integral.cc
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1// LIC// ====================================================================
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3// LIC// multi-physics finite-element library, available
4// LIC// at http://www.oomph-lib.org.
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6// LIC// Copyright (C) 2006-2022 Matthias Heil and Andrew Hazel
7// LIC//
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9// LIC// modify it under the terms of the GNU Lesser General Public
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24// LIC//
25// LIC//====================================================================
26// Static data for the Gaussian integration rules
27
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
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
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 DR 2017 to include factor 1/2
555 const double TGauss<2, 16>::Weight[52] = {
556 2 * 0.046875697427642, 2 * 0.006405878578585, 2 * 0.006405878578585,
557 2 * 0.006405878578585, 2 * 0.041710296739387, 2 * 0.041710296739387,
558 2 * 0.041710296739387, 2 * 0.026891484250064, 2 * 0.026891484250064,
559 2 * 0.026891484250064, 2 * 0.04213252276165, 2 * 0.04213252276165,
560 2 * 0.04213252276165, 2 * 0.030000266842773, 2 * 0.030000266842773,
561 2 * 0.030000266842773, 2 * 0.014200098925024, 2 * 0.014200098925024,
562 2 * 0.014200098925024, 2 * 0.003582462351273, 2 * 0.003582462351273,
563 2 * 0.003582462351273, 2 * 0.032773147460627, 2 * 0.032773147460627,
564 2 * 0.032773147460627, 2 * 0.032773147460627, 2 * 0.032773147460627,
565 2 * 0.032773147460627, 2 * 0.015298306248441, 2 * 0.015298306248441,
566 2 * 0.015298306248441, 2 * 0.015298306248441, 2 * 0.015298306248441,
567 2 * 0.015298306248441, 2 * 0.002386244192839, 2 * 0.002386244192839,
568 2 * 0.002386244192839, 2 * 0.002386244192839, 2 * 0.002386244192839,
569 2 * 0.002386244192839, 2 * 0.019084792755899, 2 * 0.019084792755899,
570 2 * 0.019084792755899, 2 * 0.019084792755899, 2 * 0.019084792755899,
571 2 * 0.019084792755899, 2 * 0.006850054546542, 2 * 0.006850054546542,
572 2 * 0.006850054546542, 2 * 0.006850054546542, 2 * 0.006850054546542,
573 2 * 0.006850054546542};
574 //------------------------------------------------------------
575 //"Full integration" weights for 2D triangles
576 // accurate up to order 15
578 // [GAUSS8X8, order 64, degree of precision 15, (essentially a product of two
579 // 8 point 1D Gauss-Legendre rules).]
580 //------------------------------------------------------------
581 const double TGauss<2, 5>::Knot[64][2] = {
582 {0.9553660447100000, 0.8862103848242247e-3},
583 {0.9553660447100000, 0.4537789678039195e-2},
584 {0.9553660447100000, 0.1058868260117431e-1},
585 {0.9553660447100000, 0.1822327082910602e-1},
586 {0.9553660447100000, 0.2641068446089399e-1},
587 {0.9553660447100000, 0.3404527268882569e-1},
588 {0.9553660447100000, 0.4009616561196080e-1},
589 {0.9553660447100000, 0.4374774490517578e-1},
590 {0.8556337429600001, 0.2866402391985981e-2},
591 {0.8556337429600001, 0.1467724979327651e-1},
592 {0.8556337429600001, 0.3424855503358430e-1},
593 {0.8556337429600001, 0.5894224214571626e-1},
594 {0.8556337429600001, 0.8542401489428375e-1},
595 {0.8556337429600001, 0.1101177020064157},
596 {0.8556337429600001, 0.1296890072467235},
597 {0.8556337429600001, 0.1414998546480140},
598 {0.7131752428600000, 0.5694926133044352e-2},
599 {0.7131752428600000, 0.2916054411712861e-1},
