integral.cc
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1// LIC// ====================================================================
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3// LIC// multi-physics finite-element library, available
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6// LIC// Copyright (C) 2006-2024 Matthias Heil and Andrew Hazel
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24// LIC//
25// LIC//====================================================================
26// Static data for the Gaussian integration rules
27
28// oomph-lib header
29#include "integral.h"
30
31namespace oomph
32{
33 // Need to define the static data here
34
35 //============================================================
36 // Define the positions and weights of the 1D Gauss points
37 //============================================================
38
39 //------------------------------------------------------------
40 // N=2
41 //------------------------------------------------------------
42 const double Gauss<1, 2>::Knot[2][1] = {{-0.577350269189626},
43 {0.577350269189626}};
44
45 const double Gauss<1, 2>::Weight[2] = {1.0, 1.0};
46
47 //------------------------------------------------------------
48 // N=3
49 //------------------------------------------------------------
50 const double Gauss<1, 3>::Knot[3][1] = {
51 {-0.774596669241483}, {0.0}, {0.774596669241483}};
52
53 const double Gauss<1, 3>::Weight[3] = {(5.0 / 9.0), (8.0 / 9.0), (5.0 / 9.0)};
54
55 //------------------------------------------------------------
56 // N=4
57 //------------------------------------------------------------
58 const double Gauss<1, 4>::Knot[4][1] = {{-0.861136311594053},
59 {-0.339981043584856},
60 {0.339981043584856},
61 {0.861136311594053}};
62
63 const double Gauss<1, 4>::Weight[4] = {
64 0.347854845137448, 0.652145154862546, 0.652145154862546, 0.347854845137448};
65
66
67 //============================================================
68 // Define the positions and weights of the 2D Gauss points
69 //============================================================
70
71 //------------------------------------------------------------
72 // N=2
73 //------------------------------------------------------------
74 const double Gauss<2, 2>::Knot[4][2] = {
75 {-0.577350269189626, -0.577350269189626},
76 {-0.577350269189626, 0.577350269189626},
77 {0.577350269189626, -0.577350269189626},
78 {0.577350269189626, 0.577350269189626}};
79 const double Gauss<2, 2>::Weight[4] = {1.0, 1.0, 1.0, 1.0};
80
81 //------------------------------------------------------------
82 // N=3
83 //------------------------------------------------------------
84 const double Gauss<2, 3>::Knot[9][2] = {
85 {-0.774596669241483, -0.774596669241483},
86 {-0.774596669241483, 0.0},
87 {-0.774596669241483, 0.774596662941483},
88 {0.0, -0.774596669241483},
89 {0.0, 0.0},
90 {0.0, 0.774596662941483},
91 {0.774596662941483, -0.774596669241483},
92 {0.774596662941483, 0.0},
93 {0.774596662941483, 0.774596662941483}};
94 const double Gauss<2, 3>::Weight[9] = {(25.0 / 81.0),
95 (40.0 / 81.0),
96 (25.0 / 81.0),
97 (40.0 / 81.0),
98 (64.0 / 81.0),
99 (40.0 / 81.0),
100 (25.0 / 81.0),
101 (40.0 / 81.0),
102 (25.0 / 81.0)};
103
104 //------------------------------------------------------------
105 // N=4
106 //------------------------------------------------------------
107 const double Gauss<2, 4>::Knot[16][2] = {
108 {-0.861136311594053, -0.861136311594053},
109 {-0.339981043584856, -0.861136311594053},
110 {0.339981043584856, -0.861136311594053},
111 {0.861136311594053, -0.861136311594053},
112
113 {-0.861136311594053, -0.339981043584856},
114 {-0.339981043584856, -0.339981043584856},
115 {0.339981043584856, -0.339981043584856},
116 {0.861136311594053, -0.339981043584856},
117
118 {-0.861136311594053, 0.339981043584856},
119 {-0.339981043584856, 0.339981043584856},
120 {0.339981043584856, 0.339981043584856},
121 {0.861136311594053, 0.339981043584856},
122
123 {-0.861136311594053, 0.861136311594053},
124 {-0.339981043584856, 0.861136311594053},
125 {0.339981043584856, 0.861136311594053},
126 {0.861136311594053, 0.861136311594053}};
127
128 // Quick sanity check: they sum to 4 :)
129 const double Gauss<2, 4>::Weight[16] = {0.1210029932855979,
130 0.2268518518518480,
131 0.2268518518518480,
132 0.1210029932855979,
133 0.2268518518518480,
134 0.4252933030106941,
135 0.4252933030106941,
136 0.2268518518518480,
137 0.2268518518518480,
138 0.4252933030106941,
139 0.4252933030106941,
140 0.2268518518518480,
141 0.1210029932855979,
142 0.2268518518518480,
143 0.2268518518518480,
144 0.1210029932855979};
145
146
147 //============================================================
148 // Define the positions and weights of the 3D Gauss points
149 // (produced with utilities/gauss_weights.cc)
150 //============================================================
151
152 //------------------------------------------------------------
153 // N=2
154 //------------------------------------------------------------
155 const double Gauss<3, 2>::Knot[8][3] = {
156 {-0.57735026918963, -0.57735026918963, -0.57735026918963},
157 {-0.57735026918963, -0.57735026918963, 0.57735026918963},
158 {-0.57735026918963, 0.57735026918963, -0.57735026918963},
159 {-0.57735026918963, 0.57735026918963, 0.57735026918963},
160 {0.57735026918963, -0.57735026918963, -0.57735026918963},
161 {0.57735026918963, -0.57735026918963, 0.57735026918963},
162 {0.57735026918963, 0.57735026918963, -0.57735026918963},
163 {0.57735026918963, 0.57735026918963, 0.57735026918963}};
164 const double Gauss<3, 2>::Weight[8] = {1, 1, 1, 1, 1, 1, 1, 1};
165
166 //------------------------------------------------------------
167 // N=3
168 //------------------------------------------------------------
169 const double Gauss<3, 3>::Knot[27][3] = {
170 {-0.77459666924148, -0.77459666924148, -0.77459666924148},
