Please use this identifier to cite or link to this item: https://hdl.handle.net/10419/338602 
Year of Publication: 
2025
Citation: 
[Journal:] Computational Optimization and Applications [ISSN:] 1573-2894 [Volume:] 93 [Issue:] 3 [Publisher:] Springer US [Place:] New York, NY [Year:] 2025 [Pages:] 1305-1353
Publisher: 
Springer US, New York, NY
Abstract: 
We introduce a framework for shape optimization in the class of simple polytopes in arbitrary dimensions. We use a parametrization via hyperplanes and provide conditions for stability of simple polytopes under small perturbations of the hyperplanes. Next, we construct a mapping based on barycentric coordinates between the perturbed and reference polytopes. This allows us to use a transport theorem to compute the derivative of integrals defined on the polytope and perform the sensitivity analysis of shape functionals. This framework is applied to a numerical study of the polygonal and polyhedral Saint–Venant inequality. We solve an optimization problem that maximizes the torsion functional over polytopes with a fixed number of facets and equal volume. Using the previously defined parametrization via hyperplanes, the problem is reduced to a finite-dimensional optimization problem. A Proximal-Perturbed Lagrangian functional is employed to handle the volume constraint. In two dimensions, our numerical results support the conjecture stating that the solution is a regular polygon. In three dimensions, the numerical solutions converge towards either regular polyhedra or irregular polyhedra, depending on the number of faces.
Subjects: 
Torsional rigidity
Shape optimization
Nonsmooth shapes
Polytopes
Persistent Identifier of the first edition: 
Creative Commons License: 
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Document Type: 
Article
Document Version: 
Published Version

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