Numerical Approach to Energy Minimization of Fluid Configurations Using Phase-Field Models
Date
2018-08
Authors
Gallo, Erika
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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.
Description
Keywords
phase field, fluid configurations, capillarity, optimization, energy minimization
Citation
Gallo, E. (2018). Numerical approach to energy minimization of fluid configurations using phase-field models (Unpublished thesis). Texas State University, San Marcos, Texas.