On Models and Methods for Global Optimization of Structural Topology

Abstract: This thesis consists of an introduction and sevenindependent, but closely related, papers which all deal withproblems in structural optimization. In particular, we considermodels and methods for global optimization of problems intopology design of discrete and continuum structures.In the first four papers of the thesis the nonconvex problemof minimizing the weight of a truss structure subject to stressconstraints is considered. First itis shown that a certainsubclass of these problems can equivalently be cast as linearprograms and thus efficiently solved to global optimality.Thereafter, the behavior of a certain well-known perturbationtechnique is studied. It is concluded that, in practice, thistechnique can not guarantee that a global minimizer is found.Finally, a convergent continuous branch-and-bound method forglobal optimization of minimum weight problems with stress,displacement, and local buckling constraints is developed.Using this method, several problems taken from the literatureare solved with a proof of global optimality for the firsttime.The last three papers of the thesis deal with topologyoptimization of discretized continuum structures. Theseproblems are usually modeled as mixed or pure nonlinear 0-1programs. First, the behavior of certain often usedpenalization methods for minimum compliance problems isstudied. It is concluded that these methods may fail to producea zero-one solution to the considered problem. To remedy this,a material interpolation scheme based on a rational functionsuch that compli- ance becomes a concave function is proposed.Finally, it is shown that a broad range of nonlinear 0-1topology optimization problems, including stress- anddisplacement-constrained minimum weight problems, canequivalently be modeled as linear mixed 0-1 programs. Thisresult implies that any of the standard methods available forgeneral linear integer programming can now be used on topologyoptimization problems.Keywords:topology optimization, global optimization,stress constraints, linear programming, mixed integerprogramming, branch-and-bound.