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6.3 CONTINUUM MECHANICS 129
Ritz method
A further procedure for the modelling of strains is the Ritz method, see for example
Bathe [19]. In this process the partial differential equation is solved and an attempt
is made to approximate an unknown displacement function, e.g. the deflection
of a beam over its length by a linear combination of n interpolation functions.
These must each correspond with the geometric boundary conditions. The n coef-
ficients of the interpolation functions are yielded by the requirement that the elastic
potential must be minimal. From this, n equations are found, which set the partial
derivative of the elastic potential with respect to the coefficients equal to zero. So
n equations are available for n coefficients. It should also be noted that the interpo-
lation functions are defined over the entire mechanical structure, which makes the
consideration of irregular structures considerably more difficult. The same applies
for nonhomogeneous distributions of mass and stiffness. For this reason the sig-
nificance of the Ritz procedure lies not so much in its direct application, but rather
in the fact that it forms the basis of the finite elements method. Nonetheless, the
direct use of the Ritz procedure can make sense in some cases.
Galerkin method
As in the finite differences approach, this method, see for example Bathe [19], also
generates a set of ordinary differential equations from a partial differential equation:
L[φ] = r (6.54)
Where L is a linear differential operator, φ the sought-after solution, and r the exci-
tation function. The solution of the problem should correspond with the following
boundary conditions B i :
(6.55)
B i [φ] = q i | at the boundary of S i
A prerequisite here is that L is both symmetrical (6.56) and positive definite (6.57).
(L[u])v · dD = (L[v])u · dD (6.56)
D D
(L[u])udD > 0 (6.57)
D
Where u and v are arbitrary functions and D is the range of the operator. The solu-
tion should now be approximated as a linear combination of weighted interpolation
functions h i :
n
φ = a i h i (6.58)
i=1
