Complete Pascal Interpolation Scheme For Approximating The Geometry Of A Quadrilateral Element

TL;DR

This paper introduces a complete Pascal polynomial-based interpolation scheme for quadrilateral geometry approximation, enhancing shape representation accuracy by incorporating additional nodal points and poles.

Contribution

It develops a complete second-order Pascal interpolation scheme for quadrilaterals, including a third-order extension, improving geometric approximation over traditional methods.

Findings

The scheme recovers Lagrangian interpolation when opposite edges are parallel.

The third-order scheme uses midpoints and nodal points for better shape reflection.

Complete Pascal polynomials provide more accurate geometry approximation.

Abstract

This paper applies a complete parametric set for approximating the geometry of a quadrilateral element. The approximation basis used is a complete Pascal polynomial of second order with six free parameters. The interpolation procedure is a natural interpolation scheme. The six free parameters are determined using the natural coordinates of the four nodal points (vertices) of the quadrilateral element and the two intersections points of the lines crossing every two opposite edges (poles). The presented scheme recovers the well known Lagrangian interpolation scheme, when every two opposite edges are parallel. A third order Pascal interpolation scheme is also presented. The four midpoints of the four edges in addition to the six nodal point from the second order case are used as significant nodal points. It is expected to reflect the geometry properties better since the shape functions are complete