Conforming polygonal finite elements

Abstract

SUMMARYIn 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.

Optimizing Voronoi Diagrams for Polygonal Finite Element Computations

“…As a consequence, generalizations of FE methods to arbitrary polygonal or polyhedral meshes have gained increasing attention, both in computational physics [34,36,37] and in computer graphics [43,25]. For instance, when cutting a tetrahedron into two pieces, polyhedral FEM can directly process the resulting two elements, while standard FEM has to remesh them into tetrahedra first.…”

“…In the following we briefly review the basic concepts of (polygonal) finite element methods, and refer the reader to the textbooks [4,20] and the survey [34] for more details on classical FEM and polygonal FEM, respectively.…”

Solving elliptic eigenvalue problems on polygonal meshes using discontinuous Galerkin composite finite element methods

“…In the continuous Galerkin setting we have the composite finite element methods (CFEs) [1,2,3,4,5], the polygonal finite element methods (PFEMs) [6,7], the extended finite element method (XFEM) [8] and the virtual finite element method (VFEM) [9]. On the other hand in the discontinuous Galerkin (DG) setting we have the interior penalty methods on polygonal and polyhedral meshes [10], the 1 agglomeration-based method [11,12,13] and the discontinuous Galerkin composite finite element methods (DGCFEMs) [14].…”

Positive Gordon–Wixom coordinates

“…Barycentric coordinates provide a method of interpolating values from the boundary of a domain over its interior and are useful in a variety of applications including finite element analysis [21,19,18], texture mapping [2], deformation [11], image compositing [3], volumetric texture synthesis [20], shading [9], and geometric modeling [13]. Barycentric coordinates can also be used to approximate solutions to Poisson problems [3].…”