<s>
In	O
mathematics	O
the	O
discrete	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
meshless	I-Algorithm
(	O
DLSM	B-Algorithm
)	O
method	O
is	O
a	O
meshless	B-Algorithm
method	I-Algorithm
based	O
on	O
the	O
least	B-Algorithm
squares	I-Algorithm
concept	O
.	O
</s>
<s>
The	O
method	O
is	O
based	O
on	O
the	O
minimization	O
of	O
a	O
least	B-Algorithm
squares	I-Algorithm
functional	O
,	O
defined	O
as	O
the	O
weighted	B-Algorithm
summation	I-Algorithm
of	O
the	O
squared	O
residual	O
of	O
the	O
governing	O
differential	O
equation	O
and	O
its	O
boundary	O
conditions	O
at	O
nodal	O
points	O
used	O
to	O
discretize	O
the	O
domain	O
and	O
its	O
boundaries	O
.	O
</s>
<s>
While	O
most	O
of	O
the	O
existing	O
meshless	B-Algorithm
methods	I-Algorithm
need	O
background	O
cells	O
for	O
numerical	B-Algorithm
integration	I-Algorithm
,	O
DLSM	B-Algorithm
did	O
not	O
require	O
a	O
numerical	B-Algorithm
integration	I-Algorithm
procedure	O
due	O
to	O
the	O
use	O
of	O
the	O
discrete	O
least	B-Algorithm
squares	I-Algorithm
method	I-Algorithm
to	O
discretize	O
the	O
governing	O
differential	O
equation	O
.	O
</s>
<s>
A	O
Moving	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
(	O
MLS	O
)	O
approximation	O
method	O
is	O
used	O
to	O
construct	O
the	O
shape	O
function	O
,	O
making	O
the	O
approach	O
a	O
fully	O
least	O
squares-based	O
approach	O
.	O
</s>
<s>
Arzani	O
and	O
Afshar	O
developed	O
the	O
DLSM	B-Algorithm
method	O
in	O
2006	O
for	O
the	O
solution	O
of	O
Poisson	O
's	O
equation	O
.	O
</s>
<s>
Firoozjaee	O
and	O
Afshar	O
proposed	O
the	O
collocated	O
discrete	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
meshless	I-Algorithm
(	O
CDLSM	O
)	O
method	O
to	O
solve	O
elliptic	O
partial	O
differential	O
equations	O
,	O
and	O
studied	O
the	O
effect	O
of	O
the	O
collocation	B-Algorithm
points	I-Algorithm
on	O
the	O
convergence	O
and	O
accuracy	O
of	O
the	O
method	O
.	O
</s>
<s>
The	O
method	O
can	O
be	O
considered	O
as	O
an	O
extension	O
the	O
earlier	O
method	O
of	O
DLSM	B-Algorithm
by	O
the	O
introduction	O
of	O
a	O
set	O
of	O
collocation	B-Algorithm
points	I-Algorithm
for	O
the	O
calculation	O
of	O
the	O
least	B-Algorithm
squares	I-Algorithm
functional	O
.	O
</s>
<s>
to	O
solve	O
elasticity	B-Algorithm
problems	O
regarding	O
the	O
irregular	O
distribution	O
of	O
nodal	O
points	O
.	O
</s>
<s>
Afshar	O
and	O
Lashckarbolok	O
used	O
the	O
CDLSM	O
method	O
for	O
the	O
adaptive	B-Algorithm
simulation	O
of	O
hyperbolic	O
problems	O
.	O
</s>
<s>
A	O
simple	O
a	O
posteriori	O
error	O
indicator	O
based	O
on	O
the	O
value	O
of	O
the	O
least	B-Algorithm
squares	I-Algorithm
functional	O
and	O
a	O
node	O
moving	O
strategy	O
was	O
used	O
and	O
tested	O
on	O
1-D	O
hyperbolic	O
problems	O
.	O
</s>
<s>
Shobeyri	O
and	O
Afshar	O
simulated	O
free	O
surface	O
problems	O
using	O
the	O
DLSM	B-Algorithm
method	O
.	O
</s>
<s>
The	O
method	O
was	O
then	O
extended	O
for	O
adaptive	B-Algorithm
simulation	O
of	O
two-dimensional	O
shocked	O
hyperbolic	O
problems	O
by	O
Afshar	O
and	O
Firoozjaee	O
.	O
</s>
<s>
Also	O
,	O
adaptive	B-Algorithm
node-moving	O
refinement	O
and	O
multi-stage	O
node	O
enrichment	O
adaptive	B-Algorithm
refinement	O
are	O
formulated	O
in	O
the	O
DLSM	B-Algorithm
for	O
the	O
solution	O
of	O
elasticity	B-Algorithm
problems	O
.	O
</s>
<s>
proposed	O
mixed	O
discrete	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
meshless	I-Algorithm
(	O
MDLSM	O
)	O
formulation	O
for	O
solution	O
of	O
planar	O
elasticity	B-Algorithm
problems	O
.	O
</s>
<s>
In	O
this	O
approach	O
,	O
the	O
differential	O
equations	O
governing	O
the	O
planar	O
elasticity	B-Algorithm
problems	O
are	O
written	O
in	O
terms	O
of	O
the	O
stresses	O
and	O
displacements	O
which	O
are	O
approximated	O
independently	O
using	O
the	O
same	O
shape	O
functions	O
.	O
</s>
<s>
Since	O
the	O
resulting	O
governing	O
equations	O
are	O
of	O
the	O
first	B-Algorithm
order	I-Algorithm
,	O
both	O
the	O
displacement	O
and	O
stress	O
boundary	O
conditions	O
are	O
of	O
the	O
Dirichlet	O
type	O
,	O
which	O
is	O
easily	O
incorporated	O
via	O
a	O
penalty	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Because	O
this	O
is	O
a	O
least	B-Algorithm
squares	I-Algorithm
based	O
algorithm	O
of	O
the	O
MDLSM	O
method	O
,	O
the	O
proposed	O
method	O
does	O
not	O
need	O
to	O
be	O
satisfied	O
by	O
the	O
Ladyzhenskaya	O
–	O
Babuška	O
–	O
Brezzi	O
(	O
LBB	O
)	O
condition	O
.	O
</s>
