Comparative Analysis of the SPH and ISPH Methods

7/31/2019 Comparative Analysis of the SPH and ISPH Methods

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Comparative Analysis of the SPH and ISPH

Methods

K.E. Afanasiev, R.S. Makarchuk, and A.Yu. Popov

Kemerovo State University, Krasnaya st. 6, 650043 Kemerovo, Russia{afa,mak,a popov}@kemsu.ru

Abstract. Free surface problems in fluid mechanics are of a great ap-plied importance, and thats why they rouse many researchers to ex-citement. Numerical simulations of such problems, using the so-called

meshless methods, become more and more popular today. One of thesubsets of these methods is the class of meshless particle methods, whichdont use any mesh during the whole numerical simulation process, andtherefore allow solving the problems with large deformations and withfailure of problem domain connectivity. These reasons cause great pop-ularity of these methods in the sphere of numerical simulation of freesurface problems.

One of the meshless particle methods is the Smoothed Particle Hydro-dynamics [1]. Numerous computations, mainly of free surface problems,carried out by different scientists, using the mentioned method, proved

its doubtless efficiency in receiving the high-quality kinematic images ofideal as well as viscous fluid flows.

However, it has a considerable drawback - it doesnt allow receivingsatisfactory images of pressure distribution.

Recently, with the purpose to deliver such drawbacks, the ISPH (In-compressible Smoothed Particle Hydrodynamics) method has been de-veloped [2,3], which is used for simulation the incompressible fluid flows,as its name shows. It uses a split step scheme for integration of basicequations of fluid dynamics.

Comparative analysis of computational results, obtained by the meth-ods, mentioned above, shows, that the ISPH gives low-grade kinematicimages of fluid flows in comparison with the classical SPH, however, itallows obtaining much better images of pressure distribution, and it willgive an opportunity to compute the hydrodynamics loads in future.

1 Introduction

At present more currency in the field of numerical simulation of free surface flows

is gained by meshless methods. Among them the subclass of particle methodsis pointed out. These methods dont require using a mesh neither at the stageof shape functions constructing, nor at the stage of integrating equations ofmotion. Their major idea lies in discretization of the computation area by a setof Lagrangian particles, which are able to move freely within the constraints,received by means of basic equations of continuum dynamics. Shape functions

E. Krause et al. (Eds.): Comp. Science & High Perf. Computing III, NNFM 101, pp. 206223, 2008.springerlink.com c Springer-Verlag Berlin Heidelberg 2008


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Comparative Analysis of the SPH and ISPH Methods 207

Table 1. Distinctions in the SPH and ISPH algorithms

SPH ISPH

Integration scheme Explicit Implicit

Pressure Equation of state Poisson equationCourant condition Sound speed Maximum speed of particlesDensity Variable Constant

Artificial viscosity Applied Not applied

Solid boundary conditions Lennard-Jones potential Morris virtual particles

in this approach are constructed for each time step using a different set of nodes(particles). The meshless nature of these methods and their easy realization and

application have caused their great popularity in the field of numerical simulationof free surface flows.The most widespread for the present moment particle methods are the

smoothed particle hydrodynamics method (SPH) [1,4,5,6,7] and the moving par-ticle semi-implicit method (MPS) [8,9]. Besides, there are many modifications ofthe SPH, i.e. the RKPM [10] and MLSPH [11], intended for optimization of itsapproximation characteristics.

This work considers the original SPH and one of its modifications the In-compressible Smoothed Particle Hydrodynamics (ISPH) [2,3], which allows, con-

trary to the original SPH method, exactly complying with the incompressibilitycondition. The most significant differences of the two methods are presented inTable 1.

2 Governing Equations

The basic equations of fluid dynamics, including the Navier-Stokes equationsand the equation of continuity, in case of the Newtonian viscous fluid, are of thefollowing form:

dva

dt= Fa

1

p

xa+

xb(Tab); (1)

d

dt=

va

xa, (2)

where a, b = 1, 2, 3 numerical indices of coordinates, va components of thevelocity vector, Fa components of the vector of volumetric forces density, ab Kronecker symbols, p and pressure and density of the fluid, correspondingly, coefficient of dynamic viscosity, while the stress components are calculatedby the formula:

Tab =va

xb+

vb

xa

2

3divv ab (3)

It should be noted, that the original SPH method is applied for simulation ofcompressible fluid flows, and a certain equation of state is used for closure the


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208 K.E. Afanasiev, R.S. Makarchuk, and A.Yu. Popov

equations (1)-(2). For computation of incompressible fluid flows, most often thebarotropic equation of state in the Theta form is applied [4]. By selection of thecoefficient of volume expansion in the equation of state we can obtain the effectof incompressible fluid. However, certain variations of particles densities occur.

