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Center for Nonlinear Studies |
An approach for solving interface problems is the diffuse interface theory, which was originally developed as methodology for modeling and approximating solid-liquid phase transitions in which the effects of surface tension and non-equilibrium thermodynamic behavior may be important at the surface. The diffuse interface model describes the interface by a mixing energy represented as a layer of small thickness. This idea can be traced to van der Waals, and is the foundation for the phase-field theory for phase transition and critical phenomena. Thus, the structure of the interface is determined by molecular forces; the tendencies for mixing and de-mixing are balanced through the non-local mixing energy. The method uses an auxiliary function (so-called phase-field function) to localize the phases, assuming distinct values in the bulk phases (for instance 1 in a phase and −1 in the other one) away from the interfacial regions over which the phase function varies smoothly.
During the seminar I will present the Cahn-Hilliard model (a classical phase field model to represent binary alloys) and the main ideas behind the derivation of different numerical schemes, showing the advantage and disadvantages of each approach. The key point is to try to preserve the properties of the original models while the numerical schemes are efficient in time. Moreover, I will present how these ideas for designing numerical schemes to approximate phase-fields models can be extended to other energy based applications, like nematic liquid crystals or mixture of fluids.
Bio: Giordano Tierra earned his M.S. and Ph.D in Mathematics from Universidad de Sevilla (Spain), under the supervision of Francisco Guillen-Gonzalez. He has held postdoctoral positions at University of Notre Dame, Charles University in Prague (Czech Republic) and Temple University. Currently he is an Assistant Professor at University of North Texas. His research focuses on modeling, numerical analysis and scientific computing for multi-physics problems with applications in life and material sciences that involve the understanding of the rheology of complex fluids.
Host: Shriram Srinivasan