171 {-0.77459666924148, -0.77459666924148, 0},
172 {-0.77459666924148, -0.77459666924148, 0.77459666924148},
173 {-0.77459666924148, 0, -0.77459666924148},
174 {-0.77459666924148, 0, 0},
175 {-0.77459666924148, 0, 0.77459666924148},
176 {-0.77459666924148, 0.77459666924148, -0.77459666924148},
177 {-0.77459666924148, 0.77459666924148, 0},
178 {-0.77459666924148, 0.77459666924148, 0.77459666924148},
179 {0, -0.77459666924148, -0.77459666924148},
180 {0, -0.77459666924148, 0},
181 {0, -0.77459666924148, 0.77459666924148},
182 {0, 0, -0.77459666924148},
183 {0, 0, 0},
184 {0, 0, 0.77459666924148},
185 {0, 0.77459666924148, -0.77459666924148},
186 {0, 0.77459666924148, 0},
187 {0, 0.77459666924148, 0.77459666924148},
188 {0.77459666924148, -0.77459666924148, -0.77459666924148},
189 {0.77459666924148, -0.77459666924148, 0},
190 {0.77459666924148, -0.77459666924148, 0.77459666924148},
191 {0.77459666924148, 0, -0.77459666924148},
192 {0.77459666924148, 0, 0},
193 {0.77459666924148, 0, 0.77459666924148},
194 {0.77459666924148, 0.77459666924148, -0.77459666924148},
195 {0.77459666924148, 0.77459666924148, 0},
196 {0.77459666924148, 0.77459666924148, 0.77459666924148}};
197
198
199 const double Gauss<3, 3>::Weight[27] = {
200 0.17146776406035, 0.27434842249657, 0.17146776406035, 0.27434842249657,
201 0.43895747599451, 0.27434842249657, 0.17146776406035, 0.27434842249657,
202 0.17146776406035, 0.27434842249657, 0.43895747599451, 0.27434842249657,
203 0.43895747599451, 0.70233196159122, 0.43895747599451, 0.27434842249657,
204 0.43895747599451, 0.27434842249657, 0.17146776406035, 0.27434842249657,
205 0.17146776406035, 0.27434842249657, 0.43895747599451, 0.27434842249657,
206 0.17146776406035, 0.27434842249657, 0.17146776406035};
207
208
209 //------------------------------------------------------------
210 // N=4
211 //------------------------------------------------------------
212 const double Gauss<3, 4>::Knot[64][3] = {
213 {-0.86113631159405, -0.86113631159405, -0.86113631159405},
214 {-0.86113631159405, -0.86113631159405, -0.33998104358486},
215 {-0.86113631159405, -0.86113631159405, 0.33998104358486},
216 {-0.86113631159405, -0.86113631159405, 0.86113631159405},
217 {-0.86113631159405, -0.33998104358486, -0.86113631159405},
218 {-0.86113631159405, -0.33998104358486, -0.33998104358486},
219 {-0.86113631159405, -0.33998104358486, 0.33998104358486},
220 {-0.86113631159405, -0.33998104358486, 0.86113631159405},
221 {-0.86113631159405, 0.33998104358486, -0.86113631159405},
222 {-0.86113631159405, 0.33998104358486, -0.33998104358486},
223 {-0.86113631159405, 0.33998104358486, 0.33998104358486},
224 {-0.86113631159405, 0.33998104358486, 0.86113631159405},
225 {-0.86113631159405, 0.86113631159405, -0.86113631159405},
226 {-0.86113631159405, 0.86113631159405, -0.33998104358486},
227 {-0.86113631159405, 0.86113631159405, 0.33998104358486},
228 {-0.86113631159405, 0.86113631159405, 0.86113631159405},
229 {-0.33998104358486, -0.86113631159405, -0.86113631159405},
230 {-0.33998104358486, -0.86113631159405, -0.33998104358486},
231 {-0.33998104358486, -0.86113631159405, 0.33998104358486},
232 {-0.33998104358486, -0.86113631159405, 0.86113631159405},
233 {-0.33998104358486, -0.33998104358486, -0.86113631159405},
234 {-0.33998104358486, -0.33998104358486, -0.33998104358486},
235 {-0.33998104358486, -0.33998104358486, 0.33998104358486},
236 {-0.33998104358486, -0.33998104358486, 0.86113631159405},
237 {-0.33998104358486, 0.33998104358486, -0.86113631159405},
238 {-0.33998104358486, 0.33998104358486, -0.33998104358486},
239 {-0.33998104358486, 0.33998104358486, 0.33998104358486},
240 {-0.33998104358486, 0.33998104358486, 0.86113631159405},
241 {-0.33998104358486, 0.86113631159405, -0.86113631159405},
242 {-0.33998104358486, 0.86113631159405, -0.33998104358486},
243 {-0.33998104358486, 0.86113631159405, 0.33998104358486},
244 {-0.33998104358486, 0.86113631159405, 0.86113631159405},
245 {0.33998104358486, -0.86113631159405, -0.86113631159405},
246 {0.33998104358486, -0.86113631159405, -0.33998104358486},
247 {0.33998104358486, -0.86113631159405, 0.33998104358486},
248 {0.33998104358486, -0.86113631159405, 0.86113631159405},
249 {0.33998104358486, -0.33998104358486, -0.86113631159405},
250 {0.33998104358486, -0.33998104358486, -0.33998104358486},
251 {0.33998104358486, -0.33998104358486, 0.33998104358486},
252 {0.33998104358486, -0.33998104358486, 0.86113631159405},
253 {0.33998104358486, 0.33998104358486, -0.86113631159405},
254 {0.33998104358486, 0.33998104358486, -0.33998104358486},
255 {0.33998104358486, 0.33998104358486, 0.33998104358486},
256 {0.33998104358486, 0.33998104358486, 0.86113631159405},
257 {0.33998104358486, 0.86113631159405, -0.86113631159405},
258 {0.33998104358486, 0.86113631159405, -0.33998104358486},
259 {0.33998104358486, 0.86113631159405, 0.33998104358486},
260 {0.33998104358486, 0.86113631159405, 0.86113631159405},
261 {0.86113631159405, -0.86113631159405, -0.86113631159405},
262 {0.86113631159405, -0.86113631159405, -0.33998104358486},
263 {0.86113631159405, -0.86113631159405, 0.33998104358486},
264 {0.86113631159405, -0.86113631159405, 0.86113631159405},
265 {0.86113631159405, -0.33998104358486, -0.86113631159405},
266 {0.86113631159405, -0.33998104358486, -0.33998104358486},
267 {0.86113631159405, -0.33998104358486, 0.33998104358486},
268 {0.86113631159405, -0.33998104358486, 0.86113631159405},
269 {0.86113631159405, 0.33998104358486, -0.86113631159405},
270 {0.86113631159405, 0.33998104358486, -0.33998104358486},
271 {0.86113631159405, 0.33998104358486, 0.33998104358486},
272 {0.86113631159405, 0.33998104358486, 0.86113631159405},
273 {0.86113631159405, 0.86113631159405, -0.86113631159405},
274 {0.86113631159405, 0.86113631159405, -0.33998104358486},
275 {0.86113631159405, 0.86113631159405, 0.33998104358486},
276 {0.86113631159405, 0.86113631159405, 0.86113631159405}};
