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Research Papers

# A Simulation of Domain Decomposition Method for Smoothed Particle HydrodynamicsOPEN ACCESS

[+] Author and Article Information
Taehyo Park

Professor
Computational Solid and Structural
Mechanics Laboratory,
Department of Civil and Environmental
Engineering,
Hanyang University,
222 Wangsimni-ro,
Seongdong-gu, Seoul 04763, South Korea
e-mail: cepark@hanyang.ac.kr

Shengjie Li

Computational Solid and Structural Mechanics
Laboratory,
Department of Civil and Environmental
Engineering,
Hanyang University,
222 Wangsimni-ro,
Seongdong-gu, Seoul 04763, South Korea
e-mail: lee901127@hanyang.ac.kr

Mina Lee

KORAIL,
240 Jungangno,
Dong-gu, Daejeon 34618, South Korea
e-mail: minalee413@naver.com

Moonho Tak

Computational Solid and Structural Mechanics
Laboratory,
Department of Civil and Environmental
Engineering,
Hanyang University,
222 Wangsimni-ro,
Seongdong-gu, Seoul 04763, South Korea
e-mail: pivotman@hanyang.ac.kr

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received June 1, 2016; final manuscript received November 2, 2016; published online February 7, 2017. Assoc. Editor: Xi Chen.

J. Eng. Mater. Technol 139(2), 021010 (Feb 07, 2017) (7 pages) Paper No: MATS-16-1163; doi: 10.1115/1.4035486 History: Received June 01, 2016; Revised November 02, 2016

## Abstract

Nowadays, the numerical method has become a very important approach for solving complex problems in engineering and science. Some grid-based methods such as the finite difference method (FDM) and finite element method (FEM) have already been widely applied to various areas; however, they still suffer from inherent difficulties which limit their applications to many problems. Therefore, a strong interest is focused on the meshfree methods such as smoothed particle hydrodynamics (SPH) to simulate fluid flow recently due to the advantages in dealing with some complicated problems. In the SPH method, a great number of particles will be used because the whole domain is represented by a set of arbitrarily distributed particles. To improve the numerical efficiency, parallelization using message-passing interface (MPI) is applied to the problems with the large computational domain. In parallel computing, the whole domain is decomposed by the parallel method for continuity of subdomain boundary under the single instruction multiple data (SIMD) and also based on the procedure of the SPH computations. In this work, a new scheme of parallel computing is employed into the SPH method to analyze SPH particle fluid. In this scheme, the whole domain is decomposed into subdomains under the SIMD process and it composes the boundary conditions to the interface particles which will improve the detection of neighbor particles near the boundary. With the method of parallel computing, the SPH method is to be more flexible and perform better.

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## Introduction

Generally, fluid flow problems are solved by employing grid-based methods such as FDM and FEM. However, when dealing with free surface, deformable boundary, large deformation, and complex mesh generation, these methods can encounter some difficulties. Therefore, a number of meshfree methods are proposed to analyze fluid flow problems [1,2].

Smoothed particle hydrodynamics (SPH) is one of the most widely applied meshfree methods. It is attractive for its adaptive and pure Lagrange nature [3,4]. SPH was first introduced by Lucy [5] and Gingold and Monaghan [6] for simulating fluid dynamics in astrophysics field and subsequently extended to the applications of continuum solid and fluid mechanics [79]. In SPH method, the problem domain is represented by a set of arbitrarily distributed particles without any grid. Each particle possesses individual mass, momentum, and other material properties, and these properties are approximated by summing up the values of the particles in the support domain with time. To enhance the computational performance, parallel computing techniques are usually applied to the SPH simulation.

Parallel computing reduces the time needed to solve a single computational problem by using parallel computers [10]. In the parallel computing, the MPI is needed to communicate with other processors. MPI is the most popular communication protocol for the parallel application programs in the network computers. It is used between processes to deliver its own data each other. At the international supercomputing conference held in 1992, the necessity of MPI came to the fore and version 1.0 of MPI was finally released in 1994 [11]. Until now, MPI has developed version 2.2, which includes dynamic process management.

