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.

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.