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Shape Optimization with Nonlinear Conjugate Gradient Methods

Sebastian Blauth

TL;DR

This work formulates and analyzes nonlinear conjugate gradient (NCG) methods for PDE-constrained shape optimization within a Riemannian framework. By leveraging a Steklov-Poincaré-type metric $g^S_\Gamma$ and the gradient deformation $\mathcal{G}$, it derives five NCG variants (FR, PR, HS, DY, HZ) and demonstrates their efficacy through two benchmark problems: a 2D Poisson-shape optimization and a 3D Navier–Stokes pipe optimization. The numerical results show that NCG methods markedly improve convergence and reduce gradient norms compared to gradient descent, and are competitive with L-BFGS while employing substantially less memory. These findings suggest NCG as a practical, memory-efficient option for large-scale industrial shape optimization tasks, supported by the cashocs implementation and rigorous Riemannian formulation with boundary-to-domain gradient relations $dJ(\Omega)[\mathcal{V}]$ and $\gamma=S^p_\Gamma g$.

Abstract

In this chapter, we investigate recently proposed nonlinear conjugate gradient (NCG) methods for shape optimization problems. We briefly introduce the methods as well as the corresponding theoretical background and investigate their performance numerically. The obtained results confirm that the NCG methods are efficient and attractive solution algorithms for shape optimization problems.

Shape Optimization with Nonlinear Conjugate Gradient Methods

TL;DR

This work formulates and analyzes nonlinear conjugate gradient (NCG) methods for PDE-constrained shape optimization within a Riemannian framework. By leveraging a Steklov-Poincaré-type metric and the gradient deformation , it derives five NCG variants (FR, PR, HS, DY, HZ) and demonstrates their efficacy through two benchmark problems: a 2D Poisson-shape optimization and a 3D Navier–Stokes pipe optimization. The numerical results show that NCG methods markedly improve convergence and reduce gradient norms compared to gradient descent, and are competitive with L-BFGS while employing substantially less memory. These findings suggest NCG as a practical, memory-efficient option for large-scale industrial shape optimization tasks, supported by the cashocs implementation and rigorous Riemannian formulation with boundary-to-domain gradient relations and .

Abstract

In this chapter, we investigate recently proposed nonlinear conjugate gradient (NCG) methods for shape optimization problems. We briefly introduce the methods as well as the corresponding theoretical background and investigate their performance numerically. The obtained results confirm that the NCG methods are efficient and attractive solution algorithms for shape optimization problems.
Paper Structure (10 sections, 1 theorem, 25 equations, 5 figures)

This paper contains 10 sections, 1 theorem, 25 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Theoretical Background
  3. Fundamentals of Shape Optimization
  4. Shape Calculus
  5. A Riemannian View on Shape Optimization and Steklov-Poincaré-Type Metrics
  6. Nonlinear CG Methods for Shape Optimization
  7. Numerical Examples
  8. Shape Optimization with a Poisson Equation
  9. Shape Optimization of Energy Dissipation in a Pipe
  10. Conclusions

Key Result

theorem 1

Let J be a shape functional which is shape differentiable at some $\Omega\subset D$ and let $\Gamma = \partial \Omega$ be compact. Further, let $k \geq 0$ be the smallest integer such that $dJ(\Omega) \colon C^\infty_0(D;\mathbb{R}^d) \to \mathbb{R};\ \mathcal{V} \mapsto dJ(\Omega)[\mathcal{V}]$ is where $n$ is the outer unit normal vector on $\Gamma$. In particular, if $g\in L^1(\Gamma)$, it hol

Figures (5)

  • Figure 1: History of the optimization methods for problem \ref{['sblauth:eq:poisson']}.
  • Figure 2: Optimized Shapes (blue) compared to the reference solution (orange) for the Poisson problem \ref{['sblauth:eq:poisson']}.
  • Figure 3: Initial and optimized geometries for problem \ref{['sblauth:eq:navier_stokes']}.
  • Figure 4: Velocity magnitude on the initial and optimized geometries, shown as slice through the middle of the geometry.
  • Figure 5: History of the optimization methods for problem \ref{['sblauth:eq:navier_stokes']}.

Theorems & Definitions (3)

  • definition 1
  • theorem 1: Structure Theorem
  • proof