Domain decomposition is an important step in parallel computing. Morris et al. [12], Wu and Tseng [13], and Holmes et al. [14] presented some suitable methods for domain decomposition in the SPH method. Liu and Liu [3] described some approaches for parallelization of the SPH method such as the particle-based decomposition, the domain-based decomposition, and the operation-based decomposition. This paper proposes a new scheme of load balance simulation of domain decomposition with interface particles located on the boundary. This scheme improves the efficiency in detecting neighboring particles and thus reduces the number of iteration significantly.

## Smoothed Particle Hydrodynamics

There are two approaches for describing the physical governing equations: the Eulerian description and the Lagrangian description [15,16]. SPH is a Lagrangian particle method whose equations of motion are based on the Navier–Stokes equations in Lagrangian form. SPH kernel approximation and particle approximation are applied to the Navier–Stokes equations to derive the equations of motion of SPH [17]. In this chapter, we will give a brief overview on formulation SPH.

###### Governing Equation.

Basically, the governing equations of fluid flow are based on three fundamental conservation laws which are conservation of mass, momentum, and energy [18]. In the approach of Lagrange description, the equations can be written as a set of partial differential equations which are known as Navier–Stokes equations. The equation of conservation of momentum in Navier–Stokes equations is Display Formula

(1)$Dv(ra)Dt=1ρa∇σ(ra)$

where $ra$ is the position vector, $v(ra)$ is the vector velocity which is the flow velocity at the point of particle a, and $ρa$ is the density of particle $a$. $∇$ represents the gradient operator and $σ(ra)$ is total stress which is composed of isotropic pressure p and viscous stress τDisplay Formula

(2)$σ(ra)=−p(ra)+τ(ra)$

###### Smoothed Particle Hydrodynamics Formulas.

The formulation of SPH consists of two steps. First step is the integral representation which also called kernel approximation and the second step is particle approximation [3,19].

Kernel approximation is started from the following identity: Display Formula

(3)$f(x)=∫f(x′)δ(x−x′)dx′$

where $f$ is the function of position vector $x$ and $δ$ is Dirac delta function. This function indicates that the value of a point $x$ can be determined by the integral of the function value at all other points of the field.

To do some approximation, the delta function is replaced by the smoothing function WDisplay Formula

(4)

where $h$ is smoothing length which represents the influence area of smoothing function.

The smoothing function $W$ should be chosen to satisfy the following conditions Display Formula

(5)$∫W(x−x′,h)dx′=1$

(6)$limh→0W(x−x′,h)=δ(x−x′)$

The piecewise quantic [20] is used in this paper Display Formula

(7)$W(R,h)=a{(3−R)5−6(2−R)5+15(1−R)50≤R<1(3−R)5−6(2−R)51≤R<2(3−R)52≤R≤3$

where factor a is 120/h, 7/478πh2, and 3/3593 in one-, two-, and three-dimensional problems, respectively. Also, R is the relative distance between two particles and written as R = r/h = |r − r′|/h where r is the distance between two particles.

The integral representation in Eq. (4) can be written in following discretized form Display Formula

(8)$f(x)=∑i=1Nmiρif(xi)W(x−xi,h)$

where $mi$ and $ρi$ are the mass and the density of particle $i$ (=1, 2,…, N), respectively, and N is the number of particles within the support domain of particle $i$. From this particle approximation, we can represent density of particle a as follows: Display Formula

(9)$ρa=∑b=1NmbWab$

where $Wab=W(ra−rb,h)$.

The momentum equation which is shown in Eq. (1) can also be rewritten by applying the SPH particle approximation to the gradient on the right-hand side of the equation Display Formula

(10)$Dv(ra)Dt=1ρa∑b=1Nmbσ(rb)ρb∇Wab$

Adding the following identity Display Formula

(11)$∑b=1Nmbσ(ra)ρaρb∇Wab=σ(ra)ρa(∑b=1Nmbρb∇Wab)=0$

to Eq. (10) leads to Display Formula

(12)$Dv(ra)Dt=∑b=1Nmbσ(ra)+σ(rb)ρaρb∇Wab$

Applying the SPH particle approximation to the gradients, we get Display Formula

(13)$Dv(ra)Dt=∑b=1Nmbρbσ(ra)ρb∇Wab+σ(ra)ρa2∑b=1Nmbρbρb∇Wab$

And with a simple arrangement, Eq. (13) can be written in the following form: Display Formula

