一维二维三维高斯积分点及权重Gaussintegrationsandweights.docx
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一维二维三维高斯积分点及权重Gaussintegrationsandweights.docx
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一维二维三维高斯积分点及权重Gaussintegrationsandweights
Gaussintegrationsandweights
(Containingtheprogram)
高斯积分点以及权重
1.1Dbarelement(p181computationmechanics)
Ordern
Accuracygrade
(2*n-1)
Locationξ
Weightw
1
1
1
0.0
2.0
2
3
1
1.0
2
1/√3
1.0
3
5
1
-
5/9
2
0.0
8/9
3
√0.6
5/9
4
7
1
0.4053
0.7454
2
0.4856
0.2546
3
-0.4856
0.2546
4
-0.4053
0.7454
5
9
1
0.8664
0.6189
2
0.5683
0.9366
3
0
0.8889
4
-0.5683
0.9366
5
-0.8664
0.6189
6
11
1
0.3152
0.9170
2
0.6265
0.8139
3
0.3197
0.2691
4
-0.3197
0.2691
5
-0.6265
0.8139
6
-0.3152
0.9170
12
23
1
-0.6732
0.
2
0.0452
0.53637
3
-0.43177
0.33586
4
-0.66143
0.30672
5
-0.81804
0.83534
6
-0.
0.34027
7
0.
0.34027
8
0.81804
0.83534
9
0.66143
0.30672
10
0.43177
0.33586
11
0.0452
0.53637
12
0.6732
0.
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/)
selectcase(element)
case('line')
selectcase(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=w
case(4)
s(1,1)=.4053;s(2,1)=.4856
s(3,1)=-s(2,1);s(4,1)=-s(1,1)
wt
(1)=.7454;wt
(2)=.2546
wt(3)=wt
(2);wt(4)=wt
(1)
case(5)
s(1,1)=.8664;s(2,1)=.5683
s(3,1)=.0;s(4,1)=-s(2,1);s(5,1)=-s(1,1)
wt
(1)=.6189;wt
(2)=.9366
wt(3)=.8889;wt(4)=wt
(2);wt(5)=wt
(1)
case(6)
s(1,1)=.3152;s(2,1)=.6265;s(3,1)=.3197
s(4,1)=-s(3,1);s(5,1)=-s(2,1);s(6,1)=-s(1,1)
wt
(1)=.9170;wt
(2)=.8139;wt(3)=.2691
wt(4)=wt(3);wt(5)=wt
(2);wt(6)=wt
(1)
casedefault
print*,"wrongnumberofintegratingpointsforaline"
endselect
%Copyright(c)2010,Thomas-PeterFries,RWTHAachenUniversity
function[xxIntRef,wwIntRef]=IntPoints1DGauss(nQ)
%SetGausspointsin1Dreferenceelementfrom[-1,1].
ifnQ==1
Data=[...
0.00000e+0002.00000e+000
];
elseifnQ==2
Data=[...
-5.62573e-0011.00000e+000
5.62573e-0011.00000e+000
];
elseifnQ==3
Data=[...
-7.48340e-0015.55558e-001
0.00000e+0008.88884e-001
7.48340e-0015.55558e-001
];
elseifnQ==4
Data=[...
-8.05257e-0013.45385e-001
-3.85626e-0016.54609e-001
3.85626e-0016.54609e-001
8.05257e-0013.45385e-001
];
elseifnQ==5
Data=[...
-9.66396e-0012.18908e-001
-5.68311e-0014.36647e-001
0.00000e+0005.88889e-001
5.68311e-0014.36647e-001
9.66396e-0012.18908e-001
];
elseifnQ==6
Data=[...
-9.15205e-0011.17036e-001
-6.26448e-0013.13861e-001
-2.19688e-0014.69104e-001
2.19688e-0014.69104e-001
6.26448e-0013.13861e-001
9.15205e-0011.17036e-001
];
elseifnQ==7
Data=[...
-9.75849e-0011.86970e-001
-7.39446e-0012.27664e-001
-4.39718e-0013.11892e-001
0.00000e+0004.46940e-001
4.39718e-0013.11892e-001
7.39446e-0012.27664e-001
9.75849e-0011.86970e-001
];
elseifnQ==8
Data=[...
-9.53618e-0011.37626e-001
-7.62673e-0012.37448e-001
-5.32899e-0013.88727e-001
-1.64978e-0013.36199e-001
1.64978e-0013.36199e-001
5.32899e-0013.88727e-001
7.62673e-0012.37448e-001
9.53618e-0011.37626e-001
];
elseifnQ==9
Data=[...
