<s>
The	O
boundary	B-Algorithm
element	I-Algorithm
method	I-Algorithm
(	O
BEM	O
)	O
is	O
a	O
numerical	O
computational	O
method	O
of	O
solving	O
linear	O
partial	O
differential	O
equations	O
which	O
have	O
been	O
formulated	O
as	O
integral	B-Algorithm
equations	I-Algorithm
(	O
i.e.	O
</s>
<s>
in	O
boundary	O
integral	O
form	O
)	O
,	O
including	O
fluid	O
mechanics	O
,	O
acoustics	O
,	O
electromagnetics	O
(	O
where	O
the	O
technique	O
is	O
known	O
as	O
method	B-Algorithm
of	I-Algorithm
moments	I-Algorithm
or	O
abbreviated	O
as	O
MoM	O
)	O
,	O
fracture	O
mechanics	O
,	O
and	O
contact	O
mechanics	O
.	O
</s>
<s>
The	O
integral	B-Algorithm
equation	I-Algorithm
may	O
be	O
regarded	O
as	O
an	O
exact	O
solution	O
of	O
the	O
governing	O
partial	O
differential	O
equation	O
.	O
</s>
<s>
The	O
boundary	B-Algorithm
element	I-Algorithm
method	I-Algorithm
attempts	O
to	O
use	O
the	O
given	O
boundary	O
conditions	O
to	O
fit	O
boundary	O
values	O
into	O
the	O
integral	B-Algorithm
equation	I-Algorithm
,	O
rather	O
than	O
values	O
throughout	O
the	O
space	O
defined	O
by	O
a	O
partial	O
differential	O
equation	O
.	O
</s>
<s>
Once	O
this	O
is	O
done	O
,	O
in	O
the	O
post-processing	O
stage	O
,	O
the	O
integral	B-Algorithm
equation	I-Algorithm
can	O
then	O
be	O
used	O
again	O
to	O
calculate	O
numerically	O
the	O
solution	O
directly	O
at	O
any	O
desired	O
point	O
in	O
the	O
interior	O
of	O
the	O
solution	O
domain	O
.	O
</s>
<s>
This	O
places	O
considerable	O
restrictions	O
on	O
the	O
range	O
and	O
generality	O
of	O
problems	O
to	O
which	O
boundary	B-Algorithm
elements	I-Algorithm
can	O
usefully	O
be	O
applied	O
.	O
</s>
<s>
Nonlinearities	O
can	O
be	O
included	O
in	O
the	O
formulation	O
,	O
although	O
they	O
will	O
generally	O
introduce	O
volume	O
integrals	O
which	O
then	O
require	O
the	O
volume	O
to	O
be	O
discretised	B-Algorithm
before	O
solution	O
can	O
be	O
attempted	O
,	O
removing	O
one	O
of	O
the	O
most	O
often	O
cited	O
advantages	O
of	O
BEM	O
.	O
</s>
<s>
The	O
technique	O
approximates	O
part	O
of	O
the	O
integrand	O
using	O
radial	B-Algorithm
basis	I-Algorithm
functions	I-Algorithm
(	O
local	O
interpolating	O
functions	O
)	O
and	O
converts	O
the	O
volume	O
integral	O
into	O
boundary	O
integral	O
after	O
collocating	O
at	O
selected	O
points	O
distributed	O
throughout	O
the	O
volume	O
domain	O
(	O
including	O
the	O
boundary	O
)	O
.	O
</s>
<s>
In	O
the	O
dual-reciprocity	O
BEM	O
,	O
although	O
there	O
is	O
no	O
need	O
to	O
discretize	B-Algorithm
the	O
volume	O
into	O
meshes	B-Algorithm
,	O
unknowns	O
at	O
chosen	O
points	O
inside	O
the	O
solution	O
domain	O
are	O
involved	O
in	O
the	O
linear	O
algebraic	O
equations	O
approximating	O
the	O
problem	O
being	O
considered	O
.	O
</s>
<s>
The	O
Green	O
's	O
function	O
elements	O
connecting	O
pairs	O
of	O
source	O
and	O
field	O
patches	O
defined	O
by	O
the	O
mesh	B-Algorithm
form	O
a	O
matrix	O
,	O
which	O
is	O
solved	O
numerically	O
.	O
</s>
<s>
The	O
form	O
