5#ifndef DUNE_LOCALFUNCTIONS_LAGRANGE_LAGRANGEPYRAMID_HH
6#define DUNE_LOCALFUNCTIONS_LAGRANGE_LAGRANGEPYRAMID_HH
15#include <dune/geometry/referenceelements.hh>
17#include <dune/localfunctions/common/localbasis.hh>
18#include <dune/localfunctions/common/localfiniteelementtraits.hh>
19#include <dune/localfunctions/common/localkey.hh>
21namespace Dune {
namespace Impl
32 template<
class D,
class R,
unsigned int k>
33 class LagrangePyramidLocalBasis
36 using Traits = LocalBasisTraits<D,3,FieldVector<D,3>,R,1,FieldVector<R,1>,FieldMatrix<R,1,3> >;
40 static constexpr std::size_t
size ()
42 std::size_t result = 0;
43 for (
unsigned int i=0; i<=k; i++)
44 result +=
power(i+1,2);
50 std::vector<typename Traits::RangeType>& out)
const
65 out[0] = (1-in[0])*(1-in[1])-in[2]*(1-in[1]);
66 out[1] = in[0]*(1-in[1])-in[2]*in[1];
67 out[2] = (1-in[0])*in[1]-in[2]*in[1];
68 out[3] = in[0]*in[1]+in[2]*in[1];
72 out[0] = (1-in[0])*(1-in[1])-in[2]*(1-in[0]);
73 out[1] = in[0]*(1-in[1])-in[2]*in[0];
74 out[2] = (1-in[0])*in[1]-in[2]*in[0];
75 out[3] = in[0]*in[1]+in[2]*in[0];
86 const R x = 2.0*in[0] + in[2] - 1.0;
87 const R y = 2.0*in[1] + in[2] - 1.0;
93 out[0] = 0.25*(x + z)*(x + z - 1)*(y - z - 1)*(y - z);
94 out[1] = -0.25*(x + z)*(y - z)*((x + z + 1)*(-y + z + 1) - 4*z) - z*(x - y);
95 out[2] = 0.25*(x + z)*(y - z)*(y - z + 1)*(x + z - 1);
96 out[3] = 0.25*(y - z)*(x + z)*(y - z + 1)*(x + z + 1);
100 out[5] = -0.5*(y - z + 1)*(x + z - 1)*(y - 1)*x;
101 out[6] = -0.5*(y - z + 1)*(((x + z + 1)*(y - 1)*x - z) + z*(2*y + 1));
102 out[7] = -0.5*(x + z - 1)*(((y - z - 1)*(x + 1)*y - z) + z*(2*x + 1));
103 out[8] = -0.5*(y - z + 1)*(x + z - 1)*(x + 1)*y;
106 out[9] = z*(x + z - 1)*(y - z - 1);
107 out[10] = -z*((x + z + 1)*(y - z - 1) + 4*z);
108 out[11] = -z*(y - z + 1)*(x + z - 1);
109 out[12] = z*(y - z + 1)*(x + z + 1);
112 out[13] = (y - z + 1)*(x + z - 1)*((y - 1)*(x + 1) + z*(x - y + z + 1));
117 out[0] = 0.25*(y + z)*(y + z - 1)*(x - z - 1)*(x - z);
118 out[1] = -0.25*(x - z)*(y + z)*(x - z + 1)*(-y - z + 1);
119 out[2] = 0.25*(x - z)*(y + z)*((x - z - 1)*(y + z + 1) + 4*z) + z*(x - y);
120 out[3] = 0.25*(y + z)*(x - z)*(x - z + 1)*(y + z + 1);
121 out[4] = z*(2*z - 1);
124 out[5] = -0.5*(y + z - 1)*(((x - z - 1)*(y + 1)*x - z) + z*(2*y + 1));
125 out[6] = -0.5*(x - z + 1)*(y + z - 1)*(y + 1)*x;
126 out[7] = -0.5*(x - z + 1)*(y + z - 1)*(x - 1)*y;
127 out[8] = -0.5*(x - z + 1)*(((y + z + 1)*(x - 1)*y - z) + z*(2*x + 1));
130 out[9] = z*(y + z - 1)*(x - z - 1);
131 out[10] = -z*(x - z + 1)*(y + z - 1);
132 out[11] = -z*((y + z + 1)*(x - z - 1) + 4*z);
133 out[12] = z*(x - z + 1)*(y + z + 1);
136 out[13] = (x - z + 1)*(y + z - 1)*((y + 1)*(x - 1) - z*(x - y - z - 1));