The ISPH method uses the split step scheme for time integration of equationsof motion, what allows calculating the particles pressure values by solving thePoisson equation without recourse to the equation of state. This approach allowsalso to neglect the artificial viscosity terms in Navier-Stokes equations, contraryto the original SPH, because of using the model of incompressible fluid.

As a result of the fact, that the ISPH uses the model of incompressible fluid,the equations (2), (3) are represented by the following formulas:

va

xa= 0; (4)

Tab =va

xb+

vb

xa. (5)

Using them we can represent the equations of motion (1) in the following form:

dva

dt= Fa

1

p

xa+

xb(

va

xb) (6)

3 Approximation of FunctionsThe first stage in the process of constructing approximation formulas for func-tions, occurring in equations of fluid dynamics, is exact presentation of thesefunctions as the integral:

f(r) =

f(r)(r r)dr, (7)

where the Dirac delta-function.Then follows substitution of the -function with a certain classic compactlysupported function, what allows receiving the integral formula of the functionapproximation on the bounded domain:

f(r) =

f(r)W(r r, h)dr, (8)

with the weight function W, which is often called the kernel function. The valueh is a size of support domain of the function W and is called a smoothing length.

At the following stage the problem domain is discretized by a finite numberof Lagrangian particles, and the integral (8) is substituted with the quadrature[4,5,6,7]:

fs(ri) =n

j=1

f(rj)mjj

W(ri rj , hi), (9)


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Fig. 1. Interaction of particles

where n number of particles, determined as nearest, in the radius hi, neighboursof the i-th particle. Two particles i and j are called neighbouring or interactingparticles, if the distance between their centres does not exceed hi+hj . ri, mi, i- radius-vector, mass and density of the i-th particle, correspondingly. As theweight W usually polynomial splines are applied.

On Fig. 1 is shown, that particles 1 and 2 are neighboring to particle i. From(9) follows, that the gradient of required function is represented by the followingexpression:

fs(ri) =n

j=1

mjj

fjW(ri rj , h) (10)

Using the formula (9), for density approximation we obtain:

i =n

j=1

mjW(ri rj , hi) (11)

It should be noted, that contrary to the original SPH, where the density value in

particles is most often determined by solving a discrete analogue of the equation(2), in the ISPH method the density is calculated just by the formula (11).

4 Kernel Function

For numerical simulation, using the considered methods, one can apply variouskernel functions, from the Gaussian function up to splines of different orders.Beside the already known kernel functions the others can be developed, but, at

that, has to follow the minimum requirements specified for them:

W(r, h) = 0, r > h;

W(r, h)dr = 1;

limh0

W(r, h) = (r),


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Fig. 2. Kernel function (left) and first derivative of kernel function (right)

where r = ||rr||. On Fig. 2 shapes of kernel function and its first derivative aresubmitted. Beside the above mentioned requirements, some additional conditions

can be imposed on the kernel function to provide best stability of the method andthe higher degree of approximation of the functions, characterizing flows. Suchadditional conditions and ways of constructing the kernel functions, subsequentupon them, have led, for instance, to development of the RKPM method. Forthe problems, considered in this work, the cubic spline from [3,12] is applied:

W(r, h) =15

7h2

2/3 q2 + q3/2, 0 q 1;

(2 q)3

/6, 1 < q 2;

0, q > 2,

(12)

where q = rh .

5 Artificial Viscosity

For additional stability of the original SPH the additional term, called artificialviscosity, is added to the right part of the Navier-Stokes equations of motion

[4,7,13]:

ij =

ij (ij c)1/2 (i + j)

,

d=1

vdi v

dj

xdi x

dj

< 0;

0, otherwise,

(13)

where dimension of the problem domain, and ij are computed by theformula:

ij =

d=1h(vdi v

dj )(x

di x

dj )

(xdi xd

j) + 2h2

; (14)

c =1

2(ci + cj), h =

1

2(hi + hj), = = 1, = 0.1 (15)

Since the ISPH method was developed for incompressible fluids it doesntneed any artificial viscosity term to be added to the approximated equations ofmotion.


