一維、二維、三維高斯積分點及權(quán)重-Gaussintegrationsandweights

Gaussintegrationsandweights

(Containingtheprogram)

高斯積分點以及權(quán)重

目錄

1.IDbarclement(pl81computationmechanics)......................................2

2.2Dtriangleelement(p230computationmechanics)...................................................................12

3.2Dquadrilateralelement(pl82computationmechanics)..........................................................15

4.3Dtetrahedronelement(p231computationmechanics)...........................................................17

5.3Dhexahedronelement(pl87computationmechanics)...........................................................19

1.IDbarelement(pl81computationmechanics)

OrdernAccuracygradeLocationgWeightw

(2*n-l)

1110.02.0

231-1/731.0

2I/V31.0

351-V0.65/9

20.08/9

3J0.65/9

4710.8611363115940530.347854845137454

20.3399810435848560.652145154862546

3-0.3399810435848560.652145154862546

4-0.8611363115940530.347854845137454

5910.9061798459386640.236926885056189

20.5384693101056830.478628670499366

300.568888888888889

4-0.5384693101056830.478628670499366

5-0.9061798459386640.236926885056189

61110.9324695142031520.171324492379170

20.6612093864662650.360761573048139

30.2386191860831970.467913934572691

4-0.2386191860831970.467913934572691

5-0.6612093864662650.360761573048139

6-0.9324695142031520.171324492379170

12231-0.9815606342467320.04717533638647547

20.9041172563704520.1069393259953637

3-0.76990267419431770.1600783285433586

4-0.58731795428661430.2031674267230672

5-0.36783149899818040.2334925365383534

60.125233408511146880.2491470458134027

70.125233408511146880.2491470458134027

80.36783149899818040.2334925365383534

90.58731795428661430.2031674267230672

100.76990267419431770.1600783285433586

110.9041172563704520.1069393259953637

120.9815606342467320.04717533638647547

root3=l./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/)

selectcase(element)

case('line')

selectcase(nip)

case(l)

s(1,1)=0.;wt(l)=2.

case(2)

s(1,1)=root3;s(2,l)=-s(lJ);wt(l)=l.;wt(2)=l.

casc(3)

s(l,l)=rl5;s(2,l)=.O;s(3,l)=-s(l,l)

wt=w

case(4)

s(l,1)=861136311594053;s(2,1)=339981043584856

s(3,l)=-s(2,l);s(4,l>-s(l,l)

wt(1)=347854845137454:wt(2)=.652145154862546

wt(3)=wt(2);wt(4)=wt(l)

case(5)

s(1,1)=906179845938664;s(2,1)=538469310105683

s(3,l)=.O;s(4,1)=-s(2,1);s(5,1)=-s(1,1)

wt(l)=236926885056189;wt(2)=.478628670499366

wt(3)=.568888888888889;wt(4)=wt(2);wt(5)=wt(l)

case(6)

s(l,l)=.932469514203152;s(2,1)=661209386466265;s(3,1)=238619186083197

s(4,l)=-s(3,l)；s(5,l)=-s(2,l);s(6,l)=-s(l,l)

wt(l)=171324492379170;wt(2)=.36O761573048139;wt(3)=.467913934572691

wt(4)=wt(3);wt(5)=wt(2);wt(6)=wt(1)

casedefault

print*,"wrongnumberofintegratingpointsforaline"

endselect

%Copyright(c)2010,Thonas-PeterFries,RWTHAachenUniversity

function[xxIntRef,wwIntRef]=IntPointslDGauss(nQ)

%SetGausspointsinIDreferenceelementfrom[-1,1].

ifnQ==1

Data=[...

0.0000000000000000e+0002.0000000000000000e+000

]；

elseifnQ==2

Data=[...

-5.7735026918962573e-0011.OOOOOOOOOOOOOOOOe+DOO

5.7735026918962573e-0011.OOOOOOOOOOOOOOOOe+ODO

]；

elseifnQ==3

Data=[...