(14)$Dv(ra)Dt=∑b=1Nmb[σ(ra)ρa2+σ(rb)ρb2]∇Wab$

As shown in Eq. (2), total stress is made up of pressure and viscous stress. Therefore, Eq. (14) can be rewritten taking into the viscous term as Display Formula

(15)$Dv(ra)Dt=−∑b=1Nmb[p(ra)ρa2+p(rb)ρb2]∇Wab+∑b=1Nmb[μ(ra)+μ(rb)ρaρbv(rab)](1|rab|∂Wab∂rab)$

where $|rab|$ is the distance between the two particles $a$ and b. The pressure of particle $a$ is derived by the artificial equation of state using the local particle density Display Formula

(16)$p(ra)=c2ρa$

where c is the speed of sound [21].

## Parallelization of Smoothed Particle Hydrodynamics

As discussed before, the particle behavior is determined by the neighboring particles and they keep on moving dynamically in the support domain. With a large number of particles that could be used in the simulation, the parallelization of SPH has been developed to improve the efficiency. Domain decomposition is a popular distribution methodology in parallel computing. In this chapter, we give a brief introduction on parallel computing, and traditional domain decomposition method in SPH is introduced. Furthermore, a new method is proposed to analyze SPH problems.

###### Parallel Computing.

The parallel programming follows Flynn's taxonomy which classifies the computer architectures as single instruction single data (SISD), SIMD, multiple instruction single data (MISD), and multiple instructions multiple data (MIMD) [22]. SISD consists of one processor and one communication channel, which are usually the bottlenecks. MISD is quite uncommon because other structures are more appropriate for parallel techniques. The MIMD system offers flexibility and scalability in the various forms of parallelism. However, this paradigm includes much more communication so users will meet more difficulties in programming. Therefore, it is proposed that SIMD processor would be used in the application of distributed procedure.

###### Traditional Domain Decomposition.

SPH method represents problem domain with a set of particles and they can interact with each other. When segmenting subdomains, the particles near the boundary of subdomains must still be allowed to interact with the calculation points in the opposite segment. Therefore, significant communications between subdomains are required in the domain decomposition of SPH method [11,19].

Figure 1 shows the domain decomposition method with ghost zone. The ghost zone is duplicated by the other side of subdomain and it is developed along the separated plane to ensure the accuracy of domain decomposition. The particles within the ghost zone possess the data, respectively, about the position, velocity, and internal energy. These data are used to calculate the forces acting on the particles near the boundary between subdomains. In this case, a great number of ghost particles are applied in the ghost zone. Furthermore, in the conventional method for searching the neighboring particles, the dimension of the subdomain is κh for smoothing kernel function, which leads to many unnecessary particles included in the adjoining cells. Dealing with these particles will occupy a lot of CPU resource without any contribution to get the result.

###### Proposed Method.

The proposed parallel method for domain decomposition is based on grouping of fluid particles and assigning each group to a particular processor. As shown in Fig. 2, the subdomain boundaries are modeled by the ghost particles called interface particles The interface particles do not have mass. They are placed right on the segment plane of the simulation domain and play a significant role in filtering the useless particles near the boundary.

In the proposed method, the fictitious Cartesian grid made up of cells is used which is fixed during the entire simulation and a group of fluid particles are contained in each cell. The distances and angles between each fluid particles and interface particle are calculated on each subdomain above all Display Formula

(17)$θa=tan−1|rint−ra|$

where $θa$ is the angle calculated using the distance $Ra=|rint−ra|$.

There are two steps in filtering the unnecessary particles. The first step is illustrated in Fig. 3(a), which is progressed by

Display Formula

(18)$Ra cos θa+Rb cos θb≤κh$

In the second step, a criterion is set on the interface particle that has circle area with a radius as follows: Display Formula

(19)$hint=κh sin θint$

where Display Formula

(20)$θint=cos−1(cos θa×Raκh)$
In this case, $θint$ is the angle between point of contact to the support domain of particle a with segment plane and particle a. For the more efficient operation, the second process of filtering is accomplished by Display Formula
(21)$Rb≤hint$

After filtering, the final zone that should be considered when computing particle a is highlighted in Fig. 3(b) by the black dashed strips. Applying the law of cosines, we can get the distance between two particles Display Formula

(22)$|rab|2=Ra2+Rb2−2RaRb cos (180 deg−θa−θb)$

## Example

Both tradition and proposed domain decomposition methods were tested on the Linux cluster of the 2.4 GHz Neahlem CPUs. Execution time and number of iteration of both methods were compared on a simple two-dimensional fluid flow model.