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6 Pressure and Viscosity

For approximation of the Poisson equation one can apply SPH formulas for thegradient of function and the vector field divergence sequentially. However, on

account of the fact, that approximation of the second-order derivative, obtainedin this case, is too sensitive to the particles disorder, approximation of the firstderivative in terms of the SPH and its finite difference analogue are usuallyapplied together [3]:

1

p

i

=n

j=1

mj8

(i + j)2

(pi pj) (ri rj) iW(ri rj , hi)

ri rj2 (16)

We can also obtain other formulas for approximation of the Poisson equation,particularly, the following springs out of the work [14]:

1

p

i

=n

j=1

mj2

ij

(pi pj) (ri rj) iW(ri rj , hi)

ri rj2

(17)

The calculations of the model problems have been executed using as the formula(16), so (17), however any essential differences in the obtained results have notbeen discovered.

The formula for approximation of viscous forces for the ISPH is obtained ina similar way and takes on the form:

2v

i

=n

j=1

mj4

(i + j)2

(i + j) (ri rj) iW(ri rj , hi)

ri rj2 (vi vj)

(18)

The pressure gradient term in the equation (1) can also be approximated indifferent ways. One can directly apply the SPH approximation formulas to thisterm, but it was found by researchers, that the most stable approximation canbe obtained using the formula:

p

=

p

+

p

2 (19)

Now one can apply the SPH approximation formulas to the right-hand side of(19) and obtain the following approximation of pressure gradient:

p

i

= n

j=1

mj

pi2i

+pj2j

iW(ri rj , hi) (20)

This approximation of pressure gradient, because of its symmetrical form, pro-vides the total momentum conservation law fulfilment.


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Based on considered approximations it is possible to rewrite equations (1)-(2)in the following form for the original SPH:

dvai

dt

= Fi n

j=1

mj pi

2i

+pj

2j

+ ijWa(ri rj , hj)++

nj=1

mj

d=1

iT

ida

2i+

jTjda

2j

Wd(ri rj , hj);

(21)

didt

=

d=1

nj=1

mj(vdi v

dj )W

d(ri rj , hj), (22)

where pi, i, i pressure, density, and coefficient of dynamic viscosity accord-

ingly for particle i. Normal and tangent components of viscous stress tensor Ti

are defined by following expressions:

Tiaa =n

j=1

mjj

[2 (vaj vai )W

a(ri rj , hj)

(vbj vbi )W

b(ri rj , hj)];(23)

Ti

ab =

nj=1

mj

j [(va

j va

i )Wb

(ri rj , hj)

(vbj vbi )W

a(ri rj , hj)](24)

7 Model of Incompressibility in the SPH

Initially method SPH was applied only to modeling strongly compressed envi-ronments. Incompressibility or weak compressibility is reached by a choice of the

suitable equation of a state to solve the pressure. The following equation, as arule, is applied to simulation of liquid dynamics [4]:

p = B

0

1

, (25)

where 0 - initial density of a fluid, = 7 - adiabatic index. For problems withfailure of a weighty fluid constant B gets out under the following expression:

B =

2000g(H y)

, (26)

where g gravity acceleration, H initial height of a fluid column, y verticalcoordinate of the particles. In case of a choice of constant B under the formula(26) density of particles during calculation differ from initial density 0 no morethan on 2-3 %, that allows to take a fluid weakly compressed.


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8 Time Integration

In the SPH method for time integration the predictor-corrector scheme is used.Predictor:

n

i =

n1/2

i + (t/2)(d

n1

i /dt);

vni = vn1/2i + (t/2)(dv

n1i /dt).

(27)

Corrector:

n+1/2i =

n1/2i + t(d

ni /dt);

vn+1/2i = v

n1/2i + t(dv

ni /dt);

xn+1i = xni + t(vn+1/2i /dt).

