Algorithms for imposition of constraints and topology changes in interactive simulations

Abstract

First, a multilevel framework that combines the efficiency of the multigrid algorithms with the ability to resolve linear projection constraints (LPC) is proposed. We present a single level modified block Gauss-Seidel (MBGS) smoother that can incorporate LPCs. This is subsequently incorporated in a standard multigrid V-cycle with corrections for constraints resulting in a modified multigrid V-cycle (MMgV). Finally, we extend the idea to the full multigrid V-cycle. Numerical tests showed that the solver is able resolve constraints while achieving the theoretical performance of multigrid algorithms.

Physically realistic simulations are increasingly being used in interactive environments such as surgical simulation, rehabilitation training and computer games. Such interactive environments, which are by definition real-time, demand numerical simulation of various physical modalities, collision handling, topology modifications and multi-sensory (visual, auditory and haptic) interactions with a very high degree of realism. An ideal candidate algorithm for such applications would have to be numerically efficient, highly scalable on modern CPU/GPU hardware and robust. Multilevel algorithms are known to have theoretically optimal computational performance and are shown to scale well with hardware capabilities. However, incorporating topology changes and collision constraints with these algorithms to make them viable for real-time applications is challenging. In this thesis we address these issues by presenting algorithms to efficiently incorporate constraints and topology changes within a multilevel framework for interactive simulations.

Second, a modified iterative constraint anticipation (MICA) algorithm as a unified solver for linear complementarity problem (LCP) and LPC is introduced. MICA performs staggered iterations of Gauss-Seidel and projected Gauss-Seidel that can be used in resolving LCP constraints from collision response along with the LPC from user interactions and Dirichlet boundary conditions. MICA is then used in designing a novel iterative predictor-corrector (IPC) approach to model static and kinetic friction during interactions with deformable objects. The proposed IPC method works within the purview of the implicit mixed linear complementarity problem (MLCP) formulation of collision response and can be used to model both asymmetric and heterogeneous type anisotropic friction models. It requires low memory and is highly tunable.

Finally, we present algorithms for cutting of deformable objects simulated using multilevel methods. Multilevel cutting is complicated by the need to reflect fine grid topology changes at the coarser grid level which requires modification of the coarse grid operators. We consider a two-level formulation and address the real-time issue by developing two separate strategies for updating the map relating the coarse and fine grid operators for structured as well as unstructured meshes. Considerable speedups of this multigrid approach compared to a preconditioned conjugate gradient (PCG) solver are observed.

Tests performed on 3-D elastic problems showed that MICA suffers from low rate of convergence for cases simulating stiff materials. To resolve this problem, we propose a modified multilevel iterative constraint anticipation (M-MgICA) as a logical extension to MICA. M-MgICA is a multilevel method that uses MICA as a relaxation technique. Coarse level complementarity spaces are formed using algebraic coarsening while elasticity operators are coarsened using well-known geometric coarsening. Results show considerable speed-up when compared to single level MICA iterations.