Finite Element Method - 2D Mesh Generator - Metfem2D
Numerical integration - Gauss's quadrature
The l.h.s integral I can be approximated by the Q - point Gauss quadrature [1,2,3,4]
|
1 |
1 |
Q |
|
I =
|
∫
|
dL1
|
∫
|
dL2 |det J| f(L1, L2, L3) ∼
|
∑
|
fq(L1, L2, L3)Wq
|
|
0 |
0 |
q=1 |
where Wq denotes the weights for q - points of the numerical integration, and can be found in the Table 5.3 in[1]. As it was already said, a set of Nk(L1, L2, L3) shape functions where k = 1, 2, 3 can be used to evaluate each f function in the interpolation series which, for instance, in the highest order 10 – nodal cubic triangular element has the following form
|
3 |
9 |
|
f(L1, L2, L3) =
|
∑
|
Nk(L1, L2, L3)fk +
|
∑
|
Nk(L1, L2, L3)Δfk +
|
N10ΔΔf10
|
|
k=1 |
k=4 |
where fk are nodal values of f function, fk denote departures from linear interpolation for mid – side nodes, and f10 is departure from both previous orders of approximation for the central nodeC[1]. For linear triangular elements only the first term is important which gives an approximation
|
f(L1, L2, L3) =
|
∑
|
Lkfk
|
|
k=1
|
Note, that the r.h.s sum does not include the jacobian j det Jj that should be incorporated by the weights Wq but it is not (in their values given in Table 5.3 from[1]). Thus let's add the triangle area to the above – recalled formula
|
|det J|/(2Δ)
|
∑
|
fq(L1, L2, L3)Wq
|
|
q=1
|
and in that way we end up with the final expression for the Q – point Gauss quadrature.
References
