Dissertation submitted in partial satisfaction
of the requirements for the degree
Doctor of Philosophy in Mathematics. University of California, Los Angeles
2013. Copyright by Alexey Dmitrievich Stomakhin 2013.
Comittee Members: Joseph Teran (Co-chair), Andrea Bertozzi (Co-chair), Stanley Osher, Jeffrey Eldredge.
In Part I the author presents a mathematical framework based on a self-exciting point process aimed at analyzing temporal patterns in the series of interaction events between agents in a social network. We develop a reconstruction model formulated as a constraint optimization problem that allows one to predict the unknown participants in a portion of those events. The results are used to predict the perpetrators of the unsolved crimes in the Los Angeles gang network.
Part II discusses the work undertaken by the author in deformable solid body simulation. We first focus on purely elastic solids and develop a method for extending an arbitrary isotropic hyperelastic energy density function to inverted configurations. This energy based extension is designed to improve robustness of elasticity simulations with extremely large deformations typical in graphics applications and demonstrates significant improvements over similar stress based techniques presented in. Moreover, it yields continuous stress and unambiguous stress derivatives in all inverted configurations. We also introduce a novel concept of a hyper-elastic model's primary contour which can be used to predict its robustness and stability. We demonstrate that our invertible energy-density-based approach outperforms the popular hyperelastic corotated model and show how to use the primary contour methodology to improve the robustness of this model to large deformations. We further develop a novel snow simulation method utilizing a user-controllable constitutive model defined by an elasto-plastic energy density function integrated with a hybrid Eulerian/Lagrangian Material Point Method (MPM). The method is continuum based and its hybrid nature allows us to use a regular Cartesian grid to automate treatment of self-collision and fracture. It also naturally allows us to derive a grid-based implicit integration scheme that has conditioning independent of the number of Lagrangian particles. We demonstrate the power of our method with a variety of snow phenomena.