一维、二维、三维高斯积分点及权重-Gauss integrations and weights

Gauss integrations and weights(Containing the program)高斯积分点以及权重目录1.1D bar element(p181 computation mechanics) (2)2.2D triangle element(p230 computation mechanics) (12)3.2D quadrilateral element(p182 computation mechanics) (15)4.3D tetrahedron element(p231 computation mechanics) (17)5.3D hexahedron element (p187 computation mechanics) (19)1.1D bar element(p181 computation mechanics)root3 = 1./sqrt(3.) ; r15 = .2*sqrt(15.)nip = ubound( s , 1 )w = (/5./9.,8./9.,5./9./); v=(/5./9.*w,8./9.*w,5./9.*w/)select case (element)case('line')select case(nip)case(1)s(1,1)=0. ; wt(1)=2.case(2)s(1,1)=root3 ; s(2,1)=-s(1,1) ; wt(1)=1. ; wt(2)=1.case(3)s(1,1)=r15 ; s(2,1)=.0 ; s(3,1)=-s(1,1)wt = wcase(4)s(1,1)=.861136311594053 ; s(2,1)=.339981043584856s(3,1)=-s(2,1) ; s(4,1)=-s(1,1)wt(1)=.347854845137454 ; wt(2)=.652145154862546wt(3)=wt(2) ; wt(4)=wt(1)case(5)s(1,1)=.906179845938664 ; s(2,1)=.538469*********s(3,1)=.0 ; s(4,1)=-s(2,1) ; s(5,1)=-s(1,1)wt(1)=.236926885056189 ; wt(2)=.478628670499366wt(3)=.568888888888889 ; wt(4)=wt(2) ; wt(5)=wt(1)case(6)s(1,1)=.932469514203152 ; s(2,1)=.661209386466265;s(3,1)=.238619186083197s(4,1)=-s(3,1) ; s(5,1)=-s(2,1) ; s(6,1)=-s(1,1)wt(1)=.171324492379170 ; wt(2)=.360761573048139;wt(3)=.467913934572691wt(4)=wt(3); wt(5)=wt(2) ; wt(6)=wt(1)case defaultprint*,"wrong number of integrating points for a line"end select% Copyright (c) 2010, Thomas-Peter Fries, RWTH Aachen University function [xxIntRef, wwIntRef] = IntPoints1DGauss(nQ)% Set Gauss points in 1D reference element from [-1, 1].if nQ == 1Data = [...0.0000000000000000e+000 2.0000000000000000e+000];elseif nQ == 2Data = [...-5.7735026918962573e-001 1.0000000000000000e+0005.7735026918962573e-001 1.0000000000000000e+000];elseif nQ == 3Data = [...-7.7459666924148340e-001 5.5555555555555558e-0010.0000000000000000e+000 8.8888888888888884e-0017.7459666924148340e-001 5.5555555555555558e-001];elseif nQ == 4Data = [...-8.6113631159405257e-001 3.4785484513745385e-001-3.3998104358485626e-001 6.5214515486254609e-0013.3998104358485626e-001 6.5214515486254609e-0018.6113631159405257e-001 3.4785484513745385e-001];elseif nQ == 5Data = [...-9.0617984593866396e-001 2.3692688505618908e-001-5.3846931010568311e-001 4.7862867049936647e-0010.0000000000000000e+000 5.6888888888888889e-0015.3846931010568311e-001 4.7862867049936647e-0019.0617984593866396e-001 2.3692688505618908e-001];elseif nQ == 6Data = [...-9.3246951420315205e-001 1.7132449237917036e-001-6.6120938646626448e-001 3.6076157304813861e-001-2.3861918608319688e-001 4.6791393457269104e-0012.3861918608319688e-001 4.6791393457269104e-0016.6120938646626448e-001 3.6076157304813861e-0019.3246951420315205e-001 1.7132449237917036e-001];elseif nQ == 7Data = [...