arXiv subjects

Paul Hüttl

Publications and source records attributed to Paul Hüttl.

4 recordsLinked to original sources

arXiv2024

A phase-field version of the Faber--Krahn theorem

We investigate a phase-field version of the Faber--Krahn theorem based on a phase-field optimization problem introduced in Garcke et al. [ESAIM Control Optim. Calc. Var. 29 (2023), Paper No. 10] formulated for the principal eigenvalue of the Dirichlet--Laplacian. The shape, that is to be optimized, is represented by a phase-field function mapping into the interval $[0,1]$. We show that any minimizer of our problem is a radially symmetric-decreasing phase-field attaining values close to $0$ and $1$ except for a thin transition layer whose thickness is of order $\varepsilon>0$. Our proof relies on radially symmetric-decreasing rearrangements and corresponding functional inequalities. Moreover, we provide a $Γ$-convergence result which allows us to recover a variant of the Faber--Krahn theorem for sets of finite perimeter in the sharp interface limit.

math.AP↗

arXiv2023

Sharp-interface limit of a multi-phase spectral shape optimization problem for elastic structures

We consider an optimization problem for the eigenvalues of a multi-material elastic structure that was previously introduced by Garcke et al. [Adv. Nonlinear Anal. 11 (2022), no. 1, 159--197]. There, the elastic structure is represented by a vector-valued phase-field variable, and a corresponding optimality system consisting of a state equation and a gradient inequality was derived. In the present paper, we pass to the sharp-interface limit in this optimality system by the technique of formally matched asymptotics. Therefore, we derive suitable Lagrange multipliers to formulate the gradient inequality as a pointwise equality. Afterwards, we introduce inner and outer expansions, relate them by suitable matching conditions and formally pass to the sharp-interface limit by comparing the leading order terms in the state equation and in the gradient equality. Furthermore, the relation between these formally derived first-order conditions and results of Allaire and Jouve [Comput. Methods Appl. Mech. Engrg., 194 (2005), pp. 3269--3290] obtained in the framework of classical shape calculus is discussed. Eventually, we provide numerical simulations for a variety of examples. In particular, we illustrate the sharp-interface limit and also consider a joint optimization problem of simultaneous compliance and eigenvalue optimization.

math.OC↗

arXiv2023

Phase-Field Methods for Spectral Shape and Topology Optimization

We optimize a selection of eigenvalues of the Laplace operator with Dirichlet or Neumann boundary conditions by adjusting the shape of the domain on which the eigenvalue problem is considered. Here, a phase-field function is used to represent the shapes over which we minimize. The idea behind this method is to modify the Laplace operator by introducing phase-field dependent coefficients in order to extend the eigenvalue problem on a fixed design domain containing all admissible shapes. The resulting shape and topology optimization problem can then be formulated as an optimal control problem with PDE constraints in which the phase-field function acts as the control. For this optimal control problem, we establish first-order necessary optimality conditions and we rigorously derive its sharp interface limit. Eventually, we present and discuss several numerical simulations for our optimization problem.

math.OC↗

arXiv2020

Shape and Topology Optimization involving the eigenvalues of an elastic structure: A multi-phase-field approach

A cost functional involving the eigenvalues of an elastic structure, that is described by a multi-phase-field equation, is optimized. This allows us to handle topology changes and multiple materials. We prove continuity and differentiability of the eigenvalues and we establish the existence of a global minimizer to our optimization problem. We further derive first-order necessary optimality conditions for local minimizers. Moreover, an optimization problem combining eigenvalue and compliance optimization is also discussed.

math.OC↗