Sensitivity-Based Topology and Shape Optimization with Application to Electric Motors
In many industrial applications, one is interested in finding an optimal layout of an object, which often leads to PDE-constrained shape optimization problems. Such problems can be approached by shape optimization methods, where a domain is altered by smooth deformation of its boundary, or by means of topology optimization methods, which in addition can alter the connectivity of the initial design. We give an overview over established topology optimization methods and focus on an approach based on the sensitivity of the cost function with respect to a topological perturbation of the domain, called the topological derivative. We illustrate a way to derive this sensitivity and discuss the additional difficulties arising in the case of a nonlinear PDE constraint. We show numerical results for the optimization of an electric motor which are obtained by a combination of two methods: a level set algorithm which is based on the topological derivative, and a shape optimization method together with a special treatment of the evolving material interface which assures accurate approximate solutions to the underlying PDE constraint as well as a smooth final design.
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