2024/10/21 by Sören Bartels, Bartels, Sören, Giuseppe Buttazzo +3 · 2 citations
Decision Sciences · Physics and Astronomy · #Electromagnetic Scattering and Analysis #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.2410.16050
openalex publication_date 2024/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
An eigenvalue problem arising in optimal insulation related to the minimization of the heat decay rate of an insulated body is adapted to enforce a positive lower bound imposed on the distribution of insulating material. We prove the existence of optimal domains among a class of convex shapes and propose a numerical scheme to approximate the eigenvalue. The stability of the shape optimization among convex, bounded domains in ℝ3 is proven for an approximation with polyhedral domains under a non-conformal convexity constraint. We prove that on the ball, symmetry breaking of the optimal insulation can be expected in general. To observe how the lower bound affects the breaking of symmetry in the optimal insulation and the shape optimization, the eigenvalue and optimal domains are approximated for several values of mass m and lower bounds ℓmin≥0. The numerical experiments suggest, that in general symmetry breaking still arises, unless m is close to a critical value m0, and ℓmin large enough such that almost all of the mass m is fixed through the lower bound. For ℓmin=0, the numerical results are consistent with previous numerical experiments on shape optimization restricted to rotationally symmetric, convex domains.