Apparently working BDMCF/E elements (#18)

bdmc element

---------

Co-authored-by: cyruscycheng21 <[email protected]>
Co-authored-by: Rob Kirby <[email protected]>
Co-authored-by: Patrick Farrell <[email protected]>
Co-authored-by: Tom Bendall <[email protected]>
Co-authored-by: Tom Bendall <[email protected]>

FIAT/__init__.py

@@ -20,6 +20,7 @@
 from FIAT.discontinuous_taylor import DiscontinuousTaylor
 from FIAT.discontinuous_raviart_thomas import DiscontinuousRaviartThomas
 from FIAT.serendipity import Serendipity
+from FIAT.brezzi_douglas_marini_cube import BrezziDouglasMariniCubeEdge, BrezziDouglasMariniCubeFace
 from FIAT.discontinuous_pc import DPC
 from FIAT.hermite import CubicHermite
 from FIAT.lagrange import Lagrange
@@ -69,6 +70,8 @@
                       "Crouzeix-Raviart": CrouzeixRaviart,
                       "Discontinuous Lagrange": DiscontinuousLagrange,
                       "S": Serendipity,
+                      "Brezzi-Douglas-Marini Cube Face": BrezziDouglasMariniCubeFace,
+                      "Brezzi-Douglas-Marini Cube Edge": BrezziDouglasMariniCubeEdge,
                       "DPC": DPC,
                       "Discontinuous Taylor": DiscontinuousTaylor,
                       "Discontinuous Raviart-Thomas": DiscontinuousRaviartThomas,

