Unification of Approximate Methods

Abstract

It seems that the approximate analysis of physical problems started with the calculus of variations, was raised with the finite differences and matured with the finite element method, depending upon the nature of a problem; however, each has its own merit and sometimes neither one alone is sufficient. Preference for one over the other two often becomes evident at the start (during the formulation of the problems). For some problems, for instance, the functional is extremely difficult to formulate. Going from the functional form to the differential form can always be achieved by the Euler-Lagrange minimization procedure; the reverse, however, is often not possible. Yet for some problems the direct formulation of the differential equation is easier. Most linear solid mechanics problems, for instance, can be handled by the direct stiffness or flexibility approach without referring to the minimization techniques explicitly.