(28)

For time integration of motion equations in the ISPH the split step scheme isapplied. Initially under consideration is only the effect of mass forces and viscousforces upon the particles, what allows receiving preliminary values of densities,velocities and coordinates of the particles. At this stage the pressure is not takeninto account, and in the formula (3), due to incompressibility of the fluid, theterm with the velocity divergence disappears. The mentioned restrictions lead

to the equations:v =

g +

2v

t;

v = vn + v;

r = rn + vt.

(29)

Here v velocity change caused by mass forces and viscous forces, v and r

preliminary values of the velocity vector and the radius-vector of the particles

centers, correspondingly, vn and rn values of the velocity vector and the radius-vector of the particles centers on the n-th time step.

Using the obtained preliminary values of the physical characteristics, we de-termine intermediate values of the particles densities i by the formula (11).Variations of the obtained particles densities from their initial values are usedlater to comply with the incompressibility condition, and the final values ofthe characteristics on the (n + 1)-th time step are determined by the followingformula:

v = 1

pn+1t;

vn+1 = vn + v;

rn+1 = rn +

vn + vn+1

2

t.

(30)


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At this stage we have to solve the pressure Poisson equation, which is of thefollowing form [3]:

1

ip

i

=0

i

0t2(31)

This equation can be finally reduced to the system of linear algebraic equations,and the conditions on the free surface, represented here by the Dirichlet condi-tion, are introduced to the matrix of coefficients of the resulted system on thesame principle, as in the finite element method [15].

The time step gets out proceeding from Courant condition [5,3]:

t Cmini

(hi)

maxi

(ci + vi)(SP H), t C

h

maxi

(vi)(ISPH), (32)

where ci - speed of sound for particle i, C (0; 1) - Courant constant. In calcu-lations it was used C [0.25 ; 0.5].

9 Solid Boundary Conditions

In the original SPH method the most frequently used way of imposing conditionsat the solid boundaries is application of virtual particles, divided into twotypes.

The first type Monaghan virtual particles [4]. These particles are locatedalong the solid boundary in a single line, dont change their characteristics intime, and effect on the fluid particles by means of a certain interaction potential.The most popular among researchers is the Lennard-Jones potential, thoughselection of this is not imperative. In his work [6] Monaghan proposes a newinteraction potential, taking into account peculiarities of the SPH method.

The most frequently used potential in a method of the smoothed particles Lennard-Jones potential:

U(r) = Dr

r0r

12

r0r

6

, (33)

where D - depth of a potential hole, r0 - distance on which the potential ofinteraction addresses in a zero. If r < r0 there are repulsive force, otherwise -attractive force. The found potential is included in the momentum equation asadditional force which turns out as a result of application of SPH approximations:

Uai =n

j=1

D(xai xaj )

r0r

12

r0r

6

r2

(34)

For simplification the potential (34) is calculated only for repulsive forces, andforces of an attraction at r > r0 can be neglected.

The second type Morris virtual particles [5]. These particles are locatedalong the solid boundary in several lines. The number of the lines depends on


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the smoothing length of particles of the fluid. This allows solving one of themain problems of the SPH method asymmetry of the kernel function near theboundaries. The effect of the Morris particles on the fluid particles differs fromthe effect of Monaghan particles by the fact, that there is no need in using any

interaction potential. Instead of this, values of the characteristics in the Morrisparticles are calculated on the basis of their values in particles of the fluid.

In the ISPH method, for imposing solid boundary conditions the Morris vir-tual particles are used. In the work [3] the approach is presented, which is appliedin solving problems by the MPS method [8,9]. For solving the pressure Poissonequation, on the solid boundaries the Neumann condition is imposed, specify-ing equality of pressures in the particles, located along the normal to the solidboundary. In this work 3 lines of virtual particles are applied for numericalsimulation. The pressure Poisson equation is solved using only one line of vir-

tual particles and also only one line is used in the gradient formula (20). Theother two lines are necessary to keep particles densities near the solid boundarythe same as for inner ones.

10 Free Surface Conditions

For identification of particles, belonging to the free surface, one can apply differ-ent ways. One of such ways is using the Dilts algorithm [11], based on the fact,

that each particle has its size, which is determined by the smoothing length inthe ISPH method. However most often, as in the MPS method, so in the ISPH,the particles, belonging to the free surfaces, are identified by their densities. Thisapproach is based on the fact, that the density of particles, belonging to the freesurface, is less than the density of inner fluid particles, because of the lack ofneighbours from one side of the boundary.

