Source code for sfepy.discrete.fem.poly_spaces

from __future__ import absolute_import
import numpy as nm
import numpy.linalg as nla

from sfepy.base.base import find_subclasses, assert_, Struct
from sfepy.linalg import combine, insert_strided_axis
from six.moves import range
from functools import reduce

# Requires fixed vertex numbering!
vertex_maps = {3 : [[0, 0, 0],
                    [1, 0, 0],
                    [1, 1, 0],
                    [0, 1, 0],
                    [0, 0, 1],
                    [1, 0, 1],
                    [1, 1, 1],
                    [0, 1, 1]],
               2 : [[0, 0],
                    [1, 0],
                    [1, 1],
                    [0, 1]],
               1 : [[0],
                    [1]]}

[docs]def transform_basis(transform, bf): """ Transform a basis `bf` using `transform` array of matrices. """ if bf.ndim == 3: nbf = nm.einsum('cij,qdj->cqdi', transform, bf, order='C') elif bf.ndim == 4: if bf.shape[0] == 1: nbf = nm.einsum('cij,qdj->cqdi', transform, bf[0], order='C') else: nbf = nm.einsum('cij,cqdj->cqdi', transform, bf, order='C') # Note: the 2nd derivatives are not supported here. # Workaround for NumPy 1.14.0 - order is ignored(?) nbf = nm.ascontiguousarray(nbf) return nbf
[docs]class LagrangeNodes(Struct): """Helper class for defining nodes of Lagrange elements."""
[docs] @staticmethod def append_edges(nodes, nts, iseq, nt, edges, order): delta = 1.0 / float(order) for ii, edge in enumerate(edges): n1 = nodes[edge[0],:].copy() n2 = nodes[edge[1],:].copy() for ie in range(order - 1): c2 = ie + 1 c1 = order - c2 nts[iseq] = [nt, ii] aux = [int(round(tmp)) for tmp in delta * (c1 * n1 + c2 * n2)] nodes[iseq,:] = aux iseq += 1 return iseq
[docs] @staticmethod def append_faces(nodes, nts, iseq, nt, faces, order): delta = 1.0 / float(order) for ii, face in enumerate(faces): n1 = nodes[face[0],:].copy() n2 = nodes[face[1],:].copy() n3 = nodes[face[2],:].copy() for i1 in range(order - 2): for i2 in range(order - 2 - i1): c3 = i1 + 1 c2 = i2 + 1 c1 = order - c3 - c2 nts[iseq] = [nt, ii] aux = [int(round(tmp)) for tmp in delta * (c1 * n1 + c2 * n2 + c3 * n3)] nodes[iseq,:] = aux iseq += 1 return iseq
[docs] @staticmethod def append_bubbles(nodes, nts, iseq, nt, order): delta = 1.0 / float(order) n1 = nodes[0,:].copy() n2 = nodes[1,:].copy() n3 = nodes[2,:].copy() n4 = nodes[3,:].copy() for i1 in range(order - 3): for i2 in range(order - 3): for i3 in range(order - 3 - i1 - i2): c4 = i1 + 1 c3 = i2 + 1 c2 = i3 + 1 c1 = order - c4 - c3 - c2 nts[iseq] = [nt, 0] aux = [int(round(tmp)) for tmp in delta * (c1 * n1 + c2 * n2 + c3 * n3 + c4 * n4)] nodes[iseq,:] = aux iseq += 1 return iseq
[docs] @staticmethod def append_tp_edges(nodes, nts, iseq, nt, edges, ao): delta = 1.0 / float(ao) for ii, edge in enumerate(edges): n1 = nodes[edge[0],:].copy() n2 = nodes[edge[1],:].copy() for ie in range(ao - 1): c2 = ie + 1 c1 = ao - c2 nts[iseq] = [nt, ii] aux = [int(round(tmp)) for tmp in delta * (c1 * n1 + c2 * n2)] nodes[iseq,:] = aux iseq += 1 return iseq
