Numerical solution of elastoplasticity and poroelasticity problems using optimization approaches and mixed finite elements
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Application of nonlinear mechanics of materials and multiphysics simulation of flow and mechanics encompasses a large class of problems in seismic analysis of structures, mechanics of fully/partially saturated soil, biomechanics of tissues, and petroleum/geothermal reservoirs. In this thesis, we explore new classes of numerical methods for two distinct aspects of such models -- (i) incremental state update of nonlinear material models; (ii) dynamic analysis of porous media. In a nonlinear analysis, the state of a spatially discretized model typically consists of displacements, velocities, stresses, and internal variables. The task in each time increment of a numerical solution is to compute the new state of the model, given the current state and the increments of external forces and boundary conditions. When non-linear material model is described using an energy approach, state update can be cast into an optimization problem. In this study, a complete framework for such analysis is developed for a general continuum and two robust numerical strategies are presented to solve the arising convex minimization problem, namely, projected Newton method and nested optimization form of displacement-based method. Various numerical examples are presented for a particular instance of material nonlinearity, that is nonlinear elastoplasticity with kinematic and isotropic hardening with von Mises yield criterion. Both methods exhibit robust solution with quadratic rate of convergence. Especially, implementation of multi-surface plasticity models in projected Newton method is no more involved than that of single-surface models. Moreover, these optimization approaches eliminate the need for some heuristics typically used in classical methods and provide insight into the working of the state update by visualization of the minimization process. Such understanding will contribute toward developing successive convex programming approaches for non-convex problems such as those involving non-associated plastic flow, softening and geometric nonlinearity.