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Pseudo-concave optimization of the first eigenvalue of elliptic operators with application to topology optimization by homogenization

Akatsuki Nishioka

Abstract

We consider optimization problems of the first eigenvalue of elliptic operators with applications to two-phase optimal design problems (also known as topology optimization problems) of conductivity and elasticity relaxed by homogenization. Under certain assumptions, we show that the first eigenvalue is a pseudo-concave function. Due to pseudo-concavity, every stationary point is a global maximizer, and there exists a global minimizer that is an extreme point (corresponding to a 0-1 solution in optimal design problems). We perform simple numerical experiments on optimal design problems to demonstrate that global optimal solutions or 0-1 solutions can be obtained by a simple gradient method.

Pseudo-concave optimization of the first eigenvalue of elliptic operators with application to topology optimization by homogenization

Abstract

We consider optimization problems of the first eigenvalue of elliptic operators with applications to two-phase optimal design problems (also known as topology optimization problems) of conductivity and elasticity relaxed by homogenization. Under certain assumptions, we show that the first eigenvalue is a pseudo-concave function. Due to pseudo-concavity, every stationary point is a global maximizer, and there exists a global minimizer that is an extreme point (corresponding to a 0-1 solution in optimal design problems). We perform simple numerical experiments on optimal design problems to demonstrate that global optimal solutions or 0-1 solutions can be obtained by a simple gradient method.

Paper Structure

This paper contains 23 sections, 11 theorems, 43 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related work
  4. Contribution
  5. Organization
  6. Notation
  7. Pseudo-concavity of the first eigenvalue
  8. Preliminary results on pseudo-concave functions
  9. Main theoretical results
  10. Applications: optimal design by homogenization
  11. Preliminary results on optimal design by homogenization
  12. Conductivity
  13. Elasticity
  14. Application to conductivity setting
  15. Density both in the denominator and numerator
  16. ...and 8 more sections

Key Result

Proposition 1

Let $X$ be a Banach space, $S\subseteq X$ be a nonempty closed convex set, and $f:S\to\mathbb{R}$ be a pseudo-concave function. Every Clarke stationary point of a pseudo-concave maximization problem is a global maximizer.

Figures (6)

  • Figure 1: An example of a pseudo-concave function that is not concave.
  • Figure 2: Numerical solutions to the pseudo-concave maximization problem \ref{['p_relaxed']}, which are supposed to be globally optimal.
  • Figure 5: Numerical solutions to the concave maximization problem \ref{['p_num_rel_max']}, which are supposed to be globally optimal. We obtained almost the same designs as those of casado22.
  • Figure 8: Numerical solutions to the pseudo-concave minimization problem \ref{['p_den_rel_min']}, which are almost a classical 0-1 designs. Solution (A) is consistent with Proposition \ref{['prp_ball']}.
  • Figure 11: A numerical solution (B) to the pseudo-concave minimization problem \ref{['p_den_rel_min']} with a non-uniform initial design (A). We obtain the same solution as Figure \ref{['fig_den_min']}(B)
  • ...and 1 more figures

Theorems & Definitions (24)

  • Definition 1: generalized directional derivative
  • Definition 2: Clarke subdifferential
  • Remark 1
  • Definition 3: pseudo-concave function penot97
  • Definition 4: Clarke stationary point
  • Proposition 1: property of solutions to pseudo-concave maximization
  • proof
  • Proposition 2: property of solutions to pseudo-concave minimization
  • proof
  • Proposition 3: Clarke subdifferential of the first eigenvalue cox95
  • ...and 14 more