一維、二維、三維高斯積分點及權重

1、Gauss integrations and weights(Containing the program)高斯積分點以及權重目錄 TOC o 1-5 h z HYPERLINK l bookmark12 o Current Document 1D bar element(p181 computation mechanics)22D triangle element(p230 computation mechanics)122D quadrilateral element(p182 computation mechanics)153D tetrahedron element(p231 comp

2、utation mechanics)173D hexahedron element (p187 computation mechanics)191. 1D bar element(p181 computation mechanics)Order nAccuracy grade (2*n-1)Location 號Weight w1110.02.0231-1/V31.021/V31.0351-V0.65/920.08/93”0.65/94710.8611363115940530.34785484513745420.3399810435848560.6521451548625463-0.339981

3、0435848560.6521451548625464-0.8611363115940530.3478548451374545910.9061798459386640.23692688505618920.5384693101056830.478628670499366300.5688888888888894-0.5384693101056830.4786286704993665-0.9061798459386640.23692688505618961110.9324695142031520.17132449237917020.6612093864662650.36076157304813930

4、.2386191860831970.4679139345726914-0.2386191860831970.4679139345726915-0.6612093864662650.3607615730481396-0.9324695142031520.17132449237917012231-0.9815606342467320.0471753363864754720.9041172563704520.10693932599536373-0.76990267419431770.16007832854335864-0.58731795428661430.20316742672306725-0.3

5、6783149899818040.23349253653835346-0.125233408511146880.249147045813402770.125233408511146880.249147045813402780.36783149899818040.233492536538353490.58731795428661430.2031674267230672100.76990267419431770.1600783285433586110.9041172563704520.1069393259953637120.9815606342467320.04717533638647547roo

6、t3 = 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 = wca

7、se(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)=.652145154862546 wt(3)=wt(2) ; wt(4)=wt(1)case(5)s(1,1)=.906179845938664 ; s(2,1)=.538469310105683s(3,1)=.0 ; s(4,1)=-s(2,1) ; s(5,1)=-s(1,1)wt(1)=.236926885056189 ; wt(2)=.47862867049

8、9366wt(3)=.568888888888889 ; wt(4)=wt(2) ; wt(5)=wt(1)case(6)s(1,1)=.932469514203152 ; s(2,1)=661209386466265;s(3,1)=238619186083197 s(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)=467913934572691 wt(4)=wt(3); wt(5)=wt(2) ; wt(6)=wt(1) case defaul

9、tprint*,wrong number of integrating points for a lineend 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+0002.0000000000000000e+000;else

10、if nQ = 2Data = .-5.7735026918962573e-0011.0000000000000000e+0005.7735026918962573e-0011.0000000000000000e+000; elseif nQ = 3Data = .-7.7459666924148340e-0015.5555555555555558e-0010.0000000000000000e+0008.8888888888888884e-0017.7459666924148340e-0015.5555555555555558e-001;elseif nQ = 4Data = .-8.611

11、3631159405257e-0013.4785484513745385e-001-3.3998104358485626e-0016.5214515486254609e-0013.3998104358485626e-0016.5214515486254609e-0018.6113631159405257e-0013.4785484513745385e-001;elseif nQ = 5Data = .-9.0617984593866396e-0012.3692688505618908e-001-5.3846931010568311e-0014.7862867049936647e-0010.00

12、00000000000000e+0005.6888888888888889e-0015.3846931010568311e-0014.7862867049936647e-0019.0617984593866396e-0012.3692688505618908e-001;elseif nQ = 6Data = .-9.3246951420315205e-0011.7132449237917036e-001-6.6120938646626448e-0013.6076157304813861e-001-2.3861918608319688e-0014.6791393457269104e-0012.3

13、861918608319688e-0014.6791393457269104e-0016.6120938646626448e-0013.6076157304813861e-0019.3246951420315205e-0011.7132449237917036e-001;123.1233.8123.61222elseif nQ = 7Data =.-9.4910791234275849e-001-7.4153118559939446e-001-4.0584515137739718e-001 TOC o 1-5 h z 0.0000000000000000e+00044.058451513773

14、9718e-00137.4153118559939446e-00129.4910791234275849e-0011；elseif nQ = 8Data = .-9.6028985649753618e-001-7.9666647741362673e-001-5.2553240991632899e-001-1.8343464249564978e-001 TOC o 1-5 h z 1.8343464249564978e-00135.2553240991632899e-00137.9666647741362673e-00129.6028985649753618e-0011；elseif nQ =

15、9Data = .-9.6816023950762609e-001-8.3603110732663577e-001-6.1337143270059036e-001-3.2425342340380892e-001 TOC o 1-5 h z 0.0000000000000000e+00033.2425342340380892e-00136.1337143270059036e-00128.3603110732663577e-00119.6816023950762609e-0018；elseif nQ = 10Data = .-9.7390652851717174e-001-8.6506336668

