor in 2D. """ return self._e.value_shape @property def discontinuous(self) -> bool: """True is element is the discontinuous variant.""" return self._e.discontinuous @property def family(self) -> ElementFamily: """Finite element family.""" return self._e.family @property def lagrange_variant(self) -> LagrangeVariant: """Lagrange variant of the element.""" return self._e.lagrange_variant @property def dpc_variant(self) -> DPCVariant: """DPC variant of the element.""" return self._e.dpc_variant @property def dof_transformations_are_permutations(self) -> bool: """True if the dof transformations are all permutations.""" return self._e.dof_transformations_are_permutations @property def dof_transformations_are_identity(self) -> bool: """True if DOF transformations are all the identity.""" return self._e.dof_transformations_are_identity @property def interpolation_is_identity(self) -> bool: """True if interpolation matrix for this element is the identity.""" return self._e.interpolation_is_identity @property def map_type(self) -> MapType: """Map type for this element.""" return self._e.map_type @property def sobolev_space(self) -> SobolevSpace: """Underlying Sobolev space for this element.""" return self._e.sobolev_space @property def points(self) -> npt.ArrayLike: """Interpolation points. Coordinates on the reference element where a function need to be evaluated in order to interpolate it in the finite element space. Shape is ``(num_points, tdim)``. """ return self._e.points @property def interpolation_matrix(self) -> npt.ArrayLike: """Interpolation points. Coordinates on the reference element where a function need to be evaluated in order to interpolate it in the finite element space. """ return self._e.interpolation_matrix @property def dual_matrix(self) -> npt.ArrayLike: """Matrix $BD^{T}$. See C++ documentation. """ return self._e.dual_matrix @property def coefficient_matrix(self) -> npt.ArrayLike: """Matrix of coefficients.""" return self._e.coefficient_matrix @property def wcoeffs(self) -> npt.ArrayLike: """Coefficients that define the polynomial set in terms of the orthonormal polynomials. See C++ documentation for details. """ return self._e.wcoeffs @property def M(self) -> list[list[npt.NDArray]]: """Interpolation matrices for each sub-entity. See C++ documentation for details. """ return self._e.M @property def x(self) -> list[list[npt.NDArray]]: """Interpolation points for each sub-entity. The indices of this data are ``(tdim, entity index, point index, dim)``. """ return self._e.x @property def has_tensor_product_factorisation(self) -> bool: """True if element has tensor-product structure.""" return self._e.has_tensor_product_factorisation @property def interpolation_nderivs(self) -> int: """Number of derivatives needed when interpolating.""" return self._e.interpolation_nderivs @property def dof_ordering(self) -> list[int]: """DOF layout.""" return self._e.dof_ordering @property def dtype(self) -> npt.DTypeLike: """Element float type.""" return np.dtype(self._e.dtype) def create_element( family: ElementFamily, celltype: CellType, degree: int, lagrange_variant: LagrangeVariant = LagrangeVariant.unset, dpc_variant: DPCVariant = DPCVariant.unset, discontinuous: bool = False, dof_ordering: typing.Optional[list[int]] = None, dtype: npt.DTypeLike = np.float64, ) -> FiniteElement: """Create a finite element. Args: family: Finite element family. celltype: Reference cell type that the element is defined on. degree: Polynomial degree of the element. lagrange_variant: Lagrange variant type. dpc_variant: DPC variant type. discontinuous: If `True` element is discontinuous. The discontinuous element will have the same DOFs as a continuous element, but the DOFs will all be associated with the interior of the cell. dof_ordering: Ordering of dofs for ``ElementDofLayout``. dtype: Element scalar type. Returns: A finite element. """ e = _create_element( family, celltype, degree, lagrange_variant, dpc_variant, discontinuous, dof_ordering if dof_ordering is not None else [], np.dtype(dtype).char, ) return FiniteElement(e) def create_custom_element( cell_type: CellType, value_shape: tuple[int, ...], wcoeffs: npt.NDArray[np.floating], x: list[list[npt.NDArray[np.floating]]], M: list[list[npt.NDArray[np.floating]]], interpolation_nderivs: int, map_type: MapType, sobolev_space: SobolevSpace, discontinuous: bool, embedded_subdegree: int, embedded_superdegree: int, poly_type: PolysetType, dtype: npt.DTypeLike = np.float64, ) -> FiniteElement: """Create a custom finite element. Args: cell_type: Element cell type. value_shape: Value shape of the element. wcoeffs: Matrices for the k-th value index containing the expansion coefficients defining a polynomial basis spanning the polynomial space for this element. Shape is ``(dim(finite element polyset), dim(Legendre polynomials))``. x: Interpolation points. Indices are ``(tdim, entity index, point index, dim)``. M: Interpolation matrices. Indices are ``(tdim, entity index, dof, vs, point_index, derivative)``. interpolation_nderivs: Number of derivatives that need to be used during interpolation. map_type: Type of map to be used to map values from the reference to a physical cell. sobolev_space: Underlying Sobolev space for the element. discontinuous: If ``True`` create the discontinuous version of the element. embedded_subdegree: Highest degree n such that a Lagrange (or vector Lagrange) element of degree n is a subspace of this element. embedded_superdegree: Degree of a polynomial in this element's polyset. poly_type: Type of polyset to use for this element. dtype: Element scalar type. Returns: A custom finite element. """ if len(x) != len(M): raise ValueError("x and M must have the same length") # Allow (eg) only three lists to be included when creating a 2D cell while len(x) < 4: x.append([]) M.append([]) if wcoeffs.dtype != dtype: wcoeffs = np.dtype(dtype).type(wcoeffs) # type: ignore x = [[np.dtype(dtype).type(j) for j in i] for i in x] # type: ignore M = [[np.dtype(dtype).type(j) for j in i] for i in M] # type: ignore # Check shape of x tdim = len(topology(cell_type)) - 1 for i in x: for j in i: if j.shape[1] != tdim: raise RuntimeError("x has a point with the wrong tdim") if len(j.shape) != 2: raise ValueError("x has the wrong dimension") # Warn if points are not inside the cell geo = geometry(cell_type) top = topology(cell_type) for points_i in x: for points_j in points_i: for p in points_j: for facet, facet_normal in zip(top[tdim - 1], facet_outward_normals(cell_type)): if np.dot(p - geo[facet[0]], facet_normal) > 0.001: warn(f"Point {p} is not in cell", UserWarning) if np.issubdtype(dtype, np.float32): _create_custom_element = _create_custom_element_float32 # type: ignore elif np.issubdtype(dtype, np.float64): _create_custom_element = _create_custom_element_float64 # type: ignore else: raise NotImplementedError(f"Type {dtype} not supported.") return FiniteElement( _create_custom_element( cell_type, value_shape, wcoeffs, x, M, interpolation_nderivs, map_type, sobolev_space, discontinuous, embedded_subdegree, embedded_superdegree, poly_type, ) ) def create_tp_element( family: ElementFamily, celltype: CellType, degree: int, lagrange_variant: LagrangeVariant = LagrangeVariant.unset, dpc_variant: DPCVariant = DPCVariant.unset, discontinuous: bool = False, dtype: npt.DTypeLike = np.float64, ) -> FiniteElement: """Create a finite element with tensor product ordering. Args: family: Finite element family. celltype: Reference cell type that the element is defined on degree: Polynomial degree of the element. lagrange_variant: Lagrange variant type. dpc_variant: DPC variant type. discontinuous: If `True` element is discontinuous. The discontinuous element will have the same DOFs as a continuous element, but the DOFs will all be associated with the interior of the cell. dtype: Element scalar type. Returns: A finite element. """ return FiniteElement( _create_tp_element( family, celltype, degree, lagrange_variant, dpc_variant, discontinuous, np.dtype(dtype).char, ) ) def tp_factors( family: ElementFamily, celltype: CellType, degree: int, lagrange_variant: LagrangeVariant = LagrangeVariant.unset, dpc_variant: DPCVariant = DPCVariant.unset, discontinuous: bool = False, dof_ordering: typing.Optional[list[int]] = None, dtype: npt.DTypeLike = np.float64, ) -> list[list[FiniteElement]]: """Elements in the tensor product factorisation of an element. If the element has no factorisation, raises a RuntimeError. Args: family: Finite element family. celltype: Reference cell type that the element is defined on degree: Polynomial degree of the element. lagrange_variant: Lagrange variant type. dpc_variant: DPC variant type. discontinuous: If `True` element is discontinuous. The discontinuous element will have the same DOFs as a continuous element, but the DOFs will all be associated with the interior of the cell. dof_ordering: Ordering of dofs for ElementDofLayout dtype: Element scalar