Solution of Field Problems Using Isoparametric Quadrilateral Elements

3 Solution of Field Problems Using Isoparametric Quadrilateral Elements. 

The previous section used the constant strain three-noded element to solve Poisson s equation with steady-state as well as transient terms. The same problems, as well as any field problems such as stress-strain and the flow momentum balance, can be formulated using isoparametric elements. With this type of element, the same (as the name suggests) shape functions used to represent the field variables are used to interpolate between the nodal coordinates and to transform from the xy coordinate system to a local element coordinate system. The first step is to discretize the domain presented in Fig. 9.12 using the isoparametric quadrilateral elements as shown in Fig. 9.15. 

If we pick an arbitrary element we can see that it is represented by the xy-coordinates of the four nodal points, as depicted in Fig. 9.16. The figure also shows a -coordinate system embedded within the element. In the r/, or local, coordinate system, we have a perfectly square element of area 2x2, where the element spreads between —1 1 and — 1 rj 1. This attribute allows us to easily allows us to use Gauss quadrature as a numerical integration scheme, where the limits vary between -1 and 1. The isoparametric element described in the //-coordinate system is presented in Fig. 9.17. 

If we assume nodal shape functions Nt that are equal 1 at ( , , r/, ) and zero at the other three nodes ( ./, r/j), we can define the field variable and the nodal coordinates in term of those shape functions and the nodal values of the parameters, i.e., 

The finite element expression for a problem using isoparametric finite elements will be similar to the ones developed in previous examples. For example, a problem governed by Poisson s equation will result in the following finite element equation, 

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