277
278
279 const double Gauss<3, 4>::Weight[64] = {
280 0.042091477490529, 0.078911515795068, 0.078911515795068, 0.042091477490529,
281 0.078911515795068, 0.14794033605678, 0.14794033605678, 0.078911515795068,
282 0.078911515795068, 0.14794033605678, 0.14794033605678, 0.078911515795068,
283 0.042091477490529, 0.078911515795068, 0.078911515795068, 0.042091477490529,
284 0.078911515795068, 0.14794033605678, 0.14794033605678, 0.078911515795068,
285 0.14794033605678, 0.27735296695391, 0.27735296695391, 0.14794033605678,
286 0.14794033605678, 0.27735296695391, 0.27735296695391, 0.14794033605678,
287 0.078911515795068, 0.14794033605678, 0.14794033605678, 0.078911515795068,
288 0.078911515795068, 0.14794033605678, 0.14794033605678, 0.078911515795068,
289 0.14794033605678, 0.27735296695391, 0.27735296695391, 0.14794033605678,
290 0.14794033605678, 0.27735296695391, 0.27735296695391, 0.14794033605678,
291 0.078911515795068, 0.14794033605678, 0.14794033605678, 0.078911515795068,
292 0.042091477490529, 0.078911515795068, 0.078911515795068, 0.042091477490529,
293 0.078911515795068, 0.14794033605678, 0.14794033605678, 0.078911515795068,
294 0.078911515795068, 0.14794033605678, 0.14794033605678, 0.078911515795068,
295 0.042091477490529, 0.078911515795068, 0.078911515795068, 0.042091477490529};
296
297 // 1D Triangles, which are just the same as 1D Gauss elements, but scaled
298 // so their coordinate runs from 0 to 1, rather than -1 to 1
299
300 //------------------------------------------------------------
301 // N=2
302 //------------------------------------------------------------
303 const double TGauss<1, 2>::Knot[2][1] = {{0.5 * (-0.577350269189626 + 1.0)},
304 {0.5 * (0.577350269189626 + 1.0)}};
305
306 const double TGauss<1, 2>::Weight[2] = {0.5, 0.5};
307
308 //------------------------------------------------------------
309 // N=3
310 //------------------------------------------------------------
311 const double TGauss<1, 3>::Knot[3][1] = {{0.5 * (-0.774596669241483 + 1.0)},
312 {0.5},
313 {0.5 * (0.774596669241483 + 1.0)}};
314
315 const double TGauss<1, 3>::Weight[3] = {
316 (5.0 / 18.0), (8.0 / 18.0), (5.0 / 18.0)};
317
318 //------------------------------------------------------------
319 // N=4
320 //------------------------------------------------------------
321 const double TGauss<1, 4>::Knot[4][1] = {{0.5 * (-0.861136311594053 + 1.0)},
322 {0.5 * (-0.339981043584856 + 1.0)},
323 {0.5 * (0.339981043584856 + 1.0)},
324 {0.5 * (0.861136311594053 + 1.0)}};
325
326 const double TGauss<1, 4>::Weight[4] = {0.5 * 0.347854845137448,
327 0.5 * 0.652145154862546,
328 0.5 * 0.652145154862546,
329 0.5 * 0.347854845137448};
330
331 //------------------------------------------------------------
332 // N=5
333 //------------------------------------------------------------
334 const double TGauss<1, 5>::Knot[5][1] = {};
335
336 const double TGauss<1, 5>::Weight[5] = {};
337
338 /// / Define the positions and weights of the 2D Gauss points for triangles
339 //
340 //------------------------------------------------------------
341 // "Full integration" weights for linear triangles
342 // accurate up to second order (Bathe p 467)
343 //------------------------------------------------------------
344 const double TGauss<2, 2>::Knot[3][2] = {{0.1666666666667, 0.1666666666667},
345 {0.6666666666667, 0.1666666666667},
346 {0.1666666666667, 0.6666666666667}};
347
348 const double TGauss<2, 2>::Weight[3] = {
349 0.1666666666667, 0.1666666666667, 0.1666666666667};
350
351 //------------------------------------------------------------
352 // "Full integration" weights for quadratic triangles
353 // accurate up to fifth order (Bathe p 467)
354 //------------------------------------------------------------
355 const double TGauss<2, 3>::Knot[7][2] = {{0.1012865073235, 0.1012865073235},
356 {0.7974269853531, 0.1012865073235},
357 {0.1012865073235, 0.7974269853531},
358 {0.4701420641051, 0.0597158717898},
359 {0.4701420641051, 0.4701420641051},
360 {0.0597158717898, 0.4701420641051},
361 {0.3333333333333, 0.3333333333333}};
362
363 const double TGauss<2, 3>::Weight[7] = {0.5 * 0.1259391805448,
364 0.5 * 0.1259391805448,
365 0.5 * 0.1259391805448,
366 0.5 * 0.1323941527885,
367 0.5 * 0.1323941527885,
368 0.5 * 0.1323941527885,
369 0.5 * 0.225};
370
371
372 //------------------------------------------------------------
373 //"Full integration" weights for cubic triangles
374 // accurate up to seventh order (Bathe p 467)
375 //------------------------------------------------------------
376 const double TGauss<2, 4>::Knot[13][2] = {{0.0651301029022, 0.0651301029022},
377 {0.8697397941956, 0.0651301029022},
378 {0.0651301029022, 0.8697397941956},
379 {0.3128654960049, 0.0486903154253},
380 {0.6384441885698, 0.3128654960049},
381 {0.0486903154253, 0.6384441885698},
382 {0.6384441885698, 0.0486903154253},
383 {0.3128654960049, 0.6384441885698},
384 {0.0486903154253, 0.3128654960049},
385 {0.2603459660790, 0.2603459660790},
386 {0.4793080678419, 0.2603459660790},
387 {0.2603459660790, 0.4793080678419},
388 {0.3333333333333, 0.3333333333333}};
389
390
391 const double TGauss<2, 4>::Weight[13] = {0.5 * 0.0533472356088,
392 0.5 * 0.0533472356088,
393 0.5 * 0.0533472356088,
394 0.5 * 0.0771137608903,
395 0.5 * 0.0771137608903,
396 0.5 * 0.0771137608903,
397 0.5 * 0.0771137608903,
398 0.5 * 0.0771137608903,
399 0.5 * 0.0771137608903,
400 0.5 * 0.1756152574332,
401 0.5 * 0.1756152574332,
402 0.5 * 0.1756152574332,
403 0.5 * -0.1495700444677};
404
405 //------------------------------------------------------------
406 //"Full integration" weights for 2D triangles
407 // accurate up to order 11
408 // http://people.sc.fsu.edu/~jburkardt/datasets/quadrature_rules_tri/quadrature_rules_tri.html
409 // [TOMS706_37, order 37, degree of precision 13, a rule from ACM TOMS
410 // algorithm #706.]
