Merge pull request #64 from firedrakeproject/rckirby/feature/macro

Implement C0/C1 macroelements

FIAT/P0.py

@@ -17,34 +17,28 @@
 class P0Dual(dual_set.DualSet):
     def __init__(self, ref_el):
         entity_ids = {}
-        nodes = []
         entity_permutations = {}
-        vs = numpy.array(ref_el.get_vertices())
-        if ref_el.get_dimension() == 0:
-            bary = ()
-        else:
-            bary = tuple(numpy.average(vs, 0))
-
-        nodes = [functional.PointEvaluation(ref_el, bary)]
-        entity_ids = {}
+        sd = ref_el.get_dimension()
         top = ref_el.get_topology()
+        if sd == 0:
+            pts = [tuple() for entity in sorted(top[sd])]
+        else:
+            pts = [tuple(numpy.average(ref_el.get_vertices_of_subcomplex(top[sd][entity]), 0))
+                   for entity in sorted(top[sd])]
+        nodes = [functional.PointEvaluation(ref_el, pt) for pt in pts]
         for dim in sorted(top):
             entity_ids[dim] = {}
             entity_permutations[dim] = {}
             sym_size = ref_el.symmetry_group_size(dim)
+            num_points = 1 if dim == sd else 0
             if isinstance(dim, tuple):
                 assert isinstance(sym_size, tuple)
-                perms = {o: [] for o in numpy.ndindex(sym_size)}
+                perms = {o: list(range(num_points)) for o in numpy.ndindex(sym_size)}
             else:
-                perms = {o: [] for o in range(sym_size)}
+                perms = {o: list(range(num_points)) for o in range(sym_size)}
             for entity in sorted(top[dim]):
-                entity_ids[dim][entity] = []
+                entity_ids[dim][entity] = [entity] if dim == sd else []
                 entity_permutations[dim][entity] = perms
-        entity_ids[dim] = {0: [0]}
-        if isinstance(dim, tuple):
-            entity_permutations[dim][0] = {o: [0] for o in numpy.ndindex(sym_size)}
-        else:
-            entity_permutations[dim][0] = {o: [0] for o in range(sym_size)}
 
         super(P0Dual, self).__init__(nodes, ref_el, entity_ids, entity_permutations)
 

FIAT/__init__.py

@@ -7,6 +7,7 @@
 # Import finite element classes
 from FIAT.finite_element import FiniteElement, CiarletElement  # noqa: F401
 from FIAT.argyris import Argyris
+from FIAT.hct import HsiehCloughTocher
 from FIAT.bernstein import Bernstein
 from FIAT.bell import Bell
 from FIAT.argyris import QuinticArgyris
@@ -30,6 +31,7 @@
 from FIAT.morley import Morley
 from FIAT.nedelec import Nedelec
 from FIAT.nedelec_second_kind import NedelecSecondKind
+from FIAT.hierarchical import Legendre, IntegratedLegendre
 from FIAT.P0 import P0
 from FIAT.raviart_thomas import RaviartThomas
 from FIAT.crouzeix_raviart import CrouzeixRaviart
@@ -48,7 +50,6 @@
 from FIAT.restricted import RestrictedElement             # noqa: F401
 from FIAT.quadrature_element import QuadratureElement     # noqa: F401
 from FIAT.kong_mulder_veldhuizen import KongMulderVeldhuizen  # noqa: F401
-from FIAT.hierarchical import Legendre, IntegratedLegendre  # noqa: F401
 from FIAT.fdm_element import FDMLagrange, FDMDiscontinuousLagrange, FDMQuadrature, FDMBrokenH1, FDMBrokenL2, FDMHermite  # noqa: F401
 
 # Important functionality
@@ -61,6 +62,7 @@
 
 # List of supported elements and mapping to element classes
 supported_elements = {"Argyris": Argyris,
+                      "HsiehCloughTocher": HsiehCloughTocher,
                       "Bell": Bell,
                       "Bernstein": Bernstein,
                       "Brezzi-Douglas-Marini": BrezziDouglasMarini,
@@ -81,6 +83,8 @@
                       "Gauss-Lobatto-Legendre": GaussLobattoLegendre,
                       "Gauss-Legendre": GaussLegendre,
                       "Gauss-Radau": GaussRadau,
+                      "Legendre": Legendre,
+                      "Integrated Legendre": IntegratedLegendre,
                       "Morley": Morley,
                       "Nedelec 1st kind H(curl)": Nedelec,
                       "Nedelec 2nd kind H(curl)": NedelecSecondKind,

