Multiresolution Green’s Function Methods for Interactive Simulation of Large-scale Elastostatic Objects

Abstract

We present a framework for low-latency interactive simulation of linear elastostatic models, and other systems arising from linear elliptic partial differential equations, which makes it feasible to interactively simulate large-scale physical models. The deformation of the models is described using precomputed Green's functions (GFs), and runtime boundary value problems (BVPs) are solved using existing Capacitance Matrix Algorithms (CMAs). Multiresolution techniques are introduced to control the amount of information input and output from the solver thus making it practical to simulate and store very large models. A key component is the efficient compressed representation of the precomputed GFs using second-generation wavelets on surfaces. This aids in reducing the large memory requirement of storing the dense GF matrix, and the fast inverse wavelet transform allows for fast summation methods to be used at runtime for response synthesis. Resulting GF compression factors are directly related to interactive simulation speedup, and examples are provided with hundredfold improvements at modest error levels. We also introduce a multiresolution constraint satisfaction technique formulated as an hierarchical CMA, so named because of its use of hierarchical GFs describing the response due to hierarchical basis constraints. This direct solution approach is suitable for hard real time simulation since it provides a mechanism for gracefully degrading to coarser resolution constraint approximations. The GFs' multiresolution displacement fields also allow for runtime adaptive multiresolution rendering.