-9.4910791234275849e-001 1.2948496616886970e-001 -7.4153118559939446e-001 2.7970539148927664e-001 -4.0584515137739718e-001 3.8183005050511892e-001 0.0000000000000000e+000 4.1795918367346940e-001 4.0584515137739718e-001 3.8183005050511892e-001 7.4153118559939446e-001 2.7970539148927664e-001 9.4910791234275849e-001 1.2948496616886970e-001 ];elseif nQ == 8Data = [...-9.6028985649753618e-001 1.0122853629037626e-001 -7.9666647741362673e-001 2.2238103445337448e-001 -5.2553240991632899e-001 3.1370664587788727e-001 -1.8343464249564978e-001 3.6268378337836199e-001 1.8343464249564978e-001 3.6268378337836199e-001 5.2553240991632899e-001 3.1370664587788727e-001 7.9666647741362673e-001 2.2238103445337448e-001 9.6028985649753618e-001 1.0122853629037626e-001 ];elseif nQ == 9Data = [...-9.6816023950762609e-001 8.1274388361574412e-002 -8.3603110732663577e-001 1.8064816069485740e-001 -6.1337143270059036e-001 2.6061069640293544e-001 -3.2425342340380892e-001 3.1234707704000286e-001 0.0000000000000000e+000 3.3023935500125978e-001 3.2425342340380892e-001 3.1234707704000286e-001 6.1337143270059036e-001 2.6061069640293544e-0018.3603110732663577e-001 1.8064816069485740e-0019.6816023950762609e-001 8.1274388361574412e-002 ];elseif nQ == 10Data = [...-9.7390652851717174e-001 6.6671344308688138e-002 -8.6506336668898454e-001 1.4945134915058059e-001 -6.7940956829902444e-001 2.1908636251598204e-001 -4.3339539412924721e-001 2.6926671930999635e-001 -1.4887433898163116e-001 2.9552422471475287e-001 1.4887433898163116e-001 2.9552422471475287e-001 4.3339539412924721e-001 2.6926671930999635e-001 6.7940956829902444e-001 2.1908636251598204e-001 8.6506336668898454e-001 1.4945134915058059e-0019.7390652851717174e-001 6.6671344308688138e-002 ];elseif nQ == 11Data = [...-9.7822865814605697e-001 5.5668567116173663e-002 -8.8706259976809532e-001 1.2558036946490461e-001 -7.3015200557404936e-001 1.8629021092773426e-001 -5.1909612920681181e-001 2.3319376459199048e-001 -2.6954315595234501e-001 2.6280454451024665e-001 0.0000000000000000e+000 2.7292508677790062e-001 2.6954315595234501e-001 2.6280454451024665e-001 5.1909612920681181e-001 2.3319376459199048e-0017.3015200557404936e-001 1.8629021092773426e-0018.8706259976809532e-001 1.2558036946490461e-0019.7822865814605697e-001 5.5668567116173663e-002 ];elseif nQ == 12Data = [...-9.8156063424671924e-001 4.7175336386511828e-002 -9.0411725637047491e-001 1.0693932599531843e-001 -7.6990267419430469e-001 1.6007832854334622e-001 -5.8731795428661737e-001 2.0316742672306592e-001 -3.6783149899818024e-001 2.3349253653835481e-001 -1.2523340851146891e-001 2.4914704581340277e-001 1.2523340851146891e-001 2.4914704581340277e-001 3.6783149899818024e-001 2.3349253653835481e-001 5.8731795428661737e-001 2.0316742672306592e-001 7.6990267419430469e-001 1.6007832854334622e-001 9.0411725637047491e-001 1.0693932599531843e-001 9.8156063424671924e-001 4.7175336386511828e-002 ];elseif nQ == 13Data = [...