Limiting Descent Directions in p-Harmonic Shape Optimization
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Recent developments in shape optimization showed improved results by using descent directions in Banach spaces, in particular by using a p-harmonic approach. Here, for each optimization step a corresponding vector-valued p-Laplace problem with boundary forcing has to be solved efficiently for possibly high-order p. Therefore, we work with an algorithm based on an interior-point method that does not require an iteration over a sequence of increasing p. Based on this algorithm, we discuss extensions to the limit setting. A key challenge in this regard is that the Lipschitz deformations obtained as solutions to the limit problem are in general non-unique. Further, the solutions feature uncommon behavior, when the forcing terms contain a switch of sign. However, this is crucial for the handling of geometric constraints in shape optimization. Thus, we focus on solutions which are in a sense limits to solutions for finite p and aim to preserve mesh quality throughout the outer optimization loop.
