Meshfree and particle methods and their applications - TAM ...

6 Li and Liu: Meshfree and particle methods and applications Appl Mech Rev vol 55, no 1, January 2002
Monte Carlo estimates of integral interpolants. The error of
random interpolation was first estimated by Niedereiter 162
as being N 1 log N n1 where N is total particle number and
n is the dimension of space. This result was further improved
by Wozniakowski 163 as being N 1 log N n1/2 . According
to reference 104, ‘‘this remarkable result was produced
by a challenge with a payoff of sixty-four dollars !’’ Twentyone
years after its invention, in 1998 Di Lisio et al 164
gave a convergence proof of smoothed particle hydrodynamics
method for regularized Euler flow equations.
Besides consistency conditions, the conservation properties
of a SPH formulation also strongly influence its performance.
This has been a critical theme throughout SPH research,
see 12,104,120,145,155,165. It is well known that
classical SPH enjoys Galilean invariance, and if certain derivative
approximations, or Golden rules as Monaghan puts
it, are chosen, the corresponding SPH formulations can preserve
some discrete conservation laws. This issue was recently
revisited by Bonet et al 166, and they set forth a
discrete variational SPH formulation, which can automatically
satisfy the balance of linear momentum and balance of
angular momentum conservation laws. Here is the basic idea.
Assume the discrete potential energy in a SPH system is
x
I
V I 0 UJ I (18)
where V I 0 is the initial volume element, and U(J I ) is the
internal energy density, which is assumed to be the function
of determinant of the Jacobian—ratio between the initial and
current volume element,
J V I
V I
0 I 0
I
(19)
where I 0 and I are pointwise density in initial configuration
and in current configuration.
For adiabatic processes, the pressure can be obtained from
U I /J p I . Thus, the stationary condition of potential energy
gives
Fig. 3
The Ghost particle approach for boundary treatment
Dv V 0 I DU I v
I
I
m I m
J
J p I
2 p J
I J
2 W I x J v I (20)
where m I is the mass associated with particle I.
On the other hand,
Dv v
I x I
I
T I •v I (21)
I
where T is the internal force summation of stress. Then
through the variational principle, one can identify,
T I m I m
I
J p I
2 p J
I J
2 W I x J (22)
and establish the discrete SPH equation of motion balance
of linear momentum,
dv I
m I
dt m I m
I
J p I
2 p J
W
I x J . (23)
I
2.2.4 Boundary conditions
SPH, and in fact particle methods in general, have difficulties
in enforcing essential boundary condition. For SPH, some
effort has been devoted to address the issue. Takeda’s image
particle method 131 is designed to satisfy the no-slip
boundary condition; it is further generalized by Morris et al
128 to satisfy boundary conditions along a curved boundary.
Based on the same philosophy, Randles and Libersky
120 proposed a so-called ghost particle approach, which is
outlined as follows: Suppose particle i is a boundary particle.
All the other particles within its support, N(i), can be divided
into three subsets:
1 I(i): all the interior points that are the neighbors of i;
2 B(i): all the boundary points that are the neighbors of i;
3 G(i): all the exterior points that are the neighbors of i, ie,
all the ghost particles.
Therefore N(i)I(i)B(i)G(i). Figure 3 illustrates such
an arrangement.
In the ghost particle approach, the boundary correction
formula for general scalar field f is given as follows
f i f bc
J
2
f j f bc V j W ij
jI„i…
1
jB„i…
V j W ij
(24)
where f bc is the prescribed boundary value at xx i . One of
the advantages of the above formula is that the sampling
formula only depends on interior particles.
2.3 Other related issues and applications
Besides resolving the above fundamental issues, there have
been some other progresses in improving the performance of
SPH, which have focused on applications as well as algorithmic
efficiency. How to choose an interpolation kernel to ensure
successful simulations is discussed in 167; how to
modify the kernel functions without correction is discussed