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// Copyright (c) 2020 Chris Richardson & Matthew Scroggs
// FEniCS Project
// SPDX-License-Identifier: MIT
#include "e-nedelec.h"
#include "e-lagrange.h"
#include "e-raviart-thomas.h"
#include "element-families.h"
#include "maps.h"
#include "math.h"
#include "mdspan.hpp"
#include "moments.h"
#include "polyset.h"
#include "quadrature.h"
#include "sobolev-spaces.h"
#include <array>
#include <concepts>
#include <vector>
using namespace basix;
namespace
{
//-----------------------------------------------------------------------------
template <std::floating_point T>
impl::mdarray_t<T, 2> create_nedelec_2d_space(int degree)
{
// Number of order (degree) vector polynomials
const std::size_t nv = degree * (degree + 1) / 2;
// Number of order (degree-1) vector polynomials
const std::size_t ns0 = (degree - 1) * degree / 2;
// Number of additional polynomials in Nedelec set
const std::size_t ns = degree;
// Tabulate polynomial set at quadrature points
const auto [_pts, wts] = quadrature::make_quadrature<T>(
quadrature::type::Default, cell::type::triangle, polyset::type::standard,
2 * degree);
impl::mdspan_t<const T, 2> pts(_pts.data(), wts.size(),
_pts.size() / wts.size());
const auto [_phi, shape] = polyset::tabulate(
cell::type::triangle, polyset::type::standard, degree, 0, pts);
impl::mdspan_t<const T, 3> phi(_phi.data(), shape);
const std::size_t psize = phi.extent(1);
// Create coefficients for order (degree-1) vector polynomials
impl::mdarray_t<T, 2> wcoeffs(nv * 2 + ns, psize * 2);
for (std::size_t i = 0; i < nv; ++i)
{
wcoeffs(i, i) = 1.0;
wcoeffs(nv + i, psize + i) = 1.0;
}
// Create coefficients for the additional Nedelec polynomials
for (std::size_t i = 0; i < ns; ++i)
{
for (std::size_t j = nv; j < psize; ++j)
{
wcoeffs(2 * nv + i, j) = 0.0;
wcoeffs(2 * nv + i, j + psize) = 0.0;
for (std::size_t k = 0; k < wts.size(); ++k)
{
T p = phi(0, ns0 + i, k);
wcoeffs(2 * nv + i, j) += wts[k] * p * pts(k, 1) * phi(0, j, k);
wcoeffs(2 * nv + i, j + psize) -= wts[k] * p * pts(k, 0) * phi(0, j, k);
}
}
}
math::orthogonalise<T>(wcoeffs, nv * 2);
return wcoeffs;
}
//-----------------------------------------------------------------------------
template <std::floating_point T>
impl::mdarray_t<T, 2> create_nedelec_3d_space(int degree)
{
// Reference tetrahedron
const std::size_t tdim = 3;
// Number of order (degree) vector polynomials
const std::size_t nv = degree * (degree + 1) * (degree + 2) / 6;
// Number of order (degree-1) vector polynomials
const std::size_t ns0 = (degree - 1) * degree * (degree + 1) / 6;
// Number of additional Nedelec polynomials that could be added
const std::size_t ns = degree * (degree + 1) / 2;
// Number of polynomials that would be included that are not
// independent so are removed
const std::size_t ns_remove = degree * (degree - 1) / 2;
// Number of dofs in the space, ie size of polynomial set
const std::size_t ndofs = 6 * degree + 4 * degree * (degree - 1)
+ (degree - 2) * (degree - 1) * degree / 2;
// Tabulate polynomial basis at quadrature points
const auto [_pts, wts] = quadrature::make_quadrature<T>(
quadrature::type::Default, cell::type::tetrahedron,
polyset::type::standard, 2 * degree);
impl::mdspan_t<const T, 2> pts(_pts.data(), wts.size(),
_pts.size() / wts.size());
const auto [_phi, shape] = polyset::tabulate(
cell::type::tetrahedron, polyset::type::standard, degree, 0, pts);