7.7459666924148340G0015.5555555555555558?001

0.0000000000000000e+0008.8888888888888884e-001

7.7459666924148340e-0015.5555555555555558e-001

]；

elseifnQ==4

Data=[...

-8.6113631159405257e-0013.4785484513745385e-001

-3.3998104358485626e-0016.5214515486254609e-001

3.3998104358485626e-0016.5214515486254609e-001

8.6113631159405257e-0013.4785484513745385e-001

]；

elseifnQ==5

Data=[...

-9.0617984593866396e-0012.3692688505618908e-001

-5.3846931010568311e-0014.7862867049936647e-001

0.0000000000000000e+0005.6888888888888889e-001

5.3846931010568311e-0014.7862867049936647e-001

9.0617984593866396e-0012.3692688505618908e-001

]；

elseifnQ==6

Data=[...

-9.3246951420315205e-0011.7132449237917036e-001

-6.6120938646626448e-0013.6076157304813861e-001

-2.3861918608319688e-0014.6791393457269104e-901

2.3861918608319688e-0014.6791393457269104e-001

6.6120938646626448e-0013.6076157304813861e-001

9.3246951420315205e-0011.7132449237917036e-001

1；

elseifnQ==7

Data=[...

-9.4910791234275849^-0011.2948496616886970e-901

-7.4153118559939446e-0012.7970539148927664e-D01

-4.0584515137739718e-0013.8183005050511892e-001

0.0000000000000000e+0004.1795918367346940e-001

4.0584515137739718e-0013.8183005050511892e-001

7.4153118559939446e-0012.7970539148927664e-001

9.4910791234275849e-0011.2948496616886970e-001

]；

elseifnQ==8

Data=[...

-9.6028985649753618e-0011.0122853629037626e-001

-7.9666647741362673e-0012.2238103445337448e-001

-5.2553240991632899e-0013.1370664587788727e-001

-1.8343464249564978e-0013.6268378337836199e-001

1.8343464249564978s0013.6268378337836199?001

5.2553240991632899e-0013.1370664587788727e-001

7.9666647741362673e-0012.2238103445337448e-001

9.6028985649753618e-0011.0122853629037626e-001

1；

elseifnQ==9

Data=[...

-9.6816023950762609e-0018.1274388361574412e-002

-8.3603110732663577e-0011.8064816069485740e-001

-6.1337143270059036e-0012.6061069640293544e-001

-3.2425342340380892e-0013.1234707704000286e-D01

0.0000000000000000e+0003.3023935500125978e-001

3.2425342340380892e-0013.1234707704000286e-001

6.1337143270059036e-0012.6061069640293544e-001

8.3603110732663577e-0011.8064816069485740e-001

9.6816023950762609e-0018.1274388361574412e-002

]；

elseifnQ==10

Data=[...

-9.7390652851717174e-0016.6671344308688138e-002

-8.6506336668898454e-0011.4945134915058059e-001

-6.7940956829902444e-0012.1908636251598204e-001

-4.3339539412924721e-0012.6926671930999635e-001

-1.4887433898163116e-0012.9552422471475287e-901

1.4887433898163116e-0012.9552422471475287e-001

4.3339539412924721e-0012.6926671930999635e-001

6.7940956829902444e-0012.1908636251598204e-001

8.6506336668898454e-0011.4945134915058059e-001

9.7390652851717174e-0016.6671344308688138e-002

1；

elaeifuQ==11

Data-[...

-9.7822865814605697e-0015.5668567116173663e-002

-8.8706259976809532e-0011.2558036946490461e-001

-7.3015200557404936e-0011.8629021092773426e-001

-5.1909612920681181e-0012.3319376459199048e-001

-2.6954315595234501e-0012.6280454451024665e-001

0.0000000000000000e+0002.7292508677790062e-001

2.69543155952345016-0012.6280454451024665e-001

5.1909612920681181e-0012.3319376459199048e-001

7.3015200557404936e-0011.8629021092773426e-001

8.8706259976809532e-0011.2558036946490461e-001

9.7822865814605697e-0015.5668567116173663e-002

1；

elseifnQ—12

Data=[...

