Analysis and Computation of Shape Sensitivities for Elliptic Interface Problems

使用数学模型来描述物理系统的工程师、数学家和其他科学家经常需要回答这样一个问题：“随着系统参数的变化，系统的响应是如何变化的？”例如，随着机翼形状的改变，机翼周围的气流如何变化，这对阻力有何影响？敏感性分析试图回答这样的问题。敏感度提供了定量的信息，可以作为一种数学工具来洞察系统的行为。这项建议涉及到对一类非常特定的问题近似敏感度的计算方法的分析和设计。研究的重点是界面问题的形状灵敏度计算。这些问题出现在对各种物理系统的分析中，如地下水通过不同类型沉积物的流动，以及制造过程中的压铸问题和合金凝固问题。例如，在压铸中，凝固部分和模具本身之间有一个界面。当分析这样的过程时，模具和零件可以被视为一种复合材料，为了优化铸造工艺，设计者需要确定整个复合材料的温度对各自组件材料厚度微小变化的敏感度。由于模具和制件由不同的材料组成，具有不同的导热性能，控制冷却过程的数学方程有一个在界面上缺乏光滑性的解。对于这些类型的问题，计算灵敏度需要不同的、更智能的近似方案，而不是通常用于确定温度的近似方案。目前的研究试图分析和利用这些问题的数学结构，并对现有的数值方法进行修改，以开发出一种准确、高效和合理实施的计算算法。关于将这类技术纳入火箭发动机设计的估计表明，设计周期时间可以从一年减少到一个月。如此重大的结果使得这种计算工具的开发对国家利益至关重要，既在设计阶段节省成本，又保持在新技术的前沿。本项目研究区域分解技术用于开发准确和高效的形状敏感度计算算法。具体地说，这项工作涉及到实现连续灵敏度方程方法(C-SEMS)，以便推导出通常采用偏微分方程组形式的无限维灵敏度方程。研究的重点是含有参数的椭圆界面问题，这些参数决定了界面的空间位置或形状。由此产生的形状敏感度在界面上表现出不连续性。这类问题的有效计算算法依赖于两个基本组成部分。首先是建立存在、唯一性和规律性等基本性质所需的数学分析。第二个部分是巧妙地选择了一种适合于求解方程的数值方法。理论分析指导了一种利用问题结构的计算方法的构建。具体地说，一种迭代的、非重叠的区域分解算法被用来准确地捕捉灵敏度变量中的不连续。

ABSTRACT.DMS-0072438Lisa G. StanleyDepartment of MathematicsUniveristy of MontanaEngineers, mathematicians and other scientists who use mathematical models to describe physical systems often need to answer the question: ``How does the system response change as system parameters change?'' For example, how does the airflow around an airplane wing change asthe shape of the wing changes, and how does this affect drag? Sensitivity analysis seeks to answer such questions. The sensitivityprovides quantitative information which can be useful as a mathematical tool to gain insight into the behavior of a system.This proposal deals with the analysis and design ofcomputational methods for approximating sensitivities for a very specific class of problems. The research focuses on shape sensitivity calculationsfor interface problems. These problems arise in the analysisof a variety of physical systems such as groundwater flow through different types of sediment as well as manufacturing processes such as die casting problems and alloy solifidification problems.In die casting, for example, there is an interfacebetween the solidifying part and the mold itself.When analyzing such a process, the mold and the part may be considered as one composite material, and in order to optimize the casting process, the designer needs to determine the sensitivity of the temperature throughout the composite material to small changes in the thickness of the respective component materials. Since the mold and the manufactured part consist of different materials which have different heat conductivity properties, the mathematical equation governing the cooling process has a solution which lacks smoothness at the interface. For these types of problems, computing the sensitivity requires a different, and more clever, approximation scheme than that which is typically used to determine the temperature. The current research attempts to analyze and exploit the mathematical structure of these problems and to modify existing numerical methods in order to develop a computational algorithm which is accurate, efficient and reasonable to implement. Estimates regarding inclusion of such techniques in the design of rocket engines show that design cycle time could be reduced from one year to one month. Resultsof this magnitude make the development of such computational tools critical for the national interestboth in cost savings during the design stage and in remainingon the forefront of new technology.This project investigates the use ofdomain decomposition techniques for the development of accurate and efficient computational algorithms for shape sensitivity calculations. Specifically, the work involvesthe implementation of Continuous Sensitivity EquationMethods (C-SEMs) in order to derive infinite dimensionalsensitivity equations which usually take the form ofpartial differential equations. The research focuseson elliptic interface problems containing parameters which determine the spatial location or the shape of the interface. The resulting shape sensitivities exhibit discontinuities across the interface. Efficient computational algorithms for thisclass of problems rely on two essential components. The first is the mathematical analysis needed to establish fundamental properties such as existence, uniqueness and regularity. The second component is the clever choice of a numerical method which is suitablefor solving the equations. The theoretical analysis guidesthe construction of a computational method which exploits the problem structure. Specifically, an iterative, nonoverlappingdomain decomposition algorithm is used to accurately capture discontinuities in the sensitivity variable.