J. Shanghai Jiaotong Univ. (Sci.), 2008, 13(2): 136–138

DOI: 10.1007/s12204-008-0136-2

A Method to Improve First Order Approximation ofSmoothed Particle Hydrodynamics

CHEN Si (陈 思), ZHOU Dai∗ (周 岱), BAO Yan (包 艳), DONG Shi-lin (董石麟)(School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University,

Shanghai 200240, China)

Abstract: Smoothed particle hydrodynamics (SPH) is a useful meshless method. The first and second ordersare the most popular derivatives of the field function in the mechanical governing equations. New methodswere proposed to improve accuracy of SPH approximation by the lemma proved. The lemma describes therelationship of functions and their SPH approximation. Finally, the error comparison of SPH method with orwithout our improvement was carried out.Key words: smoothed particle hydrodynamics; first order derivatives; accuracy; error comparisonCLC number: TU 393.3 Document code: A

Introduction

SPH was introduced by Lucy[1] and Gingold[2] whoseaim is to simulate astrophysical phenomena which in-volve large density contrasts and complicated non-symmetric geometry. The key idea of SPH which isa fully Lagrangian particle method, is that the fluid (orsolid) can be divided into arbitrarily distributed parti-cles. So SPH could achieve numerical solutions withouta grid.

As a mesh-less computational method, so far, SPHhas been successfully applied to a wide range of prob-lems. For instance, it was introduced to the area ofcomputational fluid dynamics[3] in 1978 and introducedto solid mechanics problems[4] in 1991. Moreover, it wasalso extended to free surface flows, explosion phenom-ena, heat transfer and so on[5−9].

1 Preliminary Theory

In the standard SPH method, each function is repre-sented by its integral interpolate, which is defined by

〈f(r)〉 ≡∫

f(r′)W (r − r′, h)dr′, (1)

where in 1D, 2D and 3D space, r and r′ are scalars, 2Dvectors and 3D vectors, respectively, W is the smooth-ing kernel function, h is the smoothing length definingthe influence area of the smoothing function W, thesymbol 〈 〉 was introduced to denote the convolution.

Received date: 2007-01-12Foundation item: The National Natural Science Founda-

tion of China (No. 50778111); The Key Project of Fundof Science and Technology Development of Shanghai(No. 07JC14023)

∗E-mail: zhoudai@sjtu.edu.cn

In Eq. (1), the integration is over the entire space andW is an interpolating kernel which is usually chosen tobe an even function and has the four properties[10,11].The first one is the normalization condition that states∫

W (r − r′, h)dr′ = 1. (2)

The second condition is the Delta function property,that is

limh→0

W (r − r′, h) = δ(r − r′). (3)

The third condition is the compact condition

W (r − r′, h) = 0, |r − r′| > κh, (4)

where, κ is a constant related to the smoothing func-tion for point at r, which defines the non-zero area ofthe smoothing function[10].

The fourth condition is the nonnegative condition

W (r, h) � 0. (5)

In Eq. (3), δ is Dirac function given by

δ(r) =

{+∞, r = 00, r �= 0

, (6)

and ∫δ(r)dr = 1. (7)

Using Eqs. (6) and (7), we can easily obtain

f(r) =∫

f(r′)δ(r′ − r)dr′. (8)

Then we have

limh→0

〈f(r)〉 = f(r). (9)

J. Shanghai Jiaotong Univ. (Sci.), 2008, 13(2): 136–138 137

Another key operation in the SPH method is parti-cle approximation, by which the entire system is repre-sented by a finite number of particles that carry indi-vidual mass and occupy individual space. Using parti-cle approximation, the integral interpolation is approx-imated by a summation interpolation[10]

fs(r) =n∑

j=1

mjf(rj)ρj

W (r − rj , h), (10)

where the summation index j denotes a particle label,and the summation is over all the particles.

