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

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1、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)22.2D triangle element(p230 computation mechanics)123.2D quadrilateral element(p182 computation mechanics)154.3D tetrahedron element(p231 computation mechanics)175.3D hexahedron element (p187

2、 computation mechanics)191. 1D bar element(p181 computation mechanics)Order nAccuracy gradeLocationWeight w(2*n-1)1110.02.0231-1/ 31.021/ 31.0351- 0.65/920.08/93 0.65/94710.8611363115940530.34785484513745420.3399810435848560.6521451548625463-0.3399810435848560.6521451548625464-0.8611363115940530.347

3、8548451374545910.9061798459386640.23692688505618920.5384693101056830.478628670499366300.5688888888888894-0.5384693101056830.4786286704993665-0.9061798459386640.23692688505618961110.9324695142031520.17132449237917020.6612093864662650.36076157304813930.2386191860831970.4679139345726914-0.2386191860831

4、970.4679139345726915-0.6612093864662650.3607615730481396-0.9324695142031520.17132449237917012231-0.9815606342467320.0471753363864754720.9041172563704520.10693932599536373-0.76990267419431770.16007832854335864-0.58731795428661430.20316742672306725-0.36783149899818040.23349253653835346-0.1252334085111

5、46880.249147045813402770.125233408511146880.249147045813402780.36783149899818040.233492536538353490.58731795428661430.2031674267230672100.76990267419431770.1600783285433586110.9041172563704520.1069393259953637120.9815606342467320.04717533638647547root3 = 1./sqrt(3.); r15 = .2*sqrt(15.)nip = ubound(

6、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(3,1)=-s(2,1) ; s(4,1)=-s(1,1)wt(3)=wt(2) ; wt

7、(4)=wt(1)case(5)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(4,1)=-s(3,1) ; s(5,1)=-s(2,1) ; s(6,1)=-s(1,1)wt(4)=wt(3); wt(5)=wt(2) ; wt(6)=wt(1)case defaultprint*,wrong n

8、umber 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;elseifnQ = 2Data =

9、.-5.7735026918962573e-001 1.0000000000000000e+0005.7735026918962573e-001 1.0000000000000000e+000;elseifnQ = 3Data = .-7.7459666924148340e-001 5.5555555555555558e-0010.0000000000000000e+000 8.8888888888888884e-0017.7459666924148340e-001 5.5555555555555558e-001;elseifnQ = 4Data = .;elseifnQ = 5Data =

10、.-5.3846931010568311e-001 4.7862867049936647e-0010.0000000000000000e+000 5.6888888888888889e-0015.3846931010568311e-001 4.7862867049936647e-001;elseifnQ = 6Data = .;elseifnQ = 7Data = .;elseifnQ = 8Data = .-1.8343464249564978e-001 3.6268378337836199e-0011.8343464249564978e-001 3.6268378337836199e-00

11、1;elseifnQ = 9Data = .0.0000000000000000e+000 3.3023935500125978e-001;elseifnQ = 10Data = .-4.3339539412924721e-001 2.6926671930999635e-001-1.4887433898163116e-001 2.9552422471475287e-0011.4887433898163116e-001 2.9552422471475287e-0014.3339539412924721e-001 2.6926671930999635e-001;elseifnQ = 11Data

12、= .-9.7822865814605697e-001 5.5668567116173663e-002-8.8706259976809532e-001 1.2558036946490461e-0010.0000000000000000e+000 2.7292508677790062e-0018.8706259976809532e-001 1.2558036946490461e-0019.7822865814605697e-001 5.5668567116173663e-002;elseifnQ = 12Data = .-7.6990267419430469e-001 1.60078328543

13、34622e-0017.6990267419430469e-001 1.6007832854334622e-001;elseifnQ = 13Data = .;elseif nQ = 14Data = .-9.8628380869681231e-0013.5119460331751860e-002-9.2843488366357352e-0018.0158087159760208e-002;elseifnQ = 15Data = .-9.8799251802048538e-001 3.0753241996117269e-002-9.3727339240070595e-001 7.0366047

14、488108124e-0020.0000000000000000e+000 2.0257824192556129e-0019.3727339240070595e-001 7.0366047488108124e-0029.8799251802048538e-001 3.0753241996117269e-002;elseifnQ = 16Data = .-9.4457502307323260e-001 6.2253523938647894e-002-7.5540440835500300e-001 1.2462897125553388e-0017.5540440835500300e-001 1.2

15、462897125553388e-0019.4457502307323260e-001 6.2253523938647894e-002;elseif nQ = 17Data = .-9.9057547531441736e-0012.4148302868547931e-002-9.5067552176876780e-0015.5459529373987203e-0020.0000000000000000e+000 1.7944647035620653e-0019.5067552176876780e-001 5.5459529373987203e-002;elseifnQ = 18Data = .

