Environmental Engineering Reference
In-Depth Information
d
r
a
dt
=
v
a
(13)
These equations are time-integrated in DualSPHysics using one of the two following
schemes.
2.7.1 Verlet Scheme
This algorithm was proposed by Verlet (
1967
) and has two sets of equations. The
first one which is used in most of the iterations reads as follows
r
n
+
1
a
r
a
+
n
t
2
F
a
,
=
t
v
a
+
0
.
5
v
n
+
1
a
v
n
−
1
a
t
F
a
,
=
+
2
(14)
n
+
1
n
−
1
tD
a
,
ˁ
=
ˁ
+
2
a
a
and the second one is used every certain number of steps, normally once after 50 steps
r
n
+
1
a
r
a
+
n
t
2
F
a
,
=
t
v
a
+
0
.
5
v
n
+
1
v
a
+
t
F
a
,
=
(15)
a
n
+
1
n
tD
a
.
ˁ
=
ˁ
a
+
a
This prevents time integration divergence since the equations are no longer coupled
when considering only (
14
).
2.7.2 Symplectic Scheme
Symplectic time integration algorithms are time reversible in the absence of friction
or viscous effects (Leimkuhler et al.
1996
). This method preserves geometric fea-
tures like energy time-reversal symmetry, leading to improved resolution of long term
solution behaviour. In this case, the scheme used is an explicit Symplectic scheme
of the form
n
a
a
+
t
2
d
ˁ
1
2
n
+
n
ˁ
=
ˁ
d
t
,
(16)
a
d
r
a
r
a
+
t
1
2
r
n
+
=
d
t
.
a
2
1
2
n
+
In a second time step
(
d
(ˉ
a
ˁ
a
v
a
)
)/
dt
gives the velocity and the position of the
particles at the end of each time step:
2
n
+
+
t
d
(ˉ
a
ˁ
a
v
a
)
1
2
n
+
1
n
+
(ˉ
a
ˁ
a
v
a
)
=
(ˉ
a
ˁ
a
v
)
,
(17)
2
dt
+
t
2
v
n
+
1
1
2
r
n
+
r
n
+
1
a
=
.
a
a
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