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Shape optimization via control of a shape function on a fixed domain: theory and numerical results. (English) Zbl 1268.49052

Repin, Sergei (ed.) et al., Numerical methods for differential equations, optimization, and technological problems. Dedicated to Professor P. Neittaanmäki on his 60th birthday. Selected papers based on the presentations at the ECCOMAS thematic conference computational analysis and optimization (CAO 2011), Jyväskylä, Finland, June 9–11, 2011. Dordrecht: Springer (ISBN 978-94-007-5287-0/hbk; 978-94-007-5288-7/ebook). Computational Methods in Applied Sciences (Springer) 27, 305-320 (2013).
Summary: We present a fixed-domain approach for the solution of shape optimization problems governed by linear or nonlinear elliptic partial differential state equations with Dirichlet boundary conditions, where shape optimization is facilitated via optimal control of a shape function. The method involves extending the state equation to a larger domain using regularization. Results regarding the convergence to the original problem are provided as well as differentiability properties of the control-to-state mappings. An algorithm for the numerical implementation of the method is stated and, in a series of numerical shape optimization experiments, the algorithm’s behavior is studied with regard to varying the regularization parameter and initial conditions.
For the entire collection see [Zbl 1254.65007].

MSC:

49Q10 Optimization of shapes other than minimal surfaces
49M30 Other numerical methods in calculus of variations (MSC2010)
35J60 Nonlinear elliptic equations

Software:

Triangle; PARDISO; pdelib
PDFBibTeX XMLCite
Full Text: DOI

References:

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