FIAT/barycentric_interpolation.py

@@ -11,6 +11,40 @@
 from FIAT.functional import index_iterator
 
 
+def get_lagrange_points(nodes):
+    """Extract singleton point for each node."""
+    points = []
+    for node in nodes:
+        pt, = node.get_point_dict()
+        points.append(pt)
+    return points
+
+
+def barycentric_interpolation(nodes, wts, dmat, pts, order=0):
+    """Evaluates a Lagrange basis on a line reference element
+    via the second barycentric interpolation formula. See Berrut and Trefethen (2004)
+    https://doi.org/10.1137/S0036144502417715 Eq. (4.2) & (9.4)
+    """
+    if pts.dtype == object:
+        from sympy import simplify
+        sp_simplify = numpy.vectorize(simplify)
+    else:
+        sp_simplify = lambda x: x
+    phi = numpy.add.outer(-nodes, pts.flatten())
+    with numpy.errstate(divide='ignore', invalid='ignore'):
+        numpy.reciprocal(phi, out=phi)
+        numpy.multiply(phi, wts[:, None], out=phi)
+        numpy.multiply(1.0 / numpy.sum(phi, axis=0), phi, out=phi)
+    phi[phi != phi] = 1.0
+
+    phi = sp_simplify(phi)
+    results = {(0,): phi}
+    for r in range(1, order+1):
+        phi = sp_simplify(numpy.dot(dmat, phi))
+        results[(r,)] = phi
+    return results
+
+
 def make_dmat(x):
     """Returns Lagrange differentiation matrix and barycentric weights
     associated with x[j]."""
@@ -24,83 +58,68 @@ def make_dmat(x):
 
 
 class LagrangeLineExpansionSet(expansions.LineExpansionSet):
-    """Evaluates a Lagrange basis on a line reference element
-    via the second barycentric interpolation formula. See Berrut and Trefethen (2004)
-    https://doi.org/10.1137/S0036144502417715 Eq. (4.2) & (9.4)
-    """
+    """Lagrange polynomial expansion set for given points the line."""
     def __init__(self, ref_el, pts):
         self.points = pts
-        self.x = numpy.array(pts).flatten()
-        self.dmat, self.weights = make_dmat(self.x)
+        self.x = numpy.array(pts, dtype="d").flatten()
+        self.cell_node_map = expansions.compute_cell_point_map(ref_el, pts, unique=False)
+        self.dmats = []
+        self.weights = []
+        self.nodes = []
+        for ibfs in self.cell_node_map:
+            nodes = self.x[ibfs]
+            dmat, wts = make_dmat(nodes)
+            self.dmats.append(dmat)
+            self.weights.append(wts)
+            self.nodes.append(nodes)
+
+        self.degree = max(len(wts) for wts in self.weights)-1
+        self.recurrence_order = self.degree + 1
         super(LagrangeLineExpansionSet, self).__init__(ref_el)
 
     def get_num_members(self, n):
         return len(self.points)
 
+    def get_cell_node_map(self, n):
+        return self.cell_node_map
+
     def get_points(self):
         return self.points
 
-    def get_dmats(self, degree):
-        return [self.dmat.T]
-
-    def tabulate(self, n, pts):
-        assert n == len(self.points)-1
-        results = numpy.add.outer(-self.x, numpy.array(pts).flatten())
-        with numpy.errstate(divide='ignore', invalid='ignore'):
-            numpy.reciprocal(results, out=results)
-            numpy.multiply(results, self.weights[:, None], out=results)
-            numpy.multiply(1.0 / numpy.sum(results, axis=0), results, out=results)
-
-        results[results != results] = 1.0
-        if results.dtype == object:
-            from sympy import simplify
-            results = numpy.vectorize(simplify)(results)
-        return results
-
-    def _tabulate(self, n, pts, order=0):
-        vals = self.tabulate(n, pts)
-        results = [vals]
-        for r in range(order):
-            vals = numpy.dot(self.dmat, vals)
-            if vals.dtype == object:
-                from sympy import simplify
-                vals = numpy.vectorize(simplify)(vals)
-            results.append(vals)
-        for r in range(order+1):
-            shape = results[r].shape
-            shape = shape[:1] + (1,)*r + shape[1:]
-            results[r] = numpy.reshape(results[r], shape)
-        return results
+    def get_dmats(self, degree, cell=0):
+        return [self.dmats[cell].T]
+
+    def _tabulate_on_cell(self, n, pts, order=0, cell=0, direction=None):
+        return barycentric_interpolation(self.nodes[cell], self.weights[cell], self.dmats[cell], pts, order=order)
 
