C 代码 定义格伦德曼-默勒正交规则 在 2D 三角形的内部，3D 的四面体，或 在 M 维的单纯形内部.rar

21 January 2020 08:58:02 AM SIMPLEX_GM_RULE_TEST C version Test the SIMPLEX_GM_RULE library. TEST01 SIMPLEX_UNIT_TO_GENERAL maps points in the unit simplex to a general simplex. Here we consider a simplex in 2D, a triangle. The vertices of the general triangle are: 1 1 3 1 2 5 ( XSI ETA ) ( X Y ) 0 0 1 1 1 0 3 1 0 1 2 5 0.867886 0.0254803 2.76125 1.10192 0.138259 0.210636 1.48715 1.84254 0.202708 0.329918 1.73533 2.31967 0.112803 0.689309 1.91491 3.75723 0.642508 0.198073 2.48309 1.79229 0.844955 0.014506 2.70442 1.05802 0.346508 0.63118 2.3242 3.52472 0.0242126 0.292589 1.34101 2.17036 0.372621 0.0253558 1.7706 1.10142 0.408253 0.0761431 1.89265 1.30457 TEST02 SIMPLEX_UNIT_TO_GENERAL maps points in the unit simplex to a general simplex. Here we consider a simplex in 3D, a tetrahedron. The vertices of the general tetrahedron are: 1 1 1 3 1 1 1 4 1 1 1 5 ( XSI ETA MU ) ( X Y Z ) 0 0 0 1 1 1 1 0 0 3 1 1 0 1 0 1 4 1 0 0 1 1 1 5 0.653014 0.0191719 0.0802331 2.30603 1.05752 1.32093 0.122743 0.379417 0.189469 1.24549 2.13825 1.75787 0.436322 0.0635846 0.388548 1.87264 1.19075 2.55419 0.118269 0.0364603 0.029345 1.23654 1.10938 1.11738 0.0138444 0.134129 0.301972 1.02769 1.40239 2.20789 0.0207729 0.0237097 0.286511 1.04155 1.07113 2.14604 0.288996 0.0196653 0.466915 1.57799 1.059 2.86766 0.0792463 0.536617 0.149631 1.15849 2.60985 1.59852 0.0966452 0.51108 0.0596006 1.19329 2.53324 1.2384 0.366347 0.0599075 0.203121 1.73269 1.17972 1.81248 TEST03 GM_RULE_SIZE returns N, the number of points associated with a Grundmann-Moeller quadrature rule for the unit simplex of dimension M with rule index RULE and degree of exactness DEGREE = 2*RULE+1. M RULE DEGREE N 2 0 1 1 2 1 3 4 2 2 5 10 2 3 7 20 2 4 9 35 2 5 11 56 3 0 1 1 3 1 3 5 3 2 5 15 3 3 7 35 3 4 9 70 3 5 11 126 5 0 1 1 5 1 3 7 5 2 5 28 5 3 7 84 5 4 9 210 5 5 11 462 10 0 1 1 10 1 3 12 10 2 5 78 10 3 7 364 10 4 9 1365 10 5 11 4368 TEST04 GM_UNIT_RULE_SET determines the weights and abscissas of a Grundmann-Moeller quadrature rule for the M dimensional unit simplex, using a rule of index RULE, which will have degree of exactness 2*RULE+1. Here we use M = 3 RULE = 2 DEGREE = 5 POINT W X Y Z 1 0.0507937 0.125 0.125 0.125 2 0.0507937 0.375 0.125 0.125 3 0.0507937 0.625 0.125 0.125 4 0.0507937 0.125 0.375 0.125 5 0.0507937 0.375 0.375 0.125 6 0.0507937 0.125 0.625 0.125 7 0.0507937 0.125 0.125 0.375 8 0.0507937 0.375 0.125 0.375 9 0.0507937 0.125 0.375 0.375 10 0.0507937 0.125 0.125 0.625 11 -0.0964286 0.166667 0.166667 0.166667 12 -0.0964286 0.5 0.166667 0.166667 13 -0.0964286 0.166667 0.5 0.166667 14 -0.0964286 0.166667 0.166667 0.5 15 0.0444444 0.25 0.25 0.25 TEST05 GM_UNIT_RULE_SET determines the weights and abscissas of a Grundmann-Moeller quadrature rule for the M dimensional unit simplex, using a rule of index RULE, which will have degree of exactness 2*RULE+1. In this test, we compute various rules, and simply report the number of points, and the sum of weights. M RULE N WEIGHT SUM 2 0 1 0.5 2 1 4 0.4999999999999999 2 2 10 0.4999999999999999 2 3 20 0.5000000000000006 2 4 35 0.4999999999999999 2 5 56 0.5000000000000028 3 0 1 0.1666666666666667 3 1 5 0.1666666666666667 3 2 15 0.1666666666666667 3 3 35 0.166666666666667 3 4 70 0.1666666666666664 3 5 126 0.1666666666666647 5 0 1 0.008333333333333333 5 1 7 0.008333333333333331 5 2 28 0.008333333333333331 5 3 84 0.008333333333333283 5 4 210 0.008333333333333585 5 5 462 0.008333333333332863 10 0 1 2.755731922398589e-07 10 1 12 2.755731922398589e-07 10 2 78 2.755731922398579e-07 10 3 364 2.755731922398308e-07 10 4 1365 2.755731922397991e-07 10 5 4368 2.755731922406156e-07 TEST06 GM_UNIT_RULE_SET determines the weights and abscissas of a Grundmann-Moeller quadrature rule for the M dimensional unit simplex, using a rule of index RULE, which will have degree of exactness 2*RULE+1. In this test, we write a rule to a file. Here we use M = 3 RULE = 2 DEGREE = 5 Wrote rule 2 to "gm2_3d_w.txt" and "gm2_3d_x.txt". TEST07 GM_UNIT_RULE_SET determines the weights and abscissas of a Grundmann-Moeller quadrature rule for the M dimensional unit simplex, using a rule of index RULE, which will have degree of exactness 2*RULE+1. In this test, look at all the monomials up to some maximum degree, choose a few low order rules and determine the quadrature error for each. Here we use M = 5 Rule Order Quad_Error F(X) = X1^0 * X2^0 * X3^0 * X4^0 * X5^0 0 1 0 1 7 2.22045e-16 2 28 2.22045e-16 3 84 5.9952e-15 F(X) = X1^1 * X2^0 * X3^0 * X4^0 * X5^0 0 1 1.11022e-16 1 7 3.33067e-16 2 28 4.44089e-16 3 84 6.88338e-15 F(X) = X1^0 * X2^1 * X3^0 * X4^0 * X5^0 0 1 1.11022e-16 1 7 3.33067e-16 2 28 1.11022e-16 3 84 7.54952e-15 F(X) = X1^0 * X2^0 * X3^1 *