-
Notifications
You must be signed in to change notification settings - Fork 57
/
square_felippa_rule.html
424 lines (371 loc) · 12.3 KB
/
square_felippa_rule.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
<html>
<head>
<title>
SQUARE_FELIPPA_RULE - Quadrature Rules for a Square in 2D
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SQUARE_FELIPPA_RULE <br> Quadrature Rules for a Square in 2D
</h1>
<hr>
<p>
<b>SQUARE_FELIPPA_RULE</b>
is a MATLAB library which
generates the points and weights of a Felippa quadrature rule over
the interior of a square in 2D.
</p>
<p>
Actually, the word "square" is meant to designate any quadrature region
defined by:
<pre>
A(1) <= X <= B(1)
A(2) <= Y <= B(2)
</pre>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>SQUARE_FELIPPA_RULE</b> is available in
<a href = "../../c_src/square_felippa_rule/square_felippa_rule.html">a C version</a> and
<a href = "../../cpp_src/square_felippa_rule/square_felippa_rule.html">a C++ version</a> and
<a href = "../../f77_src/square_felippa_rule/square_felippa_rule.html">a FORTRAN77 version</a> and
<a href = "../../f_src/square_felippa_rule/square_felippa_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/square_felippa_rule/square_felippa_rule.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/circle_rule/circle_rule.html">
CIRCLE_RULE</a>,
a MATLAB library which
computes quadrature rules
over the circumference of a circle in 2D.
</p>
<p>
<a href = "../../m_src/cube_arbq_rule/cube_arbq_rule.html">
CUBE_ARBQ_RULE</a>,
a MATLAB library which
computes quadrature rules
with exactness up to total degree 15,
over the interior of a cube in 3D.
</p>
<p>
<a href = "../../m_src/cube_felippa_rule/cube_felippa_rule.html">
CUBE_FELIPPA_RULE</a>,
a MATLAB library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a cube in 3D.
</p>
<p>
<a href = "../../m_src/disk_rule/disk_rule.html">
DISK_RULE</a>,
a MATLAB library which
computes quadrature rules
over the interior of a disk in 2D.
</p>
<p>
<a href = "../../m_src/pyramid_felippa_rule/pyramid_felippa_rule.html">
PYRAMID_FELIPPA_RULE</a>,
a MATLAB library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a pyramid in 3D.
</p>
<p>
<a href = "../../m_src/pyramid_rule/pyramid_rule.html">
PYRAMID_RULE</a>,
a MATLAB program which
computes a quadrature rule
over the interior of a pyramid in 3D.
</p>
<p>
<a href = "../../m_src/simplex_gm_rule/simplex_gm_rule.html">
SIMPLEX_GM_RULE</a>,
a MATLAB library which
defines Grundmann-Moeller quadrature rules
over the interior of a simplex in M dimensions.
</p>
<p>
<a href = "../../m_src/sphere_lebedev_rule/sphere_lebedev_rule.html">
SPHERE_LEBEDEV_RULE</a>,
a MATLAB library which
computes Lebedev quadrature rules
on the surface of the unit sphere in 3D.
</p>
<p>
<a href = "../../m_src/square_arbq_rule/square_arbq_rule.html">
SQUARE_ARBQ_RULE</a>,
a MATLAB library which
returns quadrature rules,
with exactness up to total degree 20,
over the interior of the symmetric square in 2D,
by Hong Xiao and Zydrunas Gimbutas.
</p>
<p>
<a href = "../../m_src/square_exactness/square_exactness.html">
SQUARE_EXACTNESS</a>,
a MATLAB library which
investigates the polynomial exactness of quadrature rules
over the interior of a cube in 3D.
</p>
<p>
<a href = "../../m_src/square_grid/square_grid.html">
SQUARE_GRID</a>,
a MATLAB library which
computes a grid of points
over the interior of a cube in 3D.
</p>
<p>
<a href = "../../m_src/square_integrals/square_integrals.html">
SQUARE_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit cube in 3D.
</p>
<p>
<a href = "../../m_src/square_monte_carlo/square_monte_carlo.html">
SQUARE_MONTE_CARLO</a>,
a MATLAB library which
applies a Monte Carlo method to estimate the integral of a function
over the interior of the unit cube in 3D;
</p>
<p>
<a href = "../../m_src/square_symq_rule/square_symq_rule.html">
SQUARE_SYMQ_RULE</a>,
a MATLAB library which
returns symmetric quadrature rules,
with exactness up to total degree 20,
over the interior of the symmetric square in 2D,
by Hong Xiao and Zydrunas Gimbutas.
</p>
<p>
<a href = "../../m_src/tetrahedron_arbq_rule/tetrahedron_arbq_rule.html">
TETRAHEDRON_ARBQ_RULE</a>,
a MATLAB library which
returns quadrature rules,
with exactness up to total degree 15,
over the interior of a tetrahedron in 3D,
by Hong Xiao and Zydrunas Gimbutas.
