A Brief Discussion Of Relationship Between Method Of Weighted Residuals And Variational Principles

Variational principle has been used in various aspects of mechanics. It is an effective method especially in mechanics, physics and other technology domains, and has important value on theory and application. Method of weighted residuals is a new type of method of calculational mechanics. It has a lot of merit, for example, lest calculation, more extractive, briefness, straightness, having no dependence on functional, and so on. Some literatures consider Method of Weighted Residuals has no relation with Variational Principle. But in this paper I point out that they have connection.In this paper, the elementary means of Method of Weighted Residuals (Point Collocation Method, Domain Collocation Method, Galerkin method, and least squares method) are discussed first of all. Then that in which conditions Ritz Method and Method of Weighted Residuals are connected is discussed. For deep studying, the maturity quality of every qualification of Variational principle is mentioned. Investigations indicate combined Variational principle will result in Method of Penalized Function.Then, using Generalized Galerkin Method, we found finite element models of conservative system. After rigorous math deducting, we find these models are equal to the finite element models that we are founded by Variational principle. It explains that Generalized Galerkin Method is effective. In the same time, it shows there are close relationship between Variational principle and Method of Weighted Residuals.After that, setting out by Laplace Equation and basic equation of elasticity mechanics, the connection between Integral Equation and Method of Weighted Residuals are discussed.In conclusion, Method of Weighted Residuals and Variational principle are connect with each other nearly. If we can combine these two theories to use in various practical questions it must be more valuable.