If the smoothing functions vanish on the surface ofits support domain

W (r − r′, h)|S = 0, (11)

the first derivative of 〈f(r)〉 is

〈f ′(r)〉 =∫

f(r′)W ′(r′ − r, h)dr′, (12)

and the first derivative of fs(r) is

f ′s(r) =

n∑j=1

mjf(rj)ρj

W ′(r − rj , h). (13)

2 Improvement of SPH Approximation

Lemma Let f(r) ∈ span(1, r, r2) and let particlesin the support domain of W (r0−r′, h) dispose symmet-rically to r0, we have

f ′s(r0) = C1(h) · f ′(r0), (14)

where the constant C1(h) is only relevant to h and thekind of kernel W (r, h).

Proof

∵ f(r) ∈ span(1, r, r2), (15)

∴ f(r) = f(r0) + (r − r0)f ′(r0) + (r − r0)2f ′′(r0).(16)

From Eq. (13), we obtain

f ′s(r0) =

n∑j=1

mjf(r0)+(rj−r0)f ′(r0)+(rj−r0)2f ′′(r0)

ρj

· W ′(rj − r0, h). (17)

Expanding Eq. (17), we have

f ′s(r0) = f(r0)

⎡⎣ n∑

j=1

mj

ρjW ′(rj − r0, h)

⎤⎦

+ f ′(r0)

⎡⎣ n∑

j=1

mj(rj − r0)ρj

W ′(rj − r0, h)

⎤⎦

+ f ′′(r0)

⎡⎣ n∑

j=1

mj(rj − r0)2

ρjW ′(rj − r0, h)

⎤⎦ . (18)

Because the particles in the support domain ofW (r0 − r′, h) dispose symmetrically to r0, we have

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

n∑j=1

mj

ρjW ′(rj − r0, h) = 0

n∑j=1

mj(rj − r0)2

ρjW ′(rj − r0, h) = 0

. (19)

Then Eq. (18) becomes

f ′s(r0) = f ′(r0)

⎡⎣ n∑

j=1

mj(rj − r0)ρj

W ′(rj − r0, h)

⎤⎦ .

(20)

Noticing that

n∑j=1

mj(rj − r0)ρj

W ′(rj − r0, h) = C1(h), (21)

where the constant C1(h) is only relevant to h and thekind of kernel W (r, h), so we get

f ′s(r0) = C1(h) · f ′(r0). (22)

To get Eq. (22), we assume that particles in the sup-port domain of W (r − ri, h) dispose symmetrically tori, which can be ensured by moving least square (MLS)method. By the computation, however, this conditioncan be omitted, if we rewrite Eq. (22) as

f ′s(r0) = C∗

1 (h, x0, y0) · f ′(r0), (23)

where

C∗1 (h, x0, y0) =

1n∑

j=1

mjxj + yj

ρjW (r − rj , h)

. (24)

3 Computation

Let set X and set Y be a table of number generatedrandomly by computer respectively, set Ω is defined asthe direct product of X and Y

Ω ≡ X × Y = {(x, y)|x ∈ X, y ∈ Y }. (25)

With each point in set Ω, there is a associated realnumber ci

ci = f(xi, yi) = exi+yi . (26)

138 J. Shanghai Jiaotong Univ. (Sci.), 2008, 13(2): 136–138

The kernel W in Eq. (12) is

W (R, h) =7

478π

×

⎧⎪⎪⎪⎨⎪⎪⎪⎩

(3−R)5−6(2−R)5+15(1−R)5, 0�R<1(3−R)5−6(2−R)5, 1�R<2(3−R)5, 2�R<30, R�3

,

(27)

where

R =

√(x − x0)2 + (y − y0)2

h. (28)

Fig. 1 The results computed by standard SPH method

Fig. 2 The results computed by the present method

The error comparisons between the standard SPHmethod and the method provided by the present paperare given in Figs. 1 and 2. In Fig.1, all the errors areless than zero. And in Fig. 2, the errors on some pointsare less than zero while on other points are great thanzero.

4 Conclusion

The accuracy of SPH approximation can improve atleast one order of magnitude by using our method. Ifthe repeated computation is needed in some mechanicsproblem, the accumulative error by the present methodshould be less than the standard SPH method.

References

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