Nets Composed of Parts of Circles for the Approximate Solution of Field Problems

Abstract

The two· dimensional differential equation describes the current flow in a sheet of conductivity cr loaded by a transverse current density , cp being the electrical potential. It is known that equation (1) can be solved approximately by a procedure in which the two· dimensional continuum is replaced by a net of straight-line bounded meshes, leading to an electrical network of conductances. The author shows that meshes bounded by "curvilinear rectangles" can be equally well dealt with and, on· the basis of different conformal transformation functimls for the individual meshes, derives the formulae required for a solution, if the mesh boundaries are circle arcs or circle arcs and straight lines. A good fit of the contours of the boundaries and equipotentials and their orthogonal trajectories can be obtained. This reduces the number of meshes without impairing the accuracy. Sharp corners at boundaries can be dealt with in a similar way. Formulae for a good accuracy computation of potential gradients and a method for changing th.e mesh size abruptly are given. Two examples using nets of only four meshes demonstrate the power of the method, the maximum errors being of the order of a few per cent.