1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
|
// Copyright (c) 2020 Chris Richardson
// FEniCS Project
// SPDX-License-Identifier: MIT
#include "e-regge.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;
namespace md = md::MDSPAN_IMPL_PROPOSED_NAMESPACE;
//-----------------------------------------------------------------------------
template <std::floating_point T>
FiniteElement<T> element::create_regge(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);
const std::size_t ndofs = basis_size * nc;
const std::size_t psize = basis_size * tdim * tdim;
impl::mdarray_t<T, 2> wcoeffs(ndofs, psize);
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;
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 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 * tdim, 0, 1));
}
// Loop over edge and higher dimension entities
for (std::size_t d = 1; d < topology.size(); ++d)
{
if (static_cast<std::size_t>(degree) + 1 < 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);
}
}
else
{
// Loop over entities of dimension dim
for (std::size_t e = 0; e < topology[d].size(); ++e)
{
// Entity coordinates
const auto [ebuffer, eshape]
= cell::sub_entity_geometry<T>(celltype, d, e);
impl::mdspan_t<const T, 2> entity_x(ebuffer.data(), eshape);
// Tabulate points in lattice
cell::type ct = cell::sub_entity_type(celltype, d, e);
const std::size_t ndofs
= polyset::dim(ct, polyset::type::standard, degree + 1 - d);
const auto [_pts, wts] = quadrature::make_quadrature<T>(
quadrature::type::Default, ct, polyset::type::standard,
degree + (degree + 1 - d));
impl::mdspan_t<const T, 2> pts(_pts.data(), wts.size(),
_pts.size() / wts.size());
FiniteElement moment_space = create_lagrange<T>(
ct, degree + 1 - d, 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);
auto& _x = x[d].emplace_back(pts.extent(0), tdim);
// Copy points
for (std::size_t p = 0; p < pts.extent(0); ++p)
{
for (std::size_t j = 0; j < entity_x.extent(1); ++j)
_x(p, j) = entity_x(0, j);
for (std::size_t i = 0; i < entity_x.extent(0) - 1; ++i)
for (std::size_t j = 0; j < entity_x.extent(1); ++j)
_x(p, j) += (entity_x(i + 1, j) - entity_x(0, j)) * pts(p, i);
}
// Store up outer(t, t) for all tangents
const std::vector<int>& vert_ids = topology[d][e];
const std::size_t ntangents = d * (d + 1) / 2;
mdex::mdarray<T, md::dextents<std::size_t, 3>> vvt(
ntangents, geometry.extent(1), geometry.extent(1));
std::vector<T> edge(geometry.extent(1));
int c = 0;
for (std::size_t s = 0; s < d; ++s)
{
for (std::size_t r = s + 1; r < d + 1; ++r)
{
for (std::size_t p = 0; p < geometry.extent(1); ++p)
edge[p] = geometry(vert_ids[r], p) - geometry(vert_ids[s], p);
// outer product v.v^T
auto [buffer, shape] = math::outer(edge, edge);
impl::mdspan_t<const T, 2> result(buffer.data(), shape);
for (std::size_t i = 0; i < vvt.extent(1); ++i)
for (std::size_t j = 0; j < vvt.extent(2); ++j)
vvt(c, i, j) = result(i, j);
++c;
}
}
auto& _M = M[d].emplace_back(ndofs * ntangents, tdim * tdim,
pts.extent(0), 1);
for (int n = 0; n < moment_space.dim(); ++n)
{
for (std::size_t j = 0; j < ntangents; ++j)
{
std::vector<T> vvt_flat;
for (std::size_t i = 0; i < vvt.extent(1); ++i)
for (std::size_t k = 0; k < vvt.extent(2); ++k)
vvt_flat.push_back(vvt(j, i, k));
for (std::size_t q = 0; q < pts.extent(0); ++q)
{
for (std::size_t i = 0; i < tdim * tdim; ++i)
{
_M(n * ntangents + j, i, q, 0)
= vvt_flat[i] * wts[q] * moment_values(0, q, n, 0);
}
}
}
}
}
}
}
// 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
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 * tdim);
xview = impl::to_mdspan(xbuffer, xshape);
Mview = impl::to_mdspan(Mbuffer, Mshape);
}
sobolev::space space
= discontinuous ? sobolev::space::L2 : sobolev::space::HEin;
return FiniteElement<T>(
element::family::Regge, celltype, polyset::type::standard, degree,
{tdim, tdim}, impl::mdspan_t<T, 2>(wcoeffs.data(), wcoeffs.extents()),
xview, Mview, 0, maps::type::doubleCovariantPiola, space, discontinuous,
-1, degree, element::lagrange_variant::unset,
element::dpc_variant::unset);
}
//-----------------------------------------------------------------------------
template FiniteElement<float> element::create_regge(cell::type, int, bool);
template FiniteElement<double> element::create_regge(cell::type, int, bool);
//-----------------------------------------------------------------------------
|
