Analysis and Computation of Shape Sensitivities for Elliptic Interface Problems

摘要。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，随着机翼形状的变化，飞机机翼周围的气流是如何变化的，这又如何影响阻力？”敏感性分析试图回答这些问题。灵敏度提供了定量信息，可以作为一种有用的数学工具来深入了解系统的行为。这一建议涉及的分析和设计的计算方法近似灵敏度的一类非常具体的问题。研究重点是界面问题的形状灵敏度计算。这些问题出现在各种物理系统的分析中，如地下水流经不同类型的沉积物，以及制造过程，如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。在分析这种工艺时，可以将模具和零件视为一种复合材料，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成材料厚度的微小变化的敏感性。由于模具和制件由具有不同导热性能的不同材料组成，控制冷却过程的数学方程的解在界面处缺乏光滑性。对于这些类型的问题，计算灵敏度需要一种不同的、更聪明的近似方案，而不是通常用来确定温度的近似方案。目前的研究试图分析和开发这些问题的数学结构，并修改现有的数值方法，以开发一种准确、高效、合理的计算算法来实现。关于在火箭发动机的设计中纳入这种技术的估计表明，设计周期时间可以从一年减少到一个月。如此巨大的结果使得这种计算工具的发展对国家利益至关重要，无论是在设计阶段节省成本还是保持在新技术的前沿。该项目研究了使用区域分解技术来开发精确和高效的形状灵敏度计算算法。具体来说，这项工作涉及实现连续灵敏度方程方法（c - sem），以推导无限维灵敏度方程，通常采用偏微分方程的形式。研究的重点是椭圆界面问题，其中包含决定界面空间位置或形状的参数。由此产生的形状灵敏度在整个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。首先是建立存在性、唯一性和规律性等基本性质所需的数学分析。第二部分是巧妙地选择了一种适合求解方程的数值方法。理论分析指导了利用问题结构的计算方法的构建。具体而言，采用迭代、非重叠域分解算法来准确捕获灵敏度变量中的不连续点。

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.