Varaiational formulation fem
Download as PPTX, PDF7 likes1,673 views
The document discusses the variational formulation and Rayleigh-Ritz method for solving systems. The variational approach involves calculating the total potential of a system and finding the stationary value where the potential is zero. The Rayleigh-Ritz method assumes the form of the unknown solution in terms of known trial functions with adjustable parameters. The functional is expressed in terms of these parameters and differentiated to obtain equations that are set to zero to solve for the parameters.
Varaiational formulation fem
- 1. Variational Formulation (Rayleigh-Ritz Method) Akshay Wadalkar (06) Tushar A Aneyrao (16)
- 2. Variational Formulations • The variational approach of establishing the governing equilibrium equations of systems involves calculation of total potential ᴨ the of system and invoke the stationary of ᴨi.e. ᴨ=0. ᴨ • Variational technique can be effective in the analysis of discrete systems. • The variational approach provides a particularly powerful mechanism for the analysis of continuous system.
- 3. Variational System • The variational method may provide a relatively easy way to construct the system governing equations. This ease of use of a variational principle depends largely o the fact in the variational formulation scalar quantities are considered rather than vector quantities. • A variational approach may lead to more directly to the system-governing equations and boundary condition.
- 4. Variational System • The variational approach provides some additional insight into a problem and gives an independent check on the formulation of the problem. • For approximate solution a larger class of trial functions can be employed in many cases if the analyst operates on the variational formulation rather than on the differential formulation of the problem.
- 5. Rayleigh-Ritz Method • In this method form of the unknown solution is assumed in terms of known functions (trial functions) with unknow adjustable parameters. • From the family of trial functions the function that renders the functional stationary are selected and substituted into the functional which is function of the function. • Thus, The functional is expressed in terms of the adjustable parameters.
- 6. Rayleigh-Ritz Method • The resulting functional is differentiated with respect to each parameter and resulting equation is set equal to zero. • If there are n unknown parameters in the functional, there will be n simultaneous equations to be solved for the parameters and best solution is obtained.
- 7. Rayleigh-Ritz Method • The main aim of Rayleigh-Ritz method is to replace the problem of finding the minima and maxima of integrals by finding the minima of functions of several variables.
- 8. Contd…. • For example – Consider search of a function L(x) that will extremize certain given functional I(L). As metioned, L(x) can be approximated by liniar combination of suitable chosen coordinate function c1(x), c2(x),…………. cn(x) – Then L(x) can be written as L(x)= g1 c1(x) + g2 c2(x) + ………………….. + gn cn(x) where gi are unknown constants to be found.
- 9. • Since each of c1(x) is an admissible function the functional I(L) becomes a function of g. By taking the diffrence of the function, unknown g can be determined as follows I (j=1,2,3,…….n) 0 gj • Using above equation n algebraic equations are obtained from which the unknown constant gj are determined.
- 10. REFERENCES • Y.M.DESAI,T.I.ELDHO,A.H.SHAH ; Finite elemnt method with application in engg. • Klaus-Jürgen Bathe; Finite element Procedures • Daryl L. Logan; Finite element Method.