[docs] @staticmethod def append_tp_faces(nodes, nts, iseq, nt, faces, ao): delta = 1.0 / (float(ao) ** 2) for ii, face in enumerate( faces ): n1 = nodes[face[0],:].copy() n2 = nodes[face[1],:].copy() n3 = nodes[face[2],:].copy() n4 = nodes[face[3],:].copy() for i1 in range(ao - 1): for i2 in range(ao - 1): c4 = i1 + 1 c3 = i2 + 1 c2 = ao - c4 c1 = ao - c3 nts[iseq] = [nt, ii] aux = [int(round(tmp)) for tmp in delta * (c1 * c2 * n1 + c2 * c3 * n2 + c3 * c4 * n3 + c4 * c1 * n4)] nodes[iseq,:] = aux iseq += 1 return iseq
[docs] @staticmethod def append_tp_bubbles(nodes, nts, iseq, nt, ao): delta = 1.0 / (float(ao) ** 3) n1 = nodes[0,:].copy() n2 = nodes[1,:].copy() n3 = nodes[2,:].copy() n4 = nodes[3,:].copy() n5 = nodes[4,:].copy() n6 = nodes[5,:].copy() n7 = nodes[6,:].copy() n8 = nodes[7,:].copy() for i1 in range(ao - 1): for i2 in range(ao - 1): for i3 in range(ao - 1): c6 = i1 + 1 c5 = i2 + 1 c4 = i3 + 1 c3 = ao - c6 c2 = ao - c5 c1 = ao - c4 nts[iseq] = [nt, 0] aux = [int(round(tmp)) for tmp in delta * (c1 * c2 * c3 * n1 + c4 * c2 * c3 * n2 + c5 * c4 * c3 * n3 + c1 * c3 * c5 * n4 + c1 * c2 * c6 * n5 + c4 * c2 * c6 * n6 + c5 * c4 * c6 * n7 + c1 * c6 * c5 * n8)] nodes[iseq,:] = aux iseq += 1 return iseq
[docs]class NodeDescription(Struct): """ Describe FE nodes defined on different parts of a reference element. """ def _describe_facets(self, ii): nts = self.node_types[ii] ik = nm.where(nts[1:,1] > nts[:-1,1])[0] if len(ik) == 0: ifacets = None n_dof = 0 else: ii = ii.astype(nm.int32) ik = nm.r_[0, ik + 1, nts.shape[0]] ifacets = [ii[ik[ir] : ik[ir+1]] for ir in range(len(ik) - 1)] n_dof = len(ifacets[0]) return ifacets, n_dof def _describe_other(self, ii): if len(ii): return ii, len(ii) else: return None, 0 def _get_facet_nodes(self, ifacets, nodes): if ifacets is None: return None else: return [nodes[ii] for ii in ifacets] def _get_nodes(self, ii, nodes): if ii is None: return None else: return nodes[ii] def __init__(self, node_types, nodes): self.node_types = node_types # Vertex nodes. ii = nm.where(node_types[:,0] == 0)[0] self.vertex, self.n_vertex_nod = self._describe_other(ii) self.vertex_nodes = self._get_nodes(self.vertex, nodes) # Edge nodes. ii = nm.where(node_types[:,0] == 1)[0] self.edge, self.n_edge_nod = self._describe_facets(ii) self.edge_nodes = self._get_facet_nodes(self.edge, nodes) # Face nodes. ii = nm.where(node_types[:,0] == 2)[0] self.face, self.n_face_nod = self._describe_facets(ii) self.face_nodes = self._get_facet_nodes(self.face, nodes) # Bubble nodes. ii = nm.where(node_types[:,0] == 3)[0] self.bubble, self.n_bubble_nod = self._describe_other(ii) self.bubble_nodes = self._get_nodes(self.bubble, nodes)
[docs] def has_extra_nodes(self): """ Return True if the element has some edge, face or bubble nodes. """ return (self.n_edge_nod + self.n_face_nod + self.n_bubble_nod) > 0
[docs]class PolySpace(Struct): """Abstract polynomial space class.""" _all = None keys = { (1, 2) : 'simplex', (2, 3) : 'simplex', (3, 4) : 'simplex', (2, 4) : 'tensor_product', (3, 8) : 'tensor_product', }
