Mathematics Colloquium: How to insulate optimally

Speaker:  Keegan Kirk, GMU

Title: How to insulate optimally

Abstract:   Given a fixed amount of insulating material, how should one coat a heat-conducting body to optimize its insulating properties? A rigorous asymptotic analysis reveals this problem can be cast as a convex variational problem with various non-smooth boundary terms depending on the mechanism of heat transfer. As these boundary terms are difficult to treat numerically, we consider equivalent (Fenchel) dual variational formulations more amenable to discretization. We propose numerical schemes to solve these dual formulations on the basis of a discrete duality theory inherited by the Raviart-Thomas and Crouzeix-Raviart finite elements, and show that the solution of the original primal problem can be reconstructed locally from the discrete dual solution. We discuss the a posteriori and a priori error analysis of our scheme, derive a posteriori estimators based on convex optimality conditions, and present numerical examples to verify theory. As applications, we consider the design of an optimally insulated home and an optimally insulated spacecraft heat shield.

Time:  Friday, August 29, 3:30pm – 4:20pm

Place:  Exploratory Hall, room 4106

Zoom and In-person