411 //------------------------------------------------------------
412 const double TGauss<2, 13>::Knot[37][2] = {
413 {0.333333333333333333333333333333, 0.333333333333333333333333333333},
414 {0.950275662924105565450352089520, 0.024862168537947217274823955239},
415 {0.024862168537947217274823955239, 0.950275662924105565450352089520},
416 {0.024862168537947217274823955239, 0.024862168537947217274823955239},
417 {0.171614914923835347556304795551, 0.414192542538082326221847602214},
418 {0.414192542538082326221847602214, 0.171614914923835347556304795551},
419 {0.414192542538082326221847602214, 0.414192542538082326221847602214},
420 {0.539412243677190440263092985511, 0.230293878161404779868453507244},
421 {0.230293878161404779868453507244, 0.539412243677190440263092985511},
422 {0.230293878161404779868453507244, 0.230293878161404779868453507244},
423 {0.772160036676532561750285570113, 0.113919981661733719124857214943},
424 {0.113919981661733719124857214943, 0.772160036676532561750285570113},
425 {0.113919981661733719124857214943, 0.113919981661733719124857214943},
426 {0.009085399949835353883572964740, 0.495457300025082323058213517632},
427 {0.495457300025082323058213517632, 0.009085399949835353883572964740},
428 {0.495457300025082323058213517632, 0.495457300025082323058213517632},
429 {0.062277290305886993497083640527, 0.468861354847056503251458179727},
430 {0.468861354847056503251458179727, 0.062277290305886993497083640527},
431 {0.468861354847056503251458179727, 0.468861354847056503251458179727},
432 {0.022076289653624405142446876931, 0.851306504174348550389457672223},
433 {0.022076289653624405142446876931, 0.126617206172027096933163647918},
434 {0.851306504174348550389457672223, 0.022076289653624405142446876931},
435 {0.851306504174348550389457672223, 0.126617206172027096933163647918},
436 {0.126617206172027096933163647918, 0.022076289653624405142446876931},
437 {0.126617206172027096933163647918, 0.851306504174348550389457672223},
438 {0.018620522802520968955913511549, 0.689441970728591295496647976487},
439 {0.018620522802520968955913511549, 0.291937506468887771754472382212},
440 {0.689441970728591295496647976487, 0.018620522802520968955913511549},
441 {0.689441970728591295496647976487, 0.291937506468887771754472382212},
442 {0.291937506468887771754472382212, 0.018620522802520968955913511549},
443 {0.291937506468887771754472382212, 0.689441970728591295496647976487},
444 {0.096506481292159228736516560903, 0.635867859433872768286976979827},
445 {0.096506481292159228736516560903, 0.267625659273967961282458816185},
446 {0.635867859433872768286976979827, 0.096506481292159228736516560903},
447 {0.635867859433872768286976979827, 0.267625659273967961282458816185},
448 {0.267625659273967961282458816185, 0.096506481292159228736516560903},
449 {0.267625659273967961282458816185, 0.635867859433872768286976979827}};
450
451 // Modified by DR 2017 to include factor 1/2
452 const double TGauss<2, 13>::Weight[37] = {
453 0.5 * 0.051739766065744133555179145422,
454 0.5 * 0.008007799555564801597804123460,
455 0.5 * 0.008007799555564801597804123460,
456 0.5 * 0.008007799555564801597804123460,
457 0.5 * 0.046868898981821644823226732071,
458 0.5 * 0.046868898981821644823226732071,
459 0.5 * 0.046868898981821644823226732071,
460 0.5 * 0.046590940183976487960361770070,
461 0.5 * 0.046590940183976487960361770070,
462 0.5 * 0.046590940183976487960361770070,
463 0.5 * 0.031016943313796381407646220131,
464 0.5 * 0.031016943313796381407646220131,
465 0.5 * 0.031016943313796381407646220131,
466 0.5 * 0.010791612736631273623178240136,
467 0.5 * 0.010791612736631273623178240136,
468 0.5 * 0.010791612736631273623178240136,
469 0.5 * 0.032195534242431618819414482205,
470 0.5 * 0.032195534242431618819414482205,
471 0.5 * 0.032195534242431618819414482205,
472 0.5 * 0.015445834210701583817692900053,
473 0.5 * 0.015445834210701583817692900053,
474 0.5 * 0.015445834210701583817692900053,
475 0.5 * 0.015445834210701583817692900053,
476 0.5 * 0.015445834210701583817692900053,
477 0.5 * 0.015445834210701583817692900053,
478 0.5 * 0.017822989923178661888748319485,
479 0.5 * 0.017822989923178661888748319485,
480 0.5 * 0.017822989923178661888748319485,
481 0.5 * 0.017822989923178661888748319485,
482 0.5 * 0.017822989923178661888748319485,
483 0.5 * 0.017822989923178661888748319485,
484 0.5 * 0.037038683681384627918546472190,
485 0.5 * 0.037038683681384627918546472190,
486 0.5 * 0.037038683681384627918546472190,
487 0.5 * 0.037038683681384627918546472190,
488 0.5 * 0.037038683681384627918546472190,
489 0.5 * 0.037038683681384627918546472190};
490
491 //------------------------------------------------------------
492 //"Full integration" weights for 2D triangles
493 // accurate up to order 16
494 // http://people.sc.fsu.edu/~jburkardt/datasets/quadrature_rules_tri/
