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.

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].…”

Robust interactive cutting based on an adaptive octree simulation mesh

“…In doing so, the elements along the cut surface are intrinsically conforming, as shown by Martin et al [17]. Their approach is based on the earlier work by Wicke et al [34] who only support convex element shapes. However, these approaches have the drawback that the update of the elemental stiffness matrices becomes very expensive.…”

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

“…To clarify this issue, let g(x) = j g j (x), with g j being q j -homogeneous. The following integral is valid: where Q is the quadrature obtained through (5). If each g j (x) is not known explicitly, then the value of the integral in (7) can not be computed.…”

“…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].…”