chapter  5
34 Pages

Numerical Integration

WithJuan C. Heinrich, Darrell W. Pepper

In the previous chapter we introduced the concept of isoparametric elements, which allows us to use elements of very general shapes. The integrands in our finite element formulations are changed from simple polynomials to rational functions that can no longer be integrated analytically (in the general case). To be able to automate the evaluation of the integrals over each element, numerical integration is required. At the same time, new issues arise that make numerical integration in finite element methodology a subject in itself. In this chapter we will discuss the most basic issues that arise with the use of numerical integration; other more specialized aspects will be introduced later when needed. First, we present some of the most commonly used families of quadrature rules, then we discuss the degree of precision necessary to guarantee convergence and optimal accuracy of the finite element approximations. The concept of reduced integration will be introduced, and we will finish the chapter with a discussion on how to evaluate the gradients of the dependent variables once the solution has been obtained.