142 DUNE_THROW(NotImplemented,
"LagrangePyramidLocalBasis::evaluateFunction for order " << k);
151 std::vector<typename Traits::JacobianType>& out)
const
158 std::fill(out[0][0].begin(), out[0][0].end(), 0);
166 out[0][0] = {-1 + in[1], -1 + in[0] + in[2], -1 + in[1]};
167 out[1][0] = { 1 - in[1], -in[0] - in[2], -in[1]};
168 out[2][0] = { -in[1], 1 - in[0] - in[2], -in[1]};
169 out[3][0] = { in[1], in[0] + in[2], in[1]};
173 out[0][0] = {-1 + in[1] + in[2], -1 + in[0], -1 + in[0]};
174 out[1][0] = { 1 - in[1] - in[2], -in[0], -in[0]};
175 out[2][0] = { -in[1] - in[2], 1 - in[0], -in[0]};
176 out[3][0] = { in[1] + in[2], in[0], in[0]};
179 out[4][0] = {0, 0, 1};
186 const R x = 2.0*in[0] + in[2] - 1.0;
187 const R y = 2.0*in[1] + in[2] - 1.0;
195 out[0][0][0] = 0.5*(y - z - 1)*(y - z)*(2*x + 2*z - 1);
196 out[0][0][1] = 0.5*(x + z)*(x + z - 1)*(2*y - 2*z - 1);
197 out[0][0][2] = 0.5*(out[0][0][0] + out[0][0][1])
198 + 0.25*((2*x + 2*z - 1)*(y - z - 1)*(y - z)
199 + (x + z)*(x + z - 1)*(-2*y + 2*z + 1));
201 out[1][0][0] = 2*(-0.25*((y - z)*((x + z + 1)*(-y + z + 1) - 4*z)
202 + (x + z)*(y - z)*(-y + z + 1)) - z);
203 out[1][0][1] = 2*(-0.25*((x + z)*((x + z + 1)*(-y + z + 1) - 4*z)
204 + (x + z)*(y - z)*(-(x + z + 1))) + z);
205 out[1][0][2] = 0.5*(out[1][0][0] + out[1][0][1])
206 - 0.25*((y - z)*((x + z + 1)*(-y + z + 1) - 4*z)
207 - (x + z)*((x + z + 1)*(-y + z + 1) - 4*z)
208 + (x + z)*(y - z)*(x - y + 2*z - 2))
211 out[2][0][0] = 0.5*(y - z)*(y - z + 1)*(2*x + 2*z - 1);
212 out[2][0][1] = 0.5*(x + z)*(2*y - 2*z + 1)*(x + z - 1);
213 out[2][0][2] = 0.5*(out[2][0][0] + out[2][0][1])
214 + 0.25*((y - x - 2*z)*(y - z + 1)*(x + z - 1)
215 + (x + z)*(y - z)*(y - x - 2*z + 2));
217 out[3][0][0] = 0.5*(y - z)*(2*x + 2*z + 1)*(y - z + 1);
218 out[3][0][1] = 0.5*(2*y - 2*z + 1)*(x + z)*(x + z + 1);
219 out[3][0][2] = 0.5*(out[3][0][0] + out[3][0][1])
220 + 0.25*((y - x - 2*z)*(y - z + 1)*(x + z + 1)
221 + (y - z)*(x + z)*(y - x - 2*z));
225 out[4][0][2] = 4*z - 1;
228 out[5][0][0] = -(y - z + 1)*(y - 1)*(2*x + z - 1);
229 out[5][0][1] = -(x + z - 1)*(y - 1)*x - (y - z + 1)*(x + z - 1)*x;
230 out[5][0][2] = 0.5*(out[5][0][0] + out[5][0][1])
231 + 0.5*(x + z - 1)*(y - 1)*x - 0.5*(y - z + 1)*(y - 1)*x;
233 out[6][0][0] = -(y - z + 1)*(2*x + z + 1)*(y - 1);
234 out[6][0][1] = -(((x + z + 1)*(y - 1)*x - z) + z*(2*y + 1)
235 + (y - z + 1)*((x + z + 1)*x + 2*z));
236 out[6][0][2] = 0.5*(out[6][0][0] + out[6][0][1])
237 - 0.5*(-(((x + z + 1)*(y - 1)*x - z) + z*(2*y + 1))
238 + (y - z + 1)*(((y - 1)*x - 1) + 2*y + 1));
240 out[7][0][0] = -(((y - z - 1)*(x + 1)*y - z) + z*(2*x + 1)
241 + (x + z - 1)*((y - z - 1)*y + 2*z));