impl::mdspan_t<const T, 3> phi(_phi.data(), shape);
const std::size_t psize = phi.extent(1);
// Create coefficients for order (degree-1) polynomials
impl::mdarray_t<T, 2> wcoeffs(ndofs, psize * tdim);
for (std::size_t i = 0; i < tdim; ++i)
for (std::size_t j = 0; j < nv; ++j)
wcoeffs(i * nv + j, i * psize + j) = 1.0;
// Create coefficients for additional Nedelec polynomials
for (std::size_t i = 0; i < ns; ++i)
{
for (std::size_t j = nv; j < psize; ++j)
{
T w = 0.0;
for (std::size_t k = 0; k < wts.size(); ++k)
w += wts[k] * phi(0, ns0 + i, k) * pts(k, 2) * phi(0, j, k);
// Don't include polynomials (*, *, 0) that are dependent
if (i >= ns_remove)
wcoeffs(tdim * nv + i - ns_remove, psize + j) = -w;
wcoeffs(tdim * nv + i + ns - ns_remove, j) = w;
}
}
for (std::size_t i = 0; i < ns; ++i)
{
for (std::size_t j = nv; j < psize; ++j)
{
T w = 0.0;
for (std::size_t k = 0; k < wts.size(); ++k)
w += wts[k] * phi(0, ns0 + i, k) * pts(k, 1) * phi(0, j, k);
wcoeffs(tdim * nv + i + ns * 2 - ns_remove, j) = -w;
// Don't include polynomials (*, *, 0) that are dependent
if (i >= ns_remove)
wcoeffs(tdim * nv + i - ns_remove, psize * 2 + j) = w;
}
}
for (std::size_t i = 0; i < ns; ++i)
{
for (std::size_t j = nv; j < psize; ++j)
{
T w = 0.0;
for (std::size_t k = 0; k < wts.size(); ++k)
w += wts[k] * phi(0, ns0 + i, k) * pts(k, 0) * phi(0, j, k);
wcoeffs(tdim * nv + i + ns - ns_remove, psize * 2 + j) = -w;
wcoeffs(tdim * nv + i + ns * 2 - ns_remove, psize + j) = w;
}
}
math::orthogonalise<T>(wcoeffs, nv * 3);
return wcoeffs;
}
//-----------------------------------------------------------------------------
} // namespace
//-----------------------------------------------------------------------------
template <std::floating_point T>
FiniteElement<T> element::create_nedelec(cell::type celltype, int degree,
lagrange_variant lvariant,
bool discontinuous)
{
if (degree < 1)
throw std::runtime_error("Degree must be at least 1");
const std::size_t tdim = cell::topological_dimension(celltype);
std::array<std::vector<impl::mdarray_t<T, 2>>, 4> x;
std::array<std::vector<impl::mdarray_t<T, 4>>, 4> M;
{
const std::size_t num_ent = cell::num_sub_entities(celltype, 0);
x[0] = std::vector(num_ent, impl::mdarray_t<T, 2>(0, tdim));
M[0] = std::vector(num_ent, impl::mdarray_t<T, 4>(0, tdim, 0, 1));
}
std::vector<T> wcoeffs;
std::array<std::size_t, 2> wshape;
switch (celltype)
{
case cell::type::triangle:
{
impl::mdarray_t<T, 2> w = create_nedelec_2d_space<T>(degree);
wshape = {w.extent(0), w.extent(1)};
wcoeffs.resize(wshape[0] * wshape[1]);
std::copy_n(w.data(), w.size(), wcoeffs.data());
break;
}
case cell::type::tetrahedron:
{
impl::mdarray_t<T, 2> w = create_nedelec_3d_space<T>(degree);
wshape = {w.extent(0), w.extent(1)};
wcoeffs.resize(wshape[0] * wshape[1]);
std::copy_n(w.data(), w.size(), wcoeffs.data());
break;
}
default:
throw std::runtime_error("Invalid celltype in Nedelec");
}
// Integral representation for the boundary (edge) dofs
{
FiniteElement edge_space = element::create_lagrange<T>(
cell::type::interval, degree - 1, lvariant, true);
auto [_x, xshape, _M, Mshape] = moments::make_tangent_integral_moments<T>(
edge_space, celltype, polyset::type::standard, tdim, 2 * degree - 1);
assert(_x.size() == _M.size());
for (std::size_t i = 0; i < _x.size(); ++i)
{
x[1].emplace_back(std::array{xshape[0], xshape[1]}, _x[i]);
