Plenary Lecture

Recent Developments in the Von Mises Transformation and its Applications in the Computational Sciences

Professor M. H. Hamdan
Department of Mathematical Sciences
University of New Brunswick
P.O. Box 5050
Saint John, New Brunswick, Canada, E2L 4L5
E-mail: hamdan@unb.ca

Abstract: The use of (φ,ψ) coordinate system has attained popularity in a number of engineering fields, including aerodynamics, groundwater, magneto-hydrodynamics, and water waves. In the computational sciences, this coordinate system offers flexibility in handling boundary value problems through the mapping a curvilinear domain into a Cartesian domain in which numerical techniques that rely on the finite differences procedure are easily applicable. In unbounded domains, this approach has received wide acceptance as a viable alternative in the study of the complicated equations of electro-magneto-hydrodynamics.
In spite of the attractiveness of the (φ,ψ) system, it introduces an extra equation to be solved for the variable φ(x,y). This is necessary for locating singularities in the computational domain and its boundaries (such as locating the leading and trailing edges of an airfoil, or the location the crest of a wave). In order to circumvent this, it is possible to judiciously replace φ by x, and replace the (φ,ψ) coordinates by the (x,ψ) coordinates which, at the outset, eliminates the need for introducing an extra equation. Furthermore, locations of singularities are either known a priori or can easily be determined using the (x,ψ) system.
The (x,ψ) coordinates are the well-known von Mises variables that were introduced by R. von Mises in 1927 to analyze the boundary layer equations. The use of this system in the study of waves with arbitrary vorticity distribution was made by T.B. Benjamin in 1961. However, the computational von Mises and its use in aerodynamics was introduced by R.M. Barron in 1986.
Over the last two decades, a number of advances have been made in the computational von Mises. These include the introduction of a time-dependent form of the coordinates; the introduction of techniques to handle the infiniteness of the Jacobian of transformation (which arises in some boundary value problems); the introduction of mapping multiple physical domains into a single computational domain; and the analysis and control of grid distortion that inevitably arises due to the use of non-orthogonal (x,ψ) system.
In this talk, we present analysis of the von Mises transformation and its computational complexities, and report on its recent developments and possible extensions to three physical dimensions. A survey of the use of the computational von Mises in the various engineering fields will be provided.

Brief Biography of the Speaker:
M. H. Hamdan received an Ordinary National Diploma in Technology-Engineering from Swindon College, U.K.; a Certificate in Negotiation, Mediation and Conflict Resolution from St. Mary’s University, Canada; a B.Sc, M.Sc., and a Ph.D in Applied Mathematics from the University of Windsor, Canada. He taught at a number of universities both as a regular faculty member and as a visiting professor, in Canada, China and the Middle East. He has been teaching at the University of New Brunswick, Canada, for 18 years, and is a pevious Chair of the Department of Mathematics, Statistics and Computer Science. His teachables span the areas Mathematics, Decision Sciences and Management Science, Mathematical Economics, and Negotiations. His research areas include computational fluid dynamics, single-phase flow through porous media, and modeling dusty gas flow through porous media. He is an International Consultant in Science and Technology Planning and in School Mathematics Curricular Development. He is the recipient of a number of teaching awards, and is listed among American Men and Women of Science; Who’s Who in Science and Engineering; Who’s Who in the World; and Two Thousand Outstanding Scientists of the 20th Century.