Hermite (Délèze) tetrahedron interpolation with 16 parameters Check integration of quadratic terms. Case 1: Unit tetrahedron ========================= x1,y1,z1 = 0.000000D+00 0.000000D+00 0.000000D+00 x2,y2,z2 = 1.000000D+00 0.000000D+00 0.000000D+00 x3,y3,z3 = 0.000000D+00 1.000000D+00 0.000000D+00 x4,y4,z4 = 0.000000D+00 0.000000D+00 1.000000D+00 Volume = 1.666667D-01 Nb. of integration points = 5 Int (1) = 0.16666665673255937 exact = 0.16666666666666666 Int (x-xm) = 1.1175870603646843E-9 exact = 0. Int (y-ym) = 1.1175870629739163E-9 exact = 0. Int (z-zm) = 1.117587070359673E-9 exact = 0. Int ((x-xm)**2) = 0.006249999648167078 exact = 0.0062499999999999995 Int ((y-ym)**2) = 0.006249999648167078 exact = 0.0062499999999999995 Int ((z-zm)**2) = 0.006249999648167079 exact = 0.0062499999999999995 Int ((x-xm)*(y-ym)) = -0.002083333216055684 exact = -0.0020833333333333333 Int ((x-xm)*(z-zm)) = -0.002083333216055684 exact = -0.0020833333333333333 Int ((y-xm)*(z-ym)) = -0.0020833332160556856 exact = -0.0020833333333333333 Nb. of integration points = 10 Int (1) = 0.16666666232049468 exact = 0.16666666666666666 Int (x-xm) = -3.686287386450715E-18 exact = 0. Int (y-ym) = -6.505213034913027E-19 exact = 0. Int (z-zm) = -3.903127820947816E-18 exact = 0. Int ((x-xm)**2) = 0.00625000042474924 exact = 0.0062499999999999995 Int ((y-ym)**2) = 0.006250000424749238 exact = 0.0062499999999999995 Int ((z-zm)**2) = 0.006250000424749237 exact = 0.0062499999999999995 Int ((x-xm)*(y-ym)) = -0.0020833334749164123 exact = -0.0020833333333333333 Int ((x-xm)*(z-zm)) = -0.0020833334749164115 exact = -0.0020833333333333333 Int ((y-xm)*(z-ym)) = -0.0020833334749164123 exact = -0.0020833333333333333 Nb. of integration points = 11 Int (1) = 0.16666666045784953 exact = 0.16666666666666666 Int (x-xm) = -6.130999785506102E-10 exact = 0. Int (y-ym) = -6.13099979678964E-10 exact = 0. Int (z-zm) = -6.130999770175864E-10 exact = 0. Int ((x-xm)**2) = 0.006249999655370075 exact = 0.0062499999999999995 Int ((y-ym)**2) = 0.006249999655370074 exact = 0.0062499999999999995 Int ((z-zm)**2) = 0.006249999655370076 exact = 0.0062499999999999995 Int ((x-xm)*(y-ym)) = -0.00208333321845669 exact = -0.0020833333333333333 Int ((x-xm)*(z-zm)) = -0.0020833332184566904 exact = -0.0020833333333333333 Int ((y-xm)*(z-ym)) = -0.00208333321845669 exact = -0.0020833333333333333 Nb. of integration points = 14 Int (1) = 0.16666666356225812 exact = 0.16666666666666666 Int (x-xm) = -6.324009386519455E-10 exact = 0. Int (y-ym) = -6.324009393024138E-10 exact = 0. Int (z-zm) = -6.324009387658185E-10 exact = 0. Int ((x-xm)**2) = 0.006249999761779734 exact = 0.0062499999999999995 Int ((y-ym)**2) = 0.0062499997617797365 exact = 0.0062499999999999995 Int ((z-zm)**2) = 0.006249999761779735 exact = 0.0062499999999999995 Int ((x-xm)*(y-ym)) = -0.0020833332539265735 exact = -0.0020833333333333333 