Analysis and Computation of Shape Sensitivities for Elliptic Interface Problems

Montana State University, Bozeman MT

Investigators

Abstract

ABSTRACT. DMS-0072438 Lisa G. Stanley Department of Mathematics Univeristy of Montana Engineers, 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 as the shape of the wing changes, and how does this affect drag? Sensitivity analysis seeks to answer such questions. The sensitivity provides 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 of computational methods for approximating sensitivities for a very specific class of problems. The research focuses on shape sensitivity calculations for interface problems. These problems arise in the analysis of 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 interface between 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. Results of this magnitude make the development of such computational tools critical for the national interest both in cost savings during the design stage and in remaining on the forefront of new technology. This project investigates the use of domain decomposition techniques for the development of accurate and efficient computational algorithms for shape sensitivity calculations. Specifically, the work involves the implementation of Continuous Sensitivity Equation Methods (C-SEMs) in order to derive infinite dimensional sensitivity equations which usually take the form of partial differential equations. The research focuses on 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 this class 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 suitable for solving the equations. The theoretical analysis guides the construction of a computational method which exploits the problem structure. Specifically, an iterative, nonoverlapping domain decomposition algorithm is used to accurately capture discontinuities in the sensitivity variable.