-9.8418305471858814e-001 4.0484004765315877e-002 -9.1759839922297792e-001 9.2121499837728452e-002 -8.0157809073330988e-001 1.3887351021978725e-001 -6.4234933944034023e-001 1.7814598076194574e-001 -4.4849275103644681e-001 2.0781604753688851e-001 -2.3045831595513477e-001 2.2628318026289723e-001 0.0000000000000000e+000 2.3255155323087390e-001 2.3045831595513477e-001 2.2628318026289723e-001 4.4849275103644681e-001 2.0781604753688851e-001 6.4234933944034023e-001 1.7814598076194574e-001 8.0157809073330988e-001 1.3887351021978725e-0019.1759839922297792e-001 9.2121499837728452e-002 9.8418305471858814e-001 4.0484004765315877e-002 ];elseif nQ == 14Data = [...-9.8628380869681231e-001 3.5119460331751860e-002 -9.2843488366357352e-001 8.0158087159760208e-002 -8.2720131506976502e-001 1.2151857068790319e-001 -6.8729290481168548e-001 1.5720316715819355e-001 -5.1524863635815410e-001 1.8553839747793782e-001 -3.1911236892788974e-001 2.0519846372129560e-001 -1.0805494870734367e-001 2.1526385346315779e-001 1.0805494870734367e-001 2.1526385346315779e-001 3.1911236892788974e-001 2.0519846372129560e-0015.1524863635815410e-001 1.8553839747793782e-0016.8729290481168548e-001 1.5720316715819355e-0018.2720131506976502e-001 1.2151857068790319e-0019.2843488366357352e-001 8.0158087159760208e-002 9.8628380869681231e-001 3.5119460331751860e-002 ];elseif nQ == 15Data = [...-9.8799251802048538e-001 3.0753241996117269e-002 -9.3727339240070595e-001 7.0366047488108124e-002 -8.4820658341042721e-001 1.0715922046717194e-001 -7.2441773136017007e-001 1.3957067792615432e-001 -5.7097217260853883e-001 1.6626920581699392e-001 -3.9415134707756339e-001 1.8616100001556221e-001 -2.0119409399743449e-001 1.9843148532711158e-001 0.0000000000000000e+000 2.0257824192556129e-0012.0119409399743449e-001 1.9843148532711158e-0013.9415134707756339e-001 1.8616100001556221e-001 5.7097217260853883e-001 1.6626920581699392e-0017.2441773136017007e-001 1.3957067792615432e-0018.4820658341042721e-001 1.0715922046717194e-0019.3727339240070595e-001 7.0366047488108124e-002 9.8799251802048538e-001 3.0753241996117269e-002 ];elseif nQ == 16Data = [...