4 - Higher Order and Isoparametric Elements

Abstract

A finite element is known as the higher order element when the interpolation polynomial is of order two or more. A higher order element can be either a complex or a multiplex element. In higher order elements, some secondary (midside and/or interior) nodes are introduced to the primary (corner) nodes to match the number of nodal degrees of freedom with the number of constants—also known as generalized coordinates―in the interpolation polynomial. It is possible to construct the nodal interpolation functions N by employing classical interpolation polynomials instead of natural coordinates. The Lagrange and Hermite interpolation polynomials are used for this purpose. For problems involving curved boundaries, a family of elements known as “isoparametric” elements is used. In isoparametric elements, the same interpolation functions used to define the element geometry are also used to describe the variation of the field variable within the element. This chapter focuses on both the higher order and isoparametric elements.