Chapter 8 - The Patch Test, Reduced Integration, and Nonconforming Elements

Abstract

This chapter describes the use of a patch of one or more elements to assess convergence of a finite element solution to that of the differential equation it approximates. The requirements are presented by a consistency and stability test on the resulting matrix problem. For elements which are numerically integrated or otherwise explicitly stated the stability test requires that the stiffness matrix of a structural problem or the conductivity matrix of a quasi-harmonic problem has proper rank. This requires the number of quadrature points to be equal or larger to some minimum value. Use of less than the minimum number is called reduced integration. Consistency requires the finite element solution to exactly represent all the derivatives up to the highest order that exists in a weak or variational expression for the problem. The chapter presents alternative methods to asses the two requirements. The patch test is an essential test to both test accuracy of the computer program used to solve problems as well as to validate formulations that violate the standard rules described in previous chapters. One such case of a non-conforming element is presented in the chapter.