Download thesis |
View on UC eScholarship
Part I: Reconstruction of Missing Data in Social
Networks Based on Temporal Patterns of Interactions,
Part II: Constitutive Modeling in Solid Mechanics for
UCLA Digital Library 2013.
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.
Professor Joseph Teran, Co-chair
Professor Andrea Bertozzi, Co-chair
Professor Stanley Osher
Professor 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 modelís 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.