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

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 the Rayleigh–Bénard instability with a melting boundary, extending previous adjoint-based Stefan problem formulations—which addressed diffusion-dominated configurations—to a convection-driven regime. 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.