Polyhedral Finite Elements Using Harmonic Basis Functions

Abstract

Finite element simulations in computer graphics are typically based on tetrahedral or hexahedral elements, which enables simple and efficient implementations, but in turn requires complicated remeshing in case of topological changes or adaptive refinement. We propose a flexible finite element method for arbitrary polyhedral elements, thereby effectively avoiding the need for remeshing. Our polyhedral finite elements are based on harmonic basis functions, which satisfy all necessary conditions for FEM simulations and seamlessly generalize both linear tetrahedral and trilinear hexahedral elements. We discretize harmonic basis functions using the method of fundamental solutions, which enables their flexible computation and efficient evaluation. The versatility of our approach is demonstrated on cutting and adaptive refinement within a simulation framework for corotated linear elasticity.

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.…”

“…Examples of generalized barycentric coordinates are Wachspress coordinates [42], mean value coordinates [12,13,18,43], Laplace interpolants [17], harmonic coordinates [21,25], and maximum entropy coordinates [31,19].…”

Frame-Based Interactive Simulation of Complex Deformable Objects

“…Multi-resolution approaches have been proposed [9,15]. In recent work, disconnected or arbitrarily-shaped elements [19,26] have been proposed to alleviate the meshing difficulties.…”

Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons

“…For example, in a twodimensional setting, with being a convex m-gon and i denoting the ith edge of the polygon, one can obtainQ i -a standard Gauss quadrature rule over the interval [−1, 1] is mapped to the line-segment i and the weights of the quadrature are multiplied by the length of the ith edge of the polygon divided by two. The approximation sign in (4) and (6) pertains to the approximation error of the quadraturesQ i , and no further approximation is introduced in the construction. In other words, beginning with quadratures that are exact for the integration of f over the faces of the region, for example a Gauss quadrature rule for polynomial f , one can obtain an exact quadrature via (6).…”

“…Conforming polygonal finite elements [1][2][3][4] and finite elements on convex polyhedra [5][6][7] require the integration of nonpolynomial basis functions. The integration of polynomials on irregular polytopes arises in the non-conforming variable-element-topology finite element method [8,9], discontinuous Galerkin finite elements [10], finite volume element method [11] and mimetic finite difference schemes [12][13][14].…”