Spatial Integration Schemes
Since the Finite Element Method (FEM) is primarily an exercise in numerical integration, it is not surprising that better integration schemes are being sought. Numerical integration schemes replace an
integral by a summation; e. g. J f(x)dx - ]T"=1 wif(xi). The choice
of the number of sampling points, i, the weights, wt, and the location of the sampling points, xh characterize the integration scheme. If f(x) is a polynomial of degree 2n-l, it can be integrated exactly with n sampling or integration points as shown by Gauss. In two dimensions, the stiffness matrix of the popular 4 node rectangular element (but not a quadrilateral) can be integrated exactly with four integration points. If only one integration point is used, the integration is approximate and spurious nodes may appear that corrupt the solution. Belytschko  discovered a way to stabilize these spurious nodes and thus reduce the integration costs by almost a factor of four for 4-node quads and almost 8 for 8-node bricks.