16、898454e-001-6.7940956829902444e-001-4.3339539412924721e-001-1.4887433898163116e-001 TOC o 1-5 h z 1.4887433898163116e-00124.3339539412924721e-00126.7940956829902444e-0012.2948496616886970e-001.7970539148927664e-001.8183005050511892e-0011795918367346940e-0018183005050511892e-0017970539148927664e-0012

17、948496616886970e-001.0122853629037626e-001.2238103445337448e-001.1370664587788727e-001.6268378337836199e-0016268378337836199e-0011370664587788727e-0012238103445337448e-0010122853629037626e-001.1274388361574412e-002.8064816069485740e-001.6061069640293544e-001.1234707704000286e-0013023935500125978e-00

18、11234707704000286e-0016061069640293544e-0018064816069485740e-0011274388361574412e-002.6671344308688138e-002.4945134915058059e-001.1908636251598204e-001.6926671930999635e-001.9552422471475287e-0019552422471475287e-0016926671930999635e-0011908636251598204e-0014945134915058059e-0018.6506336668898454e-0

19、01151122.411222.4911229.7390652851717174e-0016.6671344308688138e-002 ；elseif nQ = 11Data =.-9.7822865814605697e-001-8.8706259976809532e-001-7.3015200557404936e-001-5.1909612920681181e-001-2.6954315595234501e-001 TOC o 1-5 h z 0.0000000000000000e+00022.6954315595234501e-00125.1909612920681181e-00127.

20、3015200557404936e-00118.8706259976809532e-00119.7822865814605697e-0015;elseif nQ = 12Data = .-9.8156063424671924e-001-9.0411725637047491e-001-7.6990267419430469e-001-5.8731795428661737e-001-3.6783149899818024e-001-1.2523340851146891e-001 TOC o 1-5 h z 1.2523340851146891e-00123.6783149899818024e-0012

21、5.8731795428661737e-00127.6990267419430469e-00119.0411725637047491e-00119.8156063424671924e-0014；elseif nQ = 13Data = .-9.8418305471858814e-001-9.1759839922297792e-001-8.0157809073330988e-001-6.4234933944034023e-001-4.4849275103644681e-001-2.3045831595513477e-001 TOC o 1-5 h z 0.0000000000000000e+00

22、022.3045831595513477e-00124.4849275103644681e-00126.4234933944034023e-0011.5668567116173663e-002.2558036946490461e-001.8629021092773426e-001.3319376459199048e-001.6280454451024665e-0017292508677790062e-0016280454451024665e-0013319376459199048e-0018629021092773426e-0012558036946490461e-00156685671161

23、73663e-002.7175336386511828e-002.0693932599531843e-001.6007832854334622e-001.0316742672306592e-001.3349253653835481e-001.4914704581340277e-0014914704581340277e-0013349253653835481e-0010316742672306592e-0016007832854334622e-0010693932599531843e-0017175336386511828e-002.0484004765315877e-002.212149983

24、7728452e-002.3887351021978725e-001.7814598076194574e-001.0781604753688851e-001.2628318026289723e-0013255155323087390e-0012628318026289723e-0010781604753688851e-0017814598076194574e-0013887351021978725e-0018.0157809073330988e-00113811122.3711111.26919.1759839922297792e-00199.8418305471858814e-0014；el

25、seif nQ = 14Data =.-9.8628380869681231e-001-9.2843488366357352e-001-8.2720131506976502e-001-6.8729290481168548e-001-5.1524863635815410e-001-3.1911236892788974e-001-1.0805494870734367e-001 TOC o 1-5 h z 1.0805494870734367e-00123.1911236892788974e-00125.1524863635815410e-00116.8729290481168548e-00118.

26、2720131506976502e-00119.2843488366357352e-00189.8628380869681231e-0013；elseif nQ = 15Data = .-9.8799251802048538e-001-9.3727339240070595e-001-8.4820658341042721e-001-7.2441773136017007e-001-5.7097217260853883e-001-3.9415134707756339e-001-2.0119409399743449e-001 TOC o 1-5 h z 0.0000000000000000e+0002

27、2.0119409399743449e-00113.9415134707756339e-00115.7097217260853883e-00117.2441773136017007e-00118.4820658341042721e-00119.3727339240070595e-00179.8799251802048538e-0013；elseif nQ = 16Data = .-9.8940093499164994e-001-9.4457502307323260e-001-8.6563120238783176e-0012121499837728452e-0020484004765315877

28、e-002.5119460331751860e-002.0158087159760208e-002.2151857068790319e-001.5720316715819355e-001.8553839747793782e-001.0519846372129560e-001.1526385346315779e-0011526385346315779e-0010519846372129560e-0018553839747793782e-0015720316715819355e-0012151857068790319e-0010158087159760208e-002511946033175186

