Geometry Mapping, Complete Pascal Scheme versus Standard Bilinear Approach

This paper presents a complete Pascal interpolation scheme for use in the plane geometry mapping applied in association with numerical methods. The geometry of a domain element is approximated by a complete Pascal polynomial. The interpolation procedure is formulated in a natural coordinate system. It also presents the methodology of constructing shape functions of Pascal type and establishing a transformation relation between natural and Cartesian variables. The performance of the presented approach is investigated firstly by calculating the geometrical properties of an arbitrary quadrilateral cross-section like area and moments of inertia and comparing the results with the exact values and with those provided by the standard linear approach and a serendipity family approach. Secondly, the assessment of the scheme follows using a straight-sided, compatible quadrilateral finite element for plate bending of which geometry is approximated by a complete set of second order with six free parameters. Triangular and quadrilateral shaped plates with different boundary conditions are computed and compared with well-known results in the literature. The presented procedure is of general applicability for elements with curved edges and not limited to straight-sided edges in the framework of numerical methods.