For solving the pressure Poisson equation, in the particles, belonging to thefree boundary, the Dirichlet condition is imposed: p = 0. Besides, this conditionis also applied in calculation of the pressure gradient, used in the formula ( 20). It

is done as follows. For each particle, belonging to the free surface, all its nearestneighbours should be determined. It is evident, that the neighbours are onlyinner particles of the fluid, since there are no particles above the free surface.To keep the symmetry of support domain of kernel function for the particles,belonging to free surface, above the free surface we introduce additional ghostparticles. They are located symmetrically about the free surface to the innerparticles of the fluid, and the pressure in them is converse. In this work theapproach, used in [3], is applied.

11 Nearest Neighbours Search

Since nearest neighbours search algorithm is carried out on each time step andrequires high computing expenses it is necessary to use effective algorithms ofsearch of neighbours. In our calculations we used grid algorithm.


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Fig. 3. Direct search

Fig. 4. Grid algorithm

Fig. 5. Acceleration of computations


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To define efficiency of neighbouring particles search algorithm we compareddirect search and grid algorithm. Test calculations were carried out on uniproces-sor system: AMD Athlon 2000+ / 512 Mb RAM / Windows XP SP2 in FortranPowerStation 4.0. Time of search neighbouring particles depending on number

of particles for 1000 time steps was measured.Apparently from Fig. 3 and 4, dependence for direct search is square-law and

for grid algorithm is linear that corresponds to theoretical assumptions.Dependence of acceleration of neighbouring particles search procedure on

number of particles is resulted in the figure 5.

12 Parallelization

A great amount of problems, simulating with the ISPH, is restricted by the capa-bilities of the computers. However, thanks to the appearance of new more power-ful computers, clusters of computers and to the organization of high-performancecomputations using them, the opportunity to expand of class of problems, whichcan be simulated using the ISPH appeared.

While simulating the problems on clusters the algorithm runs as an MPI-application. MPI is a package of subprograms, giving the opportunities of com-munications between the processes, running on different CPUs.

One of the most weak places in the ISPH algorithms is a necessity to solve

high-dimensional systems of linear algebraic equations, which appear when ap-proximating the pressure Poisson equation.At first the algebraic system was solved using the standard Gauss procedure,

then the serial algorithm was parallelized.For demonstration of efficiency of obtained parallel algorithm some test com-

putations were performed. The dimensions of matrices in computations variedfrom 200 to 4600.

For comparison of computation time of serial and parallel algorithms and fordemonstration of parallel algorithm advantages the coefficients of acceleration

and efficiency are used:Sm =

T1Tm

, Em =Smm

,

where Tm is the computation time using the parallel algorithm on the clusterwith m (m>1) processors, T1 is the computation time using the serial algorithm.

Parallel computations were carried out on the cluster, consisted of units withthe characteristics: Intel Pentium 4 2.80 GHz / 1 Gb RAM / Red Hat Linuxwith Intel Fortran compiler and MPI package.

On Fig. 6 the dependencies between the dimensions of matrices and compu-

tation time for different number of processors are presented. Fig. 7 presents thedependencies between the dimensions of matrices and acceleration of computa-tions. The dependencies between the dimensions of matrices and efficiency ofparallel algorithm are presented on Fig. 8.

The developed parallel realization of the SLAE Gauss solver allows receivingconsiderable acceleration of the computations as can be seen from the Table 2. Also


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Fig. 6. Dependencies between the dimensions of matrices and computation time fordifferent number of processors

Fig. 7. Dependencies between the dimensions of matrices and acceleration of compu-tations for different number of processors

it gives an opportunity to use much more particles in computations for obtainingmore accurate results. Nevertheless, the uniprocessor PGMRES solver shows moresignificant results in computation time than parallel Gauss solver. However the fullcomputation time (Table 3) of the ISPH method remain rather great. Therefore


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Fig. 8. Dependencies between the dimensions of matrices and efficiency of parallelalgorithm for different number of processors

Table 2. Computation time of different SLAE solvers (in sec.) for one time step

Particles number PGMRES Gauss procedure Parallel Gauss procedure (32 CPU)

1409 0.032 0.984 0.031

2009 0.047 3.421 0.1082709 0.094 8.625 0.272

3509 0.203 20.203 0.638

4409 0.719 43.125 1.361

X

Y

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

2.5

3

X

Y

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

2.5

3

Fig. 9. Dam breaking problem at t = 0.0s (left - SPH, right - ISPH)


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parallelization of PGMRES algorithm is planed for future work. Possibly more ef-ficient will prove to be the defect correction multi-grid method [2].

