Chapter 4 - Higher Order and Isoparametric Elements

Abstract

This chapter deals with the higher order and isoparametric elements. For one-dimensional problems, quadratic and cubic interpolation polynomials are presented using three and four nodes, respectively. The polynomials are expressed in terms of global nodal unknowns of the element. To simplify the procedure, the quadratic and cubic elements are represented using natural coordinates. For two-dimensional problems, triangular and quadrilateral elements are considered using quadratic and cubic interpolation models in terms of natural coordinates. The geometrical interpretations of the resulting shape functions are also shown. In three dimensions, for the tetrahedron element, the quadratic and cubic models are developed using tetrahedral coordinates. The n-station Lagrange interpolation functions and general two-station Hermite interpolation functions (of zeroth and first-order functions) are defined. The C0 and C1 continuities of the interpolation model are defined. The definition of an isoparametric element is presented along with equations for a line, triangular, and quadrilateral elements. The curve-sided isoparametric elements are also defined. Finally, the details of numerical integration for different types of elements in one, two, and three dimensional spaces using Gausian quadrature are presented.