Study on derivation of shape functions in global coordinates and exact computation of element matrices for quadrilateral finite elements

This article includes a technique to derive shape functions in global co-ordinates for general quadrilateral finite elements. It identifies clearly the element geometry for which the shape functions in global co-ordinates cannot be derived and explains the way to overcome such situation. Finally, it presents formulae based on array multiplication for exact computation of different types of element matrices needed for the employment of the said element in finite element solution procedure. The computation process of element matrices require only: (1) the nodal co-ordinates of the element geometry to form a matrix G (say) and then it’s inverse matrix H (say) and (2) the values of the integral of monomials over the element. All the components of element matrices are then computed by the product of components of H with the values of the integrals. Thus, the process reduces many time consuming steps of FEM solution procedure and that substantially reduces computational effort. The accuracy and efficiency of the formulae so presented are then demonstrated through application examples.