Numerical solution of a nonlinear eigenvalue problem arising in optimal insulation

TL;DR

This paper develops a numerical method to solve a complex nonlinear eigenvalue problem related to optimal heat insulation, revealing symmetry-breaking phenomena and insights into optimal film geometries.

Contribution

It introduces a discretization and iterative solution approach for a nonlocal eigenvalue problem in optimal insulation, including numerical experiments and shape optimization insights.

Findings

Symmetry breaking occurs for small insulation masses.

Convex bodies with one axis of symmetry have favorable insulation properties.

Numerical experiments confirm theoretical predictions.

Abstract

The optimal insulation of a heat conducting body by a thin film of variable thickness can be formulated as a nondifferentiable, nonlocal eigenvalue problem. The discretization and iterative solution for the reliable computation of corresponding eigenfunctions that determine the optimal layer thickness are addressed. Corresponding numerical experiments confirm the theoretical observation that a symmetry breaking occurs for the case of small available insulation masses and provide insight in the geometry of optimal films. An experimental shape optimization indicates that convex bodies with one axis of symmetry have favorable insulation properties.