600 {0.7131752428600000, 0.6804452564827500e-1},
601 {0.7131752428600000, 0.1171055801775613},
602 {0.7131752428600000, 0.1697191769624387},
603 {0.7131752428600000, 0.2187802314917250},
604 {0.7131752428600000, 0.2576642130228714},
605 {0.7131752428600000, 0.2811298310069557},
606 {0.5451866848000000, 0.9030351006711630e-2},
607 {0.5451866848000000, 0.4623939674940125e-1},
608 {0.5451866848000000, 0.1078970888004545},
609 {0.5451866848000000, 0.1856923986620134},
610 {0.5451866848000000, 0.2691209165379867},
611 {0.5451866848000000, 0.3469162263995455},
612 {0.5451866848000000, 0.4085739184505988},
613 {0.5451866848000000, 0.4457829641932884},
614 {0.3719321645800000, 0.1247033193690498e-1},
615 {0.3719321645800000, 0.6385362269957356e-1},
616 {0.3719321645800000, 0.1489989161403976},
617 {0.3719321645800000, 0.2564292182833579},
618 {0.3719321645800000, 0.3716386171366422},
619 {0.3719321645800000, 0.4790689192796024},
620 {0.3719321645800000, 0.5642142127204264},
621 {0.3719321645800000, 0.6155975034830951},
622 {0.2143084794000000, 0.1559996151584746e-1},
623 {0.2143084794000000, 0.7987871227492103e-1},
624 {0.2143084794000000, 0.1863925811641285},
625 {0.2143084794000000, 0.3207842387034378},
626 {0.2143084794000000, 0.4649072818965623},
627 {0.2143084794000000, 0.5992989394358715},
628 {0.2143084794000000, 0.7058128083250790},
629 {0.2143084794000000, 0.7700915590841526},
630 {0.9132360790000005e-1, 0.1804183496379599e-1},
631 {0.9132360790000005e-1, 0.9238218584838476e-1},
632 {0.9132360790000005e-1, 0.2155687489628060},
633 {0.9132360790000005e-1, 0.3709968314854498},
634 {0.9132360790000005e-1, 0.5376795606145502},
635 {0.9132360790000005e-1, 0.6931076431371940},
636 {0.9132360790000005e-1, 0.8162942062516152},
637 {0.9132360790000005e-1, 0.8906345571362040},
638 {0.1777991514999999e-1, 0.1950205026019779e-1},
639 {0.1777991514999999e-1, 0.9985913490381848e-1},
640 {0.1777991514999999e-1, 0.2330157982952792},
641 {0.1777991514999999e-1, 0.4010234473667467},
642 {0.1777991514999999e-1, 0.5811966374832533},
643 {0.1777991514999999e-1, 0.7492042865547208},
644 {0.1777991514999999e-1, 0.8823609499461815},
645 {0.1777991514999999e-1, 0.9627180345898023}};
646 // Modified by DR 2017 to include factor 1/2
647 const double TGauss<2, 5>::Weight[64] = {
648 0.5 * 0.3335674062677772e-3, 0.5 * 0.7327880811491046e-3,
649 0.5 * 0.1033723454167925e-2, 0.5 * 0.1195112498415193e-2,
650 0.5 * 0.1195112498415193e-2, 0.5 * 0.1033723454167925e-2,
651 0.5 * 0.7327880811491046e-3, 0.5 * 0.3335674062677772e-3,
652 0.5 * 0.1806210919443461e-2, 0.5 * 0.3967923151181667e-2,
653 0.5 * 0.5597437146194232e-2, 0.5 * 0.6471331443180639e-2,
654 0.5 * 0.6471331443180639e-2, 0.5 * 0.5597437146194232e-2,
655 0.5 * 0.3967923151181667e-2, 0.5 * 0.1806210919443461e-2,
656 0.5 * 0.4599755803015752e-2, 0.5 * 0.1010484287526739e-1,
657 0.5 * 0.1425461651131868e-1, 0.5 * 0.1648010431039818e-1,
658 0.5 * 0.1648010431039818e-1, 0.5 * 0.1425461651131868e-1,
659 0.5 * 0.1010484287526739e-1, 0.5 * 0.4599755803015752e-2,
660 0.5 * 0.8017259531156730e-2, 0.5 * 0.1761248886287915e-1,
661 0.5 * 0.2484544071087993e-1, 0.5 * 0.2872441038508419e-1,
662 0.5 * 0.2872441038508419e-1, 0.5 * 0.2484544071087993e-1,
663 0.5 * 0.1761248886287915e-1, 0.5 * 0.8017259531156730e-2,
664 0.5 * 0.1073501897357062e-1, 0.5 * 0.2358292149331603e-1,
665 0.5 * 0.3326776143412911e-1, 0.5 * 0.3846165753898425e-1,
666 0.5 * 0.3846165753898425e-1, 0.5 * 0.3326776143412911e-1,
667 0.5 * 0.2358292149331603e-1, 0.5 * 0.1073501897357062e-1,