type. Returns: A list of finite elements. """ return [ [FiniteElement(e) for e in elements] for elements in _tp_factors( family, celltype, degree, lagrange_variant, dpc_variant, discontinuous, dof_ordering if dof_ordering is not None else [], np.dtype(dtype).char, ) ] def tp_dof_ordering( family: ElementFamily, celltype: CellType, degree: int, lagrange_variant: LagrangeVariant = LagrangeVariant.unset, dpc_variant: DPCVariant = DPCVariant.unset, discontinuous: bool = False, ) -> list[int]: """Tensor product DOF ordering for an element. This DOF ordering can be passed into create_element to create the element with DOFs ordered in a tensor product order. If the element has no factorisation, raises a RuntimeError. Args: family: Finite element family. celltype: Reference cell type that the element is defined on degree: Polynomial degree of the element. lagrange_variant: Lagrange variant type. dpc_variant: DPC variant type. discontinuous: If `True` element is discontinuous. The discontinuous element will have the same DOFs as a continuous element, but the DOFs will all be associated with the interior of the cell. Returns: The DOF ordering. """ return _tp_dof_ordering( family, celltype, degree, lagrange_variant, dpc_variant, discontinuous, ) def lex_dof_ordering( family: ElementFamily, celltype: CellType, degree: int, lagrange_variant: LagrangeVariant = LagrangeVariant.unset, dpc_variant: DPCVariant = DPCVariant.unset, discontinuous: bool = False, ) -> list[int]: """Lexicographic DOF ordering for an element. This DOF ordering can be passed into create_element to create the element with DOFs ordered in a lexicographic order. If the element contains DOFs that are not point evaluations, raises a RuntimeError. Args: family: Finite element family. celltype: Reference cell type that the element is defined on degree: Polynomial degree of the element. lagrange_variant: Lagrange variant type. dpc_variant: DPC variant type. discontinuous: If `True` element is discontinuous. The discontinuous element will have the same DOFs as a continuous element, but the DOFs will all be associated with the interior of the cell. Returns: The DOF ordering. """ return _lex_dof_ordering( family, celltype, degree, lagrange_variant, dpc_variant, discontinuous, ) def string_to_family(family: str, cell: str) -> ElementFamily: """Basix ElementFamily enum representing the family type on the given cell. Args: family: Element family as a string. cell: Cell type as a string. Returns: Element family. """ # Family names that are valid for all cells families = { "Lagrange": ElementFamily.P, "P": ElementFamily.P, "Bubble": ElementFamily.bubble, "bubble": ElementFamily.bubble, "iso": ElementFamily.iso, } # Family names that are valid on non-interval cells if cell != "interval": families.update( { "RT": ElementFamily.RT, "Raviart-Thomas": ElementFamily.RT, "N1F": ElementFamily.RT, "N1div": ElementFamily.RT, "Nedelec 1st kind H(div)": ElementFamily.RT, "N1E": ElementFamily.N1E, "N1curl": ElementFamily.N1E, "Nedelec 1st kind H(curl)": ElementFamily.N1E, "BDM": ElementFamily.BDM, "Brezzi-Douglas-Marini": ElementFamily.BDM, "N2F": ElementFamily.BDM, "N2div": ElementFamily.BDM, "Nedelec 2nd kind H(div)": ElementFamily.BDM, "N2E": ElementFamily.N2E, "N2curl": ElementFamily.N2E, "Nedelec 2nd kind H(curl)": ElementFamily.N2E, } ) # Family names that are valid for intervals if cell == "interval": families.update( { "DPC": ElementFamily.P, } ) # Family names that are valid for tensor product cells if cell in ["interval", "quadrilateral", "hexahedron"]: families.update( { "Q": ElementFamily.P, "Serendipity": ElementFamily.serendipity, "serendipity": ElementFamily.serendipity, "S": ElementFamily.serendipity, } ) # Family names that are valid for quads and hexes if cell in ["quadrilateral", "hexahedron"]: families.update( { "RTCF": ElementFamily.RT, "DPC": ElementFamily.DPC, "NCF": ElementFamily.RT, "RTCE": ElementFamily.N1E, "NCE": ElementFamily.N1E, "BDMCF": ElementFamily.BDM, "BDMCE": ElementFamily.N2E, "AAF": ElementFamily.BDM, "AAE": ElementFamily.N2E, } ) # Family names that are valid for triangles and tetrahedra if cell in ["triangle", "tetrahedron"]: families.update( { "Regge": ElementFamily.Regge, "CR": ElementFamily.CR, "Crouzeix-Raviart": ElementFamily.CR, "HHJ": ElementFamily.HHJ, "Hellan-Herrmann-Johnson": ElementFamily.HHJ, } ) try: return families[family] except KeyError: raise ValueError(f"Unknown element family: {family} with cell type {cell}")