Mesh-Free Computational Shock and Fracture

u
#
Stability of DPD (Randles ( Randles)
System: Linear, forward Euler
n 1
2
m
( ) 2 n+
1
! •
n+
1 n $ t
m = u m + "
+ m
#
n+
1
m
= # + " tC :
n
m
( ) 1 n+
! u
Spatial derivatives: MLS
n+
1
2 ( # •%
) = % • $ w
m
!
s"
N ( m)
n+
1 ( # u)
= u • $ w
s !
m"
N(
s)
Assume: Plane harmonic wave solution
u
"
n
m
n
s
= u !
o
= " !
o
n
e
ik•x
n ik x
e •
On substitution:
1
% t 2 ( 1! # ) u o + $ # " oA
m = 0
o
1
2 ( ! " ) $ + # t " C : u B = 0
1 o
o s
m
s
s
A = ( $ ( e
" % •x
Eliminate to put in terms of u0
)
[ G f(
) I]
u 0 o = • ! "
/ e
" % •x
)
g =
#
"
f
( ! )
f is function of time stepping method
G is a function of local geometry
and B = ( $ ( e
" % •x
)
/ e
" % •x
)
!