 
 class LagrangePolynomialSet(polynomial_set.PolynomialSet):
 
     def __init__(self, ref_el, pts, shape=tuple()):
-        degree = len(pts) - 1
+        if ref_el.get_shape() != reference_element.LINE:
+            raise ValueError("Invalid reference element type.")
+
+        expansion_set = LagrangeLineExpansionSet(ref_el, pts)
+        degree = expansion_set.degree
         if shape == tuple():
             num_components = 1
         else:
             flat_shape = numpy.ravel(shape)
             num_components = numpy.prod(flat_shape)
-        num_exp_functions = expansions.polynomial_dimension(ref_el, degree)
+        num_exp_functions = expansion_set.get_num_members(degree)
         num_members = num_components * num_exp_functions
         embedded_degree = degree
-        if ref_el.get_shape() == reference_element.LINE:
-            expansion_set = LagrangeLineExpansionSet(ref_el, pts)
-        else:
-            raise ValueError("Invalid reference element type.")
 
         # set up coefficients
         if shape == tuple():
-            coeffs = numpy.eye(num_members)
+            coeffs = numpy.eye(num_members, dtype="d")
         else:
             coeffs_shape = (num_members, *shape, num_exp_functions)
             coeffs = numpy.zeros(coeffs_shape, "d")
             # use functional's index_iterator function
             cur_bf = 0
             for idx in index_iterator(shape):
-                n = expansions.polynomial_dimension(ref_el, embedded_degree)
-                for exp_bf in range(n):
+                for exp_bf in range(num_exp_functions):
                     cur_idx = (cur_bf, *idx, exp_bf)
                     coeffs[cur_idx] = 1.0
                     cur_bf += 1

FIAT/check_format_variant.py

@@ -1,5 +1,22 @@
 import re
 
+from FIAT.macro import AlfeldSplit, IsoSplit
+
+# dicts mapping Lagrange variant names to recursivenodes family names
+supported_cg_variants = {
+    "spectral": "gll",
+    "chebyshev": "lgc",
+    "equispaced": "equispaced",
+    "gll": "gll"}
+
+supported_dg_variants = {
+    "spectral": "gl",
+    "chebyshev": "gc",
+    "equispaced": "equispaced",
+    "equispaced_interior": "equispaced_interior",
+    "gll": "gll",
+    "gl": "gl"}
+
 
 def check_format_variant(variant, degree):
     if variant is None:
@@ -20,3 +37,51 @@ def check_format_variant(variant, degree):
                          'or variant="integral(q)"')
 
     return variant, interpolant_degree
+
+
+def parse_lagrange_variant(variant, discontinuous=False, integral=False):
+    """Parses variant options for Lagrange elements.
+
+    variant may be a single option or comma-separated pair
+    indicating the dof type (integral, equispaced, spectral, etc)
+    and the type of splitting to give a macro-element (Alfeld, iso)
+    """
+    if variant is None:
+        variant = "integral" if integral else "equispaced"
+    options = variant.replace(" ", "").split(",")
+    assert len(options) <= 2
+
+    default = "integral" if integral else "spectral"
+    if integral:
+        supported_point_variants = {"integral": None}
+    elif discontinuous:
+        supported_point_variants = supported_dg_variants
+    else:
+        supported_point_variants = supported_cg_variants
+
+    # defaults
+    splitting = None
+    splitting_args = tuple()
+    point_variant = supported_point_variants[default]
+
+    for pre_opt in options:
+        opt = pre_opt.lower()
+        if opt == "alfeld":
+            splitting = AlfeldSplit
+        elif opt == "iso":
+            splitting = IsoSplit
+        elif opt.startswith("iso"):
+            match = re.match(r"^iso(?:\((\d+)\))?$", opt)
+            k, = match.groups()
+            call_split = IsoSplit
+            splitting_args = (int(k),)
+        elif opt in supported_point_variants:
+            point_variant = supported_point_variants[opt]
+        else:
+            raise ValueError("Illegal variant option")
+
+    if discontinuous and splitting is not None and point_variant in supported_cg_variants.values():
+        raise ValueError("Illegal variant. DG macroelements with DOFs on subcell boundaries are not unisolvent.")
+    if len(splitting_args) > 0:
+        splitting = lambda T: call_split(T, *splitting_args, point_variant or "gll")
+    return splitting, point_variant