[docs] @staticmethod def any_from_args(name, geometry, order, base='lagrange', force_bubble=False): """ Construct a particular polynomial space classes according to the arguments passed in. """ if name is None: name = PolySpace.suggest_name(geometry, order, base, force_bubble) if PolySpace._all is None: PolySpace._all = find_subclasses(globals(), [PolySpace]) table = PolySpace._all key = '%s_%s' % (base, PolySpace.keys[(geometry.dim, geometry.n_vertex)]) if (geometry.name == '1_2') and (key not in table): key = '%s_%s' % (base, 'tensor_product') if force_bubble: key += '_bubble' return table[key](name, geometry, order)
[docs] @staticmethod def suggest_name(geometry, order, base='lagrange', force_bubble=False): """ Suggest the polynomial space name given its constructor parameters. """ aux = geometry.get_interpolation_name()[:-1] if force_bubble: return aux + ('%dB' % order) else: return aux + ('%d' % order)
def __init__(self, name, geometry, order): self.name = name self.geometry = geometry self.order = order self.bbox = nm.vstack((geometry.coors.min(0), geometry.coors.max(0)))
[docs] def eval_base(self, coors, diff=0, ori=None, force_axis=False, transform=None, suppress_errors=False, eps=1e-15): """ Evaluate the basis or its first or second derivatives in points given by coordinates. The real work is done in _eval_base() implemented in subclasses. Note that the second derivative code is a work-in-progress and only `coors` and `transform` arguments are used. Parameters ---------- coors : array_like The coordinates of points where the basis is evaluated. See Notes. diff : 0, 1 or 2 If nonzero, return the given derivative. ori : array_like, optional Optional orientation of element facets for per element basis. force_axis : bool If True, force the resulting array shape to have one more axis even when `ori` is None. transform : array_like, optional The basis transform array. suppress_errors : bool If True, do not report points outside the reference domain. eps : float Accuracy for comparing coordinates. Returns ------- base : array The basis (shape (n_coor, 1, n_base)) or its first derivative (shape (n_coor, dim, n_base)) or its second derivative (shape (n_coor, dim, dim, n_base)) evaluated in the given points. An additional axis is pre-pended of length n_cell, if `ori` is given, or of length 1, if `force_axis` is True. Notes ----- If coors.ndim == 3, several point sets are assumed, with equal number of points in each of them. This is the case, for example, of the values of the volume base functions on the element facets. The indexing (of bf_b(g)) is then (ifa,iqp,:,n_ep), so that the facet can be set in C using FMF_SetCell. """ coors = nm.asarray(coors) if not coors.ndim in (2, 3): raise ValueError('coordinates must have 2 or 3 dimensions! (%d)' % coors.ndim) if (coors.ndim == 2): base = self._eval_base(coors, diff=diff, ori=ori, suppress_errors=suppress_errors, eps=eps) if (base.ndim == 3) and force_axis: base = base[None, ...] if not base.flags['C_CONTIGUOUS']: base = nm.ascontiguousarray(base) else: # Several point sets. if diff: bdim = self.geometry.dim else: bdim = 1 base = nm.empty((coors.shape[0], coors.shape[1], bdim, self.n_nod), dtype=nm.float64) for ii, _coors in enumerate(coors): base[ii] = self._eval_base(_coors, diff=diff, ori=ori, suppress_errors=suppress_errors, eps=eps) if transform is not None: base = transform_basis(transform, base) return base
