This thesis consists of an introduction and seven independent, but closely related, papers which all deal with problems in structural optimization. In particular, we consider models and methods for global optimization of problems in topology design of discrete and continuum structures.
In the first four papers of the thesis the nonconvex problem of minimizing the weight of a truss structure subject to stress constraints is considered. First it is shown that a certain subclass of these problems can equivalently be cast as linear programs and thus efficiently solved to global optimality. Thereafter, the behavior of a certain well-known perturbation technique is studied. It is concluded that, in practice, this technique can not guarantee that a global minimizer is found. Finally, a convergent continuous branch-and-bound method for global optimization of minimum weight problems with stress, displacement, and local buckling constraints is developed. Using this method, several problems taken from the literature are solved with a proof of global optimality for the first time.
The last three papers of the thesis deal with topology optimization of discretized continuum structures. These problems are usually modeled as mixed or pure nonlinear 0-1 programs. First, the behavior of certain often used penalization methods for minimum compliance problems is studied. It is concluded that these methods may fail to produce a zero-one solution to the considered problem. To remedy this, a material interpolation scheme based on a rational function such that compliance becomes a concave function is proposed. Finally, it is shown that a broad range of nonlinear 0-1 topology optimization problems, including stress- and displacement-constrained minimum weight problems, can equivalently be modeled as linear mixed 0-1 programs. This result implies that any of the standard methods available for general linear integer programming can now be used on topology optimization problems.