668 0.5 * 0.1138879740452669e-1, 0.5 * 0.2501915606814251e-1,
669 0.5 * 0.3529381699354388e-1, 0.5 * 0.4080402900378691e-1,
670 0.5 * 0.4080402900378691e-1, 0.5 * 0.3529381699354388e-1,
671 0.5 * 0.2501915606814251e-1, 0.5 * 0.1138879740452669e-1,
672 0.5 * 0.9223845391285393e-2, 0.5 * 0.2026314273544469e-1,
673 0.5 * 0.2858464328177232e-1, 0.5 * 0.3304739223149761e-1,
674 0.5 * 0.3304739223149761e-1, 0.5 * 0.2858464328177232e-1,
675 0.5 * 0.2026314273544469e-1, 0.5 * 0.9223845391285393e-2,
676 0.5 * 0.4509812715921713e-2, 0.5 * 0.9907253959306707e-2,
677 0.5 * 0.1397588340693756e-1, 0.5 * 0.1615785427783403e-1,
678 0.5 * 0.1615785427783403e-1, 0.5 * 0.1397588340693756e-1,
679 0.5 * 0.9907253959306707e-2, 0.5 * 0.4509812715921713e-2};
680
681 //------------------------------------------------------------
682 //"Full integration" weights for 2D triangles
683 // accurate up to points 19, degree of precision 8, a rule from ACM TOMS
684 // algorithm #584.
686 // TOMS589_19 a 19 point Gauss rule accurate to order 8
687 //------------------------------------------------------------
688 const double TGauss<2, 9>::Knot[19][2] = {
689 {0.3333333333333333, 0.3333333333333333},
690 {0.7974269853530872, 0.1012865073234563},
691 {0.1012865073234563, 0.7974269853530872},
692 {0.1012865073234563, 0.1012865073234563},
693 {0.0597158717897698, 0.4701420641051151},
694 {0.4701420641051151, 0.0597158717897698},
695 {0.4701420641051151, 0.4701420641051151},
696 {0.5357953464498992, 0.2321023267750504},
697 {0.2321023267750504, 0.5357953464498992},
698 {0.2321023267750504, 0.2321023267750504},
699 {0.9410382782311209, 0.0294808608844396},
700 {0.0294808608844396, 0.9410382782311209},
701 {0.0294808608844396, 0.0294808608844396},
702 {0.7384168123405100, 0.2321023267750504},
703 {0.7384168123405100, 0.0294808608844396},
704 {0.2321023267750504, 0.7384168123405100},
705 {0.2321023267750504, 0.0294808608844396},
706 {0.0294808608844396, 0.7384168123405100},
707 {0.0294808608844396, 0.2321023267750504}};
708
709 const double TGauss<2, 9>::Weight[19] = {2 * 0.0378610912003147,
710 2 * 0.0376204254131829,
711 2 * 0.0376204254131829,
712 2 * 0.0376204254131829,
713 2 * 0.0783573522441174,
714 2 * 0.0783573522441174,
715 2 * 0.0783573522441174,
716 2 * 0.1162714796569659,
717 2 * 0.1162714796569659,
718 2 * 0.1162714796569659,
719 2 * 0.0134442673751655,
720 2 * 0.0134442673751655,
721 2 * 0.0134442673751655,
722 2 * 0.0375097224552317,
723 2 * 0.0375097224552317,
724 2 * 0.0375097224552317,
725 2 * 0.0375097224552317,
726 2 * 0.0375097224552317,
727 2 * 0.0375097224552317};
728
729 //------------------------------------------------------------
730 /// / Define the positions and weights of the 3D Gauss points for tets
731
732 //------------------------------------------------------------
733 // "Full integration" weights for linear tets
734 // accurate up to second order (e.g. from German Zienkiwicz p 200)
735 //------------------------------------------------------------
736 const double TGauss<3, 2>::Knot[4][3] = {
737 {0.138196601125011, 0.138196601125011, 0.585410196624969},
738 {0.138196601125011, 0.585410196624969, 0.138196601125011},
739 {0.585410196624969, 0.138196601125011, 0.138196601125011},
740 {0.138196601125011, 0.138196601125011, 0.138196601125011}};
741
742
743 const double TGauss<3, 2>::Weight[4] = {
744 0.0416666666667, 0.0416666666667, 0.0416666666667, 0.0416666666667};
745
746
747 //------------------------------------------------------------
