Optimization problems governed by Allen-Cahn and Cahn-Hilliard variational inequalities

• Antragstellerinnen / Antragsteller: Professorin Dr. Luise Blank, von 5/2006 bis 7/2009; Professor Dr. Harald Garcke, seit 8/2009
• Fachliche Zuordnung: Mathematik
• Förderung: Förderung von 2006 bis 2015
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 25616723

The aim of this project is to develop efficient numerical methods to control interface evolution governed by Cahn-Hilliard variational inequalities. The applications ränge from quantum dot formation in crystal growth of heteroepitaxial thin films and grain growth to void evolution in microelectronic devices. In all these applications a certain location of phases or special properties of the interface distribution are of importance. The Cahn-Hilliard model is a conserved phase field model based on a diffuse (not sharp) interface. It is usually formulated äs a non-standard variational inequality of fourth Order. The semi-implicit time discretization can be viewed äs a control problem with possibly nonlinear constraints, control box constraints and a highly complex cost function. We wish to study the primal-dual active set strategy and/or a semi-smooth Newton method applied to the optimization formulation to solve the Cahn-Hillard equation. Issues äs preconditioning, adaptivity and efficient time stepping must be approached. The goal is to derive a mesh independent, superlinear convergent method where the dependence on the interfacial thickness is moderate. In practical applications the Cahn-Hilliard variational inequality has to be coupled either to an elliptic System containing an elasticity System and the Laplace equation for the electrical potential or to a nonlinear heat equation. When efficient methods are developed for Cahn-Hilliard variational inequalities these have to be generalised to the extended versions. The final goal is to solve optimal control problems in which the extended versions of the Cahn-Hilliard variational inequality act äs constraints. In addition to the highly nonlinear constraints the cost functional is often non-convex and gradient based. For the first application period we plan to study this optimal control problem analytically. We wish to derive first and second Order optimality conditions äs well äs the existence of Lagrange multipliers. In a possible second application period it is planned to derive an efficient superlinear convergent method for the optimal control problem. Also optimal design problems involving Cahn-Hilliard variational inequalities shall be an issue.

• DFG-Verfahren: Schwerpunktprogramme
• Teilprojekt zu: SPP 1253: Optimierung mit partiellen Differentialgleichungen