Numerical approximation of optimal convex and rotationally symmetric shapes for an eigenvalue problem arising in optimal insulation
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 has certain similarities to the approximation of convex functions in , we expect related difficulties. We focus on the framework of [8], which approximates optimal convex domains in and gives an example for the numerical difficulties in higher dimensions. Therefore, we restrict our domains to a class of rotationally symmetric domains in , 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 we search the optimal distribution of insulating material of total mass m, i.e. . This translates to the non-linear eigenvalue problem
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 . 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.
Section snippets
Rotationally symmetric domains and dimensional reduction
We consider a shape optimization problem that, for a given open and bounded domain , density function j, volume M and state equation , seeks a domain Ω which solves Here, the rotational symmetry is to be understood w.r.t. the -axis. We assume that and j are rotationally symmetric as well. Furthermore, we assume, that the solution of
Existence and numerical approximation of optimal domains
For an eigenvalue problem arising in a model of optimal insulation, we now discuss how to establish existence of an optimal domain.
We look at the non-linear eigenvalue problem arising in optimal insulation and follow [15] closely for this section. We try to surround a heat conducting body with an insulating material to get the best insulating properties, i.e. to minimize the heat decay rate, which is given for the thickness of the insulating layer by the principal eigenvalue of the
Discretized reduced problem
Next, we derive the dimensionally reduced problem and define the numerical scheme and point out technical difficulties in stability. Lastly, we address how this scheme can be applied to other optimization problems
Iterative computation of optimal domains
We next address the iterative numerical approximation of optimal domains. After dimensional reduction and spatial discretization, we obtain the following class of shape optimization problems.
We adopt an approach similar to [8], where the admissible domains are obtained from
First eigenvalue of the Dirichlet Laplacian
For the first eigenvalue of the Dirichlet Laplacian, it is well known that the optimal domain among open, convex shapes of a certain volume is the ball, see [24] and [25]. Therefore we will use this example to validate the shape optimization algorithm, by looking at the results for different initial domains and mesh sizes.
Similar to the eigenvalue problem in Section 3 we can derive the rotationally reduced two-dimensional eigenvalue problem:
Conclusion and discussion
We extended the framework of [8] to a three-dimensional setting with the restriction to rotationally symmetric domains. For an eigenvalue problem arising in optimal insulation, we have proven existence of optimal domains and approximated optimal domains with the proposed algorithm under the geometric restrictions of convexity and rotational symmetry. This expands the experiments from [7] and gives more conclusive results to the optimal domains concerning the eigenvalue in optimal insulation.
Acknowledgements
This work is supported by DFG grants BA2268/4-2 within the Priority Program SPP 1962 (Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization).
References (33)
- et al.
Shape optimization problems with Robin conditions on the free boundary
Ann. Inst. Henri Poincaré, Anal. Non Linéaire
(2016) - et al.
Symmetry breaking for a problem in optimal insulation
J. Math. Pures Appl.
(2017) Shape optimization of stationary Navier–Stokes equation over classes of convex domains
Nonlinear Anal., Theory Methods Appl.
(2009)- et al.
On convex functions and the finite element method
SIAM J. Numer. Anal.
(2009) - et al.
Some applications of weighted norm inequalities to the error analysis of PDE-constrained optimization problems
IMA J. Numer. Anal.
(2018) - et al.
Parametric shape optimization using the support function
- et al.
Polynomial approximation in weighted Sobolev space
C. R. Acad. Bulgare Sci.
(2001) - et al.
Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization
(2014) Numerical Approximation of Partial Differential Equations, vol. 64
(2016)- et al.
Numerical solution of a nonlinear eigenvalue problem arising in optimal insulation
Interfaces Free Bound.
(2019)