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Global convergence of $W^{1,\infty}$-steepest descent for PDE constrained shape optimisation with semilinear elliptic equations in function space

Klaus Deckelnick, Philip J. Herbert, Michael Hinze

TL;DR

It is proved global convergence in function space for the steepest descent method in shape optimisation with semilinear elliptic partial differential equations and a conditional convergence result for the resulting shapes in two space dimensions is proved.

Abstract

We prove global convergence in function space for the steepest descent method in shape optimisation with semilinear elliptic partial differential equations. Steepest descent is realized in the Lipschitz topology. In addition, we prove a conditional convergence result for the resulting shapes in two space dimensions.

Global convergence of $W^{1,\infty}$-steepest descent for PDE constrained shape optimisation with semilinear elliptic equations in function space

TL;DR

It is proved global convergence in function space for the steepest descent method in shape optimisation with semilinear elliptic partial differential equations and a conditional convergence result for the resulting shapes in two space dimensions is proved.

Abstract

We prove global convergence in function space for the steepest descent method in shape optimisation with semilinear elliptic partial differential equations. Steepest descent is realized in the Lipschitz topology. In addition, we prove a conditional convergence result for the resulting shapes in two space dimensions.
Paper Structure (8 sections, 8 theorems, 87 equations, 2 figures)

This paper contains 8 sections, 8 theorems, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Setting of the problem
  4. Descent algorithm
  5. Convergence of the descent algorithm
  6. Global convergence
  7. Convergence to a stationary shape in two dimensions
  8. Numerical experiment

Key Result

Lemma 2.1

Let $\Omega \in \mathcal{S}$. Then the problems state and adj admit unique solutions $u\in H^1_0(\Omega)\cap L^\infty(\Omega)$, and $p \in H^1_0(\Omega)$, respectively. Moreover, there exists a constant $c^\star$, such that The constant $c^*$ only depends on $d,D, f,j$ and $g$, but is independent of $\Omega$. Furthermore, we think of $u$ and $p$ as being extended by zero to $D$.

Figures (2)

  • Figure 1: Selected iterates of Algorithm \ref{['alg:Steepest']} when starting with $\Omega^0 = (-0.75,0.75)^2$ (top row), and with $\Omega^0 =(-1,1)^2$ (bottom row). Black areas indicate $\Omega$. In the top row the domain together with the grid on refinement level 1 is shown. In the bottom row we display the domains together with the grids on refinement levels 1-4 (from left to right).
  • Figure 2: The evolution of $\|D\Phi_k\|_{L^\infty(D)}$ with the iteration counter $k$ for $\Omega_0 = (-1,1)^2$ (non-degenerate case, left) and $\Omega_0 = (-0.75,0.75)^2$ (degenerate case, right). In the non-degenerate case, we see that $\|D\Phi_k\|_{L^\infty(D)}$ levels out. In the degenerate case, it is also seen that $\|D\Phi_k\|_{L^\infty(D)}$ levels out or the experiment stops due to the stopping criterion, or both.

Theorems & Definitions (14)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Definition 3.1
  • Theorem 3.3
  • ...and 4 more