Multi-Resolution MPS Method for Free Surface Flows

International Journal of Computational Methods Vol. 13, No. 4 (2016) 1641018 (17 pages) c World Scientific Publishing Company DOI: 10.1142/S0219876216410188 Multi-Resolution MPS Method for Free Surface Flows Zhenyuan Tang, Youlin Zhang and Decheng Wan State Key Laboratory of Ocean Engineering School of Naval Architecture, Ocean and Civil Engineering Shanghai Jiao Tong University Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration Shanghai 200240, P. R. China dcwan@sjtu.edu.cn Received 30 September 2015 Accepted 14 January 2016 Published 5 April 2016 A multi-resolution moving particle semi-implicit (MPS) method is applied into twodimensional (2D) free surface flows based on our in-house particle solver MLParticle- SJTU in the present work. Considering the effect of different size particles, both the influence radiuses of two adjacent particles are replaced by the arithmetic mean of their interaction radiuses. Then the modifications for kernel function of differential operator models are derived, respectively. In order to validate the present multi-resolution MPS method, two cases are carried out. Firstly, a hydrostatic case is performed. The results show that the contour of pressure field by multi-resolution MPS is quite in agreement with that by single resolution MPS. Especially, the multi-resolution MPS can still provide a relative smooth pressure together with the single resolution MPS in the vicinity of the interface between the high resolution and low resolution particles. For a long time simulation, the kinetic energy of particles by multi-resolution MPS can decrease quickly to the same level as that of single resolution MPS. In addition, a 2D dam breaking flow is simulated and the multi-resolution case can run stably during the whole simulation. The pressure by the multi-resolution MPS is in agreement with experimental data together with single resolution MPS. The contour of pressure field by the former is also similar to that by the later. Finally, the simulation by multi-resolution MPS is as accurate as the traditional MPS with fine particles distributed in the whole domain and the corresponding CPU time can be reduced. Keywords: Multi-resolution MPS (Moving Particle Semi-Implicit); modified gradient model; dam breaking; free surface flows. 1. Introduction Moving particle semi-implicit (MPS) [Koshizuka and Oka (1996); Koshizuka et al. (1998)] is one Lagrangian meshless method for incompressible flows with a free surface. Up until now, numerous nice works are carried out to improve its performance [Khayyer and Gotoh (2012); Tanaka and Masunaga (2010); Lee et al. Corresponding author. 1641018-1

Z.Tang,Y.Zhang&D.Wan (2011); Kondo and Koshizuka (2011); Zhang and Wan (2012)], which can further enhance the capability of the MPS method into violent free surface flows [Zhang and Wan (2014); Tang et al. (2015); Khayyer and Gotoh (2012); Shakibaeinia and Jin (2011); Gotoh and Sakai (2006); Khayyer and Gotoh (2008); Tang et al. (2014); Zhang et al. (2013); Shibata and Koshizuka (2012)]. Nevertheless, most of these works focus on the uniform resolution particle simulations, where only mono-mass particles are distributed in the whole computational domain. Recently, local refine techniques in the meshless methods begin to catch the massive particle practitioners attentions due to their advantages in the terms of portraying more detailed local flow field and consuming the less CPU computational time. So far, there have been several attempts to perform the spatially local refine simulations in context of particle methods. In the framework of smooth particle hydrodynamics (SPH), Omidvar et al. [2012, 2013] developed a variable mass particle distribution method employing an arbitrary Lagrangian Eulerian (ALE) formulation with an Riemann-SPH solver to study the wave body interaction. In their works, different mass particles are distributed initially and no particle splitting and coalescing occur during the following simulations. According to the particle splitting method by Feldman and Bonet [2007], Vacondio et al. [2012, 2013] presented a variable resolution SPH including a dynamic particle coalescing and splitting algorithm and the corresponding numerical tests show the great potentials of variable resolution technique. Then, Barcarolo et al. [2014] proposed an adaptive particle refinement and derefinement algorithm based on their Riemann-SPH solver, and they also dealed with the pressure discontinuities by introducing the transition zone. Most of these works are carried out based on explicit algorithm. On the other hand, the spatially local refine techniques in MPS method are reported rarely. Shibata et al. [2012] originally introduced an overlapping particle technique (OPT) and applied it into a two-dimensional (2D) green water. However, whether the total mass can be conservative is not discussed in their works. Different from Shibata s idea, Tanaka et al. [2009] presented a multi-resolution method considering the effect of the particle s diameter. In the present work, the multi-resolution technique is applied into 2D free surface flows based on improved MPS [Zhang and Wan (2012); Zhang et al. (2014)]. The paper is organized in the following way. Firstly, a brief introduction of the improved MPS method including governing equations and particle interaction models is presented. Next, the modification for kernel function of differential operator models are derived respectively considering the effect of different size particles. Finally, two cases are carried out to validate the present multi-resolution MPS method. Firstly, a hydrostatic case is performed. The pressure field and the kinetic energy for fluid particles by multi-resolution MPS are compared with that by single resolution MPS with fine particle distributed in the entire domain. In addition, a 2D dam breaking flow is simulated and the multi-resolution case can run stably during the whole simulation. The comparison among the pressure by single resolution MPS and multi-resolution MPS and experimental data is also made. 1641018-2