-9.8940093499164994e-001 2.7152459411754096e-002 -9.4457502307323260e-001 6.2253523938647894e-002 -8.6563120238783176e-001 9.5158511682492786e-002 -7.5540440835500300e-001 1.2462897125553388e-001-6.1787624440264377e-001 1.4959598881657674e-001 -4.5801677765722737e-001 1.6915651939500254e-001 -2.8160355077925892e-001 1.8260341504492358e-001 -9.5012509837637427e-002 1.8945061045506850e-001 9.5012509837637427e-002 1.8945061045506850e-001 2.8160355077925892e-001 1.8260341504492358e-001 4.5801677765722737e-001 1.6915651939500254e-0016.1787624440264377e-001 1.4959598881657674e-0017.5540440835500300e-001 1.2462897125553388e-0018.6563120238783176e-001 9.5158511682492786e-0029.4457502307323260e-001 6.2253523938647894e-002 9.8940093499164994e-001 2.7152459411754096e-002 ];elseif nQ == 17Data = [...-9.9057547531441736e-001 2.4148302868547931e-002 -9.5067552176876780e-001 5.5459529373987203e-002 -8.8023915372698591e-001 8.5036148317179178e-002 -7.8151400389680137e-001 1.1188384719340397e-001 -6.5767115921669084e-001 1.3513636846852548e-001 -5.1269053708647694e-001 1.5404576107681028e-001 -3.5123176345387630e-001 1.6800410215645004e-001 -1.7848418149584788e-001 1.7656270536699264e-0010.0000000000000000e+000 1.7944647035620653e-0011.7848418149584788e-001 1.7656270536699264e-001 3.5123176345387630e-001 1.6800410215645004e-0015.1269053708647694e-001 1.5404576107681028e-0016.5767115921669084e-001 1.3513636846852548e-0017.8151400389680137e-001 1.1188384719340397e-0018.8023915372698591e-001 8.5036148317179178e-0029.5067552176876780e-001 5.5459529373987203e-002 9.9057547531441736e-001 2.4148302868547931e-002 ];elseif nQ == 18Data = [...-9.9156516842093090e-001 2.1616013526483312e-002 -9.5582394957139771e-001 4.9714548894969797e-002 -8.9260246649755570e-001 7.6425730254889052e-002 -8.0370495897252314e-001 1.0094204410628717e-001 -6.9168704306035322e-001 1.2255520671147846e-001 -5.5977083107394754e-001 1.4064291467065065e-001 -4.1175116146284263e-001 1.5468467512626524e-001 -2.5188622569150554e-001 1.6427648374583273e-001 -8.4775013041735292e-002 1.6914238296314360e-0018.4775013041735292e-002 1.6914238296314360e-001 2.5188622569150554e-001 1.6427648374583273e-0014.1175116146284263e-001 1.5468467512626524e-0015.5977083107394754e-001 1.4064291467065065e-0016.9168704306035322e-001 1.2255520671147846e-001 8.0370495897252314e-001 1.0094204410628717e-0018.9260246649755570e-001 7.6425730254889052e-0029.5582394957139771e-001 4.9714548894969797e-002 9.9156516842093090e-001 2.1616013526483312e-002 ];elseif nQ == 19Data = [...-9.9240684384358435e-001 1.9461788229726478e-002 -9.6020815213483002e-001 4.4814226765699600e-002 -9.0315590361481790e-001 6.9044542737641226e-002 -8.2271465653714282e-001 9.1490021622449999e-002 -7.2096617733522939e-001 1.1156664554733399e-001 -6.0054530466168110e-001 1.2875396253933621e-001 -4.6457074137596099e-001 1.4260670217360660e-001 -3.1656409996362989e-001 1.5276604206585967e-001 -1.6035864564022539e-001 1.5896884339395434e-0010.0000000000000000e+000 1.6105444984878370e-0011.6035864564022539e-001 