10 2. DEFINITION OF TEST ARTICLE FINITE ELEMENT MODELS
is linearly related to the assumed displacement field employing the appropriate partial deriva-
tives. It should be noted that exact closed form solutions of partial differential equations are often
expressed in variable separable form, whenever such solutions are possible. By assuming a series
of functions that satisfy particular boundary conditions (or generally permit solution of natural
boundary conditions), substitution of Equation (2.3) into Hamilton’s principle (Equation 2.1),
the following symmetric matrix equations are deduced:
Œ
M
f
Rq
g
C
Œ
K
f
q
g
D
Œ
f
Q
g
; (2.5)
where the positive semi-definite, symmetric mass, and stiffness matrix terms are
M
mn
D
Z
R
‘‘” ‰
m
‰
n
dR; K
mn
D
Z
R
‘‘E” ‰
";m
‰
";n
dR: (2.6)
[Note: “” and “E” are representative of mass density and elastic stiffness material properties.] In
addition, the generalized forcing terms are governed by volume and surface integrals associated
with applied body forces and surface loads, respectively.
e Ritz method, outlined above, was initially employed to approximately solve difficult
problems described by partial differential equations and associated natural boundary conditions.
Ultimately, it was extensively applied in development of the Finite Element Method [8] and
Matrix Structural Analysis [9].
2.1.6 GALERKIN’S METHOD
Galerkin [10] defined an approximate method using the variable separable displacement field
and associated generalized coordinates (Equations (2.3) and (2.4)) by substitution of the as-
sumed functions into Equation (2.2) where the boundary conditions are automatically satisfied
by choice of an appropriate set of spatial functions. e general statement of Galerkin’s method
is
Z
R
.
P.D.E.
/
ıu dR D 0: (2.7)
An appealing aspect of Galerkin’s method is that it can be applied to any set of partial
differential equations (even if a suitable variational formulation is unknown). e method has
been successfully applied in the study of nonlinear dynamic systems [11].
2.1.7 TREFFTZ’S METHOD
Trefftz [12] proposed an approximate method that employs a set of assumed functions that
automatically satisfy the partial differential equations. erefore, the form of Equation (2.2)
that must be satisfied is
Z
S
.
N.B.C.
/
ıu dS D 0: (2.8)
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