An Iterative Technique for Minimising a Double Integral with Applications in Elasticity

Abstract

The solution of a problem in plane elasticity can be associated with a minimum energy principle, involving the minimisation of a double integral subject to certain boundary conditions. In 1933, Kantorovich proposed a method which reduced this problem to that of finding the minimum of a single integral. He chose part of the solution a priori (in accordance with the character of the problem) which left only one undetermined function in one of the variables. This unknown function could then be found through the solution to an ordinary differential equation. He then constructed a second approximation by introducing two unknown functions which led to the solution of two simultaneous differential equations. Again the accuracy of the solution was dependent on the initial choice for part of the function; so also was the rate of convergence.