Existence of Optimal Control for Dirichlet Boundary Optimization in a Phase Field Problem

Abstract

Phase field modeling finds utility in various areas. In optimization theory in particular, the distributed control and Neumann boundary control of phase field models have been investigated thoroughly. Dirichlet boundary control in parabolic equations is commonly addressed using the very weak formulation or an approximation by Robin boundary conditions. In this paper, the Dirichlet boundary control for a phase field model with a non-singular potential is investigated using the Dirichlet lift technique. The corresponding weak formulation is analyzed. Energy estimates and problem-specific embedding results are provided, leading to the existence and uniqueness of the solution for the state equation. These results together show that the control to state mapping is well defined and bounded. Based on the preceding findings, the optimization problem is shown to have a solution.

Focusing the latent heat release in 3D phase field simulations of dendritic crystal growth

“…The resulting term required further treatment by the Σ limiter (34) to maintain the shape of the thin diffuse interface for larger values of the initial supercooling Δu ini , which gave rise to the ΣP1-P model. Unlike the GradP model, the ΣP1-P model in its isotropic form is compatible with the numerical analysis performed in our related work [43], which provides theoretical justification of the proper function of the implemented finite volume-based numerical solvers. However, the Σ limiter can also be incorporated into the original GradP and φ 0 GradP models.…”

“…We take advantage of this flexibility and propose a new variant of the reaction term. The resulting model is compatible with the numerical analysis performed in our paper [43] for the finite volume method (as long as anisotropy is not considered) and it also achieves a very good quantitative agreement with experiments in the numerically difficult case of rapid solidification. In a number of simulations, we compare the behavior of the discussed models.…”

“…(b) The use of the computational algorithm based on the model should be justified by numerical analysis. In our recent work [43], which was performed concurrently with the research presented here, we prove convergence of the numerical solution to the unique solution of the original problem with a rather generic form of the reaction term f which is local, i.e. depending on p and u only.…”

“…The settings are the same as in figure 7 (see section 4.4.1) except for the anisotropy strength A 1 = 0, which corresponds to using the isotropic phase field equation ( 22) instead of (19). In such case, the models GradP and φ 0 GradP coincide and the ΣP1-P model has the form such that the numerical analysis we performed in [43] applies. Results obtained both without (δ = 0) and with (δ = 0.05) noise given by ( 39) are provided at the same scale, which testifies to the ability of noise to support crystal growth and branching in a seaweed pattern.…”