<s>
In	O
mathematics	O
,	O
the	O
Trefftz	B-Algorithm
method	I-Algorithm
is	O
a	O
method	O
for	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
of	O
partial	O
differential	O
equations	O
named	O
after	O
the	O
German	O
mathematician	O
Erich	O
Trefftz(de )	O
(	O
1888	O
–	O
1937	O
)	O
.	O
</s>
<s>
It	O
falls	O
within	O
the	O
class	O
of	O
finite	B-Application
element	I-Application
methods	I-Application
.	O
</s>
<s>
The	O
hybrid	O
Trefftz	O
finite-element	B-Application
method	I-Application
has	O
been	O
considerably	O
advanced	O
since	O
its	O
introduction	O
about	O
30	O
years	O
ago	O
.	O
</s>
<s>
The	O
conventional	O
method	O
of	O
finite	B-Application
element	I-Application
analysis	I-Application
involves	O
converting	O
the	O
differential	O
equation	O
that	O
governs	O
the	O
problem	O
into	O
a	O
variational	B-Algorithm
functional	O
from	O
which	O
element	O
nodal	O
properties	O
–	O
known	O
as	O
field	O
variables	O
–	O
can	O
be	O
found	O
.	O
</s>
<s>
This	O
can	O
be	O
solved	O
by	O
substituting	O
in	O
approximate	O
solutions	O
to	O
the	O
differential	O
equation	O
and	O
generating	O
the	O
finite	B-Application
element	I-Application
stiffness	B-Algorithm
matrix	I-Algorithm
which	O
is	O
combined	O
with	O
all	O
the	O
elements	O
in	O
the	O
continuum	O
to	O
obtain	O
the	O
global	O
stiffness	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Application	O
of	O
the	O
relevant	O
boundary	O
conditions	O
to	O
this	O
global	O
matrix	O
,	O
and	O
the	O
subsequent	O
solution	O
of	O
the	O
field	O
variables	O
rounds	O
off	O
the	O
mathematical	O
process	O
,	O
following	O
which	O
numerical	B-General_Concept
computations	I-General_Concept
can	O
be	O
used	O
to	O
solve	O
real	O
life	O
engineering	O
problems	O
.	O
</s>
<s>
The	O
hybrid	O
Trefftz	B-Algorithm
method	I-Algorithm
differs	O
from	O
the	O
conventional	O
finite	B-Application
element	I-Application
method	I-Application
in	O
the	O
assumed	O
displacement	O
fields	O
and	O
the	O
formulation	O
of	O
the	O
variational	B-Algorithm
functional	O
.	O
</s>
<s>
In	O
contrast	O
to	O
the	O
conventional	O
method	O
(	O
based	O
on	O
the	O
Rayleigh-Ritz	O
mathematical	O
technique	O
)	O
the	O
Trefftz	B-Algorithm
method	I-Algorithm
(	O
based	O
on	O
the	O
Trefftz	O
mathematical	O
technique	O
)	O
assumes	O
the	O
displacement	O
field	O
is	O
composed	O
of	O
two	O
independent	O
components	O
;	O
the	O
intra-element	O
displacement	O
field	O
which	O
satisfies	O
the	O
governing	O
differential	O
equation	O
and	O
is	O
used	O
to	O
approximate	O
the	O
variation	O
of	O
potential	O
within	O
the	O
element	O
domain	O
,	O
and	O
the	O
conforming	O
frame	O
field	O
which	O
specifically	O
satisfies	O
the	O
inter-element	O
continuity	O
condition	O
,	O
defined	O
on	O
the	O
boundary	O
of	O
the	O
element	O
.	O
</s>
<s>
The	O
frame	O
field	O
here	O
is	O
the	O
same	O
as	O
that	O
used	O
in	O
the	O
conventional	O
finite	B-Application
element	I-Application
method	I-Application
but	O
defined	O
strictly	O
on	O
the	O
boundary	O
of	O
the	O
element	O
–	O
hence	O
the	O
use	O
of	O
the	O
term	O
"	O
hybrid	O
"	O
in	O
the	O
method	O
's	O
nomenclature	O
.	O
</s>
<s>
The	O
variational	B-Algorithm
functional	O
must	O
thus	O
include	O
additional	O
terms	O
to	O
account	O
for	O
boundary	O
conditions	O
,	O
since	O
the	O
assumed	O
solution	O
field	O
only	O
satisfies	O
the	O
governing	O
differential	O
equation	O
.	O
</s>
<s>
The	O
main	O
advantages	O
of	O
the	O
hybrid	O
Trefftz	B-Algorithm
method	I-Algorithm
over	O
the	O
conventional	O
method	O
are	O
:	O
</s>
<s>
Since	O
its	O
mainstream	O
introduction	O
some	O
30	O
years	O
ago	O
,	O
this	O
modified	O
finite	B-Application
element	I-Application
method	I-Application
has	O
become	O
increasingly	O
popular	O
to	O
applications	O
such	O
as	O
elasticity	O
,	O
Kirchhoff	O
plates	O
,	O
thick	O
plates	O
,	O
general	O
three-dimensional	O
solid	O
mechanics	O
,	O
antisymmetric	O
solid	O
mechanics	O
,	O
potential	O
problems	O
,	O
shells	O
,	O
elastodynamic	O
problems	O
,	O
geometrically	O
nonlinear	O
plate	O
bending	O
,	O
and	O
transient	O
heat	O
conduction	O
analysis	O
among	O
various	O
others	O
.	O
</s>
<s>
The	O
hybrid	O
Trefftz	B-Algorithm
method	I-Algorithm
is	O
also	O
being	O
applied	O
to	O
some	O
fields	O
,	O
e.g.	O
</s>