242 out[7][0][1] = -(x + z - 1)*(2*y - z - 1)*(x + 1);
243 out[7][0][2] = 0.5*(out[7][0][0] + out[7][0][1])
244 - 0.5*(((y - z - 1)*(x + 1)*y - z) + z*(2*x + 1)
245 + (x + z - 1)*((-(x + 1)*y - 1) + 2*x + 1));
247 out[8][0][0] = -(y - z + 1)*(2*x + z)*y;
248 out[8][0][1] = -(2*y - z + 1)*(x + z - 1)*(x + 1);
249 out[8][0][2] = 0.5*(out[8][0][0] + out[8][0][1])
250 - 0.5*(-x + y - 2*z + 2)*(x + 1)*y;
253 out[9][0][0] = 2*z*(y - z - 1);
254 out[9][0][1] = 2*z*(x + z - 1);
255 out[9][0][2] = 0.5*(out[9][0][0] + out[9][0][1])
256 + (x + z - 1)*(y - z - 1) + z*(-x + y - 2*z);
258 out[10][0][0] = -2*z*(y - z - 1);
259 out[10][0][1] = -2*z*(x + z + 1);
260 out[10][0][2] = 0.5*(out[10][0][0] + out[10][0][1])
261 - ((x + z + 1)*(y - z - 1) + 4*z)
262 - z*(-x + y - 2*z + 2);
264 out[11][0][0] = -2*z*(y - z + 1);
265 out[11][0][1] = -2*z*(x + z - 1);
266 out[11][0][2] = 0.5*(out[11][0][0] + out[11][0][1])
267 - (y - z + 1)*(x + z - 1) - z*(-x + y - 2*z + 2);
269 out[12][0][0] = 2*z*(y - z + 1);
270 out[12][0][1] = 2*z*(x + z + 1);
271 out[12][0][2] = 0.5*(out[12][0][0] + out[12][0][1])
272 + (y - z + 1)*(x + z + 1) + z*(-x + y - 2*z);
275 out[13][0][0] = 2*((y - z + 1)*((y - 1)*(x + 1) + z*(x - y + z + 1))
276 + (y - z + 1)*(x + z - 1)*(y - 1 + z));
277 out[13][0][1] = 2*((x + z - 1)*((y - 1)*(x + 1) + z*(x - y + z + 1))
278 + (y - z + 1)*(x + z - 1)*(x + 1 - z));
279 out[13][0][2] = 0.5*(out[13][0][0] + out[13][0][1])
280 + ((-x + y - 2*z + 2)*((y - 1)*(x + 1) + z*(x - y + z + 1))
281 + (y - z + 1)*(x + z - 1)*(x - y + 2*z + 1));
286 out[0][0][0] = 0.5*(y + z)*(y + z - 1)*(2*x - 2*z - 1);
287 out[0][0][1] = 0.5*(2*y + 2*z - 1)*(x - z - 1)*(x - z);
288 out[0][0][2] = 0.5*(out[0][0][0] + out[0][0][1])
289 + 0.25*((2*y + 2*z - 1)*(x - z - 1)*(x - z)
290 + (y + z)*(y + z - 1)*(-2*x + 2*z + 1));
292 out[1][0][0] = -0.5*(y + z)*(2*x - 2*z + 1)*(-y - z + 1);
293 out[1][0][1] = -0.5*(x - z)*(x - z + 1)*(-2*y - 2*z + 1);
294 out[1][0][2] = 0.5*(out[1][0][0] + out[1][0][1])
295 - 0.25*((x - y - 2*z)*(x - z + 1)*(-y - z + 1)
296 + (x - z)*(y + z)*(-x + y + 2*z - 2));
298 out[2][0][0] = 0.5*((y + z)*((x - z - 1)*(y + z + 1) + 4*z)
299 + (x - z)*(y + z)*(y + z + 1) + 4*z);
300 out[2][0][1] = 0.5*((x - z)*((x - z - 1)*(y + z + 1) + 4*z)
301 + (x - z)*(y + z)*(x - z - 1) - 4*z);
302 out[2][0][2] = 0.5*(out[2][0][0] + out[2][0][1])
303 + 0.25*((x - y - 2*z)*((x - z - 1)*(y + z + 1) + 4*z)
304 + (x - z)*(y + z)*(x - y - 2*z + 2) + 4*(x - y));
306 out[3][0][0] = 0.5*(y + z)*(2*x - 2*z + 1)*(y + z + 1);
307 out[3][0][1] = 0.5*(x - z)*(x - z + 1)*(2*y + 2*z + 1);
308 out[3][0][2] = 0.5*(out[3][0][0] + out[3][0][1])
309 + 0.25*((x - y - 2*z)*(x - z + 1)*(y + z + 1)
310 + (y + z)*(x - z)*(x - y - 2*z));