FIAT/brezzi_douglas_marini_cube.py

@@ -0,0 +1,346 @@
+# Copyright (C) 2019 Cyrus Cheng (Imperial College London)
+#
+# This file is part of FIAT.
+#
+# FIAT is free software: you can redistribute it and/or modify
+# it under the terms of the GNU Lesser General Public License as published by
+# the Free Software Foundation, either version 3 of the License, or
+# (at your option) any later version.
+#
+# FIAT is distributed in the hope that it will be useful,
+# but WITHOUT ANY WARRANTY; without even the implied warranty of
+# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+# GNU Lesser General Public License for more details.
+#
+# You should have received a copy of the GNU Lesser General Public License
+# along with FIAT. If not, see <http://www.gnu.org/licenses/>.
+#
+# Modified by David A. Ham ([email protected]), 2019
+# Modified by Thomas Bendall ([email protected]) 2021
+
+from sympy import symbols, legendre, Array, diff, binomial, lambdify
+import numpy as np
+from FIAT.dual_set import make_entity_closure_ids
+from FIAT.finite_element import FiniteElement
+from FIAT.polynomial_set import mis
+from FIAT.reference_element import compute_unflattening_map, flatten_reference_cube
+
+x, y = symbols('x y')
+variables = (x, y)
+leg = legendre
+
+
+def triangular_number(n):
+    return int((n+1)*n/2)
+
+
+class BrezziDouglasMariniCube(FiniteElement):
+    """
+    The Brezzi-Douglas-Marini element on quadrilateral cells.
+
+    :arg ref_el: The reference element.
+    :arg k: The degree.
+    :arg mapping: A string giving the Piola mapping.
+                  Either 'contravariant Piola' or 'covariant Piola'.
+    """
+
+    def __init__(self, ref_el, degree, mapping):
+
+        # Check that ref_el and degree are appropriate
+        if degree < 1:
+            raise Exception("BDMc_k elements only valid for k >= 1")
+
+        flat_el = flatten_reference_cube(ref_el)
+        dim = flat_el.get_spatial_dimension()
+        if dim != 2:
+            raise Exception("BDMc_k elements only valid for dimension 2")
+
+        # Collect the IDs of the reference element entities
+        flat_topology = flat_el.get_topology()
+
+        entity_ids = {}
+        counter = 0
+
+        for top_dim, entities in flat_topology.items():
+            entity_ids[top_dim] = {}
+            for entity in entities:
+                entity_ids[top_dim][entity] = []
+
+        for j in sorted(flat_topology[1]):
+            entity_ids[1][j] = list(range(counter, counter + degree + 1))
+            counter += degree + 1
+
+        entity_ids[2][0] = list(range(counter, counter + 2*triangular_number(degree - 1)))
+        counter += 2*triangular_number(degree - 1)
+
+        entity_closure_ids = make_entity_closure_ids(flat_el, entity_ids)
+
+        # Set up FiniteElement
+        super(BrezziDouglasMariniCube, self).__init__(ref_el=ref_el, dual=None,
+                                                      order=degree, formdegree=1,
+                                                      mapping=mapping)
+
+        # Store unflattened entity ID dictionaries
+        topology = ref_el.get_topology()
+        unflattening_map = compute_unflattening_map(topology)
+        unflattened_entity_ids = {}
+        unflattened_entity_closure_ids = {}
+
+        for dim, entities in sorted(topology.items()):
+            unflattened_entity_ids[dim] = {}
+            unflattened_entity_closure_ids[dim] = {}
+        for dim, entities in sorted(flat_topology.items()):
+            for entity in entities:
+                unflat_dim, unflat_entity = unflattening_map[(dim, entity)]
+                unflattened_entity_ids[unflat_dim][unflat_entity] = entity_ids[dim][entity]
+                unflattened_entity_closure_ids[unflat_dim][unflat_entity] = entity_closure_ids[dim][entity]
+        self.entity_ids = unflattened_entity_ids
+        self.entity_closure_ids = unflattened_entity_closure_ids
+        self._degree = degree
+        self.flat_el = flat_el
+
+    def degree(self):
+        """Return the degree of the polynomial space."""
+        return self._degree
+
+    def get_nodal_basis(self):
+        raise NotImplementedError("get_nodal_basis not implemented for BDMCE/F elements")
+
+    def get_dual_set(self):
+        raise NotImplementedError("get_dual_set is not implemented for BDMCE/F elements")
+
+    def get_coeffs(self):
+        raise NotImplementedError("get_coeffs not implemented for BDMCE/F elements")
+
+    def tabulate(self, order, points, entity=None):
+        """Return tabulated values of derivatives up to a given order of
+        basis functions at given points.
+
