Variational formulation in FEM using Rayleigh-Ritz Method

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.