COMPUTATION OF CAPACITY, POLARIZATION AND ADDED MASS
Abstract
This thesis concerns computation of various domain constants, of two and three dimensional bounded, connected domains. The domain constants considered are the capacity, polarization, added mass and outer radius. Each of these is proportional to the value of the Dirichlet integral of a function harmonic in the exterior of the domain. We first consider the exact computation of these quantities in two dimensions or, equivalently, for infinite cylinders. We compute these quantities for various domains. In particular, we show how these quantities can be computed from the parameters in the Schwarz-Christoffel Theorem. While it is possible to obtain exact formulas for these quantities for certain geometries, no such formulas are known for most domains. For example, for the cube none of these quantities are known exactly, so we must resort to numerical methods to obtain estimates for them. Therefore, we consider a method, used by Greenspan and Silverman to find the capacity of a domain by inverting the original domain with respect to a circle or a sphere. We then extend this method to compute the polarization and the added mass. With this new formulation of the problem it is possible to compute any of the quantities from a solution of an interior problem. In fact, only the values of the solution and its gradient at a single point are needed. This formulation can be used with any method of solving Laplace's equation on a general two dimensional domain to find the capacity, polarization and added mass of infinite cylinders or of solids of revolution. For more general three dimensional domains, Laplace's equation must be solved as a three dimensional problem. This method has been implemented using ELLPACK. We next show that very accurate results can be obtained for polygonal domains, by formulating the exterior problem as an integral equation. The integral equation is solved using a collocation method, with a nonuniform partition of the boundary. An integral equation formulation is also used to find the capacity, polarization and added mass of a cube. Finally, we propose a new inequality, relating capacity, polarization and added mass, and prove it for some special cases. We also disprove a similar inequality conjectured by Schiffer and Szego.
Degree
Ph.D.
Subject Area
Mathematics
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