M[1].emplace_back(std::array{Mshape[0], Mshape[1], Mshape[2], Mshape[3]},
_M[i]);
}
}
// Face dofs
if (degree > 1)
{
FiniteElement face_space = element::create_lagrange<T>(
cell::type::triangle, degree - 2, lvariant, true);
auto [_x, xshape, _M, Mshape] = moments::make_integral_moments<T>(
face_space, celltype, polyset::type::standard, tdim, 2 * degree - 2);
assert(_x.size() == _M.size());
for (std::size_t i = 0; i < _x.size(); ++i)
{
x[2].emplace_back(std::array{xshape[0], xshape[1]}, _x[i]);
M[2].emplace_back(std::array{Mshape[0], Mshape[1], Mshape[2], Mshape[3]},
_M[i]);
}
}
else
{
const std::size_t num_ent = cell::num_sub_entities(celltype, 2);
x[2] = std::vector(num_ent, impl::mdarray_t<T, 2>(0, tdim));
M[2] = std::vector(num_ent, impl::mdarray_t<T, 4>(0, tdim, 0, 1));
}
// Volume dofs
if (tdim == 3)
{
if (degree > 2 and tdim == 3)
{
auto [_x, xshape, _M, Mshape] = moments::make_integral_moments<T>(
element::create_lagrange<T>(cell::type::tetrahedron, degree - 3,
lvariant, true),
cell::type::tetrahedron, polyset::type::standard, 3, 2 * degree - 3);
assert(_x.size() == _M.size());
for (std::size_t i = 0; i < _x.size(); ++i)
{
x[3].emplace_back(std::array{xshape[0], xshape[1]}, _x[i]);
M[3].emplace_back(
std::array{Mshape[0], Mshape[1], Mshape[2], Mshape[3]}, _M[i]);
}
}
else
{
const std::size_t num_ent = cell::num_sub_entities(celltype, 3);
x[3] = std::vector(num_ent, impl::mdarray_t<T, 2>(0, tdim));
M[3] = std::vector(num_ent, impl::mdarray_t<T, 4>(0, tdim, 0, 1));
}
}
std::array<std::vector<mdspan_t<const T, 2>>, 4> xview = impl::to_mdspan(x);
std::array<std::vector<mdspan_t<const T, 4>>, 4> Mview = impl::to_mdspan(M);
std::array<std::vector<std::vector<T>>, 4> xbuffer;
std::array<std::vector<std::vector<T>>, 4> Mbuffer;
if (discontinuous)
{
// std::tie(x, M) = element::make_discontinuous(x, M, tdim, tdim);
std::array<std::vector<std::array<std::size_t, 2>>, 4> xshape;
std::array<std::vector<std::array<std::size_t, 4>>, 4> Mshape;
std::tie(xbuffer, xshape, Mbuffer, Mshape)
= element::make_discontinuous(xview, Mview, tdim, tdim);
xview = impl::to_mdspan(xbuffer, xshape);
Mview = impl::to_mdspan(Mbuffer, Mshape);
}
sobolev::space space
= discontinuous ? sobolev::space::L2 : sobolev::space::HCurl;
return FiniteElement<T>(
element::family::N1E, celltype, polyset::type::standard, degree, {tdim},
impl::mdspan_t<T, 2>(wcoeffs.data(), wshape), xview, Mview, 0,
maps::type::covariantPiola, space, discontinuous, degree - 1, degree,
lvariant, element::dpc_variant::unset);
}
//-----------------------------------------------------------------------------
template <std::floating_point T>
FiniteElement<T> element::create_nedelec2(cell::type celltype, int degree,
lagrange_variant lvariant,
bool discontinuous)
{
if (celltype != cell::type::triangle and celltype != cell::type::tetrahedron)
throw std::runtime_error("Invalid celltype in Nedelec");
if (degree < 1)
throw std::runtime_error("Degree must be at least 1");
std::array<std::vector<impl::mdarray_t<T, 2>>, 4> x;
std::array<std::vector<impl::mdarray_t<T, 4>>, 4> M;
const std::size_t tdim = cell::topological_dimension(celltype);
{
const std::size_t num_ent = cell::num_sub_entities(celltype, 0);
x[0] = std::vector(num_ent, impl::mdarray_t<T, 2>(0, tdim));
M[0] = std::vector(num_ent, impl::mdarray_t<T, 4>(0, tdim, 0, 1));
}
// Integral representation for the edge dofs
{
FiniteElement edge_space = element::create_lagrange<T>(