29、0e-002.0753241996117269e-002.0366047488108124e-002.0715922046717194e-001.3957067792615432e-001.6626920581699392e-001.8616100001556221e-001.9843148532711158e-0010257824192556129e-0019843148532711158e-0018616100001556221e-0016626920581699392e-0013957067792615432e-0010715922046717194e-00103660474881081

30、24e-0020753241996117269e-002.7152459411754096e-002.2253523938647894e-002.5158511682492786e-002.2462897125553388e-001-7.5540440835500300e-0011111.25811111.247111111-6.1787624440264377e-001-4.5801677765722737e-001-2.8160355077925892e-001-9.5012509837637427e-002 TOC o 1-5 h z 9.5012509837637427e-00212.

31、8160355077925892e-00114.5801677765722737e-00116.1787624440264377e-00117.5540440835500300e-00118.6563120238783176e-00199.4457502307323260e-00169.8940093499164994e-0012；elseif nQ = 17Data =.-9.9057547531441736e-001-9.5067552176876780e-001-8.8023915372698591e-001-7.8151400389680137e-001-6.5767115921669

32、084e-001-5.1269053708647694e-001-3.5123176345387630e-001-1.7848418149584788e-001 TOC o 1-5 h z 0.0000000000000000e+00011.7848418149584788e-00113.5123176345387630e-00115.1269053708647694e-00116.5767115921669084e-00117.8151400389680137e-00118.8023915372698591e-00189.5067552176876780e-00159.90575475314

33、41736e-0012;elseif nQ = 18Data = .-9.9156516842093090e-001-9.5582394957139771e-001-8.9260246649755570e-001-8.0370495897252314e-001-6.9168704306035322e-001-5.5977083107394754e-001-4.1175116146284263e-001-2.5188622569150554e-001.4959598881657674e-001.6915651939500254e-001.8260341504492358e-001.8945061

34、045506850e-0018945061045506850e-0018260341504492358e-0016915651939500254e-0014959598881657674e-0012462897125553388e-0015158511682492786e-0022253523938647894e-0027152459411754096e-002.4148302868547931e-002.5459529373987203e-002.5036148317179178e-002.1188384719340397e-001.3513636846852548e-001.5404576

35、107681028e-001.6800410215645004e-001.7656270536699264e-0017944647035620653e-0017656270536699264e-0016800410215645004e-0015404576107681028e-0013513636846852548e-0011188384719340397e-0015036148317179178e-0025459529373987203e-0024148302868547931e-002.1616013526483312e-002.9714548894969797e-002.64257302

36、54889052e-002.0094204410628717e-001.2255520671147846e-001.4064291467065065e-001.5468467512626524e-001.6427648374583273e-001.6914238296314360e-001-8.4775013041735292e-002146911111.1468111111 TOC o 1-5 h z 8.4775013041735292e-00212.5188622569150554e-00114.1175116146284263e-00115.5977083107394754e-0011

37、6.9168704306035322e-00118.0370495897252314e-00118.9260246649755570e-00179.5582394957139771e-00149.9156516842093090e-0012；elseif nQ = 19Data =.-9.9240684384358435e-001-9.6020815213483002e-001-9.0315590361481790e-001-8.2271465653714282e-001-7.2096617733522939e-001-6.0054530466168110e-001-4.64570741375

38、96099e-001-3.1656409996362989e-001-1.6035864564022539e-001 TOC o 1-5 h z 0.0000000000000000e+00011.6035864564022539e-00113.1656409996362989e-00114.6457074137596099e-00116.0054530466168110e-00117.2096617733522939e-00118.2271465653714282e-00199.0315590361481790e-00169.6020815213483002e-00149.924068438

39、4358435e-0011;elseif nQ = 20Data = .-9.9312859918509488e-001-9.6397192727791381e-001-9.1223442825132595e-001-8.3911697182221878e-001-7.4633190646015080e-001-6.3605368072651502e-001-5.1086700195082702e-001-3.7370608871541955e-001-2.2778585114164507e-0016914238296314360e-0016427648374583273e-001546846

40、7512626524e-0014064291467065065e-0012255520671147846e-0010094204410628717e-0016425730254889052e-0029714548894969797e-0021616013526483312e-002.9461788229726478e-002.4814226765699600e-002.9044542737641226e-002.1490021622449999e-002.1156664554733399e-001.2875396253933621e-001.4260670217360660e-001.5276

41、604206585967e-001.5896884339395434e-0016105444984878370e-0015896884339395434e-0015276604206585967e-0014260670217360660e-0012875396253933621e-0011156664554733399e-0011490021622449999e-0029044542737641226e-0024814226765699600e-0029461788229726478e-002.7614007139152118e-002.0601429800386939e-002.267204

