Mapping isoparametric

Figure 2.17 Isoparametric mapping of an irregular quadrilateral element wilJi straight sides...

In certain types of finite element computations the application of isoparametric mapping may require transformation of second-order as well as the first-order derivatives. Isoparametric transformation of second (or higher)-order derivatives is not straightforward and requires lengthy algebraic manipulations. Details of a convenient procedure for the isoparametric transformation of second-order derivatives are given by Petera et a . (1993). 

Isoparametric mapping removes tlie geometrical inflexibility of rectangular elements and therefore they can be used to solve many types of practical problems. For example, the isoparametric C continuous rectangular Hermite element provides useful discretizations in the solution of viscoelastic flow problems. 

Isoparametric mapping described in Section 1.7 for generating curved and distorted elements is not, in general, relevant to one-dimensional problems. However, the problem solved in this section provides a simple example for the illustration of important aspects of this procedure. Consider a master element as is shown in Figure 2.23. The shape functions associated with this element are... 

In order to establish an isoparametric mapping between the master element shown in Figure 2.23 and the elements in the global domain (Figure 2.20) we use the elemental shape funetions to formulate a transformation function as... 

Figure 5.2 shows the finite element mesh corresponding to the configuration shown in Figure 5.1. This mesh consists of 225 nine-node bi-quadratic elements and its utihzation in the present model is based on the application of isoparametric mapping, described in Chapter 2.

Similar to the two-dimensional isoparametric element, for three dimensional elements we use a mapping of the normalized coordinates, , ty, C> (Li volume coordinates for a tetrahedral element), in such a way that the cartesian coordinates will appear as a curvilinear set. 

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