09:30
30 mins
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TOPOLOGY OPTIMIZATION – RECENT APPLICATIONS AND GENERALIZATIONS
Mathias Stolpe
Abstract: Topology optimization is a collection of theory, mathematical models, and numerical methods within optimal design. Topology optimization is often used in the conceptual design phase to propose innovative structures and materials. The presentation begins with an overview of topology optimization and some of the theoretical and numerical challenges in this field. Examples of some recently proposed applications for topology optimization are also presented.
Since the introduction of topology optimization several generalizations have been introduced. We focus on the powerful approach of Free Material Optimization (FMO) in which the design parameterization describes both the material distribution and the local material properties in the structure. The optimization problems in FMO are generally non convex Semi Definite Programs (SDPs). These nonstandard problems have many small matrix inequalities and special optimization methods utilizing this property have to be developed and implemented. By modifying existing interior-point methods it is possible to obtain high quality solutions to large-scale FMO problems for 3D structures using only a modest number of iterations. The main bottleneck is the computation of the search directions. This requires the solution of a very large-scale saddle point system which becomes increasingly ill-conditioned as the optimum is approached. We propose special purpose preconditioners combined with Krylov subspace methods for the saddle-point system and numerically show that they are capable of dealing with the increasing ill-conditioning. The numerical results confirm that the combination of techniques generates an efficient and robust method which is capable of routinely solving large-scale FMO problems on 3D design domains. The presentation concludes with an overview of the outstanding theoretical and numerical challenges in FMO.
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