748 //"Full integration" weights for quadratic tets
749 // accurate up to fifth order
750 // The numbers are from Keast CMAME 55 pp339-348 (1986)
751 //------------------------------------------------------------
752 const double TGauss<3, 3>::Knot[11][3] = {
753 {0.25, 0.25, 0.25},
754 {0.785714285714286, 0.071428571428571, 0.071428571428571},
755 {0.071428571428571, 0.071428571428571, 0.071428571428571},
756 {0.071428571428571, 0.785714285714286, 0.071428571428571},
757 {0.071428571428571, 0.071428571428571, 0.785714285714286},
758 {0.399403576166799, 0.399403576166799, 0.100596423833201},
759 {0.399403576166799, 0.100596423833201, 0.399403576166799},
760 {0.100596423833201, 0.399403576166799, 0.399403576166799},
761 {0.399403576166799, 0.100596423833201, 0.100596423833201},
762 {0.100596423833201, 0.399403576166799, 0.100596423833201},
763 {0.100596423833201, 0.100596423833201, 0.399403576166799}};
764
765
766 const double TGauss<3, 3>::Weight[11] = {-0.01315555555556,
767 0.00762222222222,
768 0.00762222222222,
769 0.00762222222222,
770 0.00762222222222,
771 0.02488888888889,
772 0.02488888888889,
773 0.02488888888889,
774 0.02488888888889,
775 0.02488888888889,
776 0.02488888888889};
777
778 //------------------------------------------------------------
779 //"Full integration" weights for quartic tets
780 // accurate up to eighth order
781 // The numbers are from Keast CMAME 55 pp339-348 (1986)
782 //------------------------------------------------------------
783 const double TGauss<3, 5>::Knot[45][3] = {
784 {2.50000000000000000e-01, 2.50000000000000000e-01, 2.50000000000000000e-01},
785 {1.27470936566639015e-01, 1.27470936566639015e-01, 1.27470936566639015e-01},
786 {1.27470936566639015e-01, 1.27470936566639015e-01, 6.17587190300082967e-01},
787 {1.27470936566639015e-01, 6.17587190300082967e-01, 1.27470936566639015e-01},
788 {6.17587190300082967e-01, 1.27470936566639015e-01, 1.27470936566639015e-01},
789 {3.20788303926322960e-02, 3.20788303926322960e-02, 3.20788303926322960e-02},
790 {3.20788303926322960e-02, 3.20788303926322960e-02, 9.03763508822103123e-01},
791 {3.20788303926322960e-02, 9.03763508822103123e-01, 3.20788303926322960e-02},
792 {9.03763508822103123e-01, 3.20788303926322960e-02, 3.20788303926322960e-02},
793 {4.97770956432810185e-02, 4.97770956432810185e-02, 4.50222904356718978e-01},
794 {4.97770956432810185e-02, 4.50222904356718978e-01, 4.50222904356718978e-01},
795 {4.50222904356718978e-01, 4.50222904356718978e-01, 4.97770956432810185e-02},
796 {4.50222904356718978e-01, 4.97770956432810185e-02, 4.97770956432810185e-02},
797 {4.97770956432810185e-02, 4.50222904356718978e-01, 4.97770956432810185e-02},
798 {4.50222904356718978e-01, 4.97770956432810185e-02, 4.50222904356718978e-01},
799 {1.83730447398549945e-01, 1.83730447398549945e-01, 3.16269552601450060e-01},
800 {1.83730447398549945e-01, 3.16269552601450060e-01, 3.16269552601450060e-01},
801 {3.16269552601450060e-01, 3.16269552601450060e-01, 1.83730447398549945e-01},
802 {3.16269552601450060e-01, 1.83730447398549945e-01, 1.83730447398549945e-01},
803 {1.83730447398549945e-01, 3.16269552601450060e-01, 1.83730447398549945e-01},
804 {3.16269552601450060e-01, 1.83730447398549945e-01, 3.16269552601450060e-01},
805 {2.31901089397150906e-01, 2.31901089397150906e-01, 2.29177878448171174e-02},
806 {2.31901089397150906e-01, 2.29177878448171174e-02, 5.13280033360881072e-01},
807 {2.29177878448171174e-02, 5.13280033360881072e-01, 2.31901089397150906e-01},
