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

该项目的目的是开发有效的数值方法来控制由卡恩-希利亚德变分不等式控制的界面演化。从量子点在异质外延薄膜晶体生长和晶粒生长中的应用ränge到微电子器件中的空穴演化。在所有这些应用中，相的特定位置或界面分布的特殊性质是很重要的。Cahn-Hilliard模型是基于扩散（非锐）界面的守恒相场模型。通常表示为äs一个非标准的四阶变分不等式。半隐式时间离散化可以看作äs一个可能有非线性约束、控制盒约束和高度复杂的代价函数的控制问题。我们希望研究原始-对偶活动集策略和/或半光滑牛顿方法应用于求解Cahn-Hillard方程的优化公式。问题äs预处理，适应性和有效的时间步进必须解决。目标是推导出一种网格无关的超线性收敛方法，其中对界面厚度的依赖是适度的。在实际应用中，Cahn-Hilliard变分不等式必须与包含弹性系统和拉普拉斯电势方程的椭圆系统或非线性热方程耦合。当开发出卡恩-希利亚变分不等式的有效方法时，这些方法必须推广到扩展版本。最终目标是解决最优控制问题，其中Cahn-Hilliard变分不等式的扩展版本作为äs约束。除了高度非线性约束外，代价函数通常是非凸的和基于梯度的。在第一个应用阶段，我们计划分析研究这个最优控制问题。我们希望推导出一阶和二阶最优性条件äs以及äs拉格朗日乘子的存在性。在可能的第二个应用周期内，计划推导出最优控制问题的一种有效的超线性收敛方法。此外，涉及卡恩-希利亚变分不等式的最优设计问题也是一个问题。

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.