Numerical Solution of Axisymmetrical Problems, with Applications to Electrostatics and Torsion

摘要

Numerical methods are given for solution of axisymmetrical problems involving the partial differential equation ∂2ψ ∂z2 + ∂2ψ ∂ρ2 + K ρ ∂ψ ∂ρ =0, where ρ is the radial coordinate and z the coordinate parallel to the axis. The various values of K which occur in physical situations are discussed, and common iteration methods for handling these problems are given. For Laplace''s equation, K = 1. For the Stoke''s stream function, K = - 1, but it is pointed out that for numerical work a new function, called the flow-disturbance function, having K = 3, is more tractable. Similarly a new function with K = 5, the stress-concentration function, is much easier to compute than the usual stress function (K = - 3) for the case of the torsion of a circular shaft of varying diameter. The methods are illustrated by computation of the equipotentials for an electron lens, and by a complete computation of the stresses and strains in a particular grooved circular shaft under torsion.