495 // quadrature_rules_tri.html
496 // This uses the order 16 Dunavant rule, computed using software available
497 // from
498 // https://people.sc.fsu.edu/~jburkardt/cpp_src/triangle_dunavant_rule/triangle_dunavant_rule.html
499 // This integegration scheme is accurate up to order 16 and uses 52 points
500 //------------------------------------------------------------
501 const double TGauss<2, 16>::Knot[52][2] = {
502 {0.333333333333333, 0.333333333333333},
503 {0.005238916103123, 0.497380541948438},
504 {0.497380541948438, 0.497380541948438},
505 {0.497380541948438, 0.005238916103123},
506 {0.173061122901295, 0.413469438549352},
507 {0.413469438549352, 0.413469438549352},
508 {0.413469438549352, 0.173061122901295},
509 {0.059082801866017, 0.470458599066991},
510 {0.470458599066991, 0.470458599066991},
511 {0.470458599066991, 0.059082801866017},
512 {0.518892500060958, 0.240553749969521},
513 {0.240553749969521, 0.240553749969521},
514 {0.240553749969521, 0.518892500060958},
515 {0.704068411554854, 0.147965794222573},
516 {0.147965794222573, 0.147965794222573},
517 {0.147965794222573, 0.704068411554854},
518 {0.849069624685052, 0.07546518765747399},
519 {0.07546518765747399, 0.07546518765747399},
520 {0.07546518765747399, 0.849069624685052},
521 {0.96680719475395, 0.016596402623025},
522 {0.016596402623025, 0.016596402623025},
523 {0.016596402623025, 0.96680719475395},
524 {0.103575692245252, 0.296555596579887},
525 {0.296555596579887, 0.599868711174861},
526 {0.599868711174861, 0.103575692245252},
527 {0.296555596579887, 0.103575692245252},
528 {0.599868711174861, 0.296555596579887},
529 {0.103575692245252, 0.599868711174861},
530 {0.020083411655416, 0.337723063403079},
531 {0.337723063403079, 0.6421935249415049},
532 {0.6421935249415049, 0.020083411655416},
533 {0.337723063403079, 0.020083411655416},
534 {0.6421935249415049, 0.337723063403079},
535 {0.020083411655416, 0.6421935249415049},
536 {-0.004341002614139, 0.204748281642812},
537 {0.204748281642812, 0.7995927209713271},
538 {0.7995927209713271, -0.004341002614139},
539 {0.204748281642812, -0.004341002614139},
540 {0.7995927209713271, 0.204748281642812},
541 {-0.004341002614139, 0.7995927209713271},
542 {0.04194178646801, 0.189358492130623},
543 {0.189358492130623, 0.768699721401368},
544 {0.768699721401368, 0.04194178646801},
545 {0.189358492130623, 0.04194178646801},
546 {0.768699721401368, 0.189358492130623},
547 {0.04194178646801, 0.768699721401368},
548 {0.014317320230681, 0.08528361568265699},
549 {0.08528361568265699, 0.900399064086661},
550 {0.900399064086661, 0.014317320230681},
551 {0.08528361568265699, 0.014317320230681},
552 {0.900399064086661, 0.08528361568265699},
553 {0.014317320230681, 0.900399064086661}};
554 // Modified by AR 2023 to include factor 1/2
555 const double TGauss<2, 16>::Weight[52] = {
556 0.5 * 0.046875697427642, 0.5 * 0.006405878578585, 0.5 * 0.006405878578585,
557 0.5 * 0.006405878578585, 0.5 * 0.041710296739387, 0.5 * 0.041710296739387,
558 0.5 * 0.041710296739387, 0.5 * 0.026891484250064, 0.5 * 0.026891484250064,
559 0.5 * 0.026891484250064, 0.5 * 0.04213252276165, 0.5 * 0.04213252276165,
560 0.5 * 0.04213252276165, 0.5 * 0.030000266842773, 0.5 * 0.030000266842773,
561 0.5 * 0.030000266842773, 0.5 * 0.014200098925024, 0.5 * 0.014200098925024,
562 0.5 * 0.014200098925024, 0.5 * 0.003582462351273, 0.5 * 0.003582462351273,
563 0.5 * 0.003582462351273, 0.5 * 0.032773147460627, 0.5 * 0.032773147460627,
564 0.5 * 0.032773147460627, 0.5 * 0.032773147460627, 0.5 * 0.032773147460627,
565 0.5 * 0.032773147460627, 0.5 * 0.015298306248441, 0.5 * 0.015298306248441,
566 0.5 * 0.015298306248441, 0.5 * 0.015298306248441, 0.5 * 0.015298306248441,
567 0.5 * 0.015298306248441, 0.5 * 0.002386244192839, 0.5 * 0.002386244192839,
568 0.5 * 0.002386244192839, 0.5 * 0.002386244192839, 0.5 * 0.002386244192839,
569 0.5 * 0.002386244192839, 0.5 * 0.019084792755899, 0.5 * 0.019084792755899,
570 0.5 * 0.019084792755899, 0.5 * 0.019084792755899, 0.5 * 0.019084792755899,
571 0.5 * 0.019084792755899, 0.5 * 0.006850054546542, 0.5 * 0.006850054546542,
572 0.5 * 0.006850054546542, 0.5 * 0.006850054546542, 0.5 * 0.006850054546542,
573 0.5 * 0.006850054546542};
574
575#ifdef PARANOID
576 // Initialise the warning flag that the TGauss<2, 16> scheme employs
577 // (see declaration in integral.h for more info)
578 bool TGauss<2, 16>::User_has_been_warned = false;
579#endif
580
581 //------------------------------------------------------------
582 //"Full integration" weights for 2D triangles
583 // accurate up to order 15
584 // http://people.sc.fsu.edu/~jburkardt/datasets/quadrature_rules_tri/quadrature_rules_tri.html
585 // [GAUSS8X8, order 64, degree of precision 15, (essentially a product of two
586 // 8 point 1D Gauss-Legendre rules).]
