Numerical approximation of optimal convex and rotationally symmetric shapes for an eigenvalue problem arising in optimal insulation

Abstract

We are interested in the optimization of convex domains under a PDE constraint. Due to the difficulties of approximating convex domains in ${\mathbb{R}}^3$, 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.

Introduction

Solvability of shape optimization problems relies, among other factors, on strong constraints on the geometry of the admissible domains. Since we minimize over shapes, no topology is readily available. The restriction to classes of convex domains appears attractive, since the compactness results available for convex domains let us avoid more general topological frameworks. For corresponding analytical details we refer to [12], [22], [31], [33], [11] and [14]. Therefore, we restrict the shape optimization to open, convex and bounded domains.

However, numerical approximation of convex domains is difficult in higher dimensions. Indeed, for conformal P1 finite elements we can not guarantee that a convex function can be approximated consistently (cf. [17]), and with simple examples we can show, that the nodal interpolant of a convex function is not necessarily convex itself, for such an example see [1, Figure 2.1]. To approximate convex functions, we need for example higher order conforming finite elements (cf. [32]), a weaker definition for convexity tailored to finite elements (cf. [1]), a geometric approach as in [26] or spherical harmonic decomposition (cf. [3]). Since the approximation of convex domains in ${\mathbb{R}}^3$ has certain similarities to the approximation of convex functions in ${\mathbb{R}}^2$, we expect related difficulties. We focus on the framework of [8], which approximates optimal convex domains in ${\mathbb{R}}^2$ and gives an example for the numerical difficulties in higher dimensions. Therefore, we restrict our domains to a class of rotationally symmetric domains in ${\mathbb{R}}^3$, which allows us to reduce the problem to a two-dimensional setting, for which the boundary is a convex curve.

Moreover, due to the dependence of the degrees of freedom on the dimension, the numerical approximations of the dimensionally reduced problems have a lower computational effort and therefore allow for a higher resolution of the optimization.

We are interested in the optimization under a PDE constraint, in particular in optimizing an eigenvalue occurring in a problem of optimal insulation. For more details in PDE constraint optimization we refer to [23].

A heat conducting body is to be coated by an insulating material in such a way to get the best insulating properties. For an open domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$ we search the optimal distribution of insulating material $\ell :\partial \mathrm{\Omega }\to {\mathbb{R}}_{+}$ of total mass m, i.e. ${\Vert \ell \Vert }_{L^1(\partial \mathrm{\Omega })}=m$. This translates to the non-linear eigenvalue problem

$${\lambda }_m(\mathrm{\Omega })=\min \{J_m(u):=\underset{\mathrm{\Omega }}{\int }|\mathrm{\nabla }u{|}^2\text{d}x+\frac{1}{m}{(\underset{\partial \mathrm{\Omega }}{\int }|u|\text{d}s)}^2:\underset{\mathrm{\Omega }}{\int }|u{|}^2\text{d}x=1\}.$$

From [15] we expect that in general the distribution of insulating material is asymmetric and that the ball is not optimal, in contrast to what we might expect from isoparametric inequalities for eigenvalues of the Laplacian.

The numerical framework for the approximation of the eigenvalue from [7] confirmed the expected asymmetry in two dimensions. Our goal is to perform the shape optimization for convex, rotationally symmetric domains in ${\mathbb{R}}^3$. The numerical experiments in Section 6 confirm, that the constraint to rotational symmetric domains and eigenfunctions still allows for a break in symmetry.

We focus on the existence of an optimal domain and the meaningful numerical approximations provided by the proposed algorithm. We will discuss the stability of the numerical scheme shortly, but a detailed examination lies beyond the scope of this work. In the proof of existence the geometric constraints, especially the convexity, play key roles.

First, in Section 2 we describe the dimensional reduction obtained from the rotational symmetry. Then we consider the shape optimization for the eigenvalue problem arising in the problem of optimal insulation. We prove existence of an optimal domain in Section 3 and derive the two-dimensional problem and its numerical approximation and comment on the stability of the numerical scheme in Section 4. In Section 5 we establish a framework for the numerical approximation of optimal convex domains described in [8] but adjusted for rotational symmetry, which can be applied to different shape optimization problems as well. The numerical experiments are evaluated in Section 6.