cell::type::interval, degree, lvariant, true);
auto [_x, xshape, _M, Mshape] = moments::make_tangent_integral_moments<T>(
edge_space, celltype, polyset::type::standard, tdim, 2 * degree);
assert(_x.size() == _M.size());
for (std::size_t i = 0; i < _x.size(); ++i)
{
x[1].emplace_back(std::array{xshape[0], xshape[1]}, _x[i]);
M[1].emplace_back(std::array{Mshape[0], Mshape[1], Mshape[2], Mshape[3]},
_M[i]);
}
}
if (degree > 1)
{
// Integral moments on faces
FiniteElement face_space = element::create_rt<T>(
cell::type::triangle, degree - 1, lvariant, true);
auto [_x, xshape, _M, Mshape] = moments::make_dot_integral_moments<T>(
face_space, celltype, polyset::type::standard, tdim, 2 * degree - 1);
assert(_x.size() == _M.size());
for (std::size_t i = 0; i < _x.size(); ++i)
{
x[2].emplace_back(std::array{xshape[0], xshape[1]}, _x[i]);
M[2].emplace_back(std::array{Mshape[0], Mshape[1], Mshape[2], Mshape[3]},
_M[i]);
}
}
else
{
const std::size_t num_ent = cell::num_sub_entities(celltype, 2);
x[2] = std::vector(num_ent, impl::mdarray_t<T, 2>(0, tdim));
M[2] = std::vector(num_ent, impl::mdarray_t<T, 4>(0, tdim, 0, 1));
}
if (tdim == 3)
{
if (degree > 2)
{
auto [_x, xshape, _M, Mshape] = moments::make_dot_integral_moments<T>(
element::create_rt<T>(cell::type::tetrahedron, degree - 2, lvariant,
true),
celltype, polyset::type::standard, tdim, 2 * degree - 2);
assert(_x.size() == _M.size());
for (std::size_t i = 0; i < _x.size(); ++i)
{
x[3].emplace_back(std::array{xshape[0], xshape[1]}, _x[i]);
M[3].emplace_back(
std::array{Mshape[0], Mshape[1], Mshape[2], Mshape[3]}, _M[i]);
}
}
else
{
const std::size_t num_ent = cell::num_sub_entities(celltype, 3);
x[3] = std::vector(num_ent, impl::mdarray_t<T, 2>(0, tdim));
M[3] = std::vector(num_ent, impl::mdarray_t<T, 4>(0, tdim, 0, 1));
}
}
std::array<std::vector<mdspan_t<const T, 2>>, 4> xview = impl::to_mdspan(x);
std::array<std::vector<mdspan_t<const T, 4>>, 4> Mview = impl::to_mdspan(M);
std::array<std::vector<std::vector<T>>, 4> xbuffer;
std::array<std::vector<std::vector<T>>, 4> Mbuffer;
if (discontinuous)
{
std::array<std::vector<std::array<std::size_t, 2>>, 4> xshape;
std::array<std::vector<std::array<std::size_t, 4>>, 4> Mshape;
std::tie(xbuffer, xshape, Mbuffer, Mshape)
= element::make_discontinuous(xview, Mview, tdim, tdim);
xview = impl::to_mdspan(xbuffer, xshape);
Mview = impl::to_mdspan(Mbuffer, Mshape);
}
const std::size_t psize
= polyset::dim(celltype, polyset::type::standard, degree);
return FiniteElement<T>(
element::family::N2E, celltype, polyset::type::standard, degree, {tdim},
impl::mdspan_t<T, 2>(math::eye<T>(tdim * psize).data(), tdim * psize,
tdim * psize),
xview, Mview, 0, maps::type::covariantPiola, sobolev::space::HCurl,
discontinuous, degree, degree, lvariant, element::dpc_variant::unset);
}
//-----------------------------------------------------------------------------
template FiniteElement<float> element::create_nedelec(cell::type, int,
lagrange_variant, bool);
template FiniteElement<double> element::create_nedelec(cell::type, int,
lagrange_variant, bool);
template FiniteElement<float> element::create_nedelec2(cell::type, int,
lagrange_variant, bool);
template FiniteElement<double> element::create_nedelec2(cell::type, int,
lagrange_variant, bool);
//-----------------------------------------------------------------------------
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