Smoothed Particle Hydrodynamics – Part I
Dr. Jean Luc Lacome
INTRODUCTION
Smoothed Particle Hydrodynamics (SPH) is an N-body integration scheme developed by Lucy, Gingold and
Monaghan (1977). The method was developed to avoid the limitations of mesh tangling encountered in extreme
deformation problems with the finite element method. The main difference between classical methods and SPH is
the absence of a grid. Therefore, the particles are the computational framework on which the governing equations are
resolved. This paper is devoted to the presentation of this method. The first part will present the basic principles of
the CD. This new model requires a new calculation technique, which is briefly explained in the following. In part II,
we will develop the introduction of this new element in LD-DYNA and the coupling technique used with standard
finite element.
I - Definitions
The particle approximation of a function is:
∫
−=Π dyhyxWyfxf
h
),()()(
Where
W is the kernel function.
The Kernel function
W is defined using the function θ by the relation:
)(
)(
1
),( x
x
x
θ
d
h
hW =
Where
d is the number of space dimensions and h is the so called smoothing length which varies in time and in
space and
is the location of the particle.
x
),(
hW
x should be a centrally peaked function. The most common smoothing kernel used by the SPH community
is the cubic B-spline which is defined by choosing
θ as :
≤≤−
≤+−
×=
else 0
2//1for )2(
4
1
1//0for
4
3
2
3
1
)( uu
uuu
Cuθ
Where C is a constant of normalization that depends on the number of space dimensions.
hdhuadan
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