42、8334109068e-002.3276741576704755e-002.0193011981724044e-001.1819453196151841e-001.3168863844917664e-001.4209610931838204e-001.4917298647260374e-001.5275338713072584e-001-7.6526521133497338e-0027.6526521133497338e-0021.5275338713072584e-0012.2778585114164507e-0011.4917298647260374e-0013.7370608871541

43、955e-0011.4209610931838204e-0015.1086700195082702e-0011.3168863844917664e-0016.3605368072651502e-0011.1819453196151841e-0017.4633190646015080e-0011.0193011981724044e-0018.3911697182221878e-0018.3276741576704755e-0029.1223442825132595e-0016.2672048334109068e-0029.6397192727791381e-0014.06014298003869

44、39e-0029.9312859918509488e-0011.7614007139152118e-002；elseif nQ = 21Data =.-9.9375217062038945e-0011.6017228257774335e-002-9.6722683856630631e-0013.6953789770852494e-002-9.2009933415040079e-0015.7134425426857205e-002-8.5336336458331730e-0017.6100113628379304e-002-7.6843996347567789e-0019.34444234560

45、33862e-002-6.6713880419741234e-0011.0879729916714838e-001-5.5161883588721983e-0011.2183141605372853e-001-4.2434212020743878e-0011.3226893863333747e-001-2.8802131680240106e-0011.3988739479107315e-001-1.4556185416089507e-0011.4452440398997005e-0010.0000000000000000e+0001.4608113364969041e-0011.4556185

46、416089507e-0011.4452440398997005e-0012.8802131680240106e-0011.3988739479107315e-0014.2434212020743878e-0011.3226893863333747e-0015.5161883588721983e-0011.2183141605372853e-0016.6713880419741234e-0011.0879729916714838e-0017.6843996347567789e-0019.3444423456033862e-0028.5336336458331730e-0017.61001136

47、28379304e-0029.2009933415040079e-0015.7134425426857205e-0029.6722683856630631e-0013.6953789770852494e-0029.9375217062038945e-0011.6017228257774335e-002;elseerror( The number num2str(nQ) is not implemented.) endxxIntRef = Data(:, 1);wwIntRef = Data(:, 2);% % Plot situation.% reset(cla), reset(clf), h

48、old 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)Order nLocation LocationWeight w1(a)11/31/31.03(c)10.50.51/320.50.01/330.00.51/34(d)11/31/3-27/4823/51/525/4831/51

49、/525/4841/53/525/48610.8168475729804590.0915762135097710.109951743655322*0.520.0915762135097710.8168475729804590.109951743655322*0.530.0915762135097710.0915762135097710.109951743655322*0.540.1081030181680700.4459484909159650.223381589678011*0.550.4459484909159650.1081030181680700.223381589678011*0.5

50、60.4459484909159650.4459484909159650.223381589678011*0.5121.8738219710169960.0630890144915020.050844906370207*0.52.0630890144915020.8738219710169960.050844906370207*0.53.0630890144915020.0630890144915020.050844906370207*0.54.5014265096581790.2492867451709100.116786275726379*0.55.2492867451709100.501

51、4265096581790.116786275726379*0.56.5014265096581790.5014265096581790.116786275726379*0.57.6365024991213990.3103524510337850.082851075618374*0.58.6365024991213990.0531450498448160.082851075618374*0.59.3103524510337850.6365024991213990.082851075618374*0.5100.3103524510337850.0531450498448160.082851075

52、618374*0.511.0531450498448160.6365024991213990.082851075618374*0.512.0531450498448160.3103524510337850.082851075618374*0.5case(triangle)select case(nip)case(1) ! for triangles weights multiplied by .5 s(1,1)=1./3. ; s(1,2)=1./3. ; wt(1)= .5case(3)s(1,1)=.5 ; s(1,2)=.5 ; s(2,1)=.5 s(2,2)=0.; s(3,1)=0

53、. ; s(3,2)=.5 wt(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)=445948490915965 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)=.109

54、951743655322 ;wt(2)=wt(1) ;wt(3)=wt(1)wt(4)=.223381589678011 ;wt(5)=wt(4) ;wt(6)=wt(4); wt = .5*wtcase(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

55、,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*wtcase(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)=.5

56、01426509658179 ; 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

57、(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)=.658861384496478 s(2,2)=.170569307751761 ; s(3,1)=s(2,2); s(3,2)=s(2,1)s(4,1)=s(2,2) ; s(4,

58、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)=.263112829634638 s(12,1)=s(11,1); s(

59、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)=.0950916342

60、67284 ; 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 select階仲點）高斯積分旃有就萬頊積分時取樣加Order nLocation Location nWeight w110.00.04.041-1/V3-1/V31.021/V3-1/V31.031.31.31.04-1/V31.31.091-0.6-0.625/812-