16、-8.9260246649755570e-0017.6425730254889052e-002-8.0370495897252314e-0011.0094204410628717e-001-6.9168704306035322e-0011.2255520671147846e-001-5.5977083107394754e-0011.4064291467065065e-0015.5977083107394754e-001 1.4064291467065065e-0018.0370495897252314e-001 1.0094204410628717e-0018.9260246649755570

17、e-001 7.6425730254889052e-002;elseifnQ = 19Data = .-6.0054530466168110e-001 1.2875396253933621e-001-4.6457074137596099e-001 1.4260670217360660e-001-1.6035864564022539e-001 1.5896884339395434e-0011.6035864564022539e-001 1.5896884339395434e-0014.6457074137596099e-001 1.4260670217360660e-0016.005453046

18、6168110e-001 1.2875396253933621e-001;elseifnQ = 20Data = .-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-0013.73

19、70608871541955e-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-00

20、1 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-002-7.6843996347

21、567789e-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

22、.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

23、-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);% % Plo

24、t 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 nLocation Location

25、?Weight w1(a)11/31/31.03(c)10.50.51/320.50.01/330.00.51/34(d)11/31/3-27/4823/51/525/4831/51/525/4841/53/525/48610.8168475729804590.0915762135097710.109951743655322*0.520.0915762135097710.8168475729804590.109951743655322*0.530.0915762135097710.0915762135097710.109951743655322*0.540.1081030181680700.4

26、459484909159650.223381589678011*0.550.4459484909159650.1081030181680700.223381589678011*0.560.4459484909159650.4459484909159650.223381589678011*0.5121.8738219710169960.0630890144915020.050844906370207*0.52.0630890144915020.8738219710169960.050844906370207*0.53.0630890144915020.0630890144915020.05084

27、4906370207*0.54.5014265096581790.2492867451709100.116786275726379*0.55.2492867451709100.5014265096581790.116786275726379*0.56.5014265096581790.5014265096581790.116786275726379*0.57.6365024991213990.3103524510337850.082851075618374*0.58.6365024991213990.0531450498448160.082851075618374*0.59.310352451

28、0337850.6365024991213990.082851075618374*0.5100.3103524510337850.0531450498448160.082851075618374*0.511.0531450498448160.6365024991213990.082851075618374*0.512.0531450498448160.3103524510337850.082851075618374*0.5case(triangle)select case(nip)case(1)! for triangles weights multiplied by .5s(1,1)=1./

29、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(2,1)=s(1,2); s(2,2)=s(1,1) ; s(3,1)=s(1,2); s(3,2)=s(1,2)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(2)=wt(1) ;wt(3)=wt(

30、1)wt(4)=.223381589678011 ;wt(5)=wt(4) ;wt(6)=wt(4); wt = .5*wtcase(7)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(6,1)=s(5,2) ; s(6,2)=s(5,1); s(7,1)=s(5,1); s(7,2)=s(5,1) ; 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,

31、1); s(3,1)=s(1,2) ; s(3,2)=s(1,2)s(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(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(7)=.082851075618374 ; wt(8:12)=wt(7);

32、 wt = .5*wtcase(16)s(1,1)=1./3. ; s(1,2)=1./3. ; s(2,1)=.658861384496478; 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(9,1)=s(8,2) ; s(9,2)=s(8,1); s(10,1)=s(8,2) ; s(10,2)=s(8,2)s(12,1)=s(11,

33、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(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) =

34、 wt(11) ;wt = .5*wtcase defaultprint*,wrong number of integrating points for a triangleend select3. 2D quadrilateral element(p182 computation mechanics)Order nLocation Location ?Weight w110.00.04.041-1/ 3-1/ 31.021/ 3-1/ 31.031/ 31/ 31.04-1/ 31/ 31.091- 0.6- 0.625/812- 0.60.040/813- 0.6 0.625/8140.0

35、- 0.640/8150.00.064/8160.0 0.640/817 0.6- 0.625/818 0.60.040/819 0.6 0.625/81case (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(

36、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 quadrilateralend select4. 3D tetrahedron element(p231 computation mechanics)OrderLocation rLocation sLocation tWeight wn111/41/41/41.0/641.58541020.13819660.138196

37、601/242.13819660.58541020.138196601/243.13819660.13819660.585410201/244.13819660.13819660.138196601/24511/41/41/4-4/3021/21/61/69/12031/61/21/69/12041/61/61/29/12051/61/61/69/12061-1.00.00.04/321.00.00.04/330.0-1.00.04/340.01.00.04/350.00.0-1.04/360.00.01.04/3case(tetrahedron)select case(nip)case(1)

38、! 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.

39、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.

40、 ; s(3,2)=-1. ; s(4,2)=1. ; s(5,3)=-1.case defaultprint*,wrong number of integrating points for a tetrahedronend select5. 3D hexahedron element (p187 computation mechanics)OrderLocation rLocation sLocation tn110.00.00.0811/ 31/ 31/ 321/ 31/ 3-1/331/ 3-1/31/ 341/ 3-1/3-1/35-1/31/ 31/ 36-1/3-1/31/ 37-

41、1/31/ 3-1/38-1/3-1/3-1/3case(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)= roo

42、t3s(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)=-

43、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)=-cWeight w8.01.01.01.01.01.01.01.01.0case(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,

44、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(1

45、0:16:3,1) = -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 hexahedronend select