314 out[4][0][2] = 4*z - 1;
317 out[5][0][0] = -(y + z - 1)*(2*x - z - 1)*(y + 1);
318 out[5][0][1] = -(((x - z - 1)*(y + 1)*x - z) + z*(2*y + 1)
319 + (y + z - 1)*((x - z - 1)*x + 2*z));
320 out[5][0][2] = 0.5*(out[5][0][0] + out[5][0][1])
321 - 0.5*((((x - z - 1)*(y + 1)*x - z) + z*(2*y + 1))
322 + (y + z - 1)*((-(y + 1)*x - 1) + 2*y + 1));
324 out[6][0][0] = -(2*x - z + 1)*(y + z - 1)*(y + 1);
325 out[6][0][1] = -(x - z + 1)*(2*y + z)*x;
326 out[6][0][2] = 0.5*(out[6][0][0] + out[6][0][1])
327 - 0.5*(x - y - 2*z + 2)*(y + 1)*x;
329 out[7][0][0] = -(2*x - z)*(y + z - 1)*y;
330 out[7][0][1] = -(x - z + 1)*(2*y + z - 1)*(x - 1);
331 out[7][0][2] = 0.5*(out[7][0][0] + out[7][0][1])
332 - 0.5*(x - y - 2*z + 2)*(x - 1)*y;
334 out[8][0][0] = -(((y + z + 1)*(x - 1)*y - z) + z*(2*x + 1)
335 + (x - z + 1)*((y + z + 1)*y + 2*z));
336 out[8][0][1] = -(x - z + 1)*(2*y + z + 1)*(x - 1);
337 out[8][0][2] = 0.5*(out[8][0][0] + out[8][0][1])
338 - 0.5*(-(((y + z + 1)*(x - 1)*y - z) + z*(2*x + 1))
339 + (x - z + 1)*(((x - 1)*y - 1) + 2*x + 1));
342 out[9][0][0] = 2*z*(y + z - 1);
343 out[9][0][1] = 2*z*(x - z - 1);
344 out[9][0][2] = 0.5*(out[9][0][0] + out[9][0][1])
345 + (y + z - 1)*(x - z - 1) + z*(x - y - 2*z);
347 out[10][0][0] = -2*z*(y + z - 1);
348 out[10][0][1] = -2*z*(x - z + 1);
349 out[10][0][2] = 0.5*(out[10][0][0] + out[10][0][1])
350 - (x - z + 1)*(y + z - 1) - z*(x - y - 2*z + 2);
352 out[11][0][0] = -2*z*(y + z + 1);
353 out[11][0][1] = -2*z*(x - z - 1);
354 out[11][0][2] = 0.5*(out[11][0][0] + out[11][0][1])
355 - ((y + z + 1)*(x - z - 1) + 4*z) - z*(x - y - 2*z + 2);
357 out[12][0][0] = 2*z*(y + z + 1);
358 out[12][0][1] = 2*z*(x - z + 1);
359 out[12][0][2] = 0.5*(out[12][0][0] + out[12][0][1])
360 + (x - z + 1)*(y + z + 1) + z*(x - y - 2*z);
363 out[13][0][0] = 2*((y + z - 1)*((y + 1)*(x - 1) - z*(x - y - z - 1))
364 + (x - z + 1)*(y + z - 1)*(y + 1 - z));
365 out[13][0][1] = 2*((x - z + 1)*((y + 1)*(x - 1) - z*(x - y - z - 1))
366 + (x - z + 1)*(y + z - 1)*(x - 1 + z));
367 out[13][0][2] = 0.5*(out[13][0][0] + out[13][0][1])
368 + (x - y - 2*z + 2)*((y + 1)*(x - 1) - z*(x - y - z - 1))
369 + (x - z + 1)*(y + z - 1)*(-(x - y - 2*z - 1));
375 DUNE_THROW(NotImplemented,
"LagrangePyramidLocalBasis::evaluateJacobian for order " << k);
384 void partial(
const std::array<unsigned int,3>& order,
386 std::vector<typename Traits::RangeType>& out)
const
394 evaluateFunction(in, out);
408 auto const direction = std::distance(order.begin(), std::find(order.begin(), order.end(), 1));
421 out[0] = -1 + in[0] + in[2];
422 out[1] = -in[0] - in[2];
423 out[2] = 1 - in[0] - in[2];
424 out[3] = in[0]+in[2];
435 DUNE_THROW(RangeError,
"Component out of range.");
