Introduction to Finite Element Analysis and Design, 2nd Edition

Chapter 7Isoparametric Finite Elements

7.1 INTRODUCTION

The shape functions for triangular and rectangular elements in chapter 6 are derived in the global coordinates and are dependent on the nodal coordinates of the element. Therefore, different elements have different shape functions. Knowing that hundreds of thousands of elements are often used in solving practical problems, evaluating the shape functions for individual elements might be laborious and computationally inefficient. In addition, deriving the shape functions for quadrilaterals in global coordinates is difficult as compared to rectangular elements. It is often convenient to use local coordinate systems for interpolating the field variables such as displacement or temperature fields because the shape functions are easier to derive and have simpler expressions. This involves a change of variables from the physical (x,y) or (x,y,z) coordinates to parametric (s,t) or (r,s,t) coordinates. Isoparametric elements use a parametric coordinate system that transforms the elements by scaling and deforming it. The domain in the parametric coordinate system is often referred to as the parametric space, as opposed to the space occupied by the element in the global coordinate system, which is the physical space. A mapping is established between the physical and parametric spaces. The advantage of using this approach is that more complex‐shaped elements can be constructed such as quadrilateral and hexahedral elements. In addition, ...