###### Model Geometry.

In the two-dimensional model, total number of particles is increased as 768, 1488, 2208, 2928, and 3648. The model has 48 side particles fixedly and 16, 31, 46, 61, and 76 particles on the horizontal axis (see Fig. 4). Then, one interface particle is placed on each domain boundary. Hundred time steps are calculated with each step of 0.001 s. The smoothing length $h$ is 0.05 m and the initial mesh spacing sets the minimum to $3h$. Fluid density is 1000 kg/m3 and the speed of sound is 5 m/s.

###### Result.

Parallel analysis is progressed for two cases. In case 1, we apply traditional decomposition method and in case 2 proposed method is applied. In order to express numerical efficiency, execution time of the SPH method is measured by the number of subdomains and compared with another case. Also, the number of iteration is figured out.

Figure 5 shows the case of 768 particles and two subdomains. The particles are influenced by the neighboring particles placed on the other subdomain so that the movement of particles in each subdomain has the symmetric form which verifies that our program produces a reasonable result.

Execution time with increasing particle number is plotted against the corresponding number of processors on Fig. 6. Compared with the execution time of traditional method, the execution time of proposed method decreased a little for the 3648-particles and 10-processes case, and execution time decreased from 48 s to 43 s, about 10%.

Figure 7 is the visualization of number of iteration for two cases. There is a significant reduction in the proposed method. For the 3648-particles and 10-processes case, the number of iteration is about 280,000 in the traditional method and is about 7200 in the proposed method, a remarkable reduction of 97%.

Significant decrease on execution time does not take much effect on the execution time. One of the reasons is that this method is simulated on a small-size problem with a relatively small number of particles so that the cost on filtering useless particles is not much less than the cost on computing them in tradition way. However, even such small problems can cause the large decrease, and we can expect high efficient on the large-scale simulations.

## Conclusion

Smoothed particle hydrodynamics is attractive in treating fluid flow and large deformation problems for its meshfree and adaptive nature. To improve the simulation efficiency, parallel method is adopted to the SPH method. In this study, a new load balanced domain decomposition method is proposed and compared with the traditional method by a simple example.

Traditionally, domain decomposition is applied with the help of ghost zone. It involves a lot of useless particles and would cost a lot of extra work when dealing with the ghost particles. Therefore, we propose a new domain decomposition method. The proposed method separates the problem domain into subdomains by using interface particles which are placed on the boundary without a mass. These interface particles play an important part in filtering the useless particles so that they could reduce the cost.

A simple two-dimensional fluid flow example is simulated by both tradition method and proposed method. A different number of particles are used to represent the problem domain and for each particle number, the number of central processing unit is increased from 1 to 10. For every condition, execution time and a number of iteration are checked and compared. For the proposed method, compared with tradition method, the execution time is decreased by 10% and the number of iteration is decreased by more than 95%.

Considering the notable reduction in the number of iteration, we expect a more efficient performance on large-scale problems. We will work on large-scale problems as long as more complicated three-dimensional fluid problems. Also, our idea proposed in this paper can be an experience as more and more particle methods are developed to the analysis of fluid dynamics or large deformation problems.

## Acknowledgements

This research was supported by a Grant (15CTAP-C077510-02) from Infrastructure and Transportation Tech Facilitation Research Program funded by Ministry of Land, Infrastructure, and Transport of the Korean Government.

