Introduction to Finite Element Method

Abstract

The procedure used in Chapter 15to formulate the stiffness equations of the linear triangle can be formally extended to quadrilateral elements as well as higher order triangles. But one quickly encounters technical difficulties: 1. The construction of shape functions that satisfy consistency requirements for higher order elements with curved boundaries becomes increasingly complicated. 2. Integrals that appear in the expressions of the element stiffness matrix and consistent nodal force vector can no longer be evaluated in simple closed form. These two obstacles can be overcome through the concepts of isoparametric elements and numerical quadrature, respectively. The combination of these two ideas transformed the field of finite element methods in the late 1960s. Together they support a good portion of what is presently used in production finite element programs. In the present Chapter the concept of isoparametric representation is introduced for two dimen-sional elements. This representation is illustrated on specific elements. In the next Chapter these techniques, combined with numerical integration, are applied to quadrilateral elements. 16.2. Isoparametric Representation 16.2.1. Motivation The linear triangle presented in Chapter 115is an isoparametric element although was not originally derived as such. The two key equations are (15.10), which defines the triangle geometry, and (15.16), which defines the primary variable, in this case the displacement field. These equations are reproduced here for convenience: 1 x y