Conforming polygonal finite elements

TLDR

Polygonal finite elements provide greater flexibility in mesh generation and are better‑suited for applications in solid mechanics which involve a significant change in the topology of the material domain. The study develops conforming finite elements on polygon meshes. The authors use recent advances in meshfree approximations, computational geometry, and computer graphics to construct trial and test approximations on polygonal elements, notably employing meshfree natural‑neighbour basis functions on a canonical element with an affine map to build conforming approximations on convex polygons. The numerical formulation enables construction of conforming approximations on n‑gons (n ≥ 3) and demonstrates accuracy and convergence on second‑order elliptic boundary‑value problems. © 2004 John Wiley & Sons, Ltd.

Abstract

Abstract In this paper, conforming finite elements on polygon meshes are developed. Polygonal finite elements provide greater flexibility in mesh generation and are better‐suited for applications in solid mechanics which involve a significant change in the topology of the material domain. In this study, recent advances in meshfree approximations, computational geometry, and computer graphics are used to construct different trial and test approximations on polygonal elements. A particular and notable contribution is the use of meshfree (natural‐neighbour, nn) basis functions on a canonical element combined with an affine map to construct conforming approximations on convex polygons. This numerical formulation enables the construction of conforming approximation on n ‐gons ( n ⩾3), and hence extends the potential applications of finite elements to convex polygons of arbitrary order. Numerical experiments on second‐order elliptic boundary‐value problems are presented to demonstrate the accuracy and convergence of the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.