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

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

and in that way we end up with the final expression for the Q – point Gauss quadrature.

finite element method, fem, numerical integration, differential equations, lde, 2D mesh generator, Metfem2D, taketechease


  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