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// Copyright (c) 2020 Chris Richardson
// FEniCS Project
// SPDX-License-Identifier: MIT
#include "raviart-thomas.h"
#include "dof-permutations.h"
#include "lagrange.h"
#include "moments.h"
#include "polyset.h"
#include "quadrature.h"
#include <Eigen/Dense>
#include <numeric>
#include <vector>
using namespace basix;
//----------------------------------------------------------------------------
FiniteElement basix::create_rt(cell::type celltype, int degree,
const std::string& name)
{
if (celltype != cell::type::triangle and celltype != cell::type::tetrahedron)
throw std::runtime_error("Unsupported cell type");
const int tdim = cell::topological_dimension(celltype);
const cell::type facettype
= (tdim == 2) ? cell::type::interval : cell::type::triangle;
// The number of order (degree-1) scalar polynomials
const int nv = polyset::dim(celltype, degree - 1);
// The number of order (degree-2) scalar polynomials
const int ns0 = polyset::dim(celltype, degree - 2);
// The number of additional polynomials in the polynomial basis for
// Raviart-Thomas
const int ns = polyset::dim(facettype, degree - 1);
// Evaluate the expansion polynomials at the quadrature points
auto [Qpts, Qwts] = quadrature::make_quadrature("default", celltype, 2 * degree);
Eigen::ArrayXXd Pkp1_at_Qpts
= polyset::tabulate(celltype, degree, 0, Qpts)[0];
// The number of order (degree) polynomials
const int psize = Pkp1_at_Qpts.cols();
// Create coefficients for order (degree-1) vector polynomials
Eigen::MatrixXd wcoeffs = Eigen::MatrixXd::Zero(nv * tdim + ns, psize * tdim);
for (int j = 0; j < tdim; ++j)
{
wcoeffs.block(nv * j, psize * j, nv, nv)
= Eigen::MatrixXd::Identity(nv, nv);
}
// Create coefficients for additional polynomials in Raviart-Thomas
// polynomial basis
for (int i = 0; i < ns; ++i)
{
for (int k = 0; k < psize; ++k)
{
for (int j = 0; j < tdim; ++j)
{
const double w_sum = (Qwts * Pkp1_at_Qpts.col(ns0 + i) * Qpts.col(j)
* Pkp1_at_Qpts.col(k))
.sum();
wcoeffs(nv * tdim + i, k + psize * j) = w_sum;
}
}
}
// Dual space
Eigen::MatrixXd dual = Eigen::MatrixXd::Zero(nv * tdim + ns, psize * tdim);
// quadrature degree
int quad_deg = 5 * degree;
// Add rows to dualmat for integral moments on facets
const int facet_count = tdim + 1;
const int facet_dofs = ns;
dual.block(0, 0, facet_count * facet_dofs, psize * tdim)
= moments::make_normal_integral_moments(
create_dlagrange(facettype, degree - 1), celltype, tdim, degree,
quad_deg);
// Add rows to dualmat for integral moments on interior
if (degree > 1)
{
const int internal_dofs = tdim * ns0;
// Interior integral moment
dual.block(facet_count * facet_dofs, 0, internal_dofs, psize * tdim)
= moments::make_integral_moments(create_dlagrange(celltype, degree - 2),
celltype, tdim, degree, quad_deg);
}
const std::vector<std::vector<std::vector<int>>> topology
= cell::topology(celltype);
const int ndofs = dual.rows();
int perm_count = 0;
for (int i = 1; i < tdim; ++i)
perm_count += topology[i].size() * i;
std::vector<Eigen::MatrixXd> base_permutations(
perm_count, Eigen::MatrixXd::Identity(ndofs, ndofs));
if (tdim == 2)
{
Eigen::ArrayXi edge_ref = dofperms::interval_reflection(degree);
for (int edge = 0; edge < facet_count; ++edge)
{
const int start = edge_ref.size() * edge;
for (int i = 0; i < edge_ref.size(); ++i)
{
base_permutations[edge](start + i, start + i) = 0;
base_permutations[edge](start + i, start + edge_ref[i]) = 1;
}
}
