<s>
Composite	O
methods	O
are	O
an	O
approach	O
applied	O
in	O
structural	B-Algorithm
dynamics	I-Algorithm
and	O
related	O
fields	O
.	O
</s>
<s>
The	O
existing	O
composite	O
methods	O
show	O
satisfactory	O
accuracy	O
and	O
powerful	O
numerical	O
dissipation	O
,	O
which	O
is	O
particularly	O
useful	O
for	O
solving	O
stiff	B-Algorithm
problems	I-Algorithm
and	O
differential-algebraic	O
equations	O
.	O
</s>
<s>
After	O
spatial	O
discretization	O
,	O
structural	B-Algorithm
dynamics	I-Algorithm
problems	O
are	O
generally	O
described	O
by	O
the	O
second-order	O
ordinary	O
differential	O
equation	O
:	O
</s>
<s>
Here	O
,	O
and	O
denote	O
the	O
displacement	O
,	O
velocity	O
and	O
acceleration	O
vectors	O
respectively	O
,	O
is	O
the	O
mass	B-General_Concept
matrix	I-General_Concept
,	O
is	O
the	O
damping	O
matrix	O
,	O
collects	O
the	O
internal	O
force	O
,	O
and	O
is	O
the	O
external	O
load	O
.	O
</s>
<s>
For	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
the	O
overall	O
time	O
domain	O
is	O
divided	O
into	O
a	O
series	O
of	O
time	O
steps	O
by	O
,	O
,	O
,	O
,	O
,	O
.	O
</s>
<s>
(	O
is	O
the	O
step	O
size	O
)	O
,	O
the	O
main	O
concept	O
of	O
composite	O
methods	O
is	O
to	O
subdivide	O
the	O
current	O
step	O
to	O
several	O
sub-steps	O
,	O
,	O
,	O
and	O
to	O
use	O
different	O
numerical	B-Algorithm
methods	I-Algorithm
in	O
each	O
sub-step	O
.	O
</s>
<s>
Although	O
there	O
are	O
lots	O
of	O
available	O
methods	O
,	O
see	O
the	O
review	O
,	O
the	O
existing	O
composite	O
methods	O
basically	O
employ	O
the	O
combination	O
of	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
and	O
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
In	O
the	O
first	O
sub-step	O
(	O
,	O
)	O
,	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
is	O
used	O
as	O
:	O
</s>
<s>
On	O
this	O
basis	O
,	O
to	O
acquire	O
prescribed	O
degree	O
of	O
numerical	O
dissipation	O
,	O
the	O
-Bathe	O
method	O
was	O
developed	O
by	O
replacing	O
the	O
3-point	O
Euler	B-Algorithm
backward	I-Algorithm
method	I-Algorithm
in	O
the	O
second	O
sub-step	O
with	O
a	O
general	O
formula	O
:	O
</s>
<s>
Following	O
the	O
idea	O
of	O
the	O
Bathe	O
method	O
,	O
the	O
three-sub-step	O
composite	O
methods	O
that	O
use	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
in	O
the	O
first	O
two	O
sub-steps	O
were	O
also	O
discussed	O
.	O
</s>
<s>
When	O
,	O
the	O
sub-family	O
with	O
shows	O
better	O
amplitude	B-Application
and	O
period	O
accuracy	O
than	O
the	O
-Bathe	O
method	O
under	O
the	O
same	O
computational	O
costs	O
,	O
and	O
the	O
sub-family	O
with	O
further	O
improves	O
the	O
period	O
accuracy	O
at	O
the	O
cost	O
of	O
lower	O
amplitude	B-Application
accuracy	O
.	O
</s>
<s>
Here	O
and	O
is	O
the	O
amplitude	B-Application
matrix	O
,	O
which	O
governs	O
the	O
properties	O
of	O
a	O
method	O
.	O
</s>
<s>
In	O
addition	O
to	O
the	O
accuracy	O
order	O
,	O
the	O
amplitude	B-Application
decay	O
ratio	O
and	O
period	O
elongation	O
ratio	O
are	O
also	O
usually	O
evaluated	O
to	O
measure	O
the	O
amplitude	B-Application
and	O
period	O
accuracy	O
in	O
the	O
low-frequency	O
content	O
.	O
</s>
<s>
Here	O
is	O
called	O
the	O
amplitude	B-Application
decay	O
ratio	O
,	O
and	O
(	O
)	O
is	O
called	O
the	O
period	O
elongation	O
ratio	O
.	O
</s>
<s>
With	O
and	O
,	O
the	O
spectral	O
radius	O
,	O
amplitude	B-Application
decay	O
ratio	O
,	O
and	O
period	O
elongation	O
ratio	O
are	O
shown	O
here	O
.	O
</s>
<s>
It	O
can	O
be	O
observed	O
that	O
this	O
method	O
can	O
provide	O
good	O
amplitude	B-Application
and	O
period	O
accuracy	O
in	O
the	O
low-frequency	O
content	O
,	O
while	O
strong	O
numerical	O
dissipation	O
,	O
as	O
,	O
in	O
the	O
high-frequency	O
content	O
.	O
</s>
