Geometry Mapping, Complete Pascal Scheme versus Standard Bilinear Approach

TL;DR

This paper introduces a complete Pascal interpolation scheme for plane geometry mapping in numerical methods, offering improved accuracy over standard approaches, especially for elements with curved edges.

Contribution

It develops a complete Pascal polynomial-based geometry mapping method and compares its performance with traditional linear and serendipity approaches.

Findings

Pascal scheme yields more accurate geometrical properties.

Improved results for plate bending problems with curved edges.

Applicable to elements with curved boundaries.

Abstract

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.