of	O
the	O
method	O
in	O
which	O
the	O
integrals	O
over	O
the	O
source	O
and	O
field	O
patches	O
are	O
the	O
same	O
is	O
called	O
"	O
Galerkin	B-Algorithm
's	I-Algorithm
method	I-Algorithm
"	O
.	O
</s>
<s>
Galerkin	B-Algorithm
's	I-Algorithm
method	I-Algorithm
is	O
the	O
obvious	O
approach	O
for	O
problems	O
which	O
are	O
symmetrical	O
with	O
respect	O
to	O
exchanging	O
the	O
source	O
and	O
field	O
points	O
.	O
</s>
<s>
It	O
is	O
this	O
feature	O
that	O
is	O
typically	O
employed	O
in	O
schemes	O
designed	O
to	O
accelerate	O
boundary	B-Algorithm
element	I-Algorithm
problem	O
calculations	O
.	O
</s>
<s>
Derivation	O
of	O
closed-form	O
Green	O
's	O
functions	O
is	O
of	O
particular	O
interest	O
in	O
boundary	B-Algorithm
element	I-Algorithm
method	I-Algorithm
,	O
especially	O
in	O
electromagnetics	O
.	O
</s>
<s>
For	O
a	O
robust	O
analysis	O
,	O
spatial	O
Green	O
's	O
functions	O
are	O
approximated	O
as	O
complex	O
exponentials	O
with	O
methods	O
such	O
as	O
Prony	B-Algorithm
's	I-Algorithm
method	I-Algorithm
or	O
generalized	B-Algorithm
pencil	I-Algorithm
of	I-Algorithm
function	I-Algorithm
,	O
and	O
the	O
integral	O
is	O
evaluated	O
with	O
Sommerfeld	O
identity	O
.	O
</s>
<s>
The	O
boundary	B-Algorithm
element	I-Algorithm
method	I-Algorithm
is	O
often	O
more	O
efficient	O
than	O
other	O
methods	O
,	O
including	O
finite	B-Application
elements	I-Application
,	O
in	O
terms	O
of	O
computational	O
resources	O
for	O
problems	O
where	O
there	O
is	O
a	O
small	O
surface/volume	O
ratio	O
.	O
</s>
<s>
Conceptually	O
,	O
it	O
works	O
by	O
constructing	O
a	O
"	O
mesh	B-Algorithm
"	O
over	O
the	O
modelled	O
surface	O
.	O
</s>
<s>
However	O
,	O
for	O
many	O
problems	O
boundary	B-Algorithm
element	I-Algorithm
methods	I-Algorithm
are	O
significantly	O
less	O
efficient	O
than	O
volume-discretisation	O
methods	O
(	O
finite	B-Application
element	I-Application
method	I-Application
,	O
finite	B-Algorithm
difference	I-Algorithm
method	I-Algorithm
,	O
finite	B-Algorithm
volume	I-Algorithm
method	I-Algorithm
)	O
.	O
</s>
<s>
A	O
good	O
example	O
of	O
application	O
of	O
the	O
boundary	B-Algorithm
element	I-Algorithm
method	I-Algorithm
is	O
efficient	O
calculation	O
of	O
natural	O
frequencies	O
of	O
liquid	O
sloshing	O
in	O
tanks	O
.	O
</s>
<s>
Boundary	B-Algorithm
element	I-Algorithm
method	I-Algorithm
is	O
one	O
of	O
the	O
most	O
effective	O
methods	O
for	O
numerical	O
simulation	O
of	O
contact	O
problems	O
,	O
in	O
particular	O
for	O
simulation	O
of	O
adhesive	O
contacts	O
.	O
</s>
<s>
Boundary	B-Algorithm
element	I-Algorithm
formulations	O
typically	O
give	O
rise	O
to	O
fully	O
populated	O
matrices	O
.	O
</s>
<s>
By	O
contrast	O
,	O
finite	B-Application
element	I-Application
matrices	O
are	O
typically	O
banded	O
(	O
elements	O
are	O
only	O
locally	O
connected	O
)	O
and	O
the	O
storage	O
requirements	O
for	O
the	O
system	O
matrices	O
typically	O
grow	O
quite	O
linearly	O
with	O
the	O
problem	O
size	O
.	O
</s>
