[Libmesh-devel] Numerical Integration

 [Libmesh-devel] Numerical Integration From: Ahmed EL-Sheikh - 2004-07-08 15:50:22 ```In the file http://libmesh.sourceforge.net/doxygen/quadrature__gauss__2D_8C-source.html The weights of the integration points are specified as following 00095 case THIRD: 00096 { 00097 // Exact for cubics 00098 _points.resize(4); 00099 _weights.resize(4); 00100 00101 _points[0](0) = .33333333333333333333333333333333; 00102 _points[0](1) = .33333333333333333333333333333333; 00103 00104 _points[1](0) = .2; 00105 _points[1](1) = .6; 00106 00107 _points[2](0) = .2; 00108 _points[2](1) = .2; 00109 00110 _points[3](0) = .6; 00111 _points[3](1) = .2; 00112 00113 00114 _weights[0] = -27./96.; 00115 _weights[1] = 25./96.; 00116 _weights[2] = 25./96.; 00117 _weights[3] = 25./96.; While in the Zienkiewicz book or the notes by Flaherty (www.rpi.edu/~gilade/fem6.ps page 11) The weights are doubled what is in the LibMesh. On the other hand the resulting stiffness matrices are correct. I do remember there is something about the calculation of the Jacobian for the triangles (2 * Area) but I couldn't figure what happen when the lagrangain basis are used for the Jacobian calculation. It have been a while since I took the FE course :) Thanks in advance. Ahmed ```

 [Libmesh-devel] Numerical Integration From: Ahmed EL-Sheikh - 2004-07-08 15:50:22 ```In the file http://libmesh.sourceforge.net/doxygen/quadrature__gauss__2D_8C-source.html The weights of the integration points are specified as following 00095 case THIRD: 00096 { 00097 // Exact for cubics 00098 _points.resize(4); 00099 _weights.resize(4); 00100 00101 _points[0](0) = .33333333333333333333333333333333; 00102 _points[0](1) = .33333333333333333333333333333333; 00103 00104 _points[1](0) = .2; 00105 _points[1](1) = .6; 00106 00107 _points[2](0) = .2; 00108 _points[2](1) = .2; 00109 00110 _points[3](0) = .6; 00111 _points[3](1) = .2; 00112 00113 00114 _weights[0] = -27./96.; 00115 _weights[1] = 25./96.; 00116 _weights[2] = 25./96.; 00117 _weights[3] = 25./96.; While in the Zienkiewicz book or the notes by Flaherty (www.rpi.edu/~gilade/fem6.ps page 11) The weights are doubled what is in the LibMesh. On the other hand the resulting stiffness matrices are correct. I do remember there is something about the calculation of the Jacobian for the triangles (2 * Area) but I couldn't figure what happen when the lagrangain basis are used for the Jacobian calculation. It have been a while since I took the FE course :) Thanks in advance. Ahmed ```
 [Libmesh-devel] Numerical Integration From: John Peterson - 2004-07-08 16:07:53 ```Hi, In general, the weights should add up to the area of the element. In this case, the area is 1/2 for the TRI3 and TRI6 elements In the reference you mention Eq. 6.3.7 on page 10 has the area Ae=1/2 pulled out of the summation, while we have it multiplied through. That should account for the difference. -John Ahmed EL-Sheikh writes: > In the file > http://libmesh.sourceforge.net/doxygen/quadrature__gauss__2D_8C-source.html > > The weights of the integration points are specified as following > > 00095 case THIRD: > 00096 { > 00097 // Exact for cubics > 00098 _points.resize(4); > 00099 _weights.resize(4); > 00100 > 00101 _points[0](0) = .33333333333333333333333333333333; > 00102 _points[0](1) = .33333333333333333333333333333333; > 00103 > 00104 _points[1](0) = .2; > 00105 _points[1](1) = .6; > 00106 > 00107 _points[2](0) = .2; > 00108 _points[2](1) = .2; > 00109 > 00110 _points[3](0) = .6; > 00111 _points[3](1) = .2; > 00112 > 00113 > 00114 _weights[0] = -27./96.; > 00115 _weights[1] = 25./96.; > 00116 _weights[2] = 25./96.; > 00117 _weights[3] = 25./96.; > > While in the Zienkiewicz book or the notes by Flaherty > (www.rpi.edu/~gilade/fem6.ps page 11) > > The weights are doubled what is in the LibMesh. On the other hand the > resulting > stiffness matrices are correct. > > I do remember there is something about the calculation of the Jacobian > for the > triangles (2 * Area) but I couldn't figure what happen when the > lagrangain basis > are used for the Jacobian calculation. > It have been a while since I took the FE course :) > > Thanks in advance. > Ahmed ```