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// Copyright (c) 2020 Chris Richardson
// FEniCS Project
// SPDX-License-Identifier: MIT
#include "regge.h"
#include "lattice.h"
#include "polyset.h"
#include <iostream>
using namespace basix;
namespace
{
//-----------------------------------------------------------------------------
Eigen::MatrixXd create_regge_space(cell::type celltype, int degree)
{
if (celltype != cell::type::triangle and celltype != cell::type::tetrahedron)
throw std::runtime_error("Unsupported celltype");
const int tdim = cell::topological_dimension(celltype);
const int nc = tdim * (tdim + 1) / 2;
const int basis_size = polyset::dim(celltype, degree);
const int ndofs = basis_size * nc;
const int psize = basis_size * tdim * tdim;
Eigen::ArrayXXd wcoeffs = Eigen::ArrayXXd::Zero(ndofs, psize);
int s = basis_size;
for (int i = 0; i < tdim; ++i)
{
for (int j = 0; j < tdim; ++j)
{
int xoff = i + tdim * j;
int yoff = i + j;
if (tdim == 3 and i > 0 and j > 0)
++yoff;
wcoeffs.block(yoff * s, xoff * s, s, s) = Eigen::MatrixXd::Identity(s, s);
}
}
return wcoeffs;
}
//-----------------------------------------------------------------------------
Eigen::MatrixXd create_regge_dual(cell::type celltype, int degree)
{
const int tdim = cell::topological_dimension(celltype);
const int basis_size = polyset::dim(celltype, degree);
const int ndofs = basis_size * (tdim + 1) * tdim / 2;
const int space_size = basis_size * tdim * tdim;
Eigen::ArrayXXd dualmat(ndofs, space_size);
std::vector<std::vector<std::vector<int>>> topology
= cell::topology(celltype);
const Eigen::ArrayXXd geometry = cell::geometry(celltype);
// dof counter
int dof = 0;
for (std::size_t dim = 1; dim < topology.size(); ++dim)
{
for (std::size_t i = 0; i < topology[dim].size(); ++i)
{
const Eigen::ArrayXXd entity_geom
= cell::sub_entity_geometry(celltype, dim, i);
Eigen::ArrayXd point = entity_geom.row(0);
cell::type ct = cell::sub_entity_type(celltype, dim, i);
Eigen::ArrayXXd lattice
= lattice::create(ct, degree + 2, lattice::type::equispaced, false);
Eigen::ArrayXXd pts(lattice.rows(), entity_geom.cols());
for (int j = 0; j < lattice.rows(); ++j)
{
pts.row(j) = entity_geom.row(0);
for (int k = 0; k < entity_geom.rows() - 1; ++k)
{
pts.row(j)
+= (entity_geom.row(k + 1) - entity_geom.row(0)) * lattice(j, k);
}
}
Eigen::MatrixXd basis = polyset::tabulate(celltype, degree, 0, pts)[0];
// Store up outer(t, t) for all tangents
std::vector<int>& vert_ids = topology[dim][i];
int ntangents = dim * (dim + 1) / 2;
std::vector<Eigen::MatrixXd> vvt(ntangents);
int c = 0;
for (std::size_t s = 0; s < dim; ++s)
{
for (std::size_t d = s + 1; d < dim + 1; ++d)
{
const Eigen::VectorXd edge_t
= geometry.row(vert_ids[d]) - geometry.row(vert_ids[s]);
// outer product v.v^T
vvt[c++] = edge_t * edge_t.transpose();
}
}
for (int k = 0; k < pts.rows(); ++k)
{
for (int j = 0; j < ntangents; ++j)
{
Eigen::Map<Eigen::VectorXd> vvt_flat(vvt[j].data(),
vvt[j].rows() * vvt[j].cols());
// outer product: outer(outer(t, t), basis)
const Eigen::MatrixXd vvt_b = vvt_flat * basis.row(k);
// Copy tensor values row by row into dualmat
for (int r = 0; r < vvt_b.rows(); ++r)
dualmat.block(dof, r * vvt_b.cols(), 1, vvt_b.cols())
= vvt_b.row(r);
++dof;
}
}
}
}
return dualmat;
}
//-----------------------------------------------------------------------------
} // namespace
//-----------------------------------------------------------------------------
FiniteElement basix::create_regge(cell::type celltype, int degree,
const std::string& name)
{
const int tdim = cell::topological_dimension(celltype);
Eigen::MatrixXd wcoeffs = create_regge_space(celltype, degree);
Eigen::MatrixXd dual = create_regge_dual(celltype, degree);
// TODO
const int ndofs = dual.rows();
int perm_count = tdim == 2 ? 3 : 14;
std::vector<Eigen::MatrixXd> base_permutations(
perm_count, Eigen::MatrixXd::Identity(ndofs, ndofs));
Eigen::MatrixXd coeffs = compute_expansion_coefficients(wcoeffs, dual);
// Regge has (d+1) dofs on each edge, 3d(d+1)/2 on each face
// and d(d-1)(d+1) on the interior in 3D
const std::vector<std::vector<std::vector<int>>> topology
= cell::topology(celltype);
std::vector<std::vector<int>> entity_dofs(topology.size());
entity_dofs[0].resize(topology[0].size(), 0);
entity_dofs[1].resize(topology[1].size(), degree + 1);
entity_dofs[2].resize(topology[2].size(), 3 * (degree + 1) * degree / 2);
if (tdim > 2)
entity_dofs[3] = {(degree + 1) * degree * (degree - 1)};
return FiniteElement(name, celltype, degree, {tdim, tdim}, coeffs,
entity_dofs, base_permutations, {}, {}, "double covariant piola");
}
//-----------------------------------------------------------------------------
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