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Numerical Shape Optimization Among Convex Sets

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Abstract

This article proposes a new discrete framework for approximating solutions to two dimensional shape optimization problems under convexity constraints. The numerical method, based on the support function or the gauge function, is guaranteed to generate discrete convex shapes and is easily implementable using standard optimization software. The framework can handle various objective functions ranging from geometric quantities to functionals depending on partial differential equations. Width or diameter constraints are handled using the support function. Functionals depending on a convex body and its polar body can be handled using a unified framework.

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Acknowledgements

The author thanks the authors of [5] for sharing information about the numerical optimizers from their work. The author was partially supported by the ANR Shapo (ANR-18-CE40- 0013) programme. The anonymous reviewer is warmly thanked for the careful read of the manuscript and for the suggestions that helped improve its quality.

Funding

The author was partially supported by the ANR Shapo (ANR-18-CE40-0013) programme.

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  1. CMAP, Ecole Polytechnique, route de Saclay, 91128, Palaiseau, France

    Beniamin Bogosel

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  1. Beniamin Bogosel
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Correspondence to Beniamin Bogosel.

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Code for Symbolic Computations

Code for Symbolic Computations

In order to avoid writing the tedious computations leading to the formulas (14), (16), (29), scripts performing the equivalent symbolic computations in Mathematica are provided below.

Mathematica script for the computation (14):

figure a

Mathematica script for the computation (16):

figure b

Mathematica script for the computation (29):

figure c

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Bogosel, B. Numerical Shape Optimization Among Convex Sets. Appl Math Optim 87, 1 (2023). https://doi.org/10.1007/s00245-022-09920-w

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