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

这个项目的目的是发展有效的数值方法来控制由Cahn-Hilliard变分不等式控制的界面演化。从量子点的形成在异质外延薄膜的晶体生长和晶体生长到微电子器件中的空穴演化的应用。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重要的。Cahn-Hilliard模型是一种基于扩散(而非尖锐)界面的守恒相场模型。它通常被表示成一个非标准的四阶变分不等式äS。半隐式时间离散化可以看作是一个具有可能的非线性约束、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-对偶有效集策略和/或半光滑牛顿方法在求解Cahn-Hillard方程的优化公式中的应用。S的问题是，必须探讨预条件性、适应性和有效的时间步长。我们的目标是得到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变分不等式必须耦合到包含弹性力学系统的椭圆组和电势的拉普拉斯方程，或者耦合到非线性热方程。当Cahn-Hilliard变分不等式的有效方法被开发出来时，这些方法必须被推广到扩展形式。最终目标是解决Cahn-Hilliard变分不等式的扩展形式具有S约束的最优控制问题。除了高度非线性的约束外，代价泛函通常是非凸的和基于梯度的。在第一个应用阶段，我们计划对这个最优控制问题进行解析研究。我们希望得到一阶和二阶最优性条件，S很好，S很好，拉格朗日乘子的存在性。在可能的第二个应用阶段，计划为最优控制问题推导出一种有效的超线性收敛方法。此外，涉及Cahn-Hilliard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.