587 //------------------------------------------------------------
588 const double TGauss<2, 5>::Knot[64][2] = {
589 {0.9553660447100000, 0.8862103848242247e-3},
590 {0.9553660447100000, 0.4537789678039195e-2},
591 {0.9553660447100000, 0.1058868260117431e-1},
592 {0.9553660447100000, 0.1822327082910602e-1},
593 {0.9553660447100000, 0.2641068446089399e-1},
594 {0.9553660447100000, 0.3404527268882569e-1},
595 {0.9553660447100000, 0.4009616561196080e-1},
596 {0.9553660447100000, 0.4374774490517578e-1},
597 {0.8556337429600001, 0.2866402391985981e-2},
598 {0.8556337429600001, 0.1467724979327651e-1},
599 {0.8556337429600001, 0.3424855503358430e-1},
600 {0.8556337429600001, 0.5894224214571626e-1},
601 {0.8556337429600001, 0.8542401489428375e-1},
602 {0.8556337429600001, 0.1101177020064157},
603 {0.8556337429600001, 0.1296890072467235},
604 {0.8556337429600001, 0.1414998546480140},
605 {0.7131752428600000, 0.5694926133044352e-2},
606 {0.7131752428600000, 0.2916054411712861e-1},
607 {0.7131752428600000, 0.6804452564827500e-1},
608 {0.7131752428600000, 0.1171055801775613},
609 {0.7131752428600000, 0.1697191769624387},
610 {0.7131752428600000, 0.2187802314917250},
611 {0.7131752428600000, 0.2576642130228714},
612 {0.7131752428600000, 0.2811298310069557},
613 {0.5451866848000000, 0.9030351006711630e-2},
614 {0.5451866848000000, 0.4623939674940125e-1},
615 {0.5451866848000000, 0.1078970888004545},
616 {0.5451866848000000, 0.1856923986620134},
617 {0.5451866848000000, 0.2691209165379867},
618 {0.5451866848000000, 0.3469162263995455},
619 {0.5451866848000000, 0.4085739184505988},
620 {0.5451866848000000, 0.4457829641932884},
621 {0.3719321645800000, 0.1247033193690498e-1},
622 {0.3719321645800000, 0.6385362269957356e-1},
623 {0.3719321645800000, 0.1489989161403976},
624 {0.3719321645800000, 0.2564292182833579},
625 {0.3719321645800000, 0.3716386171366422},
626 {0.3719321645800000, 0.4790689192796024},
627 {0.3719321645800000, 0.5642142127204264},
628 {0.3719321645800000, 0.6155975034830951},
629 {0.2143084794000000, 0.1559996151584746e-1},
630 {0.2143084794000000, 0.7987871227492103e-1},
631 {0.2143084794000000, 0.1863925811641285},
632 {0.2143084794000000, 0.3207842387034378},
633 {0.2143084794000000, 0.4649072818965623},
634 {0.2143084794000000, 0.5992989394358715},
635 {0.2143084794000000, 0.7058128083250790},
636 {0.2143084794000000, 0.7700915590841526},
637 {0.9132360790000005e-1, 0.1804183496379599e-1},
638 {0.9132360790000005e-1, 0.9238218584838476e-1},
639 {0.9132360790000005e-1, 0.2155687489628060},
640 {0.9132360790000005e-1, 0.3709968314854498},
641 {0.9132360790000005e-1, 0.5376795606145502},
642 {0.9132360790000005e-1, 0.6931076431371940},
643 {0.9132360790000005e-1, 0.8162942062516152},
644 {0.9132360790000005e-1, 0.8906345571362040},
645 {0.1777991514999999e-1, 0.1950205026019779e-1},
646 {0.1777991514999999e-1, 0.9985913490381848e-1},
647 {0.1777991514999999e-1, 0.2330157982952792},
648 {0.1777991514999999e-1, 0.4010234473667467},
649 {0.1777991514999999e-1, 0.5811966374832533},
650 {0.1777991514999999e-1, 0.7492042865547208},
651 {0.1777991514999999e-1, 0.8823609499461815},
652 {0.1777991514999999e-1, 0.9627180345898023}};
653 // Modified by DR 2017 to include factor 1/2
654 const double TGauss<2, 5>::Weight[64] = {
655 0.5 * 0.3335674062677772e-3, 0.5 * 0.7327880811491046e-3,
656 0.5 * 0.1033723454167925e-2, 0.5 * 0.1195112498415193e-2,
657 0.5 * 0.1195112498415193e-2, 0.5 * 0.1033723454167925e-2,
658 0.5 * 0.7327880811491046e-3, 0.5 * 0.3335674062677772e-3,
659 0.5 * 0.1806210919443461e-2, 0.5 * 0.3967923151181667e-2,
660 0.5 * 0.5597437146194232e-2, 0.5 * 0.6471331443180639e-2,
661 0.5 * 0.6471331443180639e-2, 0.5 * 0.5597437146194232e-2,
662 0.5 * 0.3967923151181667e-2, 0.5 * 0.1806210919443461e-2,
663 0.5 * 0.4599755803015752e-2, 0.5 * 0.1010484287526739e-1,
664 0.5 * 0.1425461651131868e-1, 0.5 * 0.1648010431039818e-1,
665 0.5 * 0.1648010431039818e-1, 0.5 * 0.1425461651131868e-1,
666 0.5 * 0.1010484287526739e-1, 0.5 * 0.4599755803015752e-2,
667 0.5 * 0.8017259531156730e-2, 0.5 * 0.1761248886287915e-1,
668 0.5 * 0.2484544071087993e-1, 0.5 * 0.2872441038508419e-1,
669 0.5 * 0.2872441038508419e-1, 0.5 * 0.2484544071087993e-1,
670 0.5 * 0.1761248886287915e-1, 0.5 * 0.8017259531156730e-2,
671 0.5 * 0.1073501897357062e-1, 0.5 * 0.2358292149331603e-1,
672 0.5 * 0.3326776143412911e-1, 0.5 * 0.3846165753898425e-1,
673 0.5 * 0.3846165753898425e-1, 0.5 * 0.3326776143412911e-1,
674 0.5 * 0.2358292149331603e-1, 0.5 * 0.1073501897357062e-1,
675 0.5 * 0.1138879740452669e-1, 0.5 * 0.2501915606814251e-1,
676 0.5 * 0.3529381699354388e-1, 0.5 * 0.4080402900378691e-1,
677 0.5 * 0.4080402900378691e-1, 0.5 * 0.3529381699354388e-1,
678 0.5 * 0.2501915606814251e-1, 0.5 * 0.1138879740452669e-1,
679 0.5 * 0.9223845391285393e-2, 0.5 * 0.2026314273544469e-1,
680 0.5 * 0.2858464328177232e-1, 0.5 * 0.3304739223149761e-1,
681 0.5 * 0.3304739223149761e-1, 0.5 * 0.2858464328177232e-1,
682 0.5 * 0.2026314273544469e-1, 0.5 * 0.9223845391285393e-2,
683 0.5 * 0.4509812715921713e-2, 0.5 * 0.9907253959306707e-2,
684 0.5 * 0.1397588340693756e-1, 0.5 * 0.1615785427783403e-1,
685 0.5 * 0.1615785427783403e-1, 0.5 * 0.1397588340693756e-1,
686 0.5 * 0.9907253959306707e-2, 0.5 * 0.4509812715921713e-2};
687
688 //------------------------------------------------------------
689 //"Full integration" weights for 2D triangles
690 // accurate up to points 19, degree of precision 8, a rule from ACM TOMS
691 // algorithm #584.