Further optimization of algorithms of the both presented methods lies in par-allelization of the grid algorithm for nearest neighbours search.

13 Model Problem. Dam Breaking

The equations (1)-(2) are solved. In the initial moment t = 0 the viscous fluidcolumn gets broken under the influence of gravity. The following values of phys-ical characteristics are used for this problem: = 1000 kg/m3 fluid den-sity, = 1 kg/m s2 dynamical viscosity. Fig. 9-14 presents flow pictures in

X

Y

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

2.5

3

X

Y

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

2.5

3


X

Y

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

2.5

3

X

Y

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

2.5

3



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Table 3. Full computation time (in sec.) of the SPH and ISPH methods with differentnumber of particles (2000 time steps)

Particles number SPH ISPH

1409 12.2 52.22009 17.4 111.42709 23.4 211.4

3509 30.2 436.2

4409 38.4 1476,4

X

Y

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

2.5

3

X

Y

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

2.5

3


X

Y

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

2.5

3

X

Y

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

2.5

3



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X

Y

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

2.5

3

X

Y

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

2.5

3


different moments of time with the calculation number of fluid particles 1600.The numerical results, obtained solving this problem by the SPH and MPS meth-ods, are provided in the work [16]. Computation time for the SPH and ISPHmethods with different number of particles is presented in the Table 3.

14 Conclusion

On the problems submitted in the given work in 2D-dimensional statement thebasic advantages and opportunities of the SPH and ISPH methods can be re-vealed. The shown results demonstrate wide applicability of the methods insimulation of problems with large deformations of problem domains. However,the presented methods have the following disadvantages: applying the equationof state for pressure field calculation in the original SPH doesnt allow comput-ing dynamic loads; in the obtained with the ISPH results a particle recessioncan be observed after fluid impact on solid wall. Nevertheless, the SPH showspronounced kinematics of flows. The ISPH due to using the pressure Poissonequation can be applied to calculation of dynamic loads.

References

1. Gingold, R.A., Monaghan, J.J.: Mon. Not. Royal Astron. Soc. 181, 375389 (1977)

2. Cummins, S.J., Rudman, M.: J. Comp. Phys. 152, 584607 (1999)3. Shao, S., Lo, E.Y.M.: Adv. Water Res. 26, 787800 (2003)4. Monaghan, J.J., Thompson, M.C., Hourigan, K.: J. Comput. Phys. 110, 399406

(1994)

5. Morris, J.P., Fox, P.J., Zhu, Y.: J. Comput. Phys. 136, 214226 (1997)6. Monaghan, J.J.: Rep. Prog. Phys. 68, 17031759 (2005)


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7. Liu, G.R., Liu, M.B.: Smoothed particle hydrodynamics: a meshfree particlemethod. World Scientific, Singapore (2003)

8. Koshizuka, S., Tamako, H., Oka, Y.: Comp. Fluid Dyn. 4(1), 2946 (1995)9. Koshizuka, S., Nobe, A., Oka, Y.: Int. J. Numer Methods Fluids 26, 751769 (1998)

10. Liu, W.K., Jun, S., Sihling, D.T., Chen, Y., Hao, W.: Int. J. Numer Methods Fluids24, 1 25 (1997)

11. Dilts, G.A.: Int. J. Numer Methods Eng. 48, 15031524 (2000)12. Monaghan, J.J., Lattanzio, J.C.: Astron Astrophys. 149, 135143 (1985)13. Monaghan, J.J., Poinracic, J.: Appl. Numer Math. 1, 187194 (1985)14. Brookshaw, L.A.: Proc. ASA 6, 207210 (1985)15. Connor, J.J., Brebbia, C.A.: Finite element techniques for fluid flow. Sudostroenie,

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Comparative Analysis of the SPH and ISPH Methods