-9.8156063424671924e-0014.7175336386511828e-002

-9.0411725637047491e-0011.0693932599531843e-001

-7.6990267419430469e-0011.6007832854334622e-001

-5.8731795428661737e-0012.0316742672306592e-001

-3.6783149899818024e-0012.3349253653835481e-001

-1.2523340851146891e-0012.4914704581340277e-001

1.2523340851146891e-0012.4914704581340277e-001

3.6783149899818024e-0012.3349253653835481e-001

5.8731795428661737e-0012.0316742672306592e-001

7.6990267419430469e-0011.6007832854334622e-001

9.0411725637047491e-0011.0693932599531843e-001

9.8156063424671924e-0014.7175336386511828e-0D2

]；

elseifnQ==13

Data=[...

-9.8418305471858814e-0014.0484004765315877e-002

-9.1759839922297792e-0019.2121499837728452e-002

-8.0157809073330988e-0011.3887351021978725e-001

-6.4234933944034023e-0011.7814598076194574e-001

-4.4849275103644681e-0012.0781604753688851e-001

-2.3045831595513477e-0012.2628318026289723e-001

0.0000000000000000e+0002.3255155323087390e-001

2.3045831595513477e-0012.2628318026289723e-001

4.4849275103644681e-0012.0781604753688851e-001

6.4234933944034023e-0011.7814598076194574e-001

8.0157809073330988e-0011.3887351021978725e-001

9.1759839922297792e-0019.2121499837728452e-002

9.8418305471858814e-0014.0484004765315877e-002

]；

elseifnQ==14

Data=[...

-9.8628380869681231e-0013.5119460331751860e-002

-9.2843488366357352e-0018.0158087159760208e-002

-8.2720131506976502e-0011.2151857068790319e-001

-6.8729290481168548e-0011.5720316715819355e-001

-5.1524863635815410e-0011.8553839747793782e-001

-3.1911236892788974e-0012.0519846372129560e-001

-1.0805494870734367e-0012.1526385346315779e-001

1.0805494870734367e-0012.1526385346315779e-001

3.1911236892788974e-0012.0519846372129560e-001

5.1524863635815410e-0011.8553839747793782e-001

6.8729290481168548e-0011.5720316715819355e-001

8.2720131506976502s0011.2151857068790319?001

9.2843488366357352e-0018.0158087159760208e-002

9.8628380869681231e-0013.5119460331751860e-002

]；

elseifnQ==15

Data=[...

-9.8799251802048538e-0013.0753241996117269e-002

-9.3727339240070595e-0017.0366047488108124e-002

-8.4820658341042721e-0011.0715922046717194e-001

-7.2441773136017007e-0011.3957067792615432e-001

-5.7097217260853883e-0011.6626920581699392e-D01

-3.9415134707756339e-0011.8616100001556221e-001

-2.0119409399743449e-0011.9843148532711158e-D01

0.0000000000000000e+0002.0257824192556129e-001

2.0119409399743449e-0011.9843148532711158e-001

3.9415134707756339e-0011.8616100001556221e-001

5.7097217260853883e-0011.6626920581699392e-001

7.2441773136017007e-0011.3957067792615432e-001

8.4820658341042721e-0011.0715922046717194e-0Dl

9.3727339240070595e-0017.0366047488108124e-002

9.8799251802048538e-0013.0753241996117269e-0D2

]；

elseifnQ==16

Data=[...

-9.8940093499164994e-0012.7152459411754096e-002

-9.4457502307323260e-0016.2253523938647894e-002

-8.6563120238783176e-0019.5158511682492786e-002

-7.5540440835500300e-0011.2462897125553388e-001

-6.1787624440264377e-0011.4959598881657674e-001

-4.5801677765722737e-0011.6915651939500254e-001

-2.8160355077925892^-0011.8260341504492358e-901

-9.5012509837637427e-0021.8945061045506850e-D01

9.5012509837637427e-0021.8945061045506850e-001

2.8160355077925892e-0011.8260341504492358e-001

4.5801677765722737e-0011.6915651939500254e-001

6.1787624440264377e-0011.4959598881657674e-001

7.5540440835500300e-0011.2462897125553388e-001

8.6563120238783176e-0019.5158511682492786e-002

9.4457502307323260e-0016.2253523938647894e-0D2

9.8940093499164994e-0012.7152459411754096e-002

]；

elseifnQ==17

Data=[...