Int ((x-xm)*(z-zm)) = -0.002083333253926573 exact = -0.0020833333333333333 Int ((y-xm)*(z-ym)) = -0.002083333253926574 exact = -0.0020833333333333333 Case 2: Unit rotated tetrahedron ================================ x1,y1,z1 = 1.000000D+00 0.000000D+00 0.000000D+00 x2,y2,z2 = 1.000000D+00 1.000000D+00 0.000000D+00 x3,y3,z3 = 0.000000D+00 1.000000D+00 0.000000D+00 x4,y4,z4 = 0.000000D+00 0.000000D+00 1.000000D+00 Volume = 1.666667D-01 Nb. of integration points = 5 Int (1) = 0.1666666567325593 exact = 0.16666666666666666 Int (x-xm) = -2.235174116651752E-9 exact = 0. Int (y-ym) = 2.2351741090678425E-9 exact = 0. Int (z-zm) = 1.117587058643725E-9 exact = 0. Int ((x-xm)**2) = 0.008333332864222743 exact = 0.008333333333333333 Int ((y-ym)**2) = 0.008333332864222737 exact = 0.008333333333333333 Int ((z-zm)**2) = 0.0062499996481670496 exact = 0.0062499999999999995 Int ((x-xm)*(y-ym)) = -3.4038760139242907E-17 exact = 0. Int ((x-xm)*(z-zm)) = -0.004166666432111368 exact = -0.004166666666666667 Int ((y-xm)*(z-ym)) = -0.004166666432111347 exact = -0.004166666666666667 Nb. of integration points = 10 Int (1) = 0.16666666232049462 exact = 0.16666666666666666 Int (x-xm) = 1.0842021724855044E-17 exact = 0. Int (y-ym) = 3.903127820947816E-18 exact = 0. Int (z-zm) = -3.469446951953614E-18 exact = 0. Int ((x-xm)**2) = 0.008333333899665642 exact = 0.008333333333333333 Int ((y-ym)**2) = 0.008333333899665642 exact = 0.008333333333333333 Int ((z-zm)**2) = 0.006250000424749231 exact = 0.0062499999999999995 Int ((x-xm)*(y-ym)) = 4.9873299934333204E-18 exact = 0. Int ((x-xm)*(z-zm)) = -0.004166666949832825 exact = -0.004166666666666667 Int ((y-xm)*(z-ym)) = -0.004166666949832821 exact = -0.004166666666666667 Nb. of integration points = 11 Int (1) = 0.16666666045784956 exact = 0.16666666666666666 Int (x-xm) = 1.2261999540101854E-9 exact = 0. Int (y-ym) = -1.2261999598026202E-9 exact = 0. Int (z-zm) = -6.130999748754401E-10 exact = 0. Int ((x-xm)**2) = 0.008333332873826782 exact = 0.008333333333333333 Int ((y-ym)**2) = 0.00833333287382678 exact = 0.008333333333333333 Int ((z-zm)**2) = 0.006249999655370082 exact = 0.0062499999999999995 Int ((x-xm)*(y-ym)) = -8.799454712110502E-18 exact = 0. Int ((x-xm)*(z-zm)) = -0.00416666643691339 exact = -0.004166666666666667 Int ((y-xm)*(z-ym)) = -0.004166666436913382 exact = -0.004166666666666667 Nb. of integration points = 14 Int (1) = 0.1666666635622581 exact = 0.16666666666666666 Int (x-xm) = 1.2648018697497335E-9 exact = 0. Int (y-ym) = -1.264801872891167E-9 exact = 0. Int (z-zm) = -6.324009386743389E-10 exact = 0. Int ((x-xm)**2) = 0.008333333015706315 exact = 0.008333333333333333 Int ((y-ym)**2) = 0.008333333015706313 exact = 0.008333333333333333 Int ((z-zm)**2) = 0.006249999761779734 exact = 0.0062499999999999995 Int ((x-xm)*(y-ym)) = -1.0317814209623039E-17 exact = 0. Int ((x-xm)*(z-zm)) = -0.00416666650785316 exact = -0.004166666666666667 Int ((y-xm)*(z-ym)) = -0.004166666507853148 exact = -0.004166666666666667 Case 3: Random tetrahedron =========================== x1,y1,z1 = 6.769952D+00 3.599294D+00 3.299880D+00 x2,y2,z2 = 8.474969D-01 8.960828D+00 6.568827D+00 x3,y3,z3 = 9.324702D+00 4.006439D+00 4.156038D+00 x4,y4,z4 = 6.968066D+00 6.569350D+00 8.480313D+00 Volume = 7.156732D+00 Nb. of integration points = 5 Int (1) = 7.156731179764908 exact = 7.156731606339352 Int (x-xm) = -1.5210752259424334E-7 exact = 0. Int (y-ym) = 4.1936858744025664E-7 exact = 0. Int (z-zm) = 4.465693206769756E-7 exact = 0. Int ((x-xm)**2) = 14.002106500584336 exact = 14.00210728880884 Int ((y-ym)**2) = 6.670672465521804 exact = 6.670672841035817 Int ((z-zm)**2) = 5.942823155046431 exact = 5.942823489587401 Int ((x-xm)*(y-ym)) = -8.301925644193817 exact = -8.301926111536437 Int ((x-xm)*(z-zm)) = -3.1392712092772124 exact = -3.1392713859970756 Int ((y-xm)*(z-ym)) = 4.6274236379145774 exact = 4.627423898407401 Nb. of integration points = 10 Int (1) = 7.156731419713032 exact = 7.156731606339352 Int (x-xm) = 3.5185071002585833E-16 exact = 0. Int (y-ym) = -7.13250445982588E-17 exact = 0. Int (z-zm) = 3.4758166397169665E-16 exact = 0. Int ((x-xm)**2) = 14.002108240390339 exact = 14.00210728880884 Int ((y-ym)**2) = 6.67067329437393 exact = 6.670672841035817 Int ((z-zm)**2) = 5.94282389346096 exact = 5.942823489587401 Int ((x-xm)*(y-ym)) = -8.301926675734318 exact = -8.301926111536437 Int ((x-xm)*(z-zm)) = -3.139271599341575 exact = -3.1392713859970756 Int ((y-xm)*(z-ym)) = 4.6274242128865675 exact = 4.627423898407401 Nb. of integration points = 11 Int (1) = 7.156731339730323 exact = 7.156731606339352 Int (x-xm) = 8.344506601597559E-8 exact = 0. Int (y-ym) = -2.3006250236285987E-7 exact = 0. Int (z-zm) = -2.4498462061837544E-7 exact = 0. Int ((x-xm)**2) = 14.00210651672162 exact = 14.00210728880884 Int ((y-ym)**2) = 6.6706724732096685 exact = 6.670672841035817 Int ((z-zm)**2) = 5.94282316189544 exact = 5.942823489587401 Int ((x-xm)*(y-ym)) = -8.301925653761694 exact = -8.301926111536437 Int ((x-xm)*(z-zm)) = -3.1392712128951827 exact = -3.1392713859970756 Int ((y-xm)*(z-ym)) = 4.62742364324761 exact = 4.627423898407401 Nb. of integration points = 14 Int (1) = 7.156731473034838 exact = 7.156731606339352 Int (x-xm) = 8.60719907424474E-8 exact = 0. Int (y-ym) = -2.373050828688981E-7 exact = 0. Int (z-zm) = -2.526969650673756E-7 exact = 0. Int ((x-xm)**2) = 14.002106755115106 exact = 14.00210728880884 Int ((y-ym)**2) = 6.670672586781509 exact = 6.670672841035817 Int ((z-zm)**2) = 5.942823263075248 exact = 5.942823489587401 Int ((x-xm)*(y-ym)) = -8.301925795106504 exact = -8.301926111536437 Int ((x-xm)*(z-zm)) = -3.1392712663429854 exact = -3.1392713859970756 Int ((y-xm)*(z-ym)) = 4.627423722032023 exact = 4.627423898407401