Numerical Optimization of the Dirichlet Boundary Condition in the Phase Field Model with an Application to Pure Substance Solidification

As opposed to the distributed control of parabolic PDE's, very few contributions currently exist pertaining to the Dirichlet boundary condition control for parabolic PDE's. This motivates our interest in the Dirichlet boundary condition control for the phase field model describing the solidification of a pure substance from a supercooled melt. In particular, our aim is to control the time evolution of the temperature field on the boundary of the computational domain in order to achieve the prescribed shape of the crystal at the given time. To obtain efficient means of computing the gradient of the cost functional, we derive the adjoint problem formally. The gradient is then used to perform gradient descent. The viability of the proposed optimization method is supported by several numerical experiments performed in one and two spatial dimensions. Among other things, these experiments show that a linear reaction term in the phase field equation proves to be insufficient in certain scenarios and so an alternative reaction term is considered to improve the models behavior.