[docs] def get_mtx_i(self): return self.mtx_i
[docs] def describe_nodes(self): return NodeDescription(self.nts, self.nodes)
[docs]class LagrangePolySpace(PolySpace):
[docs] def create_context(self, cmesh, eps, check_errors, i_max, newton_eps, tdim=None): from sfepy.discrete.fem.extmods.bases import CLagrangeContext ref_coors = self.geometry.coors if cmesh is not None: mesh_coors = cmesh.coors conn = cmesh.get_conn(cmesh.tdim, 0) mesh_conn = conn.indices.reshape(cmesh.n_el, -1).astype(nm.int32) if tdim is None: tdim = cmesh.tdim else: mesh_coors = mesh_conn = None if tdim is None: raise ValueError('supply either cmesh or tdim!') ctx = CLagrangeContext(order=self.order, tdim=tdim, nodes=self.nodes, ref_coors=ref_coors, mesh_coors=mesh_coors, mesh_conn=mesh_conn, mtx_i=self.get_mtx_i(), eps=eps, check_errors=check_errors, i_max=i_max, newton_eps=newton_eps) return ctx
def _eval_base(self, coors, diff=0, ori=None, suppress_errors=False, eps=1e-15): """ See :func:`PolySpace.eval_base()`. """ if diff == 2: base = self._eval_hessian(coors) else: base = self.eval_ctx.evaluate(coors, diff=diff, eps=eps, check_errors=not suppress_errors) return base
[docs]class LagrangeSimplexPolySpace(LagrangePolySpace): """Lagrange polynomial space on a simplex domain.""" name = 'lagrange_simplex' def __init__(self, name, geometry, order, init_context=True): PolySpace.__init__(self, name, geometry, order) n_v = geometry.n_vertex mtx = nm.ones((n_v, n_v), nm.float64) mtx[0:n_v-1,:] = nm.transpose(geometry.coors) self.mtx_i = nm.ascontiguousarray(nla.inv(mtx)) self.rhs = nm.ones((n_v,), nm.float64) self.nodes, self.nts, node_coors = self._define_nodes() self.node_coors = nm.ascontiguousarray(node_coors) self.n_nod = self.nodes.shape[0] if init_context: self.eval_ctx = self.create_context(None, 0, 1e-15, 100, 1e-8, tdim=n_v - 1) else: self.eval_ctx = None def _define_nodes(self): # Factorial. fac = lambda n : reduce(lambda a, b : a * (b + 1), range(n), 1) geometry = self.geometry n_v, dim = geometry.n_vertex, geometry.dim order = self.order n_nod = fac(order + dim) // (fac(order) * fac(dim)) ## print n_nod, gd nodes = nm.zeros((n_nod, n_v), nm.int32) nts = nm.zeros((n_nod, 2), nm.int32) if order == 0: nts[0,:] = [3, 0] nodes[0,:] = nm.zeros((n_v,), nm.int32) else: iseq = 0 # Vertex nodes. nts[0:n_v,0] = 0 nts[0:n_v,1] = nm.arange(n_v, dtype = nm.int32) aux = order * nm.identity(n_v, dtype = nm.int32) nodes[iseq:iseq+n_v,:] = aux iseq += n_v if dim == 1: iseq = LagrangeNodes.append_edges(nodes, nts, iseq, 3, [[0, 1]], order) elif dim == 2: iseq = LagrangeNodes.append_edges(nodes, nts, iseq, 1, geometry.edges, order) iseq = LagrangeNodes.append_faces(nodes, nts, iseq, 3, [[0, 1, 2]], order) elif dim == 3: iseq = LagrangeNodes.append_edges(nodes, nts, iseq, 1, geometry.edges, order) iseq = LagrangeNodes.append_faces(nodes, nts, iseq, 2, geometry.faces, order) iseq = LagrangeNodes.append_bubbles(nodes, nts, iseq, 