-9.62609e-0018.74412e-002
-8.63577e-0011.85740e-001
-6.59036e-0012.93544e-001
-3.80892e-0013.00286e-001
0.00000e+0003.25978e-001
3.80892e-0013.00286e-001
6.59036e-0012.93544e-001
8.63577e-0011.85740e-001
9.62609e-0018.74412e-002
];
elseifnQ==10
Data=[...
-9.17174e-0016.88138e-002
-8.98454e-0011.58059e-001
-6.02444e-0012.98204e-001
-4.24721e-0012.99635e-001
-1.63116e-0012.75287e-001
1.63116e-0012.75287e-001
4.24721e-0012.99635e-001
6.02444e-0012.98204e-001
8.98454e-0011.58059e-001
9.17174e-0016.88138e-002
];
elseifnQ==11
Data=[...
-9.05697e-0015.73663e-002
-8.09532e-0011.90461e-001
-7.04936e-0011.73426e-001
-5.81181e-0012.99048e-001
-2.34501e-0012.24665e-001
0.00000e+0002.90062e-001
2.34501e-0012.24665e-001
5.81181e-0012.99048e-001
7.04936e-0011.73426e-001
8.09532e-0011.90461e-001
9.05697e-0015.73663e-002
];
elseifnQ==12
Data=[...
-9.71924e-0014.11828e-002
-9.47491e-0011.31843e-001
-7.30469e-0011.34622e-001
-5.61737e-0012.06592e-001
-3.18024e-0012.35481e-001
-1.46891e-0012.40277e-001
1.46891e-0012.40277e-001
3.18024e-0012.35481e-001
5.61737e-0012.06592e-001
7.30469e-0011.34622e-001
9.47491e-0011.31843e-001
9.71924e-0014.11828e-002
];
elseifnQ==13
Data=[...
-9.58814e-0014.15877e-002
-9.97792e-0019.28452e-002
-8.30988e-0011.78725e-001
-6.34023e-0011.94574e-001
-4.44681e-0012.88851e-001
-2.13477e-0012.89723e-001
0.00000e+0002.87390e-001
2.13477e-0012.89723e-001
4.44681e-0012.88851e-001
6.34023e-0011.94574e-001
8.30988e-0011.78725e-001
9.97792e-0019.28452e-002
9.58814e-0014.15877e-002
];
elseifnQ==14
Data=[...
-9.81231e-0013.51860e-002
-9.57352e-0018.60208e-002
-8.76502e-0011.90319e-001
-6.68548e-0011.19355e-001
-5.15410e-0011.93782e-001
-3.88974e-0012.29560e-001
-1.34367e-0012.15779e-001
1.34367e-0012.15779e-001
3.88974e-0012.29560e-001
5.15410e-0011.93782e-001
6.68548e-0011.19355e-001
8.76502e-0011.90319e-001
9.57352e-0018.60208e-002
9.81231e-0013.51860e-002
];
elseifnQ==15
Data=[...
-9.48538e-0013.17269e-002
-9.70595e-0017.08124e-002
-8.42721e-0011.17194e-001
-7.17007e-0011.15432e-001
-5.53883e-0011.99392e-001
-3.56339e-0011.56221e-001
-2.43449e-0011.11158e-001
0.00000e+0002.56129e-001
2.43449e-0011.11158e-001
3.56339e-0011.56221e-001
5.53883e-0011.99392e-001
7.17007e-0011.15432e-001
8.42721e-0011.17194e-001
9.70595e-0017.08124e-002
9.48538e-0013.17269e-002
];
elseifnQ==16
Data=[...
-9.64994e-0012.54096e-002
-9.23260e-0016.47894e-002
-8.83176e-0019.92786e-002
-7.00300e-0011.53388e-001
-6.64377e-0011.57674e-001
-4.22737e-0011.00254e-001
-2.25892e-0011.92358e-001
-9.37427e-0021.06850e-001
9.37427e-0021.06850e-001
2.25892e-0011.92358e-001
4.22737e-0011.00254e-001
6.64377e-0011.57674e-001
7.00300e-0011.53388e-001
8.83176e-0019.92786e-002
9.23260e-0016.47894e-002
9.64994e-0012.54096e-002
];
elseifnQ==17
Data=[...
