Time integrators based on approximate discontinuous hamiltonians [electronic resource]

In this work we present a framework with which to construct time integrators for general mechanical systems. By approximating the potential energy with a nearby one, one can define a class of time integrators with several desirable properties, such as unconditional stability and conservation of invariants of the motion to machine precision. The resulting algorithms are asynchronous, explicit and can very easily be extended to handle frictionless contact with rigid boundaries. We discuss how these ideas extend to finite element discretizations of nonlinear elastic bodies and propose and specialize these methods so they scale linearly with the number of degrees of freedom in the system. Initially, we consider piecewise constant and affine approximations to the exact potential energy, and numerical verify the convergence of the resulting integrators. It is seen that while the piecewise constant integrator conserves all invariants of the problem, it is only conditionally convergent. The piecewise affine approximation converges on all numerical experiments performed, but necessarily breaks rotational invariance and hence does not conserve angular momentum. We analyze the one-dimensional piecewise affine algorithm and prove that the convergence rate of the integrator in fact depends on whether or not the exact trajectory every reaches a ``turning point, '' defined as a point where the velocity is identically zero. If no such point is reached, the integrator converges in the trajectories and velocities like O(h^2), where h> 0 is a mesh parameter. If such a point is reached, a reduced convergence rate of O(h^{3/2}) is observed. The convergence proof uses Gronwall's inequality, so convergence follows from a summation of the local truncation errors, and proves the reduced convergence rate is due to a low order error incurred near such turning points. We conclude this dissertation by proposing an algorithm for finite element systems which overcomes the shortcomings of the piecewise constant and affine integrators, at the price of a higher computational cost. This idea is based on introducing a discontinuous displacement field at nodes, similar in spirit to a discontinuous Galerkin method, and defining a piecewise quadratic potential energy for each element. With this approach we define a material frame indifferent approximate potential energy, using the edge lengths of the element in the deformed configuration. We show herein that the equations of motion for each element completely decouple, so each element can be evolved independently of other elements and hence the algorithm is extremely amenable to parallelization. In light of this feature of the algorithm, we can address the high computational cost of the integrator with a CUDA implementation on a graphics processing unit, which greatly improves the algorithm's running time. The piecewise quadratic ADH algorithm converges numerically on all examples performed while still conserving all invariants of the original problem.

Advisor: Lew Adrian primary advisor. Thesis advisor
Darve Eric advisor. Thesis advisor
Pinsky Peter M. (Peter Michael)