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ItemNonlinear Topology Optimization: Theory and Applications(Georgia Institute of Technology, 2021-12-13) Zhao, TuoTopology optimization is a powerful computational design tool that can be used to find optimal structural layouts. In the structural design of civil engineering, it has been used to generate lateral bracing systems and conceptual designs that not only have aesthetic appeal, but are also structurally efficient. In mechanical and aerospace engineering, topology optimization is applied to develop a lightweight and robust antenna bracket for Sentinel satellites. In the biomedical field, topology optimization has been utilized to design patient-specific large craniofacial segmental bone replacements. However, in most engineering applications, structural design optimization is limited by theoretical/computational developments and only considers linear elastic material and omits the nonlinearity of real-life materials. This significantly limits the scope and applicability of topology optimization in practice. Thus, the structural design of engineering is in dire need of a tailored topology optimization framework that can handle realistic nonlinear materials, which are widely used in engineering structures. This thesis develops new topology optimization formulations to incorporate nonlinear mechanics, including both material nonlinearity (e.g., surrogate nonlinear elasticity and plasticity) and geometrical nonlinearity (e.g., finite deformation involving snapping instabilities). Three types of formulations are presented consist of maximizing structural strain energy, maximizing reaction load factor, and a min-max formulation, respectively. The objective of the first two formulations is to design optimum structure with improved loading capacity; while the min-max formulation is to achieve optimized designs with programmable nonlinear structural responses, i.e., snapping instabilities. The present topology optimization framework is applied to create innovative and functional structures and systems---for example, optimum strut-and-tie modeling for reinforced concrete structures and multistable assemblages for energy-absorbing devices.
ItemTowards Understanding First Order Algorithms for Nonconvex Optimization in Machine Learning( 2019-02-11) Zhao, TuoStochastic Gradient Descent-type (SGD) algorithms have been widely applied to many non-convex optimization problems in machine learning, e.g., training deep neural networks, variational Bayesian inference and collaborative filtering. Due to current technical limit, however, establishing convergence properties of SGD for these highly complicated practical non-convex problems is generally infeasible. Therefore, we propose to analyze the behavior of the SGD-type algorithms through two simpler but non-trivial non-convex problems – (1) Streaming Principal Component Analysis and (2) Training Non-overlapping Two-layer Convolutional Neural Networks. Specifically, we prove that for both examples, SGD attains a sub-linear rate of convergence to the global optima with high probability. Our theory not only helps us better understand SGD, but also provides new insights on more complicated non-convex optimization problems in machine learning.