A Finite Element Method on Convex Polyhedra

Abstract

We present a method for animating deformable objects using a novel finite element discretization on convex polyhedra. Our finite element approach draws upon recently introduced 3D mean value coordinates to define smooth interpolants within the elements. The mathematical properties of our basis functions guarantee convergence. Our method is a natural extension to linear interpolants on tetrahedra: for tetrahedral elements, the methods are identical. For fast and robust computations, we use an elasticity model based on Cauchy strain and stiffness warping.This more flexible discretization is particularly useful for simulations that involve topological changes, such as cutting or fracture. Since splitting convex elements along a plane produces convex elements, remeshing or subdivision schemes used in simulations based on tetrahedra are not necessary, leading to less elements after such operations. We propose various operators for cutting the polyhedral discretization. Our method can handle arbitrary cut trajectories, and there is no limit on how often elements can be split.