Stable explicit integration of deformable objects by filtering high modal frequencies

Abstract

In a traditional finite element method for the simulation of deformable objects, the stiffness matrix depends on the shape of the tetrahedral elements. Ill-shaped elements containing large or small dihedral angles lead to an arbitrary large condition number of the stiffness matrix, thus slowing down the simulation. In addition, high modal frequencies cannot be simulated stably if an explicit numerical time-integration scheme is considered. We propose an approach consisting of two components to address this problem: First, we isolate the ill-shaped tetrahedra by performing an eigenvalue decomposition, and then remove their high modal frequencies by directly altering their element stiffness matrices. This makes the elements softer in some directions. To prevent element inversion, we define constraints that rigidify the object along those directions. The fast projection method, implemented as a velocity filter, is employed to enforce the constraints after the temporal evolution. With our approach, a significantly larger time step can be chosen in explicit integration methods, resulting in a faster simulation.