Multi-Resolution MPS Method for Free Surface Flows 2. Numerical Scheme 2.1. Governing equations Governing equations of the MPS method can read in Lagrangian form for incompressible fluid as following: 1 Dρ = V =0, (1) ρ Dt DV Dt = 1 ρ P + ν 2 V + g, (2) where D/Dt, ρ, V, P, ν, g and t denote the substantial derivation, the fluid density, the velocity vector, the pressure, the kinematic viscosity, gravitational acceleration vector and the time. 2.2. Particle interaction models 2.2.1. Kernel function In the MPS method, kernel function plays an important role in the discretization of differential operators. Here, we employ the kernel function [Eq. (4)] introduced by Zhang and Wan [2012] to avoid the singularity at r = 0 in the original kernel as Eq. (3) [Koshizukaet al. (1998)]. r e W (r) = r 1 0 r<r e, (3) 0 r e r, r e 1 0 r<r e, W (r) = 0.85r +0.15r e (4) 0 r e r, where r e denotes the influence radius of the target particle. According to Koshizuka s suggestion, the radius for particle number density (PND) and the gradient model is r e =2.1l 0 and it is r e =4.01l 0 for the Laplacian model, where l 0 is the initial particle space. In MPS, PND is defined as the summation of above weight function, which can be read as n i = W ( r j r i ). In the following models, Parameter n 0 indicates the constant PND. 2.2.2. Gradient model In the original MPS, the gradient operator can be discretized as Eq. (5). Then, Tanaka and Masunaga [2010] introduced a conservative form as Eq. (6). P i = D P j P i n 0 r j r i 2 (r j r i ) W ( r j r i ), (5) P i = D n 0 P j + P i r j r i 2 (r j r i ) W ( r j r i ), (6) 1641018-3

Z.Tang,Y.Zhang&D.Wan where n 0 is the initial average PND, D indicates the number of dimensions and r i presents the position vector of the ith particle. 2.2.3. Divergence model In the improved MPS method, the divergence of velocity is added into the source term of pressure Poisson equation (PPE) and can be formulated as [Yoon et al. (1999); Shakibaeinia and Jin (2012)]: V i = D (V j V i ) (r j r i ) n 0 r j r i 2 W ( r j r i ). (7) 2.2.4. Laplacian model In MPS method, the Laplacian operator is modeled as the following equations: 2 φ i = 2D n 0 (φ j φ i ) W ( r j r i ), (8) λ i λ i = W ( r j r i ) r j r i 2 W ( r. (9) j r i ) 2.3. Model of incompressibility In the present work, the mixed source ters adopted in the righthand side of the PPE [Tanaka and Masunaga (2010); Lee et al. (2011)] as following: 2 P k+1 i =(1 γ) ρ t V i γ ρ n k i n 0 t 2 n 0, (10) where the superscript k and k+1 indicate the physical quantity in the kth and k+1th time step, γ is a bending parameter between 0 and 1. In addition, γ is smaller for interface particles than that for uniform resolution particles. 2.4. Detection of free surface particles In the MPS method, there are numerous methods to identify the free surface particles [Koshizuka et al. (1998); Tanaka and Masunaga (2010); Lee et al. (2011); Khayyer et al. (2009); Zhang and Wan (2012)]. Here, we employ the detection method by Zhang and Wan [2012]. Firstly, a function according to the asymmetry distribution of neighbor particles is introduced: F i = D 1 n 0 r i r j (r i r j )W (r ij ). (11) If the absolute of the function F at target particle i is more than a threshold α, then particle i is considered as free surface particle. Where α is assigned to 0.9 F 0 and F 0 is the initial value of F for surface particle. Specially note that above condition 1641018-4

Multi-Resolution MPS Method for Free Surface Flows Fig. 1. (Color online) Particle configurations obtained by the multi-resolution MPS (red color indicates free surface particles). only adopted for particles with 0.8n 0 n i < 0.97n0 and particles with n i < 0.8n0 will be considered as free surface particles directly due to its small PND. This free surface detection method is developed for uniform resolution particles, it is expected that this method is also available for the multi-resolution MPS. Figure 1 shows the particle configurations obtained by the multi-resolution MPS in the following dam break case. 2.5. Modified gradient and Laplacian model In the single resolution MPS, the interaction radius of each particle is the same as that of its neighbor particles. However, this condition cannot be ensured in the multi-resolution MPS since both low resolution particles with larger interaction radius and high resolution particles with smaller interaction radius are distributed in the computational domain. Therefore, this may lead to situations where two interaction particles i and j with different interaction radiuses. In the other words, the influence domain of particle i contains particles j but not vice versa as shown in Fig. 2. When calculating the force between particle i and its neighbor particles j, a violation of Newton s third law may occur. To overcome this problem, the supported domain for two neighbor particles i and j is replaced by the arithmetic mean of their interaction radiuses [Tanaka et al. (2009)]. In particular, the cutoff radiuses for gradient and Laplacian models are presented as following respectively: r e = (r ei + r ej ), (12) 2 r e lap = (r ei lap + r ej lap ). (13) 2 1641018-5

Z.Tang,Y.Zhang&D.Wan Fig. 2. Different particle interaction radiuses [Tanaka et al. (2009)]. In the multi-resolution MPS, the pressure gradient in the standard MPS method may still violate the Newton s third law and the system