Applications of Stochastic Methods and Machine Learning Techniques in Computational Mechanics and Topology Optimization

Recent advances in stochastic methods and machine learning (ML) techniques have opened new pathways that can be explored for addressing uncertainty quantification and for accelerating computational tasks in various mechanics applications. However, to realize the potential of these methods, effective computational frameworks that integrate stochastic and ML methods in computational mechanics are needed. Thus, to harness the power of these tools in mechanics applications, the goal of this dissertation is to investigate and develop novel frameworks where advanced stochastic and data-driven machine learning techniques are integrated into computational platforms to address various challenges such as response simulations and uncertainty quantification in linear and nonlinear systems, design optimization under uncertainties, and multiscale modeling. In particular, probabilistic methods based on stochastic polynomial expansions are applied to quantify the uncertainties in linear and nonlinear structural responses given stochastic inputs of loading and/or material properties which are modeled by random variables or random fields. The efficiency and accuracy of these methods are then systematically investigated for linear and nonlinear mechanics problems within the context of stochastic finite elements. Based on the performance studies on uncertainty quantification methods, an efficient 2^nd-order stochastic perturbation technique is integrated with a density-based topology optimization framework for designing robust structures at finite deformations while considering load, material, and geometric uncertainties. In addition, novel data-driven machine learning models and learning techniques are proposed to create efficient and accurate surrogates for (a) linear and nonlinear dynamical systems; and (b) constitutive models of 2D and 3D microstructures in multiscale modeling. It is envisaged that the proposed probabilistic and data-driven frameworks in this dissertation will lay the foundations for the next generation of advanced computational methods in applied solid and structural mechanics.