443 out[0] = -1 + in[1] + in[2];
444 out[1] = 1 - in[1] - in[2];
445 out[2] = -in[1] - in[2];
446 out[3] = in[1] + in[2];
464 DUNE_THROW(RangeError,
"Component out of range.");
467 }
else if (totalOrder == 2)
469 if ((order[0] == 1 && order[1] == 1) ||
470 (order[1] == 1 && order[2] == 1 && in[0] > in[1]) ||
471 (order[0] == 1 && order[2] == 1 && in[0] <=in[1]))
473 out = {1, -1, -1, 1, 0};
476 out = {0, 0, 0, 0, 0};
481 out = {0, 0, 0, 0, 0};
492 const R x = 2.0*in[0] + in[2] - 1.0;
493 const R y = 2.0*in[1] + in[2] - 1.0;
496 auto const direction = std::distance(order.begin(), std::find(order.begin(), order.end(), 1));
505 out[0] = 0.5*(y - z - 1)*(y - z)*(2*x + 2*z - 1);
506 out[1] = 2*(-0.25*((y - z)*((x + z + 1)*(-y + z + 1) - 4*z) + (x + z)*(y - z)*(-y + z + 1)) - z);
507 out[2] = 0.5*(y - z)*(y - z + 1)*(2*x + 2*z - 1);
508 out[3] = 0.5*(y - z)*(2*x + 2*z + 1)*(y - z + 1);
510 out[5] = -(y - z + 1)*(2*x + z - 1)*(y - 1);
511 out[6] = -(y - z + 1)*(2*x + z + 1)*(y - 1);
512 out[7] = -(((y - z - 1)*(x + 1)*y - z) + z*(2*x + 1) + (x + z - 1)*((y - z - 1)*y + 2*z));
513 out[8] = -(y - z + 1)*(2*x + z)*y;
514 out[9] = 2*z*(y - z - 1);
515 out[10] = -2*z*(y - z - 1);
516 out[11] = -2*z*(y - z + 1);
517 out[12] = 2*z*(y - z + 1);
518 out[13] = 2*((y - z + 1)*((y - 1)*(x + 1) + z*(x - y + z + 1)) + (y - z + 1)*(x + z - 1)*(y - 1 + z));
521 out[0] = 0.5*(x + z)*(x + z - 1)*(2*y - 2*z - 1);
522 out[1] = 2*(-0.25*((x + z)*((x + z + 1)*(-y + z + 1) - 4*z) + (x + z)*(y - z)*(-(x + z + 1))) + z);
523 out[2] = 0.5*(x + z)*(2*y - 2*z + 1)*(x + z - 1);
524 out[3] = 0.5*(2*y - 2*z + 1)*(x + z)*(x + z + 1);
526 out[5] = -(x + z - 1)*(y - 1)*x - (y - z + 1)*(x + z - 1)*x;
527 out[6] = -(((x + z + 1)*(y - 1)*x - z) + z*(2*y + 1) + (y - z + 1)*((x + z + 1)*x + 2*z));
528 out[7] = -(x + z - 1)*(2*y - z - 1)*(x + 1);
529 out[8] = -(2*y - z + 1)*(x + z - 1)*(x + 1);
530 out[9] = 2*z*(x + z - 1);
531 out[10] = -2*z*(x + z + 1);
532 out[11] = -2*z*(x + z - 1);
533 out[12] = 2*z*(x + z + 1);
534 out[13] = 2*((x + z - 1)*((y - 1)*(x + 1) + z*(x - y + z + 1)) + (y - z + 1)*(x + z - 1)*(x + 1 - z));
537 out[0] = -((y - z)*(2*x + 2*z - 1)*(z - y + 1))/2;
538 out[1] = ((y - z + 1)*(y - 2*x + z + 2*x*y - 2*x*z + 2*y*z - 2*z*z))/2;
539 out[2] = ((y - z)*(2*x + 2*z - 1)*(y - z + 1))/2;
540 out[3] = ((y - z)*(2*x + 2*z + 1)*(y - z + 1))/2;
542 out[5] = (-(y - z + 1)*(2*x + z - 1)*(y - 1) - (x + z - 1)*(y - 1)*x - (y - z + 1)*(x + z - 1)*x + (x + z - 1)*(y - 1)*x - (y - z + 1)*(y - 1)*x)/2;
543 out[6] = -((y - z + 1)*(3*y - 2*x + z + 3*x*y + x*z + y*z + x*x - 1))/2;
544 out[7] = z - z*(2*x + 1) - ((2*z - y*(z - y + 1))*(x + z - 1))/2 - ((2*x - y*(x + 1))*(x + z - 1))/2 + ((x + 1)*(x + z - 1)*(z - 2*y + 1))/2 + y*(x + 1)*(z - y + 1);