Eigen::ArrayXXd edge_dir
= dofperms::interval_reflection_tangent_directions(degree);
for (int edge = 0; edge < 3; ++edge)
{
Eigen::MatrixXd directions = Eigen::MatrixXd::Identity(ndofs, ndofs);
directions.block(edge_dir.rows() * edge, edge_dir.cols() * edge,
edge_dir.rows(), edge_dir.cols())
= edge_dir;
base_permutations[edge] *= directions;
}
}
else if (tdim == 3)
{
Eigen::ArrayXi face_ref = dofperms::triangle_reflection(degree);
Eigen::ArrayXi face_rot = dofperms::triangle_rotation(degree);
for (int face = 0; face < facet_count; ++face)
{
const int start = face_ref.size() * face;
for (int i = 0; i < face_rot.size(); ++i)
{
base_permutations[6 + 2 * face](start + i, start + i) = 0;
base_permutations[6 + 2 * face](start + i, start + face_rot[i]) = 1;
base_permutations[6 + 2 * face + 1](start + i, start + i) = 0;
base_permutations[6 + 2 * face + 1](start + i, start + face_ref[i]) = -1;
}
}
}
// Raviart-Thomas has ns dofs on each facet, and ns0*tdim in the interior
std::vector<std::vector<int>> entity_dofs(topology.size());
for (int i = 0; i < tdim - 1; ++i)
entity_dofs[i].resize(topology[i].size(), 0);
entity_dofs[tdim - 1].resize(topology[tdim - 1].size(), ns);
entity_dofs[tdim] = {ns0 * tdim};
Eigen::MatrixXd coeffs = compute_expansion_coefficients(wcoeffs, dual);
return FiniteElement(name, celltype, degree, {tdim}, coeffs, entity_dofs,
base_permutations, {}, {}, "contravariant piola");
}
//-----------------------------------------------------------------------------
Eigen::MatrixXd basix::dofperms::triangle_rt_rotation(int degree)
{
const int n = degree * (degree + 2);
Eigen::MatrixXd perm = Eigen::MatrixXd::Zero(n, n);
// Permute RT functions on edges
for (int i = 0; i < degree; ++i)
{
perm(i, 2 * degree + i) = 1;
perm(2 * degree - 1 - i, i) = -1;
perm(3 * degree - 1 - i, degree + i) = -1;
}
// Rotate face
const int face_start = 3 * degree;
Eigen::ArrayXi face_rot = dofperms::triangle_rotation(degree - 1);
Eigen::ArrayXXd face_dir_rot
= dofperms::triangle_rotation_tangent_directions(degree - 1);
for (int i = 0; i < face_rot.size(); ++i)
{
for (int b = 0; b < 2; ++b)
perm(face_start + i * 2 + b, face_start + face_rot[i] * 2 + b) = 1;
}
Eigen::MatrixXd rotation = Eigen::MatrixXd::Identity(n, n);
rotation.block(face_start, face_start, face_dir_rot.rows(),
face_dir_rot.cols())
= face_dir_rot;
perm *= rotation;
return perm;
}
//-----------------------------------------------------------------------------
Eigen::MatrixXd basix::dofperms::triangle_rt_reflection(int degree)
{
const int n = degree * (degree + 2);
Eigen::MatrixXd perm = Eigen::MatrixXd::Zero(n, n);
// Permute RT functions on edges
for (int i = 0; i < degree; ++i)
{
perm(i, degree - 1 - i) = 1;
perm(degree + i, 2 * degree + i) = -1;
perm(2 * degree + i, degree + i) = -1;
}
// reflect face
const int face_start = 3 * degree;
Eigen::ArrayXi face_ref = dofperms::triangle_reflection(degree - 1);
Eigen::ArrayXXd face_dir_ref
= dofperms::triangle_reflection_tangent_directions(degree - 1);
for (int i = 0; i < face_ref.size(); ++i)
{
for (int b = 0; b < 2; ++b)
perm(face_start + i * 2 + b, face_start + face_ref[i] * 2 + b) = 1;
}
Eigen::MatrixXd reflection = Eigen::MatrixXd::Identity(n, n);
reflection.block(face_start, face_start, face_dir_ref.rows(),
face_dir_ref.cols())
= face_dir_ref;
perm *= reflection;
return perm;
}
//-----------------------------------------------------------------------------
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