692 // http://people.sc.fsu.edu/~jburkardt/datasets/quadrature_rules_tri/quadrature_rules_tri.html
693 // TOMS589_19 a 19 point Gauss rule accurate to order 8
694 //------------------------------------------------------------
695 const double TGauss<2, 9>::Knot[19][2] = {
696 {0.3333333333333333, 0.3333333333333333},
697 {0.7974269853530872, 0.1012865073234563},
698 {0.1012865073234563, 0.7974269853530872},
699 {0.1012865073234563, 0.1012865073234563},
700 {0.0597158717897698, 0.4701420641051151},
701 {0.4701420641051151, 0.0597158717897698},
702 {0.4701420641051151, 0.4701420641051151},
703 {0.5357953464498992, 0.2321023267750504},
704 {0.2321023267750504, 0.5357953464498992},
705 {0.2321023267750504, 0.2321023267750504},
706 {0.9410382782311209, 0.0294808608844396},
707 {0.0294808608844396, 0.9410382782311209},
708 {0.0294808608844396, 0.0294808608844396},
709 {0.7384168123405100, 0.2321023267750504},
710 {0.7384168123405100, 0.0294808608844396},
711 {0.2321023267750504, 0.7384168123405100},
712 {0.2321023267750504, 0.0294808608844396},
713 {0.0294808608844396, 0.7384168123405100},
714 {0.0294808608844396, 0.2321023267750504}};
715
716 const double TGauss<2, 9>::Weight[19] = {2 * 0.0378610912003147,
717 2 * 0.0376204254131829,
718 2 * 0.0376204254131829,
719 2 * 0.0376204254131829,
720 2 * 0.0783573522441174,
721 2 * 0.0783573522441174,
722 2 * 0.0783573522441174,
723 2 * 0.1162714796569659,
724 2 * 0.1162714796569659,
725 2 * 0.1162714796569659,
726 2 * 0.0134442673751655,
727 2 * 0.0134442673751655,
728 2 * 0.0134442673751655,
729 2 * 0.0375097224552317,
730 2 * 0.0375097224552317,
731 2 * 0.0375097224552317,
732 2 * 0.0375097224552317,
733 2 * 0.0375097224552317,
734 2 * 0.0375097224552317};
735
736 //------------------------------------------------------------
737 /// / Define the positions and weights of the 3D Gauss points for tets
738
739 //------------------------------------------------------------
740 // "Full integration" weights for linear tets
741 // accurate up to second order (e.g. from German Zienkiwicz p 200)
742 //------------------------------------------------------------
743 const double TGauss<3, 2>::Knot[4][3] = {
744 {0.138196601125011, 0.138196601125011, 0.585410196624969},
745 {0.138196601125011, 0.585410196624969, 0.138196601125011},
746 {0.585410196624969, 0.138196601125011, 0.138196601125011},
747 {0.138196601125011, 0.138196601125011, 0.138196601125011}};
748
749
750 const double TGauss<3, 2>::Weight[4] = {
751 0.0416666666667, 0.0416666666667, 0.0416666666667, 0.0416666666667};
752
753
754 //------------------------------------------------------------
755 //"Full integration" weights for quadratic tets
756 // accurate up to fifth order
757 // The numbers are from Keast CMAME 55 pp339-348 (1986)
758 //------------------------------------------------------------
759 const double TGauss<3, 3>::Knot[11][3] = {
760 {0.25, 0.25, 0.25},
761 {0.785714285714286, 0.071428571428571, 0.071428571428571},
762 {0.071428571428571, 0.071428571428571, 0.071428571428571},
763 {0.071428571428571, 0.785714285714286, 0.071428571428571},
764 {0.071428571428571, 0.071428571428571, 0.785714285714286},
765 {0.399403576166799, 0.399403576166799, 0.100596423833201},
766 {0.399403576166799, 0.100596423833201, 0.399403576166799},
767 {0.100596423833201, 0.399403576166799, 0.399403576166799},
768 {0.399403576166799, 0.100596423833201, 0.100596423833201},
769 {0.100596423833201, 0.399403576166799, 0.100596423833201},
770 {0.100596423833201, 0.100596423833201, 0.399403576166799}};
771
772
773 const double TGauss<3, 3>::Weight[11] = {-0.01315555555556,
774 0.00762222222222,
775 0.00762222222222,
776 0.00762222222222,
777 0.00762222222222,
778 0.02488888888889,
779 0.02488888888889,
780 0.02488888888889,
781 0.02488888888889,
782 0.02488888888889,
783 0.02488888888889};
784
785 //------------------------------------------------------------
786 //"Full integration" weights for quartic tets
787 // accurate up to eighth order
788 // The numbers are from Keast CMAME 55 pp339-348 (1986)
789 //------------------------------------------------------------
790 const double TGauss<3, 5>::Knot[45][3] = {
791 {2.50000000000000000e-01, 2.50000000000000000e-01, 2.50000000000000000e-01},
792 {1.27470936566639015e-01, 1.27470936566639015e-01, 1.27470936566639015e-01},
793 {1.27470936566639015e-01, 1.27470936566639015e-01, 6.17587190300082967e-01},
794 {1.27470936566639015e-01, 6.17587190300082967e-01, 1.27470936566639015e-01},
795 {6.17587190300082967e-01, 1.27470936566639015e-01, 1.27470936566639015e-01},
796 {3.20788303926322960e-02, 3.20788303926322960e-02, 3.20788303926322960e-02},
