Numerical Approach to Energy Minimization of Fluid Configurations Using Phase-Field Models
Abstract
We consider a fluid, under isothermal conditions and confined to a bounded container of homogeneous makeup, whose Gibbs free energy, per unit volume, is a prescribed function of its density distribution. Based on the Van der Waals-Cahn-Hilliard Theory of phase transitions, we minimize our functional, whose phase field formulation is obtained by considering an energy of the type
E∈(u) = {Ω (∈|∇u| + a u (1 − u) + uG(x) + λu dx,
where u is the phase function, G is a potential energy, and λ represents volume constraint. We know that these minimizers, EE, as E goes to 0, will Γ−converge to the minimizer of the capillary energy functional.
Although numerical approaches to this minimization exists, current approaches are unable to distinguish between local and global minimizers of the functional. I propose a mesh-grid-based optimization approach, with Dirichlet boundary conditions. Assuming convexity of our system, we utilize a logarithmic barrier optimization scheme in hopes to guarantee convergence to the global minimum of our energy functional.