-9.9057547531441736e-0012.4148302868547931e-002

9.5067552176876780G0015.5459529373987203?002

-8.8023915372698591e-0018.5036148317179178e-002

-7.8151400389680137e-0011.1188384719340397e-001

-6.5767115921669084e-0011.3513636846852548e-001

-5.1269053708647694e-0011.5404576107681028e-001

-3.5123176345387630e-0011.6800410215645004e-001

-1.7848418149584788e-0011.7656270536699264e-001

0.0000000000000000e+0001.7944647035620653e-001

1.7848418149584788e-0011.7656270536699264e-001

3.5123176345387630e-0011.6800410215645004e-001

5.1269053708647694e-0011.5404576107681028e-001

6.5767115921669084e-0011.3513636846852548e-001

7.8151400389680137e-0011.1188384719340397e-001

8.80239153726985916-0018.5036148317179178e-002

9.5067552176876780e-0015.5459529373987203e-0D2

9.9057547531441736e-0012.4148302868547931e-002

]；

elseifnQ==18

Data=[...

-9.9156516842093090e-0012.1616013526483312e-002

-9.5582394957139771e-0014.9714548894969797e-002

-8.9260246649755570e-0017.6425730254889052e-002

-8.0370495897252314e-0011.0094204410628717e-001

-6.9168704306035322e-0011.2255520671147846e-901

-5.5977083107394754e-0011.4064291467065065e-001

-4.1175116146284263e-0011.5468467512626524e-001

-2.5188622569150554e-0011.6427648374583273e-001

-8.4775013041735292e-0021.6914238296314360e-001

8.4775013041735292e-0021.6914238296314360e-001

2.5188622569150554e-0011.6427648374583273e-001

4.1175116146284263y-0011.5468467512626524e-001

5.5977083107394754e-0011.4064291467065065e-001

6.9168704306035322e-0011.2255520671147846e-001

8.0370495897252314e-0011.0094204410628717e-001

8.9260246649755570e-0017.6425730254889052e-0D2

9.5582394957139771e-0014.9714548894969797e-002

9.9156516842093090e-0012.1616013526483312e-002

]；

elseifnQ==19

Data=[...

-9.9240684384358435e-0011.9461788229726478e-002

-9.6020815213483002e-0014.4814226765699600e-002

-9.0315590361481790e-0016.9044542737641226e-002

-8.2271465653714282e-0019.1490021622449999e-002

7.2096617733522939c0011.1156664554733399e001

-6.0054530466168110e-0011.2875396253933621e-001

-4.6457074137596099e-0011.4260670217360660e-001

-3.1656409996362989e-0011.5276604206585967e-001

-1.6035864564022539e-0011.5896884339395434e-001

0.0000000000000000e+0001.6105444984878370e-001

1.6035864564022539e-0011.5896884339395434e-001

3.1656409996362989e-0011.5276604206585967e-001

4.6457074137596099e-0011.4260670217360660e-001

6.0054530466168110e-0011.2875396253933621e-001

7.2096617733522939e-0011.1156664554733399e-001

8.2271465653714282e-0019.1490021622449999e-0D2

9.0315590361481790e-0016.9044542737641226e-002

9.60208152134830026-0014.4814226765699600e-002

9.9240684384358435e-0011.9461788229726478e-0D2

1；

elseifnQ==20

Data=[...