545 out[8] = -((y - z + 1)*(y + z + 3*x*y + x*z + y*z + x*x - 1))/2;
546 out[9] = -(x + 3*z - 1)*(z - y + 1);
547 out[10] = (x + z + 1)*(z - y + 1) - 2*y*z - 6*z + 2*z*z;
548 out[11] = -(x + 3*z - 1)*(y - z + 1);
549 out[12] = (x + 3*z + 1)*(y - z + 1);
550 out[13] = (y - z + 1)*(2*y - 3*x + z + 2*x*y + 6*x*z - 2*y*z + 2*x*x + 4*z*z - 3);
553 DUNE_THROW(RangeError,
"Component out of range.");
561 out[0] = -((y + z)*(2*z - 2*x + 1)*(y + z - 1))/2;
562 out[1] = ((y + z)*(2*x - 2*z + 1)*(y + z - 1))/2;
563 out[2] = -((y + z + 1)*(y - 3*z - 2*x*y - 2*x*z + 2*y*z + 2*z*z))/2;
564 out[3] = ((y + z)*(2*x - 2*z + 1)*(y + z + 1))/2;
566 out[5] = (y + 1)*(y + z - 1)*(z - 2*x + 1);
567 out[6] = -(y + 1)*(2*x - z + 1)*(y + z - 1);
568 out[7] = -y*(2*x - z)*(y + z - 1);
569 out[8] = z - z*(2*x + 1) - (2*z + y*(y + z + 1))*(x - z + 1) - y*(x - 1)*(y + z + 1);
570 out[9] = 2*z*(y + z - 1);
571 out[10] = -2*z*(y + z - 1);
572 out[11] = -2*z*(y + z + 1);
573 out[12] = 2*z*(y + z + 1);
574 out[13] = 2*(y + z - 1)*(2*x - z + 2*x*y - 2*x*z + 2*z*z);
577 out[0] = -(x - z)*(y + z - 0.5)*(z - x + 1);
578 out[1] = ((x - z)*(2*y + 2*z - 1)*(x - z + 1))/2;
579 out[2] = -((z - x + 1)*(x + 3*z + 2*x*y + 2*x*z - 2*y*z - 2*z*z))/2;
580 out[3] = ((x - z)*(2*y + 2*z + 1)*(x - z + 1))/2;
582 out[5] = z - z*(2*y + 1) - (2*z - x*(z - x + 1))*(y + z - 1) + x*(y + 1)*(z - x + 1);
583 out[6] = -x*(2*y + z)*(x - z + 1);
584 out[7] = -(x - 1)*(x - z + 1)*(2*y + z - 1);
585 out[8] = -(x - 1)*(x - z + 1)*(2*y + z + 1);
586 out[9] = -2*z*(z - x + 1);
587 out[10] = -2*z*(x - z + 1);
588 out[11] = 2*z*(z - x + 1);
589 out[12] = 2*z*(x - z + 1);
590 out[13] = 2*(x - z + 1)*(2*x*y - z - 2*y + 2*y*z + 2*z*z);
593 out[0] = -((x - z)*(2*y + 2*z - 1)*(z - x + 1))/2;
594 out[1] = ((x - z)*(2*y + 2*z - 1)*(x - z + 1))/2;
595 out[2] = ((x - z + 1)*(x - 2*y + z + 2*x*y + 2*x*z - 2*y*z - 2*z*z))/2;
596 out[3] = ((x - z)*(2*y + 2*z + 1)*(x - z + 1))/2;
598 out[5] = z - z*(2*y + 1) - ((2*z - x*(z - x + 1))*(y + z - 1))/2 - ((2*y - x*(y + 1))*(y + z - 1))/2 + ((y + 1)*(y + z - 1)*(z - 2*x + 1))/2 + x*(y + 1)*(z - x + 1);
599 out[6] = -((x - z + 1)*(x + z + 3*x*y + x*z + y*z + y*y - 1))/2;
600 out[7] = -((x - z + 1)*(3*x*y - 4*y - z - x + x*z + y*z + y*y + 1))/2;
601 out[8] = -((x - z + 1)*(3*x - 2*y + z + 3*x*y + x*z + y*z + y*y - 1))/2;
602 out[9] = -(z - x + 1)*(y + 3*z - 1);
603 out[10] = -(x - z + 1)*(y + 3*z - 1);
604 out[11] = (y + z + 1)*(z - x + 1) - 2*x*z - 6*z + 2*z*z;
605 out[12] = (x - z + 1)*(y + 3*z + 1);
606 out[13] = (x - z + 1)*(2*x - 3*y + z + 2*x*y - 2*x*z + 6*y*z + 2*y*y + 4*z*z - 3);
609 DUNE_THROW(RangeError,
