UCLA

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 [40, 86]. 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 [13, 56] 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 consti-

tutive 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.