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Adjoint-based optimization of the Rayleigh-Bénard instability with melting boundary

Tomas Fullana, Alejandro Quirós Rodríguez, Vincent Le Chenadec, Taraneh Sayadi

TL;DR

This work develops an adjoint-based gradient optimization for the Rayleigh–Bénard instability with a melting front, introducing a cut-cell/level-set two-phase formulation and an incomplete continuous adjoint that treats the forward velocity as known to avoid a full Navier–Stokes adjoint. The method minimizes a tracking-type cost functional to control the interface shape, validated on two optimization setups and compared against a derivative-free Particle Swarm optimizer. Results show the incomplete adjoint provides gradients sufficiently accurate to steer the front while dramatically reducing the number of function evaluations, achieving near-target interface shapes at a fraction of the computational cost. The approach offers a computationally efficient pathway for shaping melting boundaries in convection-dominated phase-change problems, with clear avenues for extension to more complex geometries and three-dimensional systems.

Abstract

In this work, we propose an adjoint-based optimization procedure to control the onset of the Rayleigh-Bénard instability with a melting front. A novel cut cell method is used to solve the Navier-Stokes equations in the Boussinesq approximation and the convection-diffusion equation in the fluid layer, as well as the heat equation in the solid phase. To track the interface we use the level set method where its evolution is simply governed by an advection equation. An incomplete continuous adjoint problem is then derived by treating the velocity field obtained from the forward problem as a known variable in the adjoint convection-diffusion equation, thereby avoiding the need to solve a Navier-Stokes adjoint in the fluid phase. To the best of our knowledge, this provides the first adjoint-based optimization framework for Rayleigh-Bénard instability with a melting boundary. Two optimization problems, together with a comparison against a derivative-free particle-swarm method, demonstrate that the proposed incomplete adjoint yields gradients accurate enough to control the front shape while reducing the number of expensive function evaluations by about an order of magnitude.

Adjoint-based optimization of the Rayleigh-Bénard instability with melting boundary

TL;DR

This work develops an adjoint-based gradient optimization for the Rayleigh–Bénard instability with a melting front, introducing a cut-cell/level-set two-phase formulation and an incomplete continuous adjoint that treats the forward velocity as known to avoid a full Navier–Stokes adjoint. The method minimizes a tracking-type cost functional to control the interface shape, validated on two optimization setups and compared against a derivative-free Particle Swarm optimizer. Results show the incomplete adjoint provides gradients sufficiently accurate to steer the front while dramatically reducing the number of function evaluations, achieving near-target interface shapes at a fraction of the computational cost. The approach offers a computationally efficient pathway for shaping melting boundaries in convection-dominated phase-change problems, with clear avenues for extension to more complex geometries and three-dimensional systems.

Abstract

In this work, we propose an adjoint-based optimization procedure to control the onset of the Rayleigh-Bénard instability with a melting front. A novel cut cell method is used to solve the Navier-Stokes equations in the Boussinesq approximation and the convection-diffusion equation in the fluid layer, as well as the heat equation in the solid phase. To track the interface we use the level set method where its evolution is simply governed by an advection equation. An incomplete continuous adjoint problem is then derived by treating the velocity field obtained from the forward problem as a known variable in the adjoint convection-diffusion equation, thereby avoiding the need to solve a Navier-Stokes adjoint in the fluid phase. To the best of our knowledge, this provides the first adjoint-based optimization framework for Rayleigh-Bénard instability with a melting boundary. Two optimization problems, together with a comparison against a derivative-free particle-swarm method, demonstrate that the proposed incomplete adjoint yields gradients accurate enough to control the front shape while reducing the number of expensive function evaluations by about an order of magnitude.

Paper Structure

This paper contains 6 sections, 3 theorems, 50 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Governing equations
  3. Adjoint problem
  4. Optimization results
  5. Conclusion
  6. Shape calculus theorems

Key Result

Theorem A

Let $J(\Omega) = \int_\Gamma f ds$ be a boundary integral, the derivative is given by where $V$ is the velocity field and $\kappa$ is the mean curvature of $\Gamma$.

Figures (10)

  • Figure 1: Schematic of the melting boundary problem.
  • Figure 2: Average height and effective Rayleigh number as a function of time for $\operatorname{Ra} = 10^5, \, 8 \times10^4, \, 4 \times 10^4, \, 10^4.$
  • Figure 3: Times series of the temperature field and interface position for the $\operatorname{Ra} = 10^5$ case. The color map is the temperature field in both phases and the interface corresponding to the 0-level set is denoted in red.
  • Figure 4: Times series of the vorticity and interface position for the $\operatorname{Ra} = 10^5$ case. The interface corresponding to the 0-level set is denoted in red.
  • Figure 5: Average height and effective Rayleigh number as a function of time for the desired solution and initial guess.
  • ...and 5 more figures

Theorems & Definitions (3)

  • Theorem A: Derivative of boundary integral
  • Theorem B: Surface transport theorem
  • Corollary B: Integration by Parts in Time on a Moving Surface