## References

Liu, G. R. , and Gu, Y. T. , 2005, An Introduction to Meshfree Methods and Their Programming, Springer, Dordrecht, The Netherlands.
Liu, G. R. , 2010, Meshfree Methods: Moving Beyond the Finite Element Method, CRC Press Taylor & Francis Group, Boca Raton, FL.
Liu, G. R. , and Liu, M. B. , 2007, Smoothed Particle Hydrodynamics: A Meshfree Particle Method, World Scientific Publishing, Singapore.
Monaghan, J. J. , 2005, “ Smoothed Particle Hydrodynamics,” Rep. Prog. Phys., 68(8), pp. 1703–1759.
Lucy, L. B. , 1977, “ A Numerical Approach to the Testing of the Fission Hypothesis,” Astron. J., 82, pp. 1013–1024.
Gingold, R. A. , and Monaghan, J. J. , 1977, “ Smoothed Particle Hydrodynamics: Theory and Application to Non-Spherical Stars,” Mon. Not. R. Astron. Soc., 181(3), pp. 375–389.
Benz, W. , and Asphaug, E. , 1995, “ Simulations of Brittle Solids Using Smooth Particle Hydrodynamics,” Comput. Phys. Commun., 87(1–2), pp. 253–265.
Johnson, G. R. , and Beissel, S. R. , 1998, “ Normalized Smoothing Functions for SPH Impact Computations,” Int. J. Numer. Methods Eng., 39(16), pp. 2725–2741.
Monaghan, J. J. , 1994, “ Simulating Free Surface Flow With SPH,” J. Comput. Phys., 110(2), pp. 399–406.
Quinn, M. J. , 2003, Parallel Programming in C With MPI and OpenMP, McGraw-Hill Education, New York.
Gropp, W. , Lusk, E. , and Thakur, R. , 1999, Using MPI-2: Advanced Features of the Message-Passing Interface, The MIT Press, Cambridge, MA.
Morris, J. P. , Zhu, Y. , and Fox, P. J. , 1999, “ Parallel Simulation of Pore-Scale Flow Through Porous Media,” Comput. Geotech., 25(4), pp. 227–246.
Wu, J. S. , and Tseng, K. C. , 2005, “ Parallel DSMC Method Using Dynamic Domain Decomposition,” Int. J. Numer. Methods Eng., 63(1), pp. 37–76.
Holmes, D. W. , Williams, J. R. , and Tilke, P. , 2011, “ A Framework for Parallel Computational Physics Algorithms on Multi-Core: SPH in Parallel,” Adv. Eng. Software, 42(11), pp. 999–1008.
Batchelor, G. K. , 2000, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, UK.
Hoover, W. G. , 2006, Smooth Particle Applied Mechanics: The State of the Art, World Scientific Publishing, Singapore.
Takeda, H. , Miyama, S. M. , and Sekiya, M. , 1994, “ Numerical Simulation of Viscous Flow by Smoothed Particle Hydrodynamics,” Prog. Theor. Phys., 92(5), pp. 939–960.
Randles, P. W. , and Libersky, L. D. , 1996, “ Smoothed Particle Hydrodynamics: Some Recent Improvements and Applications,” Comput. Methods Appl. Mech. Eng., 139(1–4), pp. 375–408.
Gingold, R. A. , and Monaghan, J. J. , 1982, “ Kernel Estimates as a Basis for General Particle Methods in Hydrodynamics,” J. Comput. Phys., 46(3), pp. 429–453.
Morris, J. P. , 1996, “ Analysis of Smoothed Particle Hydrodynamics With Applications,” Ph.D. thesis, Monash University, Clayton, Australia.
Morris, J. P. , Fox, P. J. , and Zhu, Y. , 1997, “ Modeling Low Reynolds Number Incompressible Flows Using SPH,” J. Comput. Phys., 136(1), pp. 214–226.
Flynn, M. J. , 1966, “ Very High-Speed Computing Systems,” Proc. IEEE, 54(12), pp. 1901–1909.
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## References

