08-Jan-2022 10:39:53 triangle_wandzura_rule_test(): MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 Test triangle_wandzura_rule(). TRIANGLE_WANDZURA_RULE_TEST01(): WANDZURA_RULE_NUM returns the number of rules; WANDZURA_DEGREE returns the degree of a rule; WANDZURA_ORDER_NUM returns the order of a rule. Number of available rules = 6 Rule Degree Order 1 5 7 2 10 25 3 15 54 4 20 85 5 25 126 6 30 175 TRIANGLE_WANDZURA_RULE_TEST02() WANDZURA_RULE returns the points and weights of a Wandzura rule for the triangle. In this test, we simply check that the weights sum to 1. Number of available rules = 6 Rule Sum of weights 1 1.000000 2 1.000000 3 1.000000 4 1.000000 5 1.000000 6 1.000000 TRIANGLE_WANDZURA_RULE_TEST03() WANDZURA_RULE returns the points and weights of a Wandzura rule for the triangle. In this test, we simply check that, for each quadrature point, the barycentric coordinates sum to 1. Number of available rules = 6 Rule Suborder Sum of coordinates 1 3 1.000000e+00 1.000000e+00 1.000000e+00 2 7 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 3 12 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 4 19 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 5 26 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 6 36 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 TRIANGLE_WANDZURA_RULE_TEST04() WANDZURA_RULE returns the points and weights of a Wandzura rule for the unit triangle. This routine uses those rules to estimate the integral of monomomials in the unit triangle. Integrate 2.000000 * X^0 * Y^0 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 6.000000 * X^0 * Y^1 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 12.000000 * X^0 * Y^2 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 20.000000 * X^0 * Y^3 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 30.000000 * X^0 * Y^4 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 42.000000 * X^0 * Y^5 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 56.000000 * X^0 * Y^6 Rule QUAD ERROR 1 0.995414 0.004586 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 72.000000 * X^0 * Y^7 Rule QUAD ERROR 1 0.981708 0.018292 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 90.000000 * X^0 * Y^8 Rule QUAD ERROR 1 0.956602 0.043398 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 110.000000 * X^0 * Y^9 Rule QUAD ERROR 1 0.920084 0.079916 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 132.000000 * X^0 * Y^10 Rule QUAD ERROR 1 0.873671 0.126329 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 6.000000 * X^1 * Y^0 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 24.000000 * X^1 * Y^1 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 60.000000 * X^1 * Y^2 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 120.000000 * X^1 * Y^3 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 210.000000 * X^1 * Y^4 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 336.000000 * X^1 * Y^5 Rule QUAD ERROR 1 1.013757 0.013757 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 504.000000 * X^1 * Y^6 Rule QUAD ERROR 1 1.043386 0.043386 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 720.000000 * X^1 * Y^7 Rule QUAD ERROR 1 1.082131 0.082131 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 990.000000 * X^1 * Y^8 Rule QUAD ERROR 1 1.120935 0.120935 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 1320.000000 * X^1 * Y^9 Rule QUAD ERROR 1 1.152148 0.152148 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 12.000000 * X^2 * Y^0 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 60.000000 * X^2 * Y^1 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 180.000000 * X^2 * Y^2 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 420.000000 * X^2 * Y^3 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 840.000000 * X^2 * Y^4 Rule QUAD ERROR 1 0.962963 0.037037 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 1512.000000 * X^2 * Y^5 Rule QUAD ERROR 1 0.909751 0.090249 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 2520.000000 * X^2 * Y^6 Rule QUAD ERROR 1 0.866501 0.133499 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 3960.000000 * X^2 * Y^7 Rule QUAD ERROR 1 0.846090 0.153910 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 5940.000000 * X^2 * Y^8 Rule QUAD ERROR 1 0.849523 0.150477 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 20.000000 * X^3 * Y^0 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 120.000000 * X^3 * Y^1 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 420.000000 * X^3 * Y^2 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 1120.000000 * X^3 * Y^3 Rule QUAD ERROR 1 1.051146 0.051146 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 2520.000000 * X^3 * Y^4 Rule QUAD ERROR 1 1.053666 0.053666 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 5040.000000 * X^3 * Y^5 Rule QUAD ERROR 1 0.992477 0.007523 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 9240.000000 * X^3 * Y^6 Rule QUAD ERROR 1 0.896388 0.103612 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 15840.000000 * X^3 * Y^7 Rule QUAD ERROR 1 0.797080 0.202920 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 