808 {5.13280033360881072e-01, 2.31901089397150906e-01, 2.31901089397150906e-01},
809 {2.31901089397150906e-01, 5.13280033360881072e-01, 2.31901089397150906e-01},
810 {5.13280033360881072e-01, 2.31901089397150906e-01, 2.29177878448171174e-02},
811 {2.31901089397150906e-01, 2.29177878448171174e-02, 2.31901089397150906e-01},
812 {2.31901089397150906e-01, 5.13280033360881072e-01, 2.29177878448171174e-02},
813 {2.29177878448171174e-02, 2.31901089397150906e-01, 5.13280033360881072e-01},
814 {5.13280033360881072e-01, 2.29177878448171174e-02, 2.31901089397150906e-01},
815 {2.29177878448171174e-02, 2.31901089397150906e-01, 2.31901089397150906e-01},
816 {2.31901089397150906e-01, 2.31901089397150906e-01, 5.13280033360881072e-01},
817 {3.79700484718286102e-02, 3.79700484718286102e-02, 7.30313427807538396e-01},
818 {3.79700484718286102e-02, 7.30313427807538396e-01, 1.93746475248804382e-01},
819 {7.30313427807538396e-01, 1.93746475248804382e-01, 3.79700484718286102e-02},
820 {1.93746475248804382e-01, 3.79700484718286102e-02, 3.79700484718286102e-02},
821 {3.79700484718286102e-02, 1.93746475248804382e-01, 3.79700484718286102e-02},
822 {1.93746475248804382e-01, 3.79700484718286102e-02, 7.30313427807538396e-01},
823 {3.79700484718286102e-02, 7.30313427807538396e-01, 3.79700484718286102e-02},
824 {3.79700484718286102e-02, 1.93746475248804382e-01, 7.30313427807538396e-01},
825 {7.30313427807538396e-01, 3.79700484718286102e-02, 1.93746475248804382e-01},
826 {1.93746475248804382e-01, 7.30313427807538396e-01, 3.79700484718286102e-02},
827 {7.30313427807538396e-01, 3.79700484718286102e-02, 3.79700484718286102e-02},
828 {3.79700484718286102e-02,
829 3.79700484718286102e-02,
830 1.93746475248804382e-01}};
831
832 const double TGauss<3, 5>::Weight[45] = {
833 -3.93270066412926145e-02, 4.08131605934270525e-03, 4.08131605934270525e-03,
834 4.08131605934270525e-03, 4.08131605934270525e-03, 6.58086773304341943e-04,
835 6.58086773304341943e-04, 6.58086773304341943e-04, 6.58086773304341943e-04,
836 4.38425882512284693e-03, 4.38425882512284693e-03, 4.38425882512284693e-03,
837 4.38425882512284693e-03, 4.38425882512284693e-03, 4.38425882512284693e-03,
838 1.38300638425098166e-02, 1.38300638425098166e-02, 1.38300638425098166e-02,
839 1.38300638425098166e-02, 1.38300638425098166e-02, 1.38300638425098166e-02,
840 4.24043742468372453e-03, 4.24043742468372453e-03, 4.24043742468372453e-03,
841 4.24043742468372453e-03, 4.24043742468372453e-03, 4.24043742468372453e-03,
842 4.24043742468372453e-03, 4.24043742468372453e-03, 4.24043742468372453e-03,
843 4.24043742468372453e-03, 4.24043742468372453e-03, 4.24043742468372453e-03,
844 2.23873973961420164e-03, 2.23873973961420164e-03, 2.23873973961420164e-03,
845 2.23873973961420164e-03, 2.23873973961420164e-03, 2.23873973961420164e-03,
846 2.23873973961420164e-03, 2.23873973961420164e-03, 2.23873973961420164e-03,
847 2.23873973961420164e-03, 2.23873973961420164e-03, 2.23873973961420164e-03};
848
849} // namespace oomph
static const double Weight[2]
Definition: integral.h:164
static const double Knot[2][1]
Array to hold weights and knot points (defined in cc file)
Definition: integral.h:164
static const double Weight[3]
Definition: integral.h:210
static const double Knot[3][1]
Array to hold weights and knot points (defined in cc file)
Definition: integral.h:210
static const double Knot[4][1]
Array to hold weight and knot points (defined in cc file)
Definition: integral.h:256
static const double Weight[4]
Definition: integral.h:256
static const double Weight[4]
