Multiresolution Green’s Function Methods for Interactive Simulation of Large-scale Elastostatic Objects and other Physical Systems in Equilibrium
Ph.D. Thesis, Institute of Applied Mathematics, UBC, 2001.
Doug L. James
This thesis presents a framework for low-latency interactive simulation of linear elastostatic models and other systems associated with linear elliptic partial differential equations. This approach makes it feasible to interactively simulate large-scale physical models.
Linearity is exploited by formulating the boundary value problem (BVP) solution in terms of Green’s functions (GFs) which may be precomputed to provide speed and cheap lookup operations. Runtime BVPs are solved using a collection of Capacitance Matrix Algorithms (CMAs) based on the Sherman-Morrison-Woodbury formula. Temporal coherence is exploited by caching and reusing, as well as sequentially updating, previous capacitance matrix inverses.
Multiresolution enhancements make it practical to simulate and store very large models. Efficient compressed representations of precomputed GFs are obtained using second-generation wavelets defined on surfaces. Fast inverse wavelet transforms allow fast summation methods to be used to accelerate runtime BVP solution. Wavelet GF compression factors are directly related to interactive simulation speedup, and examples are provided with hundredfold improvements at modest error levels. Furthermore, hierarchical constraints are defined using hierarchical basis functions, and related hierarchical GFs are then used to construct an hierarchical CMA. This direct solution approach is suitable for hard real time simulation since it provides a mechanism for gracefully degrading to coarser resolution approximations, and the wavelet representations allow for runtime adaptive multiresolution rendering.
These GF CMAs are well-suited to interactive haptic applications since GFs allow random access to solution components and the capacitance matrix is the contact compliance used for high-fidelity force feedback rendering. Examples are provided for distributed and point-like interactions.
Precomputed multizone kinematic GF models are also considered, with examples provided for character animation in computer graphics.
Finally, we briefly discuss the generation of multiresolution GF models using either numerical precomputation methods or reality-based robotic measurement.