Browsing by Author "Munro, Dirk Pieter."
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- ItemA direct approach to structural topology optimization(Stellenbosch : Stellenbosch University, 2017-03) Munro, Dirk Pieter.; Groenwold, Albert A.; Stellenbosch University. Faculty of Engineering. Dept. of Mechanical and Mechatronic Engineering.ENGLISH ABSTRACT: This dissertation addresses various topics that emerge from the unification of conventional structural optimization—based on ‘sequential approximate optimization’ (SAO)—with the alternative ‘direct’—or ‘simultaneous analysis and design’ (SAND)—formulation of the structural topology design problem. Structural topology optimization—in the form of a ‘material distribution problem’—is a generalisation of structural optimization, encompassing and simultaneously addressing al the aspects of structural design. In structural optimization, SAO techniques are preferred because the number of structural analyses—which are expensive, computationally speaking—are ostensibly minimised. However, particularly in the presence of local state-based constraints—e.g. local stress constraints—the sensitivity analyses which accompany traditional ‘nested analysis an design’ (NAND) methods require a prohibitive number of structural analysis runs per design iteration. In the alternative SAND setting, structural analysis is conducted approximately and sequentially: the finite element (FE) equilibrium equations are retained as a set of nonlinear equality constraints and the state variables—i.e. displacements—form part of the overall set of primal variables. Therefore, the FE equilibrium equations may only be satisfied at convergence of the optimization algorithm, and the complex and expensive sensitivity analyses associated with state-based constraints, simplify to the calculation of partial derivatives. Moreover, the equivalent of a single structural analysis only is required per design iteration, notwithstanding the imposition of a large number of state-based constraints. Based on a dual method in theory, we propose a separable and strictly convex quadratic Lagrange-Newton approximate subproblem for use in SAO of the SAND formulated topology design problem. In classical (simply-constrained) minimum compliance design, the dual statement of the subproblem is equivalent to the ever-popular optimality criteria (OC) approach—a class of NAND methods. This relates, in turn, to the known equivalence between dual SAONAND algorithms based on intervening variables and the OC method. Due to the presence of nonlinear equality constraints, the classical SAO procedure (exclusively geared, traditionally, for inequality constrained problems) is extended to a general, nonlinear and nonconvex, mathematical programming framework. It turns out that conventional techniques of enforced convergence and termination in traditional NAND-based SAO may be transplanted into the SAND setting with only minor complications. It is demonstrated that the compounded issues of existence of solutions, mesh-dependence, local minima, and macro-scale manufacturability, may be addressed in a computationally efficient manner by the imposition of so-called ‘slope constraints’—point-wise bounds on the gradient of the material distribution function. For global optimization, random multistart strategies may be pursued. A specialized version of ‘linear independence constraint qualification’ (LICQ) may hold in many practical situations, and because the global stiffness matrix is not inverted per se, material density variables are permitted a value of zero on the lower bound. Hence, singular local minima are feasible and available—in both simply-constrained and local stress-constrained problems—and may be converged to with standard gradient-based optimization methods without having to resort to any relaxation or perturbation techniques whatsoever.