When
12:30 – 1:30 p.m., April 16, 2026
Where
Speaker: Keegan Kirk, George Mason University
Title: Optimal Thermal Insulation on Non-Smooth Domains: Analysis and Algorithms
Abstract: Given a fixed amount of insulating material, how should one coat a heat-conducting body to optimize its thermal performance? For smooth bodies, Buttazzo and collaborators gave a rigorous asymptotic treatment in the 1980s. Starting from a coupled bulk–layer heat transfer model with a thin coating of variable thickness, they derived, via Γ-convergence as the thickness tends to zero, an effective boundary energy governing the optimal distribution of insulation. Their analysis requires Lipschitz regularity of the outward unit normal, an assumption that fails for nonsmooth geometries of engineering interest with edges, corners, and other singular features. We extend the thin-insulation framework to Lipschitz domains with piecewise-flat boundary by replacing the outward normal with a globally transversal Lipschitz vector field that remains well defined across edges and corners. The resulting Γ-limit is a nonsmooth, nonlocal convex variational problem whose boundary term encodes the type of heat transfer. We also study the Fenchel–Rockafellar dual formulation, which admits a natural interpretation as an optimal heat flux problem and provides complementary analytical and computational perspectives. Building on these formulations, we develop finite element algorithms for the numerical solution of the optimal insulation problem. Numerical examples verify the convergence theory and illustrate how optimal insulation distributions respond to geometry, boundary conditions, and the heat-loss mechanism.