-9.9312859918509488e-0011.7614007139152118e-002

-9.6397192727791381e-0014.0601429800386939e-002

-9.1223442825132595e-0016.2672048334109068e-002

-8.3911697182221878e-0018.3276741576704755e-002

-7.4633190646015080e-0011.0193011981724044e-001

-6.3605368072651502e-0011.181945319615184le-901

-5.1086700195082702e-0011.3168863844917664e-001

-3.7370608871541955e-0011.4209610931838204e-001

-2.2778585114164507e-0011.4917298647260374e-001

-7.6526521133497338e-0021.5275338713072584e-001

7.6526521133497338e-0021.5275338713072584e-001

2.2778585114164507e-0011.4917298647260374e-001

3.7370608871541955t!-0011.4209610931838204e-091

5.1086700195082702e-0011.3168863844917664e-001

6.3605368072651502e-0011.1819453196151841e-001

7.4633190646015080e-0011.0193011981724044e-001

8.3911697182221878e-0018.3276741576704755e-0D2

9.1223442825132595e-0016.2672048334109068e-002

9.6397192727791381e-0014.0601429800386939e-002

9.9312859918509488e-0011.7614007139152118e-002

]；

elseifnQ==21

Data=[?.?

-9.9375217062038945e-0011.6017228257774335e-002

-9.6722683856630631e-0013.6953789770852494e-002

-9.2009933415040079e-0015.7134425426857205e-002

8.5336336458331730G0017.6100113628379304e902

-7.6843996347567789e-0019.3444423456033862e-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-001

0.0000000000000000e+0001.460811336496904le-001

1.4556185416089507e-0011.4452440398997005e-001

2.8802131680240106e-0011.3988739479107315e-001

4.2434212020743878e-0011.3226893863333747e-001

5.5161883588721983e-0011.2183141605372853e-001

6.6713880419741234e-0011.0879729916714838e-001

7.6843996347567789e-0019.3444423456033862e-002

8.5336336458331730e-0017.6100113628379304e-0D2

9.2009933415040079e-0015.7134425426857205e-002

9.6722683856630631e-0013.6953789770852494e-002

9.93752170620389456-0011.6017228257774335e-002

]；

else

error([*Thenumber'num2str(nQ)isnotimplemented.*])

end

xxIntRef=Data(:,1),;

wwIntRef=Data(:,2)1;

%%Plotsituation.

%reset(cla),reset(elf),holdon

%a=plot(xxIntRef,zeros(size(xxIntRef)),1k**);

%set(a,*Linewidth,,2,'MarkerSize*,15)

%a=liny([-11]z[00]);

%set(a,*Linewidth,,2,'Color'z'k')

%boxon,axisequal,axis([-11-0.10.1])

2.2Dtriangleelement(p230computationmechanics)

(a)(b)(c)

OrdernLocationJLocationi\Weightw

1(a)11/31/31.0

3(c)10.50.51/3

20.50.01/3

30.00.51/3

4(d)11/31/3-27/48

23/51/525/48

31/51/525/48

41/53/525/48

610.8168475729804590.0915762135097710.109951743655322*0.5

20.0915762135097710.8168475729804590.109951743655322*0.5

30.0915762135097710.0915762135097710.109951743655322*0.5

40.1081030181680700.4459484909159650.223381589678011*0.5

50.4459484909159650.1081030181680700.223381589678011*0.5

60.4459484909159650.4459484909159650.223381589678011*0.5

121.8738219710169960.0630890144915020.050844906370207*0.5

2.0630890144915020.8738219710169960.050844906370207*0.5

3.0630890144915020.0630890144915020.050844906370207*0.5

4.5014265096581790.2492867451709100116786275726379*0.5

5.2492867451709100.5014265096581790.116786275726379*0.5

6.5014265096581790.5014265096581790116786275726379*0.5

7.6365024991213990.3103524510337850.082851075618374*0.5

8.6365024991213990.0531450498448160.082851075618374*0.5

9.3103524510337850.6365024991213990.082851075618374*0.5

100.3103524510337850.0531450498448160.082851075618374*0.5

11.05314504984^8160.6365024991213990.082851075618374*0.5

12.0531450498-U8160.3103524510337850.082851075618374*0.5

case('triangle')

selectcase(nip)

case(1)!fortrianglesweightsmultipliedby.5

s(l,l)=l./3.;s(l,2)=l./3.;wt(l)=.5

case(3)