+        :arg order: The maximum order of derivative.
+        :arg points: An iterable of points.
+        :arg entity: Optional (dimension, entity number) pair
+                     indicating which topological entity of the
+                     reference element to tabulate on.  If ``None``,
+                     tabulated values are computed by geometrically
+                     approximating which facet the points are on.
+        """
+        if entity is None:
+            entity = (self.ref_el.get_dimension(), 0)
+
+        entity_dim, entity_id = entity
+        transform = self.ref_el.get_entity_transform(entity_dim, entity_id)
+        points = np.asarray(list(map(transform, points)))
+        npoints, pointdim = points.shape
+
+        # Turn analytic basis functions into python functions via lambdify
+        basis_callable = {(0, 0): numpy_lambdify(variables, self.basis[(0, 0)],
+                                                 modules="numpy")}
+
+        # Dictionary of values of functions
+        phivals = {}
+        for o in range(order+1):
+            alphas = mis(2, o)
+            # Collect basis and derivatives of basis
+            for alpha in alphas:
+                try:
+                    callable = basis_callable[alpha]
+                except KeyError:
+                    diff_basis = diff(self.basis[(0, 0)], *zip(variables, alpha))
+                    callable = numpy_lambdify(variables, diff_basis, modules="numpy")
+                    self.basis[alpha] = diff_basis
+
+                # tabulate by passing points through all lambdified functions
+                # resulting array has shape (len(self.basis), spatial_dim, npoints)
+                T = np.array([[[func_component(point) for point in points]
+                               for func_component in func] for func in callable])
+
+                phivals[alpha] = T
+
+        return phivals
+
+    def entity_dofs(self):
+        """Return the map of topological entities to degrees of
+        freedom for the finite element."""
+        return self.entity_ids
+
+    def entity_closure_dofs(self):
+        """Return the map of topological entities to degrees of
+        freedom on the closure of those entities for the finite element."""
+        return self.entity_closure_ids
+
+    def value_shape(self):
+        """Return the value shape of the finite element functions."""
+        return np.shape(self.basis[(0, 0)][0])
+
+    def dmats(self):
+        raise NotImplementedError
+
+    def get_num_members(self, arg):
+        raise NotImplementedError
+
+    def space_dimension(self):
+        """Return the dimension of the finite element space."""
+        return int(len(self.basis[(0, 0)])/2)
+
+
+def bdmce_edge_basis(deg, dx, dy, x_mid, y_mid):
+    """Returns the basis functions associated with DoFs on the
+    edges of elements for the HCurl Brezz-Douglas-Marini element on
+    quadrilateral cells.
+
+    These were introduced by Brezzi, Douglas, Marini (1985)
+    "Two families of mixed finite elements for Second Order Elliptic Problems"
+
+    Following, e.g. Brezzi, Douglas, Fortin, Marini (1987)
+    "Efficient rectangular mixed finite elements in two and three space variables"
+    For rectangle K and degree j:
+    BDM_j(K) = [P_j(K)^2 + Span(curl(xy^{j+1}, x^{j+1}y))] x P_{j-1}(K)
+
+    The resulting basis functions all have a curl whose polynomials are
+    of degree (j - 1).
+
+    :arg deg: The element degree.
+    :arg dx: A tuple of sympy expressions, expanding the interval in the
+             x direction. Probably (1-x, x).
+    :arg dy: A tuple of sympy expressions, expanding the interval in the
+             y direction. Probably (1-y, y).
+    :arg x_mid: A sympy expression, probably 2*x-1.
+    :arg y_mid: A sympy expression, probably 2*y-1.
+    """
+
+    # For some functions, we need to multiply by a coefficient to ensure that
+    # the result curl is of the correct degree. The coefficient of the
+    # highest-order term of leg(deg, 2x-1), is binomial(2*deg, deg)
+    coeff = binomial(2*deg, deg) / ((deg+1)*binomial(2*deg-2, deg-1))
+
+    basis = tuple([(0, -leg(j, y_mid)*dx[0]) for j in range(deg)] +
+                  [(coeff*-leg(deg-1, y_mid)*dy[0]*dy[1], -leg(deg, y_mid)*dx[0])] +
+                  [(0, -leg(j, y_mid)*dx[1]) for j in range(deg)] +
+                  [(coeff*leg(deg-1, y_mid)*dy[0]*dy[1], -leg(deg, y_mid)*dx[1])] +
+                  [(-leg(j, x_mid)*dy[0], 0) for j in range(deg)] +
+                  [(-leg(deg, x_mid)*dy[0], coeff*-leg(deg-1, x_mid)*dx[0]*dx[1])] +
+                  [(-leg(j, x_mid)*dy[1], 0) for j in range(deg)] +