30.000000 * X^4 * Y^0 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 210.000000 * X^4 * Y^1 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 840.000000 * X^4 * Y^2 Rule QUAD ERROR 1 0.962963 0.037037 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 2520.000000 * X^4 * Y^3 Rule QUAD ERROR 1 1.053666 0.053666 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 6300.000000 * X^4 * Y^4 Rule QUAD ERROR 1 1.137746 0.137746 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 13860.000000 * X^4 * Y^5 Rule QUAD ERROR 1 1.142490 0.142490 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 27720.000000 * X^4 * Y^6 Rule QUAD ERROR 1 1.069173 0.069173 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 42.000000 * X^5 * Y^0 Rule QUAD ERROR 1 1.000000 0.000000 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 336.000000 * X^5 * Y^1 Rule QUAD ERROR 1 1.013757 0.013757 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 1512.000000 * X^5 * Y^2 Rule QUAD ERROR 1 0.909751 0.090249 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 5040.000000 * X^5 * Y^3 Rule QUAD ERROR 1 0.992477 0.007523 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 13860.000000 * X^5 * Y^4 Rule QUAD ERROR 1 1.142490 0.142490 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 33264.000000 * X^5 * Y^5 Rule QUAD ERROR 1 1.239579 0.239579 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 56.000000 * X^6 * Y^0 Rule QUAD ERROR 1 0.995414 0.004586 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 504.000000 * X^6 * Y^1 Rule QUAD ERROR 1 1.043386 0.043386 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 2520.000000 * X^6 * Y^2 Rule QUAD ERROR 1 0.866501 0.133499 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 9240.000000 * X^6 * Y^3 Rule QUAD ERROR 1 0.896388 0.103612 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 27720.000000 * X^6 * Y^4 Rule QUAD ERROR 1 1.069173 0.069173 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 72.000000 * X^7 * Y^0 Rule QUAD ERROR 1 0.981708 0.018292 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 720.000000 * X^7 * Y^1 Rule QUAD ERROR 1 1.082131 0.082131 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 3960.000000 * X^7 * Y^2 Rule QUAD ERROR 1 0.846090 0.153910 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 15840.000000 * X^7 * Y^3 Rule QUAD ERROR 1 0.797080 0.202920 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 90.000000 * X^8 * Y^0 Rule QUAD ERROR 1 0.956602 0.043398 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 990.000000 * X^8 * Y^1 Rule QUAD ERROR 1 1.120935 0.120935 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 5940.000000 * X^8 * Y^2 Rule QUAD ERROR 1 0.849523 0.150477 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 110.000000 * X^9 * Y^0 Rule QUAD ERROR 1 0.920084 0.079916 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 1320.000000 * X^9 * Y^1 Rule QUAD ERROR 1 1.152148 0.152148 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 Integrate 132.000000 * X^10 * Y^0 Rule QUAD ERROR 1 0.873671 0.126329 2 1.000000 0.000000 3 1.000000 0.000000 4 1.000000 0.000000 5 1.000000 0.000000 6 1.000000 0.000000 TRIANGLE_WANDZURA_RULE_TEST05() REFERENCE_TO_PHYSICAL_T3 transforms a rule on the unit (reference) triangle to a rule on an arbitrary (physical) triangle. The reference triangle: 1 0.000000 0.000000 2 1.000000 0.000000 3 0.000000 1.000000 Rule 2 for reference triangle with area = 0.500000 X Y W 1 0.333333 0.333333 0.083523 2 0.004269 0.497865 0.007230 3 0.497865 0.497865 0.007230 4 0.497865 0.004269 0.007230 5 0.143975 0.428012 0.074492 6 0.428012 0.428012 0.074492 7 0.428012 0.143975 0.074492 8 0.630487 0.184756 0.078646 9 0.184756 0.184756 0.078646 10 0.184756 0.630487 0.078646 11 0.959038 0.020481 0.006928 12 0.020481 0.020481 0.006928 13 0.020481 0.959038 0.006928 14 0.035003 0.136574 0.029518 15 0.136574 0.828423 0.029518 16 0.828423 0.035003 0.029518 17 0.136574 0.035003 0.029518 18 0.828423 0.136574 0.029518 19 0.035003 0.828423 0.029518 20 0.037549 0.332744 0.039579 21 0.332744 0.629707 0.039579 22 0.629707 0.037549 0.039579 23 0.332744 0.037549 0.039579 24 0.629707 0.332744 0.039579 25 0.037549 0.629707 0.039579 The physical triangle: 1 1.000000 2.000000 2 1.000000 1.000000 3 3.000000 2.000000 Rule 2 for physical triangle with area = 1.000000 X Y W 1 1.666667 1.666667 0.083523 2 1.995731 1.995731 0.007230 3 1.995731 1.502135 0.007230 4 1.008538 1.502135 0.007230 5 1.856025 1.856025 0.074492 6 1.856025 1.571988 0.074492 7 1.287950 1.571988 0.074492 8 1.369513 1.369513 0.078646 9 1.369513 1.815244 0.078646 10 2.260974 1.815244 0.078646 11 1.040962 1.040962 0.006928 12 1.040962 1.979519 0.006928 13 2.918075 1.979519 0.006928 14 1.273147 1.964997 0.029518 15 2.656847 1.863426 0.029518 16 1.070006 1.171577 0.029518 17 1.070006 1.863426 0.029518 18 1.273147 1.171577 0.029518 19 2.656847 1.964997 0.029518 20 1.665487 1.962451 0.039579 21 2.259415 1.667256 0.039579 22 1.075098 1.370293 0.039579 23 1.075098 1.667256 0.039579 24 1.665487 1.370293 0.039579 25 2.259415 1.962451 0.039579 triangle_wandzura_rule_test(): Normal end of execution. 08-Jan-2022 10:39:53