"Component out of range.");
613 DUNE_THROW(NotImplemented,
"Desired derivative order is not implemented");
619 DUNE_THROW(NotImplemented,
"LagrangePyramidLocalBasis::partial for order " << k);
623 static constexpr unsigned int order ()
633 template<
unsigned int k>
634 class LagrangePyramidLocalCoefficients
638 LagrangePyramidLocalCoefficients ()
643 localKeys_[0] = LocalKey(0,0,0);
649 for (std::size_t i=0; i<
size(); i++)
650 localKeys_[i] = LocalKey(i,3,0);
657 localKeys_[0] = LocalKey(0,3,0);
658 localKeys_[1] = LocalKey(1,3,0);
659 localKeys_[2] = LocalKey(2,3,0);
660 localKeys_[3] = LocalKey(3,3,0);
661 localKeys_[4] = LocalKey(4,3,0);
664 localKeys_[5] = LocalKey(0,2,0);
665 localKeys_[6] = LocalKey(1,2,0);
666 localKeys_[7] = LocalKey(2,2,0);
667 localKeys_[8] = LocalKey(3,2,0);
668 localKeys_[9] = LocalKey(4,2,0);
669 localKeys_[10] = LocalKey(5,2,0);
670 localKeys_[11] = LocalKey(6,2,0);
671 localKeys_[12] = LocalKey(7,2,0);
674 localKeys_[13] = LocalKey(0,1,0);
680 DUNE_THROW(NotImplemented,
"LagrangePyramidLocalCoefficients for order " << k);
685 static constexpr std::size_t
size ()
687 std::size_t result = 0;
688 for (
unsigned int i=0; i<=k; i++)
689 result +=
power(i+1,2);
694 const LocalKey& localKey (std::size_t i)
const
696 return localKeys_[i];
700 std::vector<LocalKey> localKeys_;
707 template<
class LocalBasis>
708 class LagrangePyramidLocalInterpolation
719 template<
typename F,
typename C>
720 void interpolate (
const F& f, std::vector<C>& out)
const
722 constexpr auto k = LocalBasis::order();
723 using D =
typename LocalBasis::Traits::DomainType;
724 using DF =
typename LocalBasis::Traits::DomainFieldType;
726 out.resize(LocalBasis::size());
739 for (
unsigned int i=0; i<LocalBasis::size(); i++)
750 out[0] = f( D( {0.0, 0.0, 0.0} ) );
751 out[1] = f( D( {1.0, 0.0, 0.0} ) );
752 out[2] = f( D( {0.0, 1.0, 0.0} ) );
753 out[3] = f( D( {1.0, 1.0, 0.0} ) );
754 out[4] = f( D( {0.0, 0.0, 1.0} ) );
755 out[5] = f( D( {0.0, 0.5, 0.0} ) );
756 out[6] = f( D( {1.0, 0.5, 0.0} ) );
757 out[7] = f( D( {0.5, 0.0, 0.0} ) );
758 out[8] = f( D( {0.5, 1.0, 0.0} ) );
759 out[9] = f( D( {0.0, 0.0, 0.5} ) );
760 out[10] = f( D( {0.5, 0.0, 0.5} ) );
761 out[11] = f( D( {0.0, 0.5, 0.5} ) );
762 out[12] = f( D( {0.5, 0.5, 0.5} ) );
763 out[13] = f( D( {0.5, 0.5, 0.0} ) );
768 DUNE_THROW(NotImplemented,
"LagrangePyramidLocalInterpolation not implemented for order " << k);
807 template<
class D,
class R,
int k>
814 Impl::LagrangePyramidLocalCoefficients<k>,
815 Impl::LagrangePyramidLocalInterpolation<Impl::LagrangePyramidLocalBasis<D,R,k> > >;
828 return coefficients_;
835 return interpolation_;
839 static constexpr std::size_t
size ()