</p>
<p>
<a href = "../../m_src/tetrahedron_felippa_rule/tetrahedron_felippa_rule.html">
TETRAHEDRON_FELIPPA_RULE</a>,
a MATLAB library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/tetrahedron_keast_rule/tetrahedron_keast_rule.html">
TETRAHEDRON_KEAST_RULE</a>,
a MATLAB library which
defines ten quadrature rules, with exactness degrees 0 through 8,
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/tetrahedron_ncc_rule/tetrahedron_ncc_rule.html">
TETRAHEDRON_NCC_RULE</a>,
a MATLAB library which
defines Newton-Cotes closed quadrature rules
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/tetrahedron_nco_rule/tetrahedron_nco_rule.html">
TETRAHEDRON_NCO_RULE</a>,
a MATLAB library which
defines Newton-Cotes open quadrature rules
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/triangle_dunavant_rule/triangle_dunavant_rule.html">
TRIANGLE_DUNAVANT_RULE</a>,
a MATLAB library which
defines Dunavant rules for quadrature
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_fekete_rule/triangle_fekete_rule.html">
TRIANGLE_FEKETE_RULE</a>,
a MATLAB library which
defines Fekete rules for interpolation or quadrature
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_felippa_rule/triangle_felippa_rule.html">
TRIANGLE_FELIPPA_RULE</a>,
a MATLAB library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_lyness_rule/triangle_lyness_rule.html">
TRIANGLE_LYNESS_RULE</a>,
a MATLAB library which
returns Lyness-Jespersen quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_monte_carlo/triangle_monte_carlo.html">
TRIANGLE_MONTE_CARLO</a>,
a MATLAB program which
uses the Monte Carlo method to estimate integrals
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_ncc_rule/triangle_ncc_rule.html">
TRIANGLE_NCC_RULE</a>,
a MATLAB library which
defines Newton-Cotes Closed (NCC) quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_nco_rule/triangle_nco_rule.html">
TRIANGLE_NCO_RULE</a>,
a MATLAB library which
defines Newton-Cotes Open (NCO) quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_symq_rule/triangle_symq_rule.html">
TRIANGLE_SYMQ_RULE</a>,
a MATLAB library which
returns efficient symmetric quadrature rules,
with exactness up to total degree 50,
over the interior of an arbitrary triangle in 2D,
by Hong Xiao and Zydrunas Gimbutas.
</p>
<p>
<a href = "../../m_src/triangle_wandzura_rule/triangle_wandzura_rule.html">
TRIANGLE_WANDZURA_RULE</a>,
a MATLAB library which
defines Wandzura rules for quadrature
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/wedge_felippa_rule/wedge_felippa_rule.html">
WEDGE_FELIPPA_RULE</a>,
a MATLAB library which
returns quadratures rules for approximating integrals
over the interior of the unit wedge in 3D.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Carlos Felippa,<br>
A compendium of FEM integration formulas for symbolic work,<br>
Engineering Computation,<br>
Volume 21, Number 8, 2004, pages 867-890.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "comp_next.m">comp_next.m</a>,
computes the compositions of the integer N into K parts.
</li>
<li>
<a href = "line_unit_o01.m">line_unit_o01.m</a>,
returns a 1 point quadrature rule for the unit line.
</li>
<li>
<a href = "line_unit_o02.m">line_unit_o02.m</a>,
returns a 2 point quadrature rule for the unit line.
</li>
<li>
<a href = "line_unit_o03.m">line_unit_o03.m</a>,
returns a 3 point quadrature rule for the unit line.
</li>
<li>
<a href = "line_unit_o04.m">line_unit_o04.m</a>,
returns a 4 point quadrature rule for the unit line.
</li>
<li>
<a href = "line_unit_o05.m">line_unit_o05.m</a>,
returns a 5 point quadrature rule for the unit line.
</li>
<li>
<a href = "monomial_value.m">monomial_value.m</a>,
evaluates a monomial.
</li>
<li>
<a href = "square_monomial.m">square_monomial.m</a>,
returns the exact integral of a monomial in a square in 2D.
</li>
<li>
<a href = "square_monomial_test.m">square_monomial_test.m</a>,
tests SQUARE_MONOMIAL.
</li>
<li>
<a href = "square_quad_test.m">square_quad_test.m</a>,
tests the quadrature rules for a square in 2D.
</li>
<li>
<a href = "square_rule.m">square_rule.m</a>,
returns a quadrature rule for a square in 2D;
</li>
<li>
<a href = "square_volume.m">square_volume.m</a>,
returns the volume of a square in 2D;
</li>
<li>
<a href = "r8vec_direct_product.m">r8vec_direct_product.m</a>,
creates a direct product of R8VEC's.
</li>
<li>
<a href = "r8vec_direct_product2.m">r8vec_direct_product2.m</a>,
creates a direct product of R8VEC's.
</li>
<li>
<a href = "subcomp_next.m">subcomp_next.m</a>,
computes the next subcomposition of N into K parts.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the YMDHMS date as a timestamp;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "square_felippa_rule_test.m">square_felippa_rule_test.m</a>,
runs all the tests.
</li>
<li>
<a href = "square_felippa_rule_test_output.txt">
square_felippa_rule_test_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 07 September 2014.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>