Variational principles and approximate methods
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The variational calculus can be applied to continuous and discrete systems, especially to discretized systems with higher comfort. The exact solutions requires the satisfaction of compatibility and equilibrium conditions to be satisfied completely. When the exact solution does not exist, the approximate solution close to the exact is made possible by Rayleigh-Ritz method. The basis for this method is the minimum potential energy principle. More the number of Ritz constants in trial function more accurate is the solution. The Lagrange-Euler formulation of variational principle aids in establishing the governing differential equation of equilibrium describing the physical phenomenon. The physical problems like heat conduction and fluid flow cannot be addressed by Rayleigh-Ritz method as it is based on stationarity condition in potential energy functional. Such processes have the answer in weighted residual methods. Several weighted residual methods are - point collocation, subdomain collocation, least square and Galerkin's method. The Galerkin's method is popularly used in finite element method as it can be comfortably employed for elasticity and fluid/ heat related problems. The virtual displacement approach gives suitable base for FE formulation of static analysis problems. The Newton's laws of motion, the work-energy principle are the basis for the principle of virtual work which helps to derive expressions for stress, strain and displacement in an elastic structure. This chapter deals with variational approach, minimum potential energy principle, Rayleigh-Ritz method, weighted residual methods, and principle of virtual displacement and virtual work with suitable illustrative examples from elasticity and heat transfer.