1.5896884339395434e-0013.1656409996362989e-001 1.5276604206585967e-0014.6457074137596099e-001 1.4260670217360660e-0016.0054530466168110e-001 1.2875396253933621e-0017.2096617733522939e-001 1.1156664554733399e-0018.2271465653714282e-001 9.1490021622449999e-0029.0315590361481790e-001 6.9044542737641226e-002 9.6020815213483002e-001 4.4814226765699600e-002 9.9240684384358435e-001 1.9461788229726478e-002 ];elseif nQ == 20Data = [...-9.9312859918509488e-001 1.7614007139152118e-002 -9.6397192727791381e-001 4.0601429800386939e-002 -9.1223442825132595e-001 6.2672048334109068e-002 -8.3911697182221878e-001 8.3276741576704755e-002 -7.4633190646015080e-001 1.0193011981724044e-001 -6.3605368072651502e-001 1.1819453196151841e-001 -5.1086700195082702e-001 1.3168863844917664e-001 -3.7370608871541955e-001 1.4209610931838204e-001 -2.2778585114164507e-001 1.4917298647260374e-001 -7.6526521133497338e-002 1.5275338713072584e-0017.6526521133497338e-002 1.5275338713072584e-0012.2778585114164507e-001 1.4917298647260374e-0013.7370608871541955e-001 1.4209610931838204e-0015.1086700195082702e-001 1.3168863844917664e-0016.3605368072651502e-001 1.1819453196151841e-0017.4633190646015080e-001 1.0193011981724044e-0018.3911697182221878e-001 8.3276741576704755e-0029.1223442825132595e-001 6.2672048334109068e-0029.6397192727791381e-001 4.0601429800386939e-0029.9312859918509488e-001 1.7614007139152118e-002];elseif nQ == 21Data = [...-9.9375217062038945e-001 1.6017228257774335e-002-9.6722683856630631e-001 3.6953789770852494e-002-9.2009933415040079e-001 5.7134425426857205e-002-8.5336336458331730e-001 7.6100113628379304e-002-7.6843996347567789e-001 9.3444423456033862e-002-6.6713880419741234e-001 1.0879729916714838e-001-5.5161883588721983e-001 1.2183141605372853e-001-4.2434212020743878e-001 1.3226893863333747e-001-2.8802131680240106e-001 1.3988739479107315e-001-1.4556185416089507e-001 1.4452440398997005e-0010.0000000000000000e+000 1.4608113364969041e-0011.4556185416089507e-001 1.4452440398997005e-0012.8802131680240106e-001 1.3988739479107315e-0014.2434212020743878e-001 1.3226893863333747e-0015.5161883588721983e-001 1.2183141605372853e-0016.6713880419741234e-001 1.0879729916714838e-0017.6843996347567789e-001 9.3444423456033862e-0028.5336336458331730e-001 7.6100113628379304e-0029.2009933415040079e-001 5.7134425426857205e-0029.6722683856630631e-001 3.6953789770852494e-0029.9375217062038945e-001 1.6017228257774335e-002];elseerror(['The number ' num2str(nQ) ' is not implemented.']) endxxIntRef = Data(:, 1)';wwIntRef = Data(:, 2)';% % Plot situation.