s(l,l)=.5;s(l,2)=.5:s(2,l)=.5

s(2,2)=0.;s(3,1)=0.;s(3,2)=.5

wt(1)=1./3.;wt(2)=wt(l);wt(3)=wt(l);wt=.5*wt

case(6)

s(l,1)=816847572980459;s(l,2)=.O91576213509771

s(2,l)=s(l,2);s(2,2)=s(l,l);s(3,l)=s(l,2);s(3,2)=s(l,2)

s(4,l)=108103018168070;s(4,2)=.445948490915965

s(5,1)=s(4,2);s(5,2)=s(4,l);s(6,l)=s(4,2);s(6,2)=s(4,2)

wt(l)=.l09951743655322;wt(2)=wt(l);wt(3)=wt(l)

wt(4)=.223381589678011;wt(5)=wt(4);wt(6)=wt(4);wt=.5*wt

case(7)

s(l,l)=l./3.;s(1,2)=1./3.;s(2,1)=.797426985353087;s(2,2)=.101286507323456

s(3,l)=s(2,2);s(3,2)=s(2,l);s(4,l)=s(2,2);s(4,2)=s(2,2)

s(5,l)=470142064105115;s(5,2)=059715871789770

s(6,1)=s(5,2);s(6,2)=s(5,l);s(7,l)=s(5,l);s(7,2)=s(5,l)

wt(1)=225;wt(2)=125939180544827;wt(3)=wt(2);wt(4)=wl(2)

wt(5)=.132394152788506;wt(6)=wt(5);wt(7)=wt(5);wt=.5*wt

case(12)

s(1,1)=873821971016996;s(l,2)=.063089014491502

s(2,l)=s(l,2);s(2,2)=s(l,l);s(3,l)=s(l,2);s(3,2)=s(1,2)

s(4,1)=501426509658179;s(4,2)=249286745170910

s(5,l)=s(4,2);s(5,2)=s(4,l);s(6,l)=s(4,2);s(6,2)=s(4,2)

s(7,l)=.636502499121399;s(7,2)=.310352451033785

s(8,l)=s(7,l);s(8,2)=,053145049844816;s(9,l)=s(7,2);s(9,2)=s(7,1)

s(10,l)=s(7,2);s(10,2)=s(8,2);s(ll,l)=s(8,2);s(ll,2)=s(7J)

s(12,l)=s(8,2);s(12,2)=s(7,2)

wt(l)=.050844906370207;wt(2)=w:(l);wt(3)=wt(1)

wt(4)=.l16786275726379;wt(5)=wt(4);wt(6)=wt(4)

wt(7)=.082851075618374;wt(8:12)=wt(7);wt=.5*wt

case(16)

s(l,l)=l./3.;s(l,2)=l./3.;s(2,1)=.658861384496478

s(2,2)=170569307751761；s(3,l)=s(2,2)；s(3,2)=s(2,1)

s(4,l)=s(2,2);s(4,2)=s(2,2)

s(5,l)=.898905543365938;s(5,2)=.050547228317031

s(6,1)=s(5,2);s(6,2)=s(5J);s(7,l)=s(5,2);s(7,2)=s(5,2)

s(8,1)=081414823414554;s(8,2)=459292588292723

s(9,l)=s(8,2);s(9,2)=s(8,l);s(IO,l)=s(8,2);s(10,2)=s(8,2)

s(l1J)=.008394777409958;s(l1,2)=263112829634638

s(12,l)=s(ll,l);s(l2,2)=728492392955404

s(13,l)=s(ll,2);s(13,2)=s(ll,l);s(14,l)=s(ll,2);s(14,2)=s(12,2)

s(15,l)=s(12,2);s(15,2)=s(l1,1);s(16,1)=s(12,2);s(16,2)=s(11,2)

wt(l)=144315607677787;wt(2)=.103217370534718;wt(3)=wt(2);wt(4)=wt(2)

wt(5)=.032458497623198;wt(6)=w:(5);wt(