Conformal mappings onto multiply connected regions with specified boundary shapes: A preliminary discussion of computer implementation

Abstract

Abstract : The author describes a method of calculating conformal mappings of any given finitely connected region onto a region with arbitrarily specified boundary shapes. If the specified shapes are rectangles, than this method can be used to generate conformal grids which should be useful for numerical solution of many partial differential equations, for example in calculating the airflow past an airflow with flaps or the flow of cooling water past fuel pins in a nuclear reactor. He has proved that there exists a conformal mapping of any given finitely connected region onto a region with arbitrarily specified boundary shapes. The construction in the proof has been adapted for computer implementation. Some examples have been worked to determine the region bounded by circles which is the image of a given region in the extended complex plane under a conformal mapping taking infinity to infinity. A region whose outer boundary is a rectangle is mapped conformally to a region with all rectangular boundaries, and the vertices of the outer boundaries correspond. Mappings onto regions bounded by rectangles should be of considerable use in grid generation for numerical solution of partial differential equations. Grids have been calculated for one simple example. (Author)