Abstract
We present a worst-case approach to topology optimization (TO) for maximum stiffness under boundary displacement parametrized by a matrix-valued scaling function times an uncertain vector giving its direction. The objective function in the TO problem is the minimum of the potential energy maximized over the set of boundary displacements, which in the absence of prescribed loads means maximizing the reaction loads arising from enforcing the boundary displacement. It is shown that the TO problem can be cast as the minimization of the maximum eigenvalue of a matrix depending on solutions to a small number of (linear elastic) state problems. Numerical solution of this potentially non-smooth problem using algorithms for smooth optimization, a non-linear semi-definite programming reformulation, and a non-smooth bundle method is discussed and tested.
Highlights
Topology optimization (TO) has become a popular tool for design of mechanical structures [10,13,24]
Some numerical examples are given here to show the effect of accounting for uncertain boundary displacement and give an indication as to how the numerical solution methods discussed in Section 4 works
We have presented and showed well-posedness of a worstcase approach for TO under uncertain boundary displacement in a formulation leading to an optimization problem with a maximum eigenvalue as the objective function
Summary
Topology optimization (TO) has become a popular tool for design of mechanical structures [10,13,24]. To achieve a robust structure which is a stiff as possible we seek a design which minimize the minimum (average) reaction load obtained as the boundary displacement vary in an uncertainty set (see problem (6) below) This approach is related to some earlier work on worst-case load-uncertainty, see for example [56,33,58], but prescribing displacements rather than loads make the derivation and analysis somewhat different. Detailed and accurate probability distributions may be difficult to obtain, so this speaks in favour of using worst-case oriented methods Such methods guarantee structural performance for all data in the uncertainty set, whereas a corresponding stochastic method can only give the same guarantee with a certain probability. We try solving a smooth, nonlinear semidefinite programming formulation and present some preliminary results from applying a simple non-smooth bundle method
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