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Topology optimization for microfluidic mixers by a phase field method

Zongyuan Liu, Jiajie Li, Shengfeng Zhu

TL;DR

This work introduces a multiphysics phase-field topology-optimization framework for microfluidic mixers by deriving a modified Ginzburg–Landau free-energy functional and solving an Allen–Cahn gradient flow to evolve the design variable φ. The method couples incompressible Navier–Stokes flow with Poisson–Boltzmann electrostatics and convection–diffusion, enabling design in fluid, solid, and porous regimes via α(φ) and ε(φ). Sensitivity analysis with adjoint equations yields a gradient-based scheme, and a higher-order interpolation term g(φ) directs optimization toward physically meaningful interfaces; a dissipation-based term is added to mitigate inlet blockage in 3D. Numerical experiments in 2D and 3D, including electrokinetically enhanced mixing, demonstrate stable energy decay, monotone convergence, and effective mixing with controlled solid-volume constraints. The approach offers a robust framework for complex multiphysics micromixer design with potential extensions to broader multiphysics topology optimization problems.

Abstract

We investigate multi-physical topology optimization for microfluidic mixers employing the phase-field model. The optimization problem is formulated using a modified Ginzburg-Landau free energy functional. To eliminate fluid blockage in microfluidic mixers, we incorporate the coupled Navier-Stokes, convection-diffusion and Poisson-Boltzmann equations. An Allen-Cahn type gradient flow method is proposed based on sensitivity analysis. The algorithm is validated for its computational effectiveness through numerical simulations of benchmark problems in 2D and 3D.

Topology optimization for microfluidic mixers by a phase field method

TL;DR

This work introduces a multiphysics phase-field topology-optimization framework for microfluidic mixers by deriving a modified Ginzburg–Landau free-energy functional and solving an Allen–Cahn gradient flow to evolve the design variable φ. The method couples incompressible Navier–Stokes flow with Poisson–Boltzmann electrostatics and convection–diffusion, enabling design in fluid, solid, and porous regimes via α(φ) and ε(φ). Sensitivity analysis with adjoint equations yields a gradient-based scheme, and a higher-order interpolation term g(φ) directs optimization toward physically meaningful interfaces; a dissipation-based term is added to mitigate inlet blockage in 3D. Numerical experiments in 2D and 3D, including electrokinetically enhanced mixing, demonstrate stable energy decay, monotone convergence, and effective mixing with controlled solid-volume constraints. The approach offers a robust framework for complex multiphysics micromixer design with potential extensions to broader multiphysics topology optimization problems.

Abstract

We investigate multi-physical topology optimization for microfluidic mixers employing the phase-field model. The optimization problem is formulated using a modified Ginzburg-Landau free energy functional. To eliminate fluid blockage in microfluidic mixers, we incorporate the coupled Navier-Stokes, convection-diffusion and Poisson-Boltzmann equations. An Allen-Cahn type gradient flow method is proposed based on sensitivity analysis. The algorithm is validated for its computational effectiveness through numerical simulations of benchmark problems in 2D and 3D.
Paper Structure (14 sections, 1 theorem, 39 equations, 17 figures, 1 algorithm)

This paper contains 14 sections, 1 theorem, 39 equations, 17 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Topology optimization in microfluidic mixer
  3. Governing equations
  4. Phase-field models
  5. Phase field method
  6. Sensitivity analysis
  7. Gradient flow method
  8. Numerical simulation
  9. Energy dissipation problem
  10. Microfluidic mixer problem
  11. 2D mixer problem
  12. 3D mixer problem
  13. Conclusion
  14. Derivation of the Ginzburg–Landau free energy from the Ising Model

Key Result

Lemma 3.1

Let $\phi\in \mathcal{O}_{\rm ad}$ and $(\bm u, p, \psi, c)\in \textbf{H}^1_d(\Omega) \times L^2_0(\Omega)\times H^1_\psi(\Omega) \times H^1_c(\Omega)$ be the solution of the coupled Navier-Stokes equations weakNSPBC. Then, the weak formulation of the adjoint system is given by: Find $(\hbox{\boldma

Figures (17)

  • Figure 1: The design domain consisting of fluid $\Omega_1$ and solid $\Omega_1$ regions.
  • Figure 2: Ginzburg-Landau free energy under different conditions
  • Figure 3: Example 1 for the energy dissipation problem
  • Figure 4: Results for the energy dissipation problem: Example 2.
  • Figure 5: Results for the energy dissipation problem: Example 3.
  • ...and 12 more figures

Theorems & Definitions (2)

  • Lemma 3.1
  • proof