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
195
196
197
198
199
200
201
202
203
204
|
// Copyright (c) 2020 Chris Richardson & Matthew Scroggs
// FEniCS Project
// SPDX-License-Identifier: MIT
#include "e-bubble.h"
#include "element-families.h"
#include "lattice.h"
#include "maps.h"
#include "polyset.h"
#include "quadrature.h"
#include "sobolev-spaces.h"
#include <array>
#include <vector>
using namespace basix;
//----------------------------------------------------------------------------
template <std::floating_point T>
FiniteElement<T> basix::element::create_bubble(cell::type celltype, int degree,
bool discontinuous)
{
if (discontinuous)
throw std::runtime_error("Cannot create a discontinuous bubble element.");
switch (celltype)
{
case cell::type::interval:
if (degree < 2)
throw std::runtime_error(
"Bubble element on an interval must have degree at least 2");
break;
case cell::type::triangle:
if (degree < 3)
throw std::runtime_error(
"Bubble element on a triangle must have degree at least 3");
break;
case cell::type::tetrahedron:
if (degree < 4)
throw std::runtime_error(
"Bubble element on a tetrahedron must have degree at least 4");
break;
case cell::type::quadrilateral:
if (degree < 2)
throw std::runtime_error("Bubble element on a quadrilateral interval "
"must have degree at least 2");
break;
case cell::type::hexahedron:
if (degree < 2)
throw std::runtime_error(
"Bubble element on a hexahedron must have degree at least 2");
break;
default:
throw std::runtime_error("Unsupported cell type");
}
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;
for (std::size_t i = 0; i < tdim; ++i)
{
const std::size_t num_ent = cell::num_sub_entities(celltype, i);
x[i] = std::vector(num_ent, impl::mdarray_t<T, 2>(0, tdim));
M[i] = std::vector(num_ent, impl::mdarray_t<T, 4>(0, 1, 0, 1));
}
// Evaluate the expansion polynomials at the quadrature points
const auto [_pts, wts] = quadrature::make_quadrature<T>(
quadrature::type::Default, celltype, 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(celltype, polyset::type::standard, degree, 0, pts);
impl::mdspan_t<const T, 3> phi(_phi.data(), shape);
// The number of order (degree) polynomials
const std::size_t psize = phi.extent(1);
// Create points at nodes on interior
std::size_t ndofs = 0;
{
const auto [points, pshape] = lattice::create<T>(
celltype, degree, lattice::type::equispaced, false);
ndofs = pshape[0];
x[tdim].emplace_back(pshape, points);
}
auto create_phi1 = [](auto& phi, auto& buffer)
{
buffer.resize(phi.extent(1) * phi.extent(2));
impl::mdspan_t<T, 2> phi1(buffer.data(), phi.extent(1), phi.extent(2));
for (std::size_t i = 0; i < phi1.extent(0); ++i)
for (std::size_t j = 0; j < phi1.extent(1); ++j)
phi1(i, j) = phi(0, i, j);
return phi1;
};
// Create coefficients for order (degree-1) vector polynomials
std::vector<T> phi1_buffer;
impl::mdspan_t<T, 2> phi1;
std::vector<T> bubble;
switch (celltype)
{
case cell::type::interval:
{
const auto [_phi1, shape] = polyset::tabulate(
celltype, polyset::type::standard, degree - 2, 0, pts);
impl::mdspan_t<const T, 3> p1(_phi1.data(), shape);
phi1 = create_phi1(p1, phi1_buffer);
for (std::size_t i = 0; i < pts.extent(0); ++i)
{
T x0 = pts(i, 0);
bubble.push_back(x0 * (1.0 - x0));
}
break;
}
case cell::type::triangle:
{
const auto [_phi1, shape] = polyset::tabulate(
celltype, polyset::type::standard, degree - 3, 0, pts);
impl::mdspan_t<const T, 3> p1(_phi1.data(), shape);
phi1 = create_phi1(p1, phi1_buffer);
for (std::size_t i = 0; i < pts.extent(0); ++i)
{
T x0 = pts(i, 0);
T x1 = pts(i, 1);
bubble.push_back(x0 * x1 * (1.0 - x0 - x1));
}
break;
}
case cell::type::tetrahedron:
{
const auto [_phi1, shape] = polyset::tabulate(
celltype, polyset::type::standard, degree - 4, 0, pts);
impl::mdspan_t<const T, 3> p1(_phi1.data(), shape);
phi1 = create_phi1(p1, phi1_buffer);
for (std::size_t i = 0; i < pts.extent(0); ++i)
{
T x0 = pts(i, 0);
T x1 = pts(i, 1);
T x2 = pts(i, 2);
bubble.push_back(x0 * x1 * x2 * (1 - x0 - x1 - x2));
}
break;
}
case cell::type::quadrilateral:
{
const auto [_phi1, shape] = polyset::tabulate(
celltype, polyset::type::standard, degree - 2, 0, pts);
impl::mdspan_t<const T, 3> p1(_phi1.data(), shape);
phi1 = create_phi1(p1, phi1_buffer);
for (std::size_t i = 0; i < pts.extent(0); ++i)
{
T x0 = pts(i, 0);
T x1 = pts(i, 1);
bubble.push_back(x0 * (1 - x0) * x1 * (1 - x1));
}
break;
}
case cell::type::hexahedron:
{
const auto [_phi1, shape] = polyset::tabulate(
celltype, polyset::type::standard, degree - 2, 0, pts);
impl::mdspan_t<const T, 3> p1(_phi1.data(), shape);
phi1 = create_phi1(p1, phi1_buffer);
for (std::size_t i = 0; i < pts.extent(0); ++i)
{
T x0 = pts(i, 0);
T x1 = pts(i, 1);
T x2 = pts(i, 2);
bubble.push_back(x0 * (1 - x0) * x1 * (1 - x1) * x2 * (1 - x2));
}
break;
}
default:
throw std::runtime_error("Unknown cell type.");
}
impl::mdarray_t<T, 2> wcoeffs(ndofs, psize);
for (std::size_t i = 0; i < phi1.extent(0); ++i)
for (std::size_t j = 0; j < psize; ++j)
for (std::size_t k = 0; k < wts.size(); ++k)
wcoeffs(i, j) += wts[k] * phi1(i, k) * bubble[k] * phi(0, j, k);
math::orthogonalise<T>(wcoeffs);
auto& _M = M[tdim].emplace_back(ndofs, 1, ndofs, 1);
for (std::size_t i = 0; i < _M.extent(0); ++i)
_M(i, 0, i, 0) = 1.0;
impl::mdspan_t<T, 2> wview(wcoeffs.data(), wcoeffs.extents());
sobolev::space space
= discontinuous ? sobolev::space::L2 : sobolev::space::H1;
return FiniteElement<T>(
element::family::bubble, celltype, polyset::type::standard, degree, {},
wview, impl::to_mdspan(x), impl::to_mdspan(M), 0, maps::type::identity,
space, discontinuous, -1, degree, element::lagrange_variant::unset,
element::dpc_variant::unset);
}
//-----------------------------------------------------------------------------
template FiniteElement<float> element::create_bubble(cell::type, int, bool);
template FiniteElement<double> element::create_bubble(cell::type, int, bool);
//-----------------------------------------------------------------------------
|
