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10.5.5
General Remarks
The most fundamental decisions involved in using the direct integration method concern
the choice of (1) the appropriate integration method and (2) the appropriate integration
step. These choices play an important role in the control of the cost of the analysis. The
efficiency, stability, and accuracy should be taken into account for the effective use of a
direct integration method.
Efficiency
The basic difference between the explicit and implicit methods is that the explicit integra-
tion method evaluates the equilibrium at t
=
n
t, while the implicit integration method
= (
+
)
.
evaluates the equilibrium at t
n
1
t
Both of these methods need to solve a set of
simultaneous linear equations KV n + 1 =
P n + 1 , where K is the effective stiffness matrix. No
iteration at a particular time step is needed. In the implicit method, the stiffness matrix K
is in K , while in the explicit method, K is not a function of K .Ifthefinite element model is
very large with a large bandwidth, it can be more efficient to use the explicit method with a
lumped mass matrix, so that the effective stiffness matrix does not need to be triangularized.
Of course, the time step should be smaller than the critical time step.
Stability
Some remarks about the stability of the different integration methods are provided in
Sections 10.5.1 through 10.5.4. In these discussions, only a single DOF system is analyzed;
however, the results are assumed to apply to a multi-DOF system. The validity of this
assumption follows from the relationship between the mode superposition and the direct
integration methods. For a system governed by the equations
M V
C V
+
+
KV
=
P
(10.126)
and with a proportional damping matrix, i.e., C
= α
M
+ β
K , the principal coordinate
transformation leads to
+
˙
+
=
q
q
Λq
P
(10.127a)
or
2
q i
+
2
ζ
ω
q i
˙
+ ω
i q i
=
P i ,
i
=
1 , 2 ,
...
,n d
(10.127b)
i
i
i , respectively.
If these “single-DOF-like” equations are integrated directly, the results would be the same
as for the integration of Eq. (10.126) with the same time step. It is reasoned then that the
stability criteria (developed in Section 10.5.1 for single DOF systems) for Eq. (10.127) apply
to Eq. (10.126) as well. For the central difference method, the integration is conditionally
stable. The integration time step should be smaller than the critical time step
ζ
ω
ω
where and Λ are diagonal matrices with the diagonal elements 2
i and
i
t crit =
2
max ,
2 max is the largest diagonal element in Λ
where
ω
.
For the implicit integration methods, the
integration is unconditionally stable.
Accuracy
The analysis of accuracy provides the rationale for selecting the time step
t . “Accuracy”
refers to the difference between the numerical solution and the exact solution when the
numerical solution process is stable. For the central difference method, the time step
t must
be smaller than
t crit , the
result of the integration is usually fairly accurate, i.e., the difference between the numerical
solution and exact solution is small. For the implicit integration, which is numerically
t crit in order for the integration scheme to be stable. When
t
<
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