Generation of shape functions by optimization

Abstract

In boundary value problems, the solution region is always discreti-zed into finite elements. The polynomial chosen to interpolate the field vari-ables over the element are called shape functions. The shape functions establish the relationship between the displacement at any point in the element with the nodal displacement of the element. However, the polynomial cannot guarantee the shape function of all the transition elements as the inverse of the matrix generated from some of the transition elements are not feasible. This paper of-fers an insight into the derivation of shape function using minimization theory. In the case of irregular elements, such as transition elements, improvements are made regarding the derivation so as to capture the peculiarities of the so-called transition elements. All the shape functions derived using minimization approach are validated according to interpolation properties.