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Continuation methods for higher-order topology optimization

P. Gangl, M. Winkler

TL;DR

This work tackles nonconvex density-based topology optimization with PDE constraints and box-density bounds by proposing a barrier-augmented homotopy continuation strategy to globally converge to a stationary point. It combines a global homotopy that traces a zero-curve with a primal-dual barrier formulation to implicitly enforce $0 \leq \rho \leq 1$ and to regularize the optimization problem. The PDE-extension embeds the discretized Lagrangian into the barrier-homotopy map, yielding a robust Newton solve along the continuation path. Numerical experiments on a linear-elasticity compliance problem demonstrate feasibility preservation and convergence to a locally optimal design from an unbiased initial guess, while highlighting parameter sensitivities and potential extensions to multi-objective Pareto optimization.

Abstract

We aim to solve a topology optimization problem where the distribution of material in the design domain is represented by a density function. To obtain candidates for local minima, we want to solve the first order optimality system via Newton's method. This requires the initial guess to be sufficiently close to the a priori unknown solution. Introducing a stepsize rule often allows for less restrictions on the initial guess while still preserving convergence. In topology optimization one typically encounters nonconvex problems where this approach might fail. We therefore opt for a homotopy (continuation) approach which is based on solving a sequence of parametrized problems to approach the solution of the original problem. In the density based framework the values of the design variable are constrained by 0 from below and 1 from above. Coupling the homotopy method with a barrier strategy enforces these constraints to be satisified. The numerical results for a PDE-constrained compliance minimization problem demonstrate that this combined approach maintains feasibility of the density function and converges to a (candidate for a) locally optimal design without a priori knowledge of the solution.

Continuation methods for higher-order topology optimization

TL;DR

This work tackles nonconvex density-based topology optimization with PDE constraints and box-density bounds by proposing a barrier-augmented homotopy continuation strategy to globally converge to a stationary point. It combines a global homotopy that traces a zero-curve with a primal-dual barrier formulation to implicitly enforce and to regularize the optimization problem. The PDE-extension embeds the discretized Lagrangian into the barrier-homotopy map, yielding a robust Newton solve along the continuation path. Numerical experiments on a linear-elasticity compliance problem demonstrate feasibility preservation and convergence to a locally optimal design from an unbiased initial guess, while highlighting parameter sensitivities and potential extensions to multi-objective Pareto optimization.

Abstract

We aim to solve a topology optimization problem where the distribution of material in the design domain is represented by a density function. To obtain candidates for local minima, we want to solve the first order optimality system via Newton's method. This requires the initial guess to be sufficiently close to the a priori unknown solution. Introducing a stepsize rule often allows for less restrictions on the initial guess while still preserving convergence. In topology optimization one typically encounters nonconvex problems where this approach might fail. We therefore opt for a homotopy (continuation) approach which is based on solving a sequence of parametrized problems to approach the solution of the original problem. In the density based framework the values of the design variable are constrained by 0 from below and 1 from above. Coupling the homotopy method with a barrier strategy enforces these constraints to be satisified. The numerical results for a PDE-constrained compliance minimization problem demonstrate that this combined approach maintains feasibility of the density function and converges to a (candidate for a) locally optimal design without a priori knowledge of the solution.
Paper Structure (8 sections, 22 equations, 33 figures, 3 algorithms)

This paper contains 8 sections, 22 equations, 33 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Model problem
  3. Homotopy method
  4. Barrier method
  5. A barrier homotopy approach
  6. Extension to PDE constraints
  7. Numerical Results
  8. Conclusion and Outlook

Figures (33)

  • Figure 1: The homotopy map for $t=0, 0.4, 0.65$ and $0.9$ in contrast to the graph of $F$
  • Figure 2: Visualization of tracing the zero curve of $H$
  • Figure 3: The barrier function for different parameters $\mu=2.9,1.1,0.4,0.1$
  • Figure 4: The design domain $\Omega=[0,2.4] \times [0,0.8]$
  • Figure 5: $t=0.000000$
  • ...and 28 more figures

Theorems & Definitions (6)

  • Example 3.1
  • Remark 3.2
  • Example 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 5.1