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// Copyright (c) 2022 Matthew Scroggs
// FEniCS Project
// SPDX-License-Identifier: MIT
#include "e-hhj.h"
#include "e-lagrange.h"
#include "element-families.h"
#include "maps.h"
#include "math.h"
#include "polyset.h"
#include "quadrature.h"
#include "sobolev-spaces.h"
#include <cmath>
using namespace basix;
//-----------------------------------------------------------------------------
template <std::floating_point T>
FiniteElement<T> basix::element::create_hhj(cell::type celltype, int degree,
bool discontinuous)
{
if (celltype != cell::type::triangle and celltype != cell::type::tetrahedron)
throw std::runtime_error("Unsupported celltype");
const std::size_t tdim = cell::topological_dimension(celltype);
const int nc = tdim * (tdim + 1) / 2;
const int basis_size
= polyset::dim(celltype, polyset::type::standard, degree);
impl::mdarray_t<T, 2> wcoeffs(basis_size * nc, basis_size * tdim * tdim);
for (std::size_t i = 0; i < tdim; ++i)
{
for (std::size_t j = 0; j < tdim; ++j)
{
int xoff = i + tdim * j;
int yoff = i + j;
if (tdim == 3 and i > 0 and j > 0)
++yoff;
const std::size_t s = basis_size;
for (std::size_t k = 0; k < s; ++k)
wcoeffs(yoff * s + k, xoff * s + k) = i == j ? 1.0 : std::sqrt(0.5);
}
}
const std::vector<std::vector<std::vector<int>>> topology
= cell::topology(celltype);
const auto [gbuffer, gshape] = cell::geometry<T>(celltype);
impl::mdspan_t<const T, 2> geometry(gbuffer.data(), gshape);
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 facet_dim = tdim - 1;
// No DOFs on entities of lower dimension than facets
for (std::size_t d = 0; d < facet_dim; ++d)
{
for (std::size_t e = 0; e < topology[d].size(); ++e)
{
x[d].emplace_back(0, tdim);
M[d].emplace_back(0, tdim * tdim, 0, 1);
}
}
// DOFs on facets
// These dofs are defined by l(F) = integral(n^t @ F @ n * p),
// where n is normal to the facet, p is polynomial on the facet
// of degree `degree`
auto [_data, _shape] = cell::scaled_facet_normals<T>(celltype);
impl::mdspan_t<const T, 2> normals(_data.data(), _shape);
for (std::size_t e = 0; e < topology[facet_dim].size(); ++e)
{
cell::type ct = cell::sub_entity_type(celltype, facet_dim, e);
const std::size_t ndofs = polyset::dim(ct, polyset::type::standard, degree);
const auto [ptsbuffer, wts] = quadrature::make_quadrature<T>(
quadrature::type::Default, ct, polyset::type::standard, 2 * degree);
impl::mdspan_t<const T, 2> pts(ptsbuffer.data(), wts.size(), facet_dim);
FiniteElement<T> moment_space = create_lagrange<T>(
ct, degree, element::lagrange_variant::legendre, true);
const auto [phib, phishape] = moment_space.tabulate(0, pts);
impl::mdspan_t<const T, 4> moment_values(phib.data(), phishape);
// Entity coordinates
const auto [entity_x_buffer, eshape]
= cell::sub_entity_geometry<T>(celltype, facet_dim, e);
std::span<const T> x0(entity_x_buffer.data(), eshape[1]);
impl::mdspan_t<const T, 2> entity_x(entity_x_buffer.data(), eshape);
// Copy points
auto& _x = x[facet_dim].emplace_back(pts.extent(0), tdim);
for (std::size_t p = 0; p < pts.extent(0); ++p)
{
for (std::size_t k = 0; k < _x.extent(1); ++k)
_x(p, k) = x0[k];
for (std::size_t k0 = 0; k0 + 1 < entity_x.extent(0); ++k0)
for (std::size_t k1 = 0; k1 < _x.extent(1); ++k1)
_x(p, k1) += (entity_x(k0 + 1, k1) - x0[k1]) * pts(p, k0);
}
auto& _M = M[facet_dim].emplace_back(ndofs, tdim * tdim, pts.extent(0), 1);
for (int n = 0; n < moment_space.dim(); ++n)
{
for (std::size_t q = 0; q < pts.extent(0); ++q)
