Environmental Engineering Reference
In-Depth Information
2.1 SPH Interpolant
The main principle of the SPHmethod is that any given function
A
(
r
)
can be approx-
imated by the integral interpolant:
r
)
r
,
d
r
A
(
r
)
=
A
(
W
(
r
−
h
)
(1)
r
,
where
h
is the smoothing length and
W
is the weighting function known
as the kernel. Applying a Lagrangian approximation to the previous integral leads to
a discrete notation of the interpolation at a given point:
(
r
−
h
)
b
m
b
A
b
ˁ
b
A
(
r
)
=
W
(
r
−
r
b
,
h
),
(2)
with the summation index (
b
) running over all the particleswithin the region delimited
by the compact support of the kernel function.
m
b
and
ˁ
b
are the mass and the density
of the particle
b
, respectively,
V
b
=
m
b
/ˁ
b
represents the volume of a particle,
r
b
its
position vector, and
W
(
r
a
−
r
b
,
h
)
is the weighting function or the kernel referred
to particles
a
and
b
.
2.2 The Smoothing Kernel
The choice of the kernel is crucial for the performance of the SPH method. The
weighting functionmust satisfy the following conditions: positivity, compact support,
and normalization. Another property to be satisfied is that the selected function
must be monotonically decreasing with the distance from particle
a
. The kernels are
expressed as functions of the non-dimensional quantity
q
h
, where
r
is the
distance between particles
a
and
b
. The parameter
h
controls the size of the area
around particle
a
where the contribution of a given particle
b
cannot be neglected.
The different kernels that the user can choose in DualSPHysics are:
(a) Cubic-Spline:
=
r
/
⊧
⊨
3
2
q
2
3
4
q
3
1
−
+
0
≤
q
≤
1
,
1
3
W
(
r
,
h
)
=
ʱ
D
4
(
2
−
q
)
1
<
q
≤
2
,
(3)
⊩
>
0
q
2
(b) Wendland (Wendland
1995
):
)
=
ʱ
D
1
4
q
2
W
(
r
,
h
−
(
2
q
+
1
)
0
≤
q
≤
2
.
(4)
ʱ
D
values for the different kernel functions are shown in Table
1
.
The
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