Variational Formulation (Rayleigh-Ritz Method)

Akshay Wadalkar (06)Tushar A Aneyrao (16)

Variational Formulations

• The variational approach of establishing the governing equilibrium equations of systems involves calculation of total potential ᴨ of the system and invoke the stationary of ᴨ i.e. ᵹᴨ=0.

• 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.

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.

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.

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.

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.

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.

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.

• 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

(j=1,2,3,…….n)

• Using above equation n algebraic equations are obtained from which the unknown constant gj are determined.

j

0g

I

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.