841 return Impl::LagrangePyramidLocalBasis<D,R,k>::size();
852 Impl::LagrangePyramidLocalBasis<D,R,k> basis_;
853 Impl::LagrangePyramidLocalCoefficients<k> coefficients_;
854 Impl::LagrangePyramidLocalInterpolation<Impl::LagrangePyramidLocalBasis<D,R,k> > interpolation_;
Unique label for each type of entities that can occur in DUNE grids.
Definition: type.hh:114
Lagrange finite element for 3d pyramids with compile-time polynomial order.
Definition: lagrangepyramid.hh:809
const Traits::LocalCoefficientsType & localCoefficients() const
Returns the assignment of the degrees of freedom to the element subentities.
Definition: lagrangepyramid.hh:826
static constexpr std::size_t size()
The number of shape functions.
Definition: lagrangepyramid.hh:839
static constexpr GeometryType type()
The reference element that the local finite element is defined on.
Definition: lagrangepyramid.hh:846
const Traits::LocalBasisType & localBasis() const
Returns the local basis, i.e., the set of shape functions.
Definition: lagrangepyramid.hh:819
const Traits::LocalInterpolationType & localInterpolation() const
Returns object that evaluates degrees of freedom.
Definition: lagrangepyramid.hh:833
Implements a matrix constructed from a given type representing a field and compile-time given number ...
Implements a vector constructed from a given type representing a field and a compile-time given size.
static void interpolate(const GridFunction &u, DiscreteFunction &v)
perform native interpolation of a discrete function space
Definition: interpolate.hh:54
#define DUNE_THROW(E, m)
Definition: exceptions.hh:218
constexpr GeometryType pyramid
GeometryType representing a 3D pyramid.
Definition: type.hh:522
constexpr GeometryType vertex
GeometryType representing a vertex.
Definition: type.hh:492
constexpr T accumulate(Range &&range, T value, F &&f)
Accumulate values.
Definition: hybridutilities.hh:279
Some useful basic math stuff.
Dune namespace.
Definition: alignedallocator.hh:13
constexpr std::integral_constant< std::size_t, sizeof...(II)> size(std::integer_sequence< T, II... >)
Return the size of the sequence.
Definition: integersequence.hh:75
constexpr Base power(Base m, Exponent p)
Power method for integer exponents.
Definition: math.hh:75
static const ReferenceElement & general(const GeometryType &type)
get general reference elements
Definition: referenceelements.hh:156
D DomainType
domain type
Definition: localbasis.hh:43
traits helper struct
Definition: localfiniteelementtraits.hh:13
LB LocalBasisType
Definition: localfiniteelementtraits.hh:16
LC LocalCoefficientsType
Definition: localfiniteelementtraits.hh:20
LI LocalInterpolationType
Definition: localfiniteelementtraits.hh:24