% reset(cla), reset(clf), hold on% a = plot(xxIntRef, zeros(size(xxIntRef)), 'k*'); % set(a, 'LineWidth', 2, 'MarkerSize', 15)% a = line([-1 1], [0 0]);% set(a, 'LineWidth', 2, 'Color', 'k')% box on, axis equal, axis([-1 1 -0.1 0.1])2.2D triangle element(p230 computation mechanics)Order n Location ξLocation ηWeight wcase('triangle')select case(nip)case(1) ! for triangles weights multiplied by .5s(1,1)=1./3. ; s(1,2)=1./3. ; wt(1)= .5case(3)s(1,1)=.5 ; s(1,2)=.5 ; s(2,1)=.5s(2,2)=0.; s(3,1)=0. ; s(3,2)=.5wt(1)=1./3. ; wt(2)=wt(1) ; wt(3)=wt(1) ; wt = .5*wtcase(6)s(1,1)=.816847572980459 ; s(1,2)=.091576213509771s(2,1)=s(1,2); s(2,2)=s(1,1) ; s(3,1)=s(1,2); s(3,2)=s(1,2)s(4,1)=.108103018168070 ; s(4,2)=.445948*********s(5,1)=s(4,2) ; s(5,2)=s(4,1) ; s(6,1)=s(4,2) ; s(6,2)=s(4,2)wt(1)=.109951743655322 ; wt(2)=wt(1) ; wt(3)=wt(1)wt(4)=.223381589678011 ; wt(5)=wt(4) ; wt(6)=wt(4) ; wt = .5*wt case(7)s(1,1)=1./3. ; s(1,2)=1./3.;s(2,1)=.797426985353087 ;s(2,2)=.101286507323456 s(3,1)=s(2,2) ; s(3,2)=s(2,1) ; s(4,1)=s(2,2) ; s(4,2)=s(2,2)s(5,1)=.470142064105115 ; s(5,2)=.059715871789770s(6,1)=s(5,2) ; s(6,2)=s(5,1); s(7,1)=s(5,1); s(7,2)=s(5,1)wt(1)=.225 ; wt(2)=.125939180544827 ; wt(3)=wt(2); wt(4)=wt(2)wt(5)=.132394152788506; wt(6)=wt(5) ; wt(7)=wt(5) ;wt = .5*wt case(12)s(1,1)=.873821971016996 ; s(1,2)=.063089014491502s(2,1)=s(1,2) ; s(2,2)=s(1,1); s(3,1)=s(1,2) ; s(3,2)=s(1,2)s(4,1)=.501426509658179 ; s(4,2)=.249286745170910s(5,1)=s(4,2); s(5,2)=s(4,1) ; s(6,1)=s(4,2) ; s(6,2)=s(4,2)s(7,1)=.636502499121399 ; s(7,2)=.310352451033785s(8,1)=s(7,1) ; s(8,2)=.053145049844816 ; s(9,1)=s(7,2) ; s(9,2)=s(7,1)s(10,1)=s(7,2) ; s(10,2)=s(8,2) ; s(11,1)=s(8,2); s(11,2)=s(7,1)s(12,1)=s(8,2) ; s(12,2)=s(7,2)wt(1)=.050844906370207 ; wt(2)=wt(1); wt(3)=wt(1)wt(4)=.116786275726379 ; wt(5)=wt(4); wt(6)=wt(4)wt(7)=.082851075618374 ; wt(8:12)=wt(7) ; wt = .5*wtcase(16)s(1,1)=1./3. ; s(1,2)=1./3. ; s(2,1)=.658861384496478s(2,2)=.170569307751761 ; s(3,1)=s(2,2) ; s(3,2)=s(2,1)s(4,1)=s(2,2) ; s(4,2)=s(2,2)s(5,1)=.898905543365938 ; s(5,2)=.050547228317031s(6,1)=s(5,2); s(6,2)=s(5,1) ; s(7,1)=s(5,2) ; s(7,2)=s(5,2)s(8,1)=.081414823414554; s(8,2)=.459292588292723s(9,1)=s(8,2) ; s(9,2)=s(8,1); s(10,1)=s(8,2) ; s(10,2)=s(8,2)s(11,1)=.008394777409958; s(11,2)=.263112829634638s(12,1)=s(11,1) ; s(12,2)=.728492392955404s(13,1)=s(11,2) ; s(13,2)=s(11,1) ; s(14,1)=s(11,2); s(14,2)=s(12,2)s(15,1)=s(12,2) ; s(15,2)=s(11,1) ; s(16,1)=s(12,2) ; s(16,2)=s(11,2)wt(1)=.144315607677787 ; wt(2)=.103217370534718 ; wt(3)=wt(2); wt(4)=wt(2) wt(5)=.032458497623198 ; wt(6)=wt(5) ; wt(7)=wt(5)wt(8)=.095091634267284 ; wt(9)=wt(8) ; wt(10)=wt(8)wt(11)=.027230314174435 ; wt(12:16) = wt(11) ; wt = .5*wtcase defaultprint*,"wrong number of integrating points for a triangle"end select3.2D quadrilateral element(p182 computation mechanics)case ('quadrilateral')select case (nip)case(1)s(1,1) = .0 ; wt(1) = 4.case(4)s(1,1)=-root3; s(1,2)= root3s(2,1)= root3; s(2,2)= root3s(3,1)=-root3; s(3,2)=-root3s(4,1)= root3; s(4,2)=-root3wt = 1.0case(9)s(1:7:3,1) = -r15; s(2:8:3,1) = .0s(3:9:3,1) = r15; s(1:3,2) = r15s(4:6,2) = .0 ; s(7:9,2) =-r15wt= vcase defaultprint*,"wrong number of integrating points for a quadrilateral"end select4.3D tetrahedron element(p231 computation mechanics)case('tetrahedron')select case(nip)case(1) ! for tetrahedra weights multiplied by 1/6s(1,1)=.25 ; s(1,2)=.25 ; s(1,3)=.25 ; wt(1)=1./6.case(4)s(1,1)=.58541020 ; s(1,2)=.13819660 ; s(1,3)=s(1,2)s(2,2)=s(1,1) ; s(2,3)=s(1,2) ; s(2,1)=s(1,2)s(3,3)=s(1,1) ; s(3,1)=s(1,2) ; s(3,2)=s(1,2)s(4,1)=s(1,2) ; s(4,2)=s(1,2) ; s(4,3)=s(1,2) ; wt(1:4)=.25/6.case(5)s(1,1)=.25 ; s(1,2)=.25 ; s(1,3)=.25 ; s(2,1)=.5s(2,2)=1./6. ; s(2,3)=s(2,2); s(3,2)=.5s(3,3)=1./6. ; s(3,1)=s(3,3) ; s(4,3)=.5s(4,1)=1./6. ; s(4,2)=s(4,1); s(5,1)=1./6.s(5,2)=s(5,1) ; s(5,3)=s(5,1)wt(1)=-.8 ; wt(2)=9./20. ; wt(3:5)=wt(2) ; wt =wt/6.case(6)wt = 4./3. ; s(6,3) = 1.s(1,1)=-1. ;s(2,1)=1. ; s(3,2)=-1. ; s(4,2)=1. ; s(5,3)=-1.case defaultprint*,"wrong number of integrating points for a tetrahedron"end select5.3D hexahedron element (p187 computation mechanics)case('hexahedron')select case ( nip )case(1)s(1,1) = .0 ; wt(1) = 8.case(8)s(1,1)= root3;s(1,2)= root3;s(1,3)= root3s(2,1)= root3;s(2,2)= root3;s(2,3)=-root3s(3,1)= root3;s(3,2)=-root3;s(3,3)= root3s(4,1)= root3;s(4,2)=-root3;s(4,3)=-root3s(5,1)=-root3;s(5,2)= root3;s(5,3)= root3s(6,1)=-root3;s(6,2)=-root3;s(6,3)= root3s(7,1)=-root3;s(7,2)= root3;s(7,3)=-root3s(8,1)=-root3;s(8,2)=-root3;s(8,3)=-root3wt = 1.0case(14)b=0.795822426 ; c=0.758786911wt(1:6)=0.886426593 ; wt(7:) = 0.335180055s(1,1)=-b ; s(2,1)=b ; s(3,2)=-b ; s(4,2)=bs(5,3)=-b ; s(6,3)=bs(7:,:) = cs(7,1)=-c ; s(7,2)=-c ; s(7,3)=-c ; s(8,2)=-c ; s(8,3)=-cs(9,1)=-c ; s(9,3)=-c ; s(10,3)=-c; s(11,1)=-cs(11,2)=-c ; s(12,2)=-c ; s(13,1)=-ccase(15)b=1. ; c=0.674199862wt(1)=1.564444444 ; wt(2:7)=0.355555556 ; wt(8:15)=0.537777778 s(2,1)=-b ; s(3,1)=b ; s(4,2)=-b ; s(5,2)=bs(6,3)=-b ; s(7,3)=b ; s(8:,:)=c ; s(8,1)=-cs(8,2)=-c ; s(8,3)=-c ; s(9,2)=-c ; s(9,3)=-cs(10,1)=-c ; s(10,3)=-c ; s(11,3)=-c ; s(12,1)=-cs(12,2)=-c ; s(13,2)=-c ; s(14,1)=-ccase(27)wt = (/5./9.*v,8./9.*v,5./9.*v/)s(1:7:3,1) = -r15; s(2:8:3,1) = .0s(3:9:3,1) = r15; s(1:3,3) = r15s(4:6,3) = .0 ; s(7:9,3) =-r15s(1:9,2) = -r15s(10:16:3,1) = -r15; s(11:17:3,1) = .0s(12:18:3,1) = r15; s(10:12,3) = r15s(13:15,3) = .0 ; s(16:18,3) =-r15s(10:18,2) = .0s(19:25:3,1) = -r15; s(20:26:3,1) = .0s(21:27:3,1) = r15; s(19:21,3) = r15s(22:24,3) = .0 ; s(25:27,3) =-r15s(19:27,2) = r15case defaultprint*,"wrong number of integrating points for a hexahedron"end select。