A linear view on shape optimization. (English) Zbl 1523.49050
The authors deal with numerical methods of the shape optimization. In order to make them efficient they linearize the problem using the deformations as representatives of shapes. They use the standard shape calculus and show the relation to standard vector space algorithms leading to a novel interpretation of shape Newton methods. A standard shape optimization algorithms applies the shape deformation \(T_t\) from the shape derivative in order to generate iterated shapes via \(\Omega^{k+1}=T_{t^k}^{V^k}(\Omega^k)\), where \(\{V^k\}\) are sufficiently smooth vector fields related to shape derivatives.
The main achievements are the analysis and the realization of standard descending shape optimization algorithms from the point of view of iterating over deformations rather than geometries.
The main achievements are the analysis and the realization of standard descending shape optimization algorithms from the point of view of iterating over deformations rather than geometries.
Reviewer: Igor Bock (Bratislava)
MSC:
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 49M15 | Newton-type methods |
| 49M41 | PDE constrained optimization (numerical aspects) |
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