On Methods for Discrete Topology Optimization of Continuum Structures

Abstract: This thesis consists of an introduction and seven appended papers. The purpose of the introduction is to give an overview of the field of topology optimization of discretized load carrying continuum structures. It is assumed that the design domain has been discretized by the finite element method and that the design variable vector is a binary vector indicating presence or absence of material in the various finite elements. Common to all papers is the incorporation of von Mises stresses in the problem formulations.In the first paper the design variables are binary but it is assumed that the void structure can actually take some load. This is equivalent to adding a small positive value, epsilon, to all design variables, both those that are void and those that are filled with material. With this small positive lower bound the stiffness matrix becomes positive definite for all designs. If only one element is changed (from material to void or from void to material) the new global stiffness matrix is just a low rank modification of the old one and thus the Sherman-Morrison-Woodbury formula can be used to compute the displacements in the neighbouring designs efficiently. These efficient sensitivity calculations can then be applied in the context of a neighbourhood search method. Since the computed displacements are exact in the 1-neighbourhood (when one design variable is changed) the neighbourhood search method will find a local optimum with respect to the 1-neighbourhood.The second paper presents globally optimal zero-one solutions to some small scale topology optimization problems defined on discretized continuum design domains. The idea is that these solutions can be used as benchmarks when testing new algorithms for finding pure zero-one solutions to topology optimization problems.In the third paper the results from the first paper are extended to include also the case where there is no epsilon>0. In this case the stiffness matrix will no longer be positive definite which means that the Sherman-Morrison-Woodbury formula can no longer be applied. The changing of one or two binary design variables to their opposite binary values will still result in a low rank change, but the size of the reduced stiffness matrix will change with the design. It turns out, however, that it is possible to compute the effect of these low rank changes efficiently also without the positive lower bound. These efficient sensitivity calculations can then be used in the framework of a neighbourhood search method. In this case the complete 1-neighbourhood and a subset of the 2-neighbourhood is investigated in the search for a locally optimal solution.In the fourth paper the sensitivity calculations developed in the third paper are used to generate first and partial second order approximations of the nonlinear functions usually present in topology optimization problems. These approximations are then used to generate subproblems in two different sequential integer programming methods (SLIP and SQIP, respectively). Both these methods generate a sequence of iteration points that can be proven to converge to a local optimum with respect to the 1-neighbourhood. The methods are tested on some different topology optimization problems.The fifth paper demonstrates that the SLIP method developed in the previous paper can be applied also to the mechanism design problem with stress constraints. In order to generate the subproblems in a fast way small displacements are assumed, which implies that the efficient sensitivity calculations derived in the third paper can be used. The numerical results indicate that the method can be used to lower the stresses and still get a functional mechanism.In the sixth paper the SLIP method developed in the fourth paper is used as a post processor to obtain locally optimal zero-one solutions starting from a rounded solution to the corresponding continuous problem. The numerical results indicate that the method can perform well as a post processor.The seventh paper is a theoretical paper that investigates the validity of the commonly used positive lower bound epsilon on the design variables when stating and solving topology optimization problems defined on discretized load carrying continuum structures. The main result presented here is that an optimal "epsilon-1" solution to an "epsilon-perturbed" discrete minimum weight problem with constraints on compliance, von Mises stresses and strain energy densities, is optimal, after rounding to zero-one, to the corresponding "unperturbed" discrete problem. This holds if the constraints in the perturbed problem are carefully defined and epsilon>0 is sufficiently small.