One Formula Generates Nth Order Shape Functions

Abstract

The paper presents a generalized Lagrangian interpolation formula which generates polynomial shape functions of any desired order. Examples are given to show that polynomial shape functions can be generated for simple one to three dimensional elements commonly used in finite element analysis. The higher order functions are of interest in problems where large strain gradients may be expected and are frequently used in isolated ``super elements'' as in stress intensity calculations. Computational advantages exist due to better matching of the physical phenomena being modeled and consequent reduction in number of elements for a given level of accuracy. The formula presented facilitates selection of appropriate polynomials to represent shape functions associated with a given element type and physical problem.