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Geometric Interpretation of Physical Systems for Improved Elasticity Simulations

The physics of most mechanical systems can be described from a geometric viewpoint; i.e., by defining variational principles that the system obeys and the properties that are being preserved (often referred to as invariants). The methods that arise from properly discretizing such principles preserve corresponding discrete invariants of the mechanical system, even at very coarse resolutions, yielding robust and efficient algorithms. In this thesis geometric interpretations of physical systems are used to develop algorithms for discretization of both space (including proper material discretization) and time. The effectiveness of these algorithms is demonstrated by their application to the simulation of elastic bodies. Time discretization is performed using variational time integrators that, unlike many of the standard integrators (e.g., Explicit Euler, Implicit Euler, Runge-Kutta), do not introduce artificial numerical energy decrease (damping) or increase. A new physical damping model that does not depend on timestep size is proposed for finite viscoelasticity simulation. When used in conjunction with variational time integrators, this model yields simulations that physically damp the energy of the system, even when timesteps of different sizes are used. The usual root-finding procedure for time update is replaced with an energy minimization procedure, allowing for more precise step size control inside a non-linear solver. Additionally, a study of variational and time-reversible methods for adapting timestep size during the simulation is presented. Spatial discretization is performed using a finite element approach for finite (non-linear) or linear elasticity. A new method for the coarsening of elastic properties of heterogeneous linear materials is proposed. The coarsening is accomplished through a precomputational procedure that converts the heterogeneous elastic coefficients of the very fine mesh into anisotropic elastic coefficients of the coarse mesh. This method does not depend on the material structure of objects, allowing for complex and non-uniform material structures. Simulation on the coarse mesh, equipped with the resulting elastic coefficients, can then be performed at interactive rates using existing linear elasticity solvers and, if desired, co-rotational methods. A time-reversible integrator is used to improve time integration of co-rotated linear elasticity.