Liu, G. R. , and Gu, Y. T. , 2005, An Introduction to Meshfree Methods and Their Programming, Springer, Dordrecht, The Netherlands.
Liu, G. R. , 2010, Meshfree Methods: Moving Beyond the Finite Element Method, CRC Press Taylor & Francis Group, Boca Raton, FL.
Liu, G. R. , and Liu, M. B. , 2007, Smoothed Particle Hydrodynamics: A Meshfree Particle Method, World Scientific Publishing, Singapore.
Monaghan, J. J. , 2005, “ Smoothed Particle Hydrodynamics,” Rep. Prog. Phys., 68(8), pp. 1703–1759.
Lucy, L. B. , 1977, “ A Numerical Approach to the Testing of the Fission Hypothesis,” Astron. J., 82, pp. 1013–1024.
Gingold, R. A. , and Monaghan, J. J. , 1977, “ Smoothed Particle Hydrodynamics: Theory and Application to Non-Spherical Stars,” Mon. Not. R. Astron. Soc., 181(3), pp. 375–389.
Benz, W. , and Asphaug, E. , 1995, “ Simulations of Brittle Solids Using Smooth Particle Hydrodynamics,” Comput. Phys. Commun., 87(1–2), pp. 253–265.
Johnson, G. R. , and Beissel, S. R. , 1998, “ Normalized Smoothing Functions for SPH Impact Computations,” Int. J. Numer. Methods Eng., 39(16), pp. 2725–2741.
Monaghan, J. J. , 1994, “ Simulating Free Surface Flow With SPH,” J. Comput. Phys., 110(2), pp. 399–406.
Quinn, M. J. , 2003, Parallel Programming in C With MPI and OpenMP, McGraw-Hill Education, New York.
Gropp, W. , Lusk, E. , and Thakur, R. , 1999, Using MPI-2: Advanced Features of the Message-Passing Interface, The MIT Press, Cambridge, MA.
Morris, J. P. , Zhu, Y. , and Fox, P. J. , 1999, “ Parallel Simulation of Pore-Scale Flow Through Porous Media,” Comput. Geotech., 25(4), pp. 227–246.
Wu, J. S. , and Tseng, K. C. , 2005, “ Parallel DSMC Method Using Dynamic Domain Decomposition,” Int. J. Numer. Methods Eng., 63(1), pp. 37–76.
Holmes, D. W. , Williams, J. R. , and Tilke, P. , 2011, “ A Framework for Parallel Computational Physics Algorithms on Multi-Core: SPH in Parallel,” Adv. Eng. Software, 42(11), pp. 999–1008.
Batchelor, G. K. , 2000, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, UK.
Hoover, W. G. , 2006, Smooth Particle Applied Mechanics: The State of the Art, World Scientific Publishing, Singapore.
Takeda, H. , Miyama, S. M. , and Sekiya, M. , 1994, “ Numerical Simulation of Viscous Flow by Smoothed Particle Hydrodynamics,” Prog. Theor. Phys., 92(5), pp. 939–960.
Randles, P. W. , and Libersky, L. D. , 1996, “ Smoothed Particle Hydrodynamics: Some Recent Improvements and Applications,” Comput. Methods Appl. Mech. Eng., 139(1–4), pp. 375–408.
Gingold, R. A. , and Monaghan, J. J. , 1982, “ Kernel Estimates as a Basis for General Particle Methods in Hydrodynamics,” J. Comput. Phys., 46(3), pp. 429–453.
Morris, J. P. , 1996, “ Analysis of Smoothed Particle Hydrodynamics With Applications,” Ph.D. thesis, Monash University, Clayton, Australia.
Morris, J. P. , Fox, P. J. , and Zhu, Y. , 1997, “ Modeling Low Reynolds Number Incompressible Flows Using SPH,” J. Comput. Phys., 136(1), pp. 214–226.
Flynn, M. J. , 1966, “ Very High-Speed Computing Systems,” Proc. IEEE, 54(12), pp. 1901–1909.

## Figures

Fig. 1

Domain decomposition with ghost zone

Fig. 2

Domain decomposition with interface particle: (a) problem domain and (b) separated subdomains

Fig. 7

Number of iteration in two cases: (a) 768 particles, (b) 1448 particles, (c) 2208 particles, (d) 2928 particles, and (e) 3648 particles

Fig. 3

Filtering the particles: (a) step 1 and (b) step 2

Fig. 4

Domain decomposition

Fig. 5

Domain decomposition using two CPUs on 768 particles: (a) at time 0 s, (b) at time 0.01 s, (c) at time 0.03 s, (d) at time 0.05 s, (e) at time 0.07 s, and (f) at time 0.9 s

Fig. 6

Execution time in two cases: (a) 768 particles, (b) 1448 particles, (c) 2208 particles, (d) 2928 particles, and (e) 3648 particles

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