Definition: integral.h:302
static const double Knot[4][2]
Array to hold the weight and know points (defined in cc file)
Definition: integral.h:302
static const double Weight[9]
Definition: integral.h:348
static const double Knot[9][2]
Array to hold the weights and knots (defined in cc file)
Definition: integral.h:348
static const double Weight[16]
Definition: integral.h:394
static const double Knot[16][2]
Array to hold the weights and knots (defined in cc file)
Definition: integral.h:394
static const double Knot[8][3]
Array to hold the weights and knots (defined in cc file)
Definition: integral.h:441
static const double Weight[8]
Definition: integral.h:441
static const double Knot[27][3]
Array to hold the weights and knots (defined in cc file)
Definition: integral.h:487
static const double Weight[27]
Definition: integral.h:487
static const double Knot[64][3]
Array to hold the weights and knots (defined in cc file)
Definition: integral.h:533
static const double Weight[64]
Definition: integral.h:533
static const double Knot[2][1]
Array to hold the weights and knots (defined in cc file)
Definition: integral.h:646
static const double Weight[2]
Definition: integral.h:646
static const double Knot[3][1]
Array to hold the weights and knots (defined in cc file)
Definition: integral.h:692
static const double Weight[3]
Definition: integral.h:692
static const double Knot[4][1]
Array to hold the weights and knots (defined in cc file)
Definition: integral.h:739
static const double Weight[4]
Definition: integral.h:739
static const double Weight[5]
Definition: integral.h:778
static const double Knot[5][1]
Array to hold the weights and knots (defined in cc file)
Definition: integral.h:778
static const double Knot[37][2]
Array to hold the weights and knots (defined in cc file)
Definition: integral.h:963
static const double Weight[37]
Definition: integral.h:963
static const double Knot[52][2]
Array to hold the weights and knots (defined in cc file)
Definition: integral.h:1056
static const double Weight[52]
Definition: integral.h:1056
static const double Weight[3]
Definition: integral.h:826
static const double Knot[3][2]
Array to hold the weights and knots (defined in cc file)
Definition: integral.h:826
static const double Weight[7]
Definition: integral.h:872
static const double Knot[7][2]
Array to hold the weights and knots (defined in cc file)
Definition: integral.h:872
static const double Knot[13][2]
Array to hold the weights and knots (defined in cc file)
Definition: integral.h:919
static const double Weight[13]
Definition: integral.h:919
static const double Weight[64]
Definition: integral.h:1100
static const double Knot[64][2]
Array to hold the weights and knots (defined in cc file)
Definition: integral.h:1100
static const double Weight[19]
Definition: integral.h:1008
static const double Knot[19][2]
Array to hold the weights and knots (defined in cc file)
Definition: integral.h:1008
static const double Knot[4][3]
Array to hold the weights and knots (defined in cc file)
Definition: integral.h:1146
static const double Weight[4]
Definition: integral.h:1146
static const double Knot[11][3]
Array to hold the weights and knots (defined in cc file)
Definition: integral.h:1194
static const double Weight[11]
Definition: integral.h:1194
static const double Knot[45][3]
Array to hold the weights and knots (defined in cc file)
Definition: integral.h:1242
static const double Weight[45]
Definition: integral.h:1242
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