Isoparametric interpolation

In order to increase the accuracy of the element and therefore reduce the number of boundary elements we must increase the order of the interpolation functions across the element. If we choose a linear element, similar to FEM, we can approximate any variable within the element with the use of a linear isoparametric interpolation as follows... 

The value of any variable at any point within the element is defined in terms of the node s values according to the isoparametric interpolation. As with finite elements, the coordinates and the velocity field for each element can be written as follows... 

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. 

The results reported below were obtained with conventional "mixed interpolation" on isoparametric rectangles (12), using nine-node biquadratic basis functions 1 for the velocity components and four-node bilinear basis functions for the pres-... 

During and after their creation, characteristic curves can be created on surfaces according to their type. Figure 7-10 shows examples of curves extracted from surfaces as an isoparametric curve (Figure 7-lOa), an interpolation curve on a lofted surface (Figure 7-lOb), and a section curve on a swept surface... 

Using the above terms the weak form of the momentum equations are obtained. Then the convective terms are treated explicitly and diffusive terms are evaluated implicitly. All variables are interpolated using the bilinear isoparametric shape functions for a generic function /(x, z) as ... 

In isoparametric elements, the same interpolation functions are used to define the variation of the primary variable across the element and the shape of the element. In other words, [hJ] is used to determine the element shape from the nodal coordinates [x ] (hence the term shape function, which takes on added significance for isoparametric elements) and to determine element displacement from nodal displacements. 

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