+                  [(-leg(deg, x_mid)*dy[1], coeff*leg(deg-1, x_mid)*dx[0]*dx[1])])
+
+    return basis
+
+
+def bdmce_face_basis(deg, dx, dy, x_mid, y_mid):
+    """Returns the basis functions associated with DoFs on the
+    faces of elements for the HCurl Brezz-Douglas-Marini element on
+    quadrilateral cells.
+
+    These were introduced by Brezzi, Douglas, Marini (1985)
+    "Two families of mixed finite elements for Second Order Elliptic Problems"
+
+    Following, e.g. Brezzi, Douglas, Fortin, Marini (1987)
+    "Efficient rectangular mixed finite elements in two and three space variables"
+    For rectangle K and degree j:
+    BDM_j(K) = [P_j(K)^2 + Span(curl(xy^{j+1}, x^{j+1}y))] x P_{j-1}(K)
+
+    The resulting basis functions all have a curl whose polynomials are
+    of degree (j - 1).
+
+    :arg deg: The element degree.
+    :arg dx: A tuple of sympy expressions, expanding the interval in the
+             x direction. Probably (1-x, x).
+    :arg dy: A tuple of sympy expressions, expanding the interval in the
+             y direction. Probably (1-y, y).
+    :arg x_mid: A sympy expression, probably 2*x-1.
+    :arg y_mid: A sympy expression, probably 2*y-1.
+    """
+
+    basis = []
+    for k in range(2, deg+1):
+        for j in range(k-1):
+            basis += [(0, leg(j, x_mid)*leg(k-2-j, y_mid)*dx[0]*dx[1])]
+            basis += [(leg(k-2-j, x_mid)*leg(j, y_mid)*dy[0]*dy[1], 0)]
+
+    return tuple(basis)
+
+
+class BrezziDouglasMariniCubeEdge(BrezziDouglasMariniCube):
+    """
+    The Brezzi-Douglas-Marini HCurl element on quadrilateral cells.
+
+    :arg ref_el: The reference element.
+    :arg k: The degree.
+    """
+
+    def __init__(self, ref_el, degree):
+
+        bdmce_list = construct_bdmce_basis(ref_el, degree)
+        self.basis = {(0, 0): Array(bdmce_list)}
+
+        super(BrezziDouglasMariniCubeEdge, self).__init__(ref_el=ref_el, degree=degree,
+                                                          mapping="covariant piola")
+
+
+class BrezziDouglasMariniCubeFace(BrezziDouglasMariniCube):
+    """
+    The Brezzi-Douglas-Marini HDiv element on quadrilateral cells.
+
+    :arg ref_el: The reference element.
+    :arg k: The degree.
+    """
+
+    def __init__(self, ref_el, degree):
+
+        bdmce_list = construct_bdmce_basis(ref_el, degree)
+
+        # BDMCF functions are rotations of BDMCE functions
+        bdmcf_list = [[-a[1], a[0]] for a in bdmce_list]
+        self.basis = {(0, 0): Array(bdmcf_list)}
+
+        super(BrezziDouglasMariniCubeFace, self).__init__(ref_el=ref_el, degree=degree,
+                                                          mapping="contravariant piola")
+
+
+def construct_bdmce_basis(ref_el, degree):
+    """
+    Return the basis functions for a particular BDMCE space as a list.
+
+    :arg ref_el: The reference element.
+    :arg k: The degree.
+    """
+
+    # Extract the vertices from the reference element
+    flat_el = flatten_reference_cube(ref_el)
+    verts = flat_el.get_vertices()
+
+    # dx on reference quad is (1-x, x)
+    dx = ((verts[-1][0] - x) / (verts[-1][0] - verts[0][0]),
+          (x - verts[0][0]) / (verts[-1][0] - verts[0][0]))
+    # dy on reference quad is (1-y, y)
+    dy = ((verts[-1][1] - y) / (verts[-1][1] - verts[0][1]),
+          (y - verts[0][1]) / (verts[-1][1] - verts[0][1]))
+
+    # x_mid and y_mid are (2x-1) and (2y-1)
+    x_mid = 2*x-(verts[-1][0] + verts[0][0])
+    y_mid = 2*y-(verts[-1][1] + verts[0][1])
+
+    # Compute basis functions for BDMcE
+    edge_basis = bdmce_edge_basis(degree, dx, dy, x_mid, y_mid)
+    face_basis = bdmce_face_basis(degree, dx, dy, x_mid, y_mid) if degree > 1 else ()
+
+    return edge_basis + face_basis
+
+
+def numpy_lambdify(X, F, modules="numpy", dummify=False):
+    '''Unfortunately, SymPy's own lambdify() doesn't work well with
+    NumPy in that simple functions like
+        lambda x: 1.0,
+    when evaluated with NumPy arrays, return just "1.0" instead of
+    an array of 1s with the same shape as x. This function does that.
+    '''
+    try:
+        lambda_x = [numpy_lambdify(X, f, modules=modules, dummify=dummify) for f in F]
+    except TypeError:  # 'function' object is not iterable
+        # SymPy's lambdify also works on functions that return arrays.
+        # However, use it componentwise here so we can add 0*x to each
+        # component individually. This is necessary to maintain shapes
+        # if evaluated with NumPy arrays.
+        lmbd_tmp = lambdify(X, F, modules=modules, dummify=dummify)
+        lambda_x = lambda u: lmbd_tmp(*[v for v in u]) + 0 * u[0]
+
+    return lambda_x