3, order) else: raise NotImplementedError ## print nm.concatenate((nts, nodes), 1) # Check orders. orders = nm.sum(nodes, 1) if not nm.all(orders == order): raise AssertionError('wrong orders! (%d == all of %s)' % (order, orders)) # Coordinates of the nodes. if order == 0: tmp = nm.ones((n_nod, n_v), nm.int32) node_coors = nm.dot(tmp, geometry.coors) / n_v else: node_coors = nm.dot(nodes, geometry.coors) / order return nodes, nts, node_coors def _eval_hessian(self, coors): """ Evaluate the second derivatives of the basis. """ def get_bc(coor): rhs = nm.concatenate((coor, [1])) bc = nm.dot(self.mtx_i, rhs) return bc def get_val(bc, node, omit=[]): val = nm.ones(1, nm.float64) for i1 in range(bc.shape[0]): if i1 in omit: continue for i2 in range(node[i1]): val *= (self.order * bc[i1] - i2) / (i2 + 1.0) return val def get_der(bc1, node1, omit=[]): val = nm.zeros(1, nm.float64) for i1 in range(node1): if i1 in omit: continue aux = nm.ones(1, nm.float64) for i2 in range(node1): if (i1 == i2) or (i2 in omit): continue aux *= (self.order * bc1 - i2) / (i2 + 1.0) val += aux * self.order / (i1 + 1.0) return val n_v = self.mtx_i.shape[0] dim = n_v - 1 mi = self.mtx_i[:, :dim] bfgg = nm.zeros((coors.shape[0], dim, dim, self.n_nod), dtype=nm.float64) for ic, coor in enumerate(coors): bc = get_bc(coor) for ii, node in enumerate(self.nodes): for ig1, bc1 in enumerate(bc): # 1. derivative w.r.t. bc1. for ig2, bc2 in enumerate(bc): # 2. derivative w.r.t. bc2. if ig1 == ig2: val = get_val(bc, node, omit=[ig1]) vv = 0.0 for i1 in range(node[ig1]): aux = get_der(bc2, node[ig2], omit=[i1]) vv += aux * self.order / (i1 + 1.0) val *= vv else: val = get_val(bc, node, omit=[ig1, ig2]) val *= get_der(bc1, node[ig1]) val *= get_der(bc2, node[ig2]) bfgg[ic, :, :, ii] += val * mi[ig1] * mi[ig2][:, None] return bfgg
[docs]class LagrangeSimplexBPolySpace(LagrangeSimplexPolySpace): """Lagrange polynomial space with forced bubble function on a simplex domain.""" name = 'lagrange_simplex_bubble' def __init__(self, name, geometry, order, init_context=True): LagrangeSimplexPolySpace.__init__(self, name, geometry, order, init_context=False) nodes, nts, node_coors = self.nodes, self.nts, self.node_coors shape = [nts.shape[0] + 1, 2] nts = nm.resize(nts, shape) nts[-1,:] = [3, 0] shape = [nodes.shape[0] + 1, nodes.shape[1]] nodes = nm.resize(nodes, shape) # Make a 'hypercubic' (cubic in 2D) node. nodes[-1,:] = 1 n_v = self.geometry.n_vertex tmp = nm.ones((n_v,), nm.int32) node_coors = nm.vstack((node_coors, nm.dot(tmp, self.geometry.coors) / n_v)) self.nodes, self.nts = nodes, nts self.node_coors = nm.ascontiguousarray(node_coors) self.bnode = nodes[-1:,:] self.n_nod = self.nodes.shape[0] if init_context: self.eval_ctx = self.create_context(None, 0, 1e-15, 100, 1e-8, tdim=n_v - 1) else: self.eval_ctx = None
[docs] def create_context(self, *args, **kwargs): ctx = LagrangePolySpace.create_context(self, *args, **kwargs) ctx.is_bubble = 1 return ctx
