@misc{BartelsWachsmuth, author = {Bartels, S{\"o}ren and Wachsmuth, Gerd}, title = {Numerical Approximation of Optimal Convex Shapes}, series = {SIAM Journal on Scientific Computing}, volume = {42}, journal = {SIAM Journal on Scientific Computing}, number = {2}, issn = {1095-7197}, doi = {10.1137/19M1256853}, pages = {A1226 -- A1244}, abstract = {This article investigates the numerical approximation of shape optimization problems with PDE constraint on classes of convex domains. The convexity constraint provides a compactness property which implies well posedness of the problem. Moreover, we prove the convergence of discretizations in two-dimensional situations. A numerical algorithm is devised that iteratively solves the discrete formulation. Numerical experiments show that optimal convex shapes are generally nonsmooth and that three-dimensional problems require an appropriate relaxation of the convexity condition.}, language = {en} } @misc{KellerBartelsWachsmuth, author = {Keller, Hedwig and Bartels, S{\"o}ren and Wachsmuth, Gerd}, title = {Numerical Approximation of Optimal Convex and Rotationally Symmetric Shapes for an Eigenvalue Problem arising in Optimal Insulation}, pages = {34}, abstract = {We are interested in the optimization of convex domains under a PDE constraint. Due to the difficulties of approximating convex domains in R^3, the restriction to rotationally symmetric domains is used to reduce shape optimization problems to a two-dimensional setting. For the optimization of an eigenvalue arising in a problem of optimal insulation, the existence of an optimal domain is proven. An algorithm is proposed that can be applied to general shape optimization problems under the geometric constraints of convexity and rotational symmetry. The approximated optimal domains for the eigenvalue problem in optimal insulation are discussed.}, language = {en} } @misc{BartelsKellerWachsmuth, author = {Bartels, S{\"o}ren and Keller, Hedwig and Wachsmuth, Gerd}, title = {Numerical Approximation of Optimal Convex Shapes in ℝ³}, series = {arXiv}, journal = {arXiv}, doi = {10.48550/arXiv.2311.13386}, pages = {1 -- 17}, language = {en} } @misc{KellerBartelsWachsmuth, author = {Keller, Hedwig and Bartels, S{\"o}ren and Wachsmuth, Gerd}, title = {Numerical Approximation of Optimal Convex and Rotationally Symmetric Shapes for an Eigenvalue Problem arising in Optimal Insulation}, series = {Computers \& Mathematics with Applications}, volume = {119}, journal = {Computers \& Mathematics with Applications}, issn = {1873-7668}, doi = {10.1016/j.camwa.2022.05.026}, pages = {327 -- 339}, language = {en} }