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Energy stable gradient flow schemes for shape and topology optimization in Navier-Stokes flows

Jiajie Li, Shengfeng Zhu

TL;DR

The paper addresses topology optimization of incompressible Navier–Stokes flows using a phase-field formulation and develops stabilized, energy-dissipative gradient-flow schemes of Allen–Cahn and Cahn–Hilliard types. It demonstrates unconditional energy stability in both continuous and discrete settings by incorporating stabilization terms, and provides a finite-element realization with a robust NS–adjoint–gradient-flow coupling. The main contributions are the derivation of energy-dissipating gradient flows for a phase-field topology optimization problem under nonlinear PDE constraints, and a complete numerical framework (MINI elements, BV projection, and stable semi-implicit schemes) validated by 2D and 3D CFD experiments. The work offers a practical and scalable approach to energy-stable topology optimization in fluid flows, with potential extensions to other nonlinear PDE-constrained design problems.

Abstract

We study topology optimization governed by the incompressible Navier-Stokes flows using a phase field model. Novel stabilized semi-implicit schemes for the gradient flows of Allen-Cahn and Cahn-Hilliard types are proposed for solving the resulting optimal control problem. Unconditional energy stability is shown for the gradient flow schemes in continuous and discrete spaces. Numerical experiments of computational fluid dynamics in 2d and 3d show the effectiveness and robustness of the optimization algorithms proposed.

Energy stable gradient flow schemes for shape and topology optimization in Navier-Stokes flows

TL;DR

The paper addresses topology optimization of incompressible Navier–Stokes flows using a phase-field formulation and develops stabilized, energy-dissipative gradient-flow schemes of Allen–Cahn and Cahn–Hilliard types. It demonstrates unconditional energy stability in both continuous and discrete settings by incorporating stabilization terms, and provides a finite-element realization with a robust NS–adjoint–gradient-flow coupling. The main contributions are the derivation of energy-dissipating gradient flows for a phase-field topology optimization problem under nonlinear PDE constraints, and a complete numerical framework (MINI elements, BV projection, and stable semi-implicit schemes) validated by 2D and 3D CFD experiments. The work offers a practical and scalable approach to energy-stable topology optimization in fluid flows, with potential extensions to other nonlinear PDE-constrained design problems.

Abstract

We study topology optimization governed by the incompressible Navier-Stokes flows using a phase field model. Novel stabilized semi-implicit schemes for the gradient flows of Allen-Cahn and Cahn-Hilliard types are proposed for solving the resulting optimal control problem. Unconditional energy stability is shown for the gradient flow schemes in continuous and discrete spaces. Numerical experiments of computational fluid dynamics in 2d and 3d show the effectiveness and robustness of the optimization algorithms proposed.
Paper Structure (9 sections, 11 theorems, 89 equations, 14 figures, 2 tables, 2 algorithms)

This paper contains 9 sections, 11 theorems, 89 equations, 14 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model problem
  3. Governing equations
  4. Phase-field model
  5. Gradient flow
  6. Numerical realization
  7. Finite element discretization
  8. Numerical experiments
  9. Conclusion

Key Result

Lemma 2.1

The trilinear form $b(\cdot, \cdot, \cdot)$ is well-defined and continuous in the space $\textbf{H}^1_0(\Omega)\times \textbf{H}^1(\Omega)\times\textbf{H}^1_0(\Omega)$. The following estimate holds where with $|\Omega|$ being the Lebesgue measure of $\Omega$. Furthermore, the following properties hold:

Figures (14)

  • Figure 1: Illustrations of design domain with subdomains represented implicitly by phase field function (left) and phases with diffuse layer (right).
  • Figure 2: Design domains for different Examples in 2d space.
  • Figure 3: Convergence histories of energy by Algorithm 1 ($\mu=0.01, 0.005$ left and $0.1$ middle) and volume errors (right) for Example 1.
  • Figure 4: Optimized distributions (line 1) and velocity fields (line 2) for Example 1: $\mu=0.1$, $0.01$, and $0.005$ from left to right.
  • Figure 5: Optimal distributions by Algorithm \ref{['algCH']} left ($\mu=0.1$), middle ($\mu=0.01$) and $\tau =0.2$ right.
  • ...and 9 more figures

Theorems & Definitions (23)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • ...and 13 more