[docs]class LagrangeTensorProductPolySpace(LagrangePolySpace): """Lagrange polynomial space on a tensor product domain.""" name = 'lagrange_tensor_product' def __init__(self, name, geometry, order, init_context=True): PolySpace.__init__(self, name, geometry, order) g1d = Struct(n_vertex = 2, dim = 1, coors = self.bbox[:,0:1].copy()) self.ps1d = LagrangeSimplexPolySpace('P_aux', g1d, order, init_context=False) self.nodes, self.nts, node_coors = self._define_nodes() self.node_coors = nm.ascontiguousarray(node_coors) self.n_nod = self.nodes.shape[0] if init_context: tdim = int(nm.sqrt(geometry.n_vertex)) self.eval_ctx = self.create_context(None, 0, 1e-15, 100, 1e-8, tdim=tdim) else: self.eval_ctx = None def _define_nodes(self): geometry = self.geometry order = self.order n_v, dim = geometry.n_vertex, geometry.dim vertex_map = order * nm.array(vertex_maps[dim], dtype=nm.int32) n_nod = (order + 1) ** dim nodes = nm.zeros((n_nod, 2 * dim), nm.int32) nts = nm.zeros((n_nod, 2), nm.int32) if order == 0: nts[0,:] = [3, 0] nodes[0,:] = nm.zeros((n_nod,), nm.int32) else: iseq = 0 # Vertex nodes. nts[0:n_v,0] = 0 nts[0:n_v,1] = nm.arange( n_v, dtype = nm.int32 ) order * nm.identity( n_v, dtype = nm.int32 ) if dim == 3: for ii in range(n_v): i1, i2, i3 = vertex_map[ii] nodes[iseq,:] = [order - i1, i1, order - i2, i2, order - i3, i3] iseq += 1 elif dim == 2: for ii in range(n_v): i1, i2 = vertex_map[ii] nodes[iseq,:] = [order - i1, i1, order - i2, i2] iseq += 1 else: for ii in range(n_v): i1 = vertex_map[ii][0] nodes[iseq,:] = [order - i1, i1] iseq += 1 if dim == 1: iseq = LagrangeNodes.append_tp_edges(nodes, nts, iseq, 3, [[0, 1]], order) elif dim == 2: iseq = LagrangeNodes.append_tp_edges(nodes, nts, iseq, 1, geometry.edges, order) iseq = LagrangeNodes.append_tp_faces(nodes, nts, iseq, 3, [[0, 1, 2, 3]], order) elif dim == 3: iseq = LagrangeNodes.append_tp_edges(nodes, nts, iseq, 1, geometry.edges, order) iseq = LagrangeNodes.append_tp_faces(nodes, nts, iseq, 2, geometry.faces, order) iseq = LagrangeNodes.append_tp_bubbles(nodes, nts, iseq, 3, order) else: raise NotImplementedError # Check orders. orders = nm.sum(nodes, 1) if not nm.all(orders == order * dim): raise AssertionError('wrong orders! (%d == all of %s)' % (order * dim, orders)) # Coordinates of the nodes. if order == 0: tmp = nm.ones((n_nod, n_v), nm.int32) node_coors = nm.dot(tmp, geometry.coors) / n_v else: c_min, c_max = self.bbox[:,0] cr = nm.arange(2 * dim) node_coors = (nodes[:,cr[::2]] * c_min + nodes[:,cr[1::2]] * c_max) / order return nodes, nts, node_coors def _eval_base_debug(self, coors, diff=False, ori=None, suppress_errors=False, eps=1e-15): """Python version of eval_base().""" dim = self.geometry.dim ev = self.ps1d.eval_base if diff: base = nm.ones((coors.shape[0], dim, self.n_nod), dtype=nm.float64) for ii in range(dim): self.ps1d.nodes = self.nodes[:,2*ii:2*ii+2].copy() self.ps1d.n_nod = self.n_nod for iv in range(dim): if ii == iv: base[:,iv:iv+1,:] *= ev(coors[:,ii:ii+1].copy(), diff=True, suppress_errors=suppress_errors, eps=eps) else: base[:,iv:iv+1,:] *= ev(coors[:,ii:ii+1].copy(), diff=False, suppress_errors=suppress_errors, eps=eps) else: base = nm.ones((coors.shape[0], 1, self.n_nod), dtype=nm.float64) for ii in range(dim): self.ps1d.nodes = self.nodes[:,2*ii:2*ii+2].copy() self.ps1d.n_nod = self.n_nod base *= ev(coors[:,ii:ii+1].copy(), diff=diff, suppress_errors=suppress_errors, eps=eps) return base def _eval_hessian(self, coors): """ Evaluate the second derivatives of the basis. """ evh = self.ps1d.eval_base dim = self.geometry.dim bfgg = nm.zeros((coors.shape[0], dim, dim, self.n_nod), dtype=nm.float64) v0s = [] v1s = [] v2s = [] for ii in range(dim): self.ps1d.nodes = self.nodes[:,2*ii:2*ii+2].copy() self.ps1d.n_nod = self.n_nod ev = self.ps1d.create_context(None, 0, 1e-15, 100, 1e-8, tdim=1).evaluate v0s.append(ev(coors[:, ii:ii+1].copy())[:, 0, :]) v1s.append(ev(coors[:, ii:ii+1].copy(), diff=1)[:, 0, :]) v2s.append(evh(coors[:, ii:ii+1], diff=2)[:, 0, 0, :]) for ir in range(dim): vv = v2s[ir] # Destroys v2s! for ik in range(dim): if ik == ir: continue vv *= v0s[ik] bfgg[:, ir, ir, :] = vv for ic in range(dim): if ic == ir: continue val = v1s[ir] * v1s[ic] for ik in range(dim): if (ik == ir) or (ik == ic): continue val *= v0s[ik] bfgg[:, ir, ic, :] += val return bfgg
[docs] def get_mtx_i(self): return self.ps1d.mtx_i
[docs]class LobattoTensorProductPolySpace(PolySpace): """ Hierarchical polynomial space using Lobatto functions. Each row of the `nodes` attribute defines indices of Lobatto functions that need to be multiplied together to evaluate the corresponding shape function. This defines the ordering of basis functions on the reference element. """ name = 'lobatto_tensor_product' def __init__(self, name, geometry, order): PolySpace.__init__(self, name, geometry, order) aux = self._define_nodes() self.nodes, self.nts, node_coors, self.face_axes, self.sfnodes = aux self.node_coors = nm.ascontiguousarray(node_coors) self.n_nod = self.nodes.shape[0] aux = nm.where(self.nodes > 0, self.nodes, 1) self.node_orders = nm.prod(aux, axis=1) self.edge_indx = nm.where(self.nts[:, 0] == 1)[0] self.face_indx = nm.where(self.nts[:, 0] == 2)[0] self.face_axes_nodes = self._get_face_axes_nodes(self.face_axes) def _get_counts(self): order = self.order dim = self.geometry.dim n_nod = (order + 1) ** dim n_per_edge = (order - 1) n_per_face = (order - 1) ** (dim - 1) n_bubble = (order - 1) ** dim return n_nod, n_per_edge, n_per_face, n_bubble def _define_nodes(self): geometry = self.geometry order = self.order n_v, dim = geometry.n_vertex, geometry.dim n_nod, n_per_edge, n_per_face, n_bubble = self._get_counts() nodes = nm.zeros((n_nod, dim), nm.int32) nts = nm.zeros((n_nod, 2), nm.int32) # Vertex nodes. nts[0:n_v, 0] = 0 nts[0:n_v, 1] = nm.arange(n_v, dtype=nm.int32) nodes[0:n_v] = nm.array(vertex_maps[dim], dtype=nm.int32) ii = n_v # Edge nodes. if (dim > 1) and (n_per_edge > 0): ik = nm.arange(2, order + 1, dtype=nm.int32) zo = nm.zeros((n_per_edge, 2), dtype=nm.int32) zo[:, 1] = 1 for ie, edge in enumerate(geometry.edges): n1, n2 = nodes[edge] ifix = nm.where(n1 == n2)[0] irun = nm.where(n1 != n2)[0][0] ic = n1[ifix] nodes[ii:ii + n_per_edge, ifix] = zo[:, ic] nodes[ii:ii + n_per_edge, irun] = ik nts[ii:ii + n_per_edge] = [[1, ie]] ii += n_per_edge # 3D face nodes. face_axes = [] sfnodes = None