# 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. 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. 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). 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

1. ^ O. C. Zienkiewicz, R. L. Taylor and J. Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, Sixth edition., Elsevier 2005
2. ^R. Radau, Etude sur les formules d'approximation qui servent calculer la valeur d'une integrate definie, Journ. de Math. 6(3), pp. 283-336, 1880
3. ^P. C. Hammer and O. J. Marlowe and A. H. Stroud, Numerical Integration Over Simplexes and Cones, Math. Tables Aids Comp., 10, pp. 130-137, 1956
4. ^F. R. Cowper, Gaussian quadrature formulas for triangles, Int. J. Numer. Meth. Eng., 7, pp. 405-408, 1973