Adjoint‐Based Optimal Boundary Control of a Stefan Problem System Fully Coupled with Navier‐Stokes Equations

Problems featuring moving interfaces appear in many applications. They can model solidification and melting of pure materials, crystal growth and other multi-phase problems. The control of the moving interface enables to, for example, influence production processes and, thus, the product material quality.We consider the two-phase Stefan problem that models a solid and a liquid phase separated by the moving interface. In the liquid phase, the heat distribution is characterized by a convection-diffusion equation. The fluid flow in the liquid phase is described by the Navier-Stokes equations which introduces a differential algebraic structure to the system. The interface movement is coupled with the temperature through the Stefan condition, which adds additional algebraic constraints. Our formulation uses a sharp interface representation and we define a quadratic tracking-type cost functional as a target of a control input.We compute an open loop optimal control for the Stefan problem using an adjoint system. For a feedback representation, we linearize the system about the trajectory defined by the open loop control. This results in a linear-quadratic regulator problem, for which we formulate the differential Riccati equation with time varying coefficients. This Riccati equation defines the corresponding feedback gain.Further, we present the feedback formulation that takes into account the structure and the differential algebraic components of the problem. Also, we discuss how the complexities that come, for example, with mesh movements, can be handled in a feedback setting.We aim to control and stabilize a coupled nonlinear system. As exemplary coupled nonlinear system we use a two-phase two-dimensional Stefan problem. We compute a reference solution (x,ũ) with an open loop control as from [1]. Since the open loop controlũ is not robust against perturbations and uncertainties, it should be stabilized. We choose a Riccati feedback strategy to achieve the stabilization. The system is linearized around (x,ũ) which leads to the following linear-quadratic control problem:Here, x is the state, u the input, and y the output of the system. Details regarding the linearization and the matrices M, A, B, and C are given in Section 2. We use mesh movement techniques to represent the interface and resolve the discontinuity of the temperature gradient along the interface. The spatial discretization mesh is updated in every time step with the mesh velocity V mesh from equations (2b) and (2c). Thus, not only the functions in equation (1) are time dependent, but also the matrices. We omit the time dependence "(t)" for the sake of brevity in what follows. The optimal control u = −Kx can be computed with the feedback matrix K = B T XM [2]. The matrix X is the solution of the generalized differential Riccati equation with time dependent coefficients: −M TẊ M = C T C + (Ṁ + A) T XM + M T X(Ṁ + A) − M T XBB

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“…For a linearization of a Stefan problem, we use known reference trajectories (T ,Ṽ mesh ,ṽ) from open loop optimal control [1]:…”

Section: Feedback Control Of a Stefan Problemmentioning

confidence: 99%

“…As exemplary coupled nonlinear system we use a two-phase two-dimensional Stefan problem. We compute a reference solution (x,ũ) with an open loop control as from [1]. Since the open loop controlũ is not robust against perturbations and uncertainties, it should be stabilized.…”

mentioning

confidence: 99%