{
for (std::size_t k0 = 0; k0 < tdim; ++k0)
{
for (std::size_t k1 = 0; k1 < tdim; ++k1)
{
_M(n, tdim * k0 + k1, q, 0) = normals(e, k0) * normals(e, k1)
* wts[q] * moment_values(0, q, n, 0);
}
}
}
}
}
// Interior
// These DOFs are defined as in section 5.2 of
// https://doi.org/10.1007/s00211-017-0933-3 (Pechstein & Schoberl, 2017)
if (tdim == 2)
{
if (degree == 0)
{
x[tdim].emplace_back(0, tdim);
M[tdim].emplace_back(0, tdim * tdim, 0, 1);
}
else
{
const std::size_t ndofs
= polyset::dim(celltype, polyset::type::standard, degree - 1);
const auto [ptsbuffer, wts] = quadrature::make_quadrature<T>(
quadrature::type::Default, celltype, polyset::type::standard,
2 * degree - 1);
impl::mdspan_t<const T, 2> pts(ptsbuffer.data(), wts.size(), tdim);
// Copy points
auto& _x = x[tdim].emplace_back(pts.extent(0), tdim);
for (std::size_t p = 0; p < pts.extent(0); ++p)
{
for (std::size_t k = 0; k < pts.extent(1); ++k)
_x(p, k) += pts(p, k);
}
auto& _M = M[tdim].emplace_back(ndofs * 3, tdim * tdim, pts.extent(0), 1);
FiniteElement<T> moment_space = create_lagrange<T>(
celltype, degree - 1, element::lagrange_variant::legendre, true);
const auto [phib, phishape] = moment_space.tabulate(0, pts);
impl::mdspan_t<const T, 4> moment_values(phib.data(), phishape);
for (int n = 0; n < moment_space.dim(); ++n)
{
for (std::size_t q = 0; q < pts.extent(0); ++q)
{
// [0, 1], [1, 0]
_M(n * 3, 1, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 3, 2, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
// [-2, 1], [1, 0]
_M(n * 3 + 1, 0, q, 0) = -2.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 3 + 1, 1, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 3 + 1, 2, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
// [0, 1], [1, -2]
_M(n * 3 + 2, 1, q, 0) = -1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 3 + 2, 2, q, 0) = -1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 3 + 2, 3, q, 0) = 2.0 * wts[q] * moment_values(0, q, n, 0);
}
}
}
}
else
{
const std::size_t ndofs0
= degree == 0
? 0
: polyset::dim(celltype, polyset::type::standard, degree - 1);
const std::size_t ndofs1
= polyset::dim(celltype, polyset::type::standard, degree);
const auto [ptsbuffer, wts]
= quadrature::make_quadrature<T>(quadrature::type::Default, celltype,
polyset::type::standard, 2 * degree);
impl::mdspan_t<const T, 2> pts(ptsbuffer.data(), wts.size(), tdim);
// Copy points
auto& _x = x[tdim].emplace_back(pts.extent(0), tdim);
for (std::size_t p = 0; p < pts.extent(0); ++p)
{
for (std::size_t k = 0; k < pts.extent(1); ++k)
_x(p, k) += pts(p, k);
}
auto& _M = M[tdim].emplace_back(ndofs0 * 4 + ndofs1 * 2, tdim * tdim,
pts.extent(0), 1);
if (degree > 0)
{
FiniteElement<T> moment_space = create_lagrange<T>(
celltype, degree - 1, element::lagrange_variant::legendre, true);
const auto [phib, phishape] = moment_space.tabulate(0, pts);
impl::mdspan_t<const T, 4> moment_values(phib.data(), phishape);
for (int n = 0; n < moment_space.dim(); ++n)
{
for (std::size_t q = 0; q < pts.extent(0); ++q)
{
// [0, 1, 1], [1, 0, 1], [1, 1, 0]
_M(n * 4, 1, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4, 2, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4, 3, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4, 5, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4, 6, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4, 7, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
// [-6, 1, 1], [1, 0, 1], [1, 1, 0]
