Relaxation methods applied to engineering problems V. Conformal transformation of a region in plane space

Abstract

1. Part III of this series (Christopherson and Southwell 1938) brought within the scope of relaxation methods problems governed by Poisson’s or Laplace’s equation in two variables. Its first example (torsion of an isotropic cylinder) was a standard problem in potential theory, calling for the evaluation within a given region of a planeharmonic function defined as having specified values at the boundary. This was solved without difficulty, and with more than sufficient accuracy for practical purposes. Conformal transformation, the problem considered here, although essentially similar presents some questions of detail which were not encountered in Part III. Regarded as a weapon of the analyst, it is a means whereby problems relating to specified regions in plane space may be reduced to problems which concern regions of simpler shape (e.g. circles or rectangles) and can be solved in terms of known functions of ordinary (e.g. polar or Cartesian) co-ordinates. Contours of these co-ordinates by intersection divide the simpler region into rectangles: conformal transformation entails a similar division, or “ mapping”, of the specified region by intersecting contours of two conjugate planeharmonic functions, , which in turn serve as co-ordinates to define the position of any point. Orthodox mathematics presents the transformation in an equation of the type which, when the form of is known, expresses a one-to-one relation between points in the first region and in the second; but this functional relation is of no concern to the practical computer provided that he can construct “ maps” of which it is the mathematical expression, and for this it is only necessary to have and evaluated at nodal points of some regular “ net” . The procedure whereby contours are constructed is so obvious as not to require description.