UBC Theses and Dissertations
Integrators for elastodynamic simulation with stiffness and stiffening Chen, Yu Ju
The main goal of this thesis is to develop effective numerical algorithms for stiff elastodynamic simulation, a key procedure in computer graphics applications. To enable such simulations, the governing differential system is discretized in 3D space using a finite element method (FEM) and then integrated forward in discrete time steps. To perform such simulations at a low cost, coarse spatial discretization and large time steps are desirable. However, using a coarse spatial mesh can introduce numerical stiffening that impede visual accuracy. Moreover, to enable large time steps while maintaining stability, the semi-implicit backward Euler method (SI) is often used; but this method causes uncontrolled damping and makes simulation appear less lively. To improve the dynamic consistency and accuracy as the spatial mesh resolution is coarsened, we propose and demonstrate, for both linear and nonlinear force models, a new method called EigenFit. This method applies a partial spectral decomposition, solving a generalized eigenvalue problem in the leading mode subspace and then replacing the first several eigenvalues of the coarse mesh by those of the fine one at rest. We show its efficacy on a number of objects with both homogeneous and heterogeneous material distribution. To develop efficient time integrators, we first demonstrate that an exponential Rosenbrock-Euler (ERE) integrator can avoid excessive numerical damping while being relatively inexpensive to apply for moderately stiff elastic material. This holds even in challenging circumstances involving non-convex elastic energies. Finally, we design a hybrid, semi-implicit exponential integrator, SIERE, that allows SI and ERE to each perform what they are good at. To achieve this we apply ERE in a small subspace constructed from the leading modes in the partial spectral decomposition, and the remaining system is handled (i.e., effectively damped out) by SI. We show that the resulting method maintains stability and produces lively simulations at a low cost, regardless of the stiffness parameter used.
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