_M(n * 4 + 1, 0, q, 0) = -6.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4 + 1, 1, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4 + 1, 2, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4 + 1, 3, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4 + 1, 5, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4 + 1, 6, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4 + 1, 7, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
// [0, 1, 1], [1, -6, 1], [1, 1, 0]
_M(n * 4 + 2, 1, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4 + 2, 2, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4 + 2, 3, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4 + 2, 4, q, 0) = -6.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4 + 2, 5, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4 + 2, 6, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4 + 2, 7, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
// [0, 1, 1], [1, 0, 1], [1, 1, -6]
_M(n * 4 + 3, 1, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4 + 3, 2, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4 + 3, 3, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4 + 3, 5, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4 + 3, 6, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4 + 3, 7, q, 0) = 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(n * 4 + 3, 8, q, 0) = -6.0 * wts[q] * moment_values(0, q, n, 0);
}
}
}
FiniteElement<T> moment_space = create_lagrange<T>(
celltype, degree, element::lagrange_variant::legendre, true);
const auto [phib, phishape] = moment_space.tabulate(0, pts);
impl::mdspan_t<const T, 4> moment_values(phib.data(), phishape);
for (int n = 0; n < moment_space.dim(); ++n)
{
for (std::size_t q = 0; q < pts.extent(0); ++q)
{
// [0, 0, -1], [0, 0, 1], [-1, 1, 0]
_M(ndofs0 * 4 + n * 2, 2, q, 0)
= -1.0 * wts[q] * moment_values(0, q, n, 0);
_M(ndofs0 * 4 + n * 2, 5, q, 0)
= 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(ndofs0 * 4 + n * 2, 6, q, 0)
= -1.0 * wts[q] * moment_values(0, q, n, 0);
_M(ndofs0 * 4 + n * 2, 7, q, 0)
= 1.0 * wts[q] * moment_values(0, q, n, 0);
// [0, -1, 0], [-1, 0, 1], [0, 1, 0]
_M(ndofs0 * 4 + n * 2 + 1, 1, q, 0)
= -1.0 * wts[q] * moment_values(0, q, n, 0);
_M(ndofs0 * 4 + n * 2 + 1, 3, q, 0)
= -1.0 * wts[q] * moment_values(0, q, n, 0);
_M(ndofs0 * 4 + n * 2 + 1, 5, q, 0)
= 1.0 * wts[q] * moment_values(0, q, n, 0);
_M(ndofs0 * 4 + n * 2 + 1, 7, q, 0)
= 1.0 * wts[q] * moment_values(0, q, n, 0);
}
}
}
std::array<std::vector<impl::mdspan_t<const T, 2>>, 4> xview
= impl::to_mdspan(x);
std::array<std::vector<impl::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 * tdim);
xview = impl::to_mdspan(xbuffer, xshape);
Mview = impl::to_mdspan(Mbuffer, Mshape);
}
sobolev::space space
= discontinuous ? sobolev::space::L2 : sobolev::space::HDivDiv;
return FiniteElement<T>(
element::family::HHJ, celltype, polyset::type::standard, degree,
{tdim, tdim}, impl::mdspan_t<T, 2>(wcoeffs.data(), wcoeffs.extents()),
xview, Mview, 0, maps::type::doubleContravariantPiola, space,
discontinuous, -1, degree, element::lagrange_variant::unset,
element::dpc_variant::unset);
}
//-----------------------------------------------------------------------------
template FiniteElement<float> element::create_hhj(cell::type, int, bool);
template FiniteElement<double> element::create_hhj(cell::type, int, bool);
//-----------------------------------------------------------------------------
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