-9.41736e-0012.47931e-002
-9.76780e-0015.87203e-002
-8.98591e-0018.79178e-002
-7.80137e-0011.40397e-001
-6.69084e-0011.52548e-001
-5.47694e-0011.81028e-001
-3.87630e-0011.45004e-001
-1.84788e-0011.99264e-001
0.00000e+0001.20653e-001
1.84788e-0011.99264e-001
3.87630e-0011.45004e-001
5.47694e-0011.81028e-001
6.69084e-0011.52548e-001
7.80137e-0011.40397e-001
8.98591e-0018.79178e-002
9.76780e-0015.87203e-002
9.41736e-0012.47931e-002
];
elseifnQ==18
Data=[...
-9.93090e-0012.83312e-002
-9.39771e-0014.69797e-002
-8.55570e-0017.89052e-002
-8.52314e-0011.28717e-001
-6.35322e-0011.47846e-001
-5.94754e-0011.65065e-001
-4.84263e-0011.26524e-001
-2.50554e-0011.83273e-001
-8.35292e-0021.14360e-001
8.35292e-0021.14360e-001
2.50554e-0011.83273e-001
4.84263e-0011.26524e-001
5.94754e-0011.65065e-001
6.35322e-0011.47846e-001
8.52314e-0011.28717e-001
8.55570e-0017.89052e-002
9.39771e-0014.69797e-002
9.93090e-0012.83312e-002
];
elseifnQ==19
Data=[...
-9.58435e-0011.26478e-002
-9.83002e-0014.99600e-002
-9.81790e-0016.41226e-002
-8.14282e-0019.49999e-002
-7.22939e-0011.33399e-001
-6.68110e-0011.33621e-001
-4.96099e-0011.60660e-001
-3.62989e-0011.85967e-001
-1.22539e-0011.95434e-001
0.00000e+0001.78370e-001
1.22539e-0011.95434e-001
3.62989e-0011.85967e-001
4.96099e-0011.60660e-001
6.68110e-0011.33621e-001
7.22939e-0011.33399e-001
8.14282e-0019.49999e-002
9.81790e-0016.41226e-002
9.83002e-0014.99600e-002
9.58435e-0011.26478e-002
];
elseifnQ==20
Data=[...
-9.09488e-0011.52118e-002
-9.91381e-0014.86939e-002
-9.32595e-0016.09068e-002
-8.21878e-0018.04755e-002
-7.15080e-0011.24044e-001
-6.51502e-0011.51841e-001
-5.82702e-0011.17664e-001
-3.41955e-0011.38204e-001
-2.64507e-0011.60374e-001
-7.97338e-0021.72584e-001
7.97338e-0021.72584e-001
2.64507e-0011.60374e-001
3.41955e-0011.38204e-001
5.82702e-0011.17664e-001
6.51502e-0011.51841e-001
7.15080e-0011.24044e-001
8.21878e-0018.04755e-002
9.32595e-0016.09068e-002
9.91381e-0014.86939e-002
9.09488e-0011.52118e-002
];
elseifnQ==21
Data=[...
-9.38945e-0011.74335e-002
-9.30631e-0013.52494e-002
-9.40079e-0015.57205e-002
-8.31730e-0017.79304e-002
-7.67789e-0019.33862e-002
-6.41234e-0011.14838e-001
-5.21983e-0011.72853e-001
-4.43878e-0011.33747e-001
-2.40106e-0011.07315e-001
-1.89507e-0011.97005e-001
0.00000e+0001.69041e-001
1.89507e-0011.97005e-001
2.40106e-0011.07315e-001
4.43878e-0011.33747e-001
5.21983e-0011.72853e-001
6.41234e-0011.14838e-001
7.67789e-0019.33862e-002
8.31730e-0017.79304e-002
9.40079e-0015.57205e-002
9.30631e-0013.52494e-002
9.38945e-0011.74335e-002
];
else
error(['Thenumber'num2str(nQ)'isnotimplemented.'])
end
xxIntRef=Data(:
1)';
wwIntRef=Data(:
2)';
%%Plotsituation.
%reset(cla),reset(clf),holdon
%a=plot(xxIntRef,zeros(size(xxIntRef)),'k*');
%set(a,'LineWidth',2,'MarkerSize',15)
%a=line([-11],[00]);
%set(a,'LineWidth',2,'Color','k')
%boxon,axisequal,axis([-11-0.10.1])
2.2Dtriangleelement(p230computationmechanics)
Ordern
Locationξ
Location𝛈
Weightw
1(a)
1
1/3
1/3
1.0
3(c)
1
0.5
0.5
1/3
2
0.5
0.0
1/3
3
0.0
0.5
1/3
4(d)
1
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