Models and Methods for Free Material Optimization

Free Material Optimization (FMO) is a powerful approach for structural optimization in which the design parametrization allows the entire elastic stiffness tensor to vary freely at each point of the design domain. The only requirement imposed on the stiffness tensor lies on its mild necessary conditions for physical attainability, in the context that, it has to be symmetric and positive semidefinite. FMO problems have been studied for the last two decades in many articles that led to the development of a wide range of models, methods, and theories. As the design variables in FMO are the local material properties any results using coarse finite element discretization are not essentially predictive. Besides the variables are the entries of matrices at each point of the design domain. Thus, we face large-scale problems that are modeled as nonlinear and mostly non convex semidefinite programming. These problems are more difficult to solve and demand higher computational efforts than the standard optimization problems. The focus of today’s development of solution methods for FMO problems is based on first-order methods that require a large number of iterations to obtain optimal solutions. The scope of the formulations in most of the studies is indeed limited to FMO models for two- and three-dimensional structures. To the best of our knowledge, such models are not proposed for general laminated shell structures which nowadays have extensive industrial applications. This thesis has two main goals. The first goal is to propose an efficient optimization method for FMO that exploits the sparse structures arising from the many small matrix inequality constraints. It is developed by coupling secondorder primal dual interior point solution techniques for the standard nonlinear optimization problems and linear semidefinite programs. The method has successfully obtained solutions to large-scale classical FMO problems of simultaneous analysis and design, nested and dual formulations. The second goal is to extend the method and the FMO problem formulations to general laminated shell structures. The thesis additionally addresses FMO problem formulations with stress constraints. These problems are highly nonlinear and lead to the so-called singularity phenomenon. The method described in the thesis has successfully solved these problems. In the numerical experiments the stress constraints have been satisfied with high feasibility tolerances. The thesis further includes some preliminary numerical progresses on solving FMO problems using iterative solvers.