if (dim == 3) and (n_per_face > 0): n_face = len(geometry.faces) sfnodes = nm.zeros((n_per_face * n_face, dim), nm.int32) ii0 = ii ik = nm.arange(2, order + 1, dtype=nm.int32) zo = nm.zeros((n_per_face, 2), dtype=nm.int32) zo[:, 1] = 1 for ifa, face in enumerate(geometry.faces): ns = nodes[face] diff = nm.diff(ns, axis=0) asum = nm.abs(diff).sum(axis=0) ifix = nm.where(asum == 0)[0][0] ic = ns[0, ifix] irun1 = nm.where(asum == 2)[0][0] irun2 = nm.where(asum == 1)[0][0] iy, ix = nm.meshgrid(ik, ik) nodes[ii:ii + n_per_face, ifix] = zo[:, ic] nodes[ii:ii + n_per_face, irun1] = ix.ravel() nodes[ii:ii + n_per_face, irun2] = iy.ravel() nts[ii:ii + n_per_face] = [[2, ifa]] ij = ii - ii0 sfnodes[ij:ij + n_per_face, ifix] = zo[:, ic] sfnodes[ij:ij + n_per_face, irun1] = iy.ravel() sfnodes[ij:ij + n_per_face, irun2] = ix.ravel() face_axes.append([irun1, irun2]) ii += n_per_face face_axes = nm.array(face_axes) # Bubble nodes. if n_bubble > 0: ik = nm.arange(2, order + 1, dtype=nm.int32) nodes[ii:] = nm.array([aux for aux in combine([ik] * dim)]) nts[ii:ii + n_bubble] = [[3, 0]] ii += n_bubble assert_(ii == n_nod) # Coordinates of the "nodes". All nodes on a facet have the same # coordinates - the centre of the facet. c_min, c_max = self.bbox[:, 0] node_coors = nm.zeros(nodes.shape, dtype=nm.float64) node_coors[:n_v] = nodes[:n_v] if (dim > 1) and (n_per_edge > 0): ie = nm.where(nts[:, 0] == 1)[0] node_coors[ie] = node_coors[geometry.edges[nts[ie, 1]]].mean(1) if (dim == 3) and (n_per_face > 0): ifa = nm.where(nts[:, 0] == 2)[0] node_coors[ifa] = node_coors[geometry.faces[nts[ifa, 1]]].mean(1) if n_bubble > 0: ib = nm.where(nts[:, 0] == 3)[0] node_coors[ib] = node_coors[geometry.conn].mean(0) return nodes, nts, node_coors, face_axes, sfnodes def _get_face_axes_nodes(self, face_axes): if not len(face_axes): return None nodes = self.nodes[self.face_indx] n_per_face = self._get_counts()[2] anodes = nm.tile(nodes[:n_per_face, face_axes[0]], (6, 1)) return anodes def _eval_base(self, coors, diff=False, ori=None, suppress_errors=False, eps=1e-15): """ See PolySpace.eval_base(). """ from .extmods.lobatto_bases import eval_lobatto_tensor_product as ev c_min, c_max = self.bbox[:, 0] base = ev(coors, self.nodes, c_min, c_max, self.order, diff) if ori is not None: ebase = nm.tile(base, (ori.shape[0], 1, 1, 1)) if self.edge_indx.shape[0]: # Orient edge functions. ie, ii = nm.where(ori[:, self.edge_indx] == 1) ii = self.edge_indx[ii] ebase[ie, :, :, ii] *= -1.0 if self.face_indx.shape[0]: # Orient face functions. fori = ori[:, self.face_indx] # ... normal axis order ie, ii = nm.where((fori == 1) | (fori == 2)) ii = self.face_indx[ii] ebase[ie, :, :, ii] *= -1.0 # ... swapped axis order sbase = ev(coors, self.sfnodes, c_min, c_max, self.order, diff) sbase = insert_strided_axis(sbase, 0, ori.shape[0]) # ...overwrite with swapped axes basis. ie, ii = nm.where(fori >= 4) ii2 = self.face_indx[ii] ebase[ie, :, :, ii2] = sbase[ie, :, :, ii] # ...deal with orientation. ie, ii = nm.where((fori == 5) | (fori == 6)) ii = self.face_indx[ii] ebase[ie, :, :, ii] *= -1.0 base = ebase return base