<s>
The	O
finite	B-Application
element	I-Application
method	I-Application
(	O
FEM	O
)	O
is	O
a	O
powerful	O
technique	O
originally	O
developed	O
for	O
numerical	O
solution	O
of	O
complex	O
problems	O
in	O
structural	O
mechanics	O
,	O
and	O
it	O
remains	O
the	O
method	O
of	O
choice	O
for	O
complex	O
systems	O
.	O
</s>
<s>
In	O
the	O
FEM	O
,	O
the	O
structural	O
system	O
is	O
modeled	O
by	O
a	O
set	O
of	O
appropriate	O
finite	B-Application
elements	I-Application
interconnected	O
at	O
discrete	O
points	O
called	O
nodes	O
.	O
</s>
<s>
Finite	B-Application
element	I-Application
concepts	O
were	O
developed	O
based	O
on	O
engineering	O
methods	O
in	O
1950s	O
.	O
</s>
<s>
The	O
finite	B-Application
element	I-Application
method	I-Application
obtained	O
its	O
real	O
impetus	O
in	O
the	O
1960s	O
and	O
1970s	O
by	O
John	O
Argyris	O
,	O
and	O
co-workers	O
;	O
at	O
the	O
University	O
of	O
Stuttgart	O
,	O
by	O
Ray	O
W	O
.	O
Clough	O
;	O
at	O
the	O
University	O
of	O
California	O
,	O
Berkeley	O
,	O
by	O
Olgierd	O
Zienkiewicz	O
,	O
and	O
co-workers	O
Ernest	O
Hinton	O
,	O
Bruce	O
Irons	O
;	O
at	O
the	O
University	O
of	O
Swansea	O
,	O
by	O
Philippe	O
G	O
.	O
Ciarlet	O
;	O
at	O
the	O
University	O
of	O
Paris	O
;	O
at	O
Cornell	O
University	O
,	O
by	O
Richard	O
Gallagher	O
and	O
co-workers	O
.	O
</s>
<s>
The	O
original	O
works	O
such	O
as	O
those	O
by	O
Argyris	O
and	O
Clough	O
became	O
the	O
foundation	O
for	O
today	O
’s	O
finite	B-Application
element	I-Application
structural	B-Algorithm
analysis	I-Algorithm
methods	O
.	O
</s>
<s>
Nodes	O
will	O
have	O
nodal	O
(	O
vector	O
)	O
displacements	O
or	O
degrees	O
of	O
freedom	O
which	O
may	O
include	O
translations	O
,	O
rotations	O
,	O
and	O
for	O
special	O
applications	O
,	O
higher	O
order	O
derivatives	B-Algorithm
of	O
displacements	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
displacements	O
of	O
any	O
points	O
in	O
the	O
element	O
will	O
be	O
interpolated	B-Algorithm
from	O
the	O
nodal	O
displacements	O
,	O
and	O
this	O
is	O
the	O
main	O
reason	O
for	O
the	O
approximate	O
nature	O
of	O
the	O
solution	O
.	O
</s>
<s>
While	O
the	O
theory	O
of	O
FEM	O
can	O
be	O
presented	O
in	O
different	O
perspectives	O
or	O
emphases	O
,	O
its	O
development	O
for	O
structural	B-Algorithm
analysis	I-Algorithm
follows	O
the	O
more	O
traditional	O
approach	O
via	O
the	O
virtual	O
work	O
principle	O
or	O
the	O
minimum	O
total	O
potential	O
energy	O
principle	O
.	O
</s>
<s>
The	O
latter	O
would	O
result	O
in	O
an	O
intractable	O
problem	O
,	O
hence	O
the	O
utility	O
of	O
the	O
finite	B-Application
element	I-Application
method	I-Application
.	O
</s>
<s>
The	O
displacements	O
at	O
any	O
other	O
point	O
of	O
the	O
element	O
may	O
be	O
found	O
by	O
the	O
use	O
of	O
interpolation	B-Algorithm
functions	O
as	O
,	O
symbolically	O
:	O
</s>
<s>
=	O
matrix	O
of	O
shape	O
functions	O
serving	O
as	O
interpolation	B-Algorithm
functions	O
.	O
</s>
<s>
These	O
matrices	O
are	O
usually	O
evaluated	O
numerically	O
using	O
Gaussian	B-Algorithm
quadrature	I-Algorithm
for	O
numerical	B-Algorithm
integration	I-Algorithm
.	O
</s>
<s>
Again	O
,	O
numerical	B-Algorithm
integration	I-Algorithm
is	O
convenient	O
for	O
their	O
evaluation	O
.	O
</s>
<s>
This	O
direct	O
addition	O
of	O
into	O
gives	O
the	O
procedure	O
the	O
name	O
Direct	B-Algorithm
Stiffness	I-Algorithm
Method	I-Algorithm
.	O
</s>
