Date of Award
Doctor of Philosophy
Numerical methods for a class of free and moving boundary problems are considered. The class involves the solution of Laplace's equation on a domain which is changing shape with time. The position of the boundary is described by an evolution equation. With the time fixed, a boundary approximation method is employed to solve the potential problem. The boundary location at the next time is determined from the evolution equation using standard techniques and the process is repeated.;Two boundary methods are examined. Both are characterized by representing the approximate solution of the potential problem as a series of known basis functions, chosen from a complete set of particular solutions to the Laplace equation. In the first approach, the parameters, to be determined from the boundary data, appear linearly in the trial solution. The basis functions are closely related to the well studied harmonic polynomials and this permits an extensive analysis of the linear method. In particular, convergence of the method is demonstrated and some estimates on the degree of convergence are derived. In the second approach, the parameters appear nonlinearly. This approach is new, but derives from classical results on complex rational approximation and may be interpreted as an acceleration of the convergence of the linear technique.;The linear method is applied to a number of electrochemical machining examples and performs well for relatively smooth boundaries. A nonlinear approach is tested on the inverse machining problem with excellent results.;Both the linear and nonlinear methods are applied to several challenging examples of Hele-Shaw flow. In all instances, the nonlinear scheme outperforms the linear. The ease of programming, efficiency and concomitant accuracy of the nonlinear scheme make it an attractive choice for the numerical integration of a class of free and moving boundary problems.
Meredith, David Gerald, "Boundary Approximation Methods For Some Free And Moving Boundary Problems" (1989). Digitized Theses. 1845.