CONCLUSIONS As the7 •Same shape functions are used to interpolate nodal coordinates and displacements •Shape functions are defined for an idealized mapped element (e.g. square for any quadrilateral element) •Advantages include more flexible shapes and compatibility •We pay the price in complexity and require numerical integration to calculate stiffness matrices and equivalent loads The developments presented in this chapter show how interpolation functions for one-, two-, and three-dimensional elements can be obtained via a systematic procedure. Also, the algebraically tedious procedure can often be bypassed using intuition and logic when natural coordinates are used. The interpolation functions discussed are standard polynomial forms but by no means exhaustive of the interpolation functions that have been developed for use in finite element analysis. As the objective of this text is to present the fundamentals of finite element analysis, the material of this chapter is intended to cover the basic concepts of interpolation functions without proposing to be comprehensive. The treatment here is intended to form a basis for formulation of finite element models of various physical problems in following chapters. In general, every element and the associated interpolation functions discussed here can be applied to specific problems, as is illustrated in the remainder of the text. CONCLUSIONS