797 {3.20788303926322960e-02, 3.20788303926322960e-02, 9.03763508822103123e-01},
798 {3.20788303926322960e-02, 9.03763508822103123e-01, 3.20788303926322960e-02},
799 {9.03763508822103123e-01, 3.20788303926322960e-02, 3.20788303926322960e-02},
800 {4.97770956432810185e-02, 4.97770956432810185e-02, 4.50222904356718978e-01},
801 {4.97770956432810185e-02, 4.50222904356718978e-01, 4.50222904356718978e-01},
802 {4.50222904356718978e-01, 4.50222904356718978e-01, 4.97770956432810185e-02},
803 {4.50222904356718978e-01, 4.97770956432810185e-02, 4.97770956432810185e-02},
804 {4.97770956432810185e-02, 4.50222904356718978e-01, 4.97770956432810185e-02},
805 {4.50222904356718978e-01, 4.97770956432810185e-02, 4.50222904356718978e-01},
806 {1.83730447398549945e-01, 1.83730447398549945e-01, 3.16269552601450060e-01},
807 {1.83730447398549945e-01, 3.16269552601450060e-01, 3.16269552601450060e-01},
808 {3.16269552601450060e-01, 3.16269552601450060e-01, 1.83730447398549945e-01},
809 {3.16269552601450060e-01, 1.83730447398549945e-01, 1.83730447398549945e-01},
810 {1.83730447398549945e-01, 3.16269552601450060e-01, 1.83730447398549945e-01},
811 {3.16269552601450060e-01, 1.83730447398549945e-01, 3.16269552601450060e-01},
812 {2.31901089397150906e-01, 2.31901089397150906e-01, 2.29177878448171174e-02},
813 {2.31901089397150906e-01, 2.29177878448171174e-02, 5.13280033360881072e-01},
814 {2.29177878448171174e-02, 5.13280033360881072e-01, 2.31901089397150906e-01},
815 {5.13280033360881072e-01, 2.31901089397150906e-01, 2.31901089397150906e-01},
816 {2.31901089397150906e-01, 5.13280033360881072e-01, 2.31901089397150906e-01},
817 {5.13280033360881072e-01, 2.31901089397150906e-01, 2.29177878448171174e-02},
818 {2.31901089397150906e-01, 2.29177878448171174e-02, 2.31901089397150906e-01},
819 {2.31901089397150906e-01, 5.13280033360881072e-01, 2.29177878448171174e-02},
820 {2.29177878448171174e-02, 2.31901089397150906e-01, 5.13280033360881072e-01},
821 {5.13280033360881072e-01, 2.29177878448171174e-02, 2.31901089397150906e-01},
822 {2.29177878448171174e-02, 2.31901089397150906e-01, 2.31901089397150906e-01},
823 {2.31901089397150906e-01, 2.31901089397150906e-01, 5.13280033360881072e-01},
824 {3.79700484718286102e-02, 3.79700484718286102e-02, 7.30313427807538396e-01},
825 {3.79700484718286102e-02, 7.30313427807538396e-01, 1.93746475248804382e-01},
826 {7.30313427807538396e-01, 1.93746475248804382e-01, 3.79700484718286102e-02},
827 {1.93746475248804382e-01, 3.79700484718286102e-02, 3.79700484718286102e-02},
828 {3.79700484718286102e-02, 1.93746475248804382e-01, 3.79700484718286102e-02},
829 {1.93746475248804382e-01, 3.79700484718286102e-02, 7.30313427807538396e-01},
830 {3.79700484718286102e-02, 7.30313427807538396e-01, 3.79700484718286102e-02},
831 {3.79700484718286102e-02, 1.93746475248804382e-01, 7.30313427807538396e-01},
832 {7.30313427807538396e-01, 3.79700484718286102e-02, 1.93746475248804382e-01},
833 {1.93746475248804382e-01, 7.30313427807538396e-01, 3.79700484718286102e-02},
834 {7.30313427807538396e-01, 3.79700484718286102e-02, 3.79700484718286102e-02},
835 {3.79700484718286102e-02,
836 3.79700484718286102e-02,
837 1.93746475248804382e-01}};
838
839 const double TGauss<3, 5>::Weight[45] = {
840 -3.93270066412926145e-02, 4.08131605934270525e-03, 4.08131605934270525e-03,
841 4.08131605934270525e-03, 4.08131605934270525e-03, 6.58086773304341943e-04,
842 6.58086773304341943e-04, 6.58086773304341943e-04, 6.58086773304341943e-04,
843 4.38425882512284693e-03, 4.38425882512284693e-03, 4.38425882512284693e-03,
844 4.38425882512284693e-03, 4.38425882512284693e-03, 4.38425882512284693e-03,
845 1.38300638425098166e-02, 1.38300638425098166e-02, 1.38300638425098166e-02,
846 1.38300638425098166e-02, 1.38300638425098166e-02, 1.38300638425098166e-02,
847 4.24043742468372453e-03, 4.24043742468372453e-03, 4.24043742468372453e-03,
848 4.24043742468372453e-03, 4.24043742468372453e-03, 4.24043742468372453e-03,
849 4.24043742468372453e-03, 4.24043742468372453e-03, 4.24043742468372453e-03,
850 4.24043742468372453e-03, 4.24043742468372453e-03, 4.24043742468372453e-03,
851 2.23873973961420164e-03, 2.23873973961420164e-03, 2.23873973961420164e-03,
852 2.23873973961420164e-03, 2.23873973961420164e-03, 2.23873973961420164e-03,
853 2.23873973961420164e-03, 2.23873973961420164e-03, 2.23873973961420164e-03,
854 2.23873973961420164e-03, 2.23873973961420164e-03, 2.23873973961420164e-03};
855
856} // namespace oomph
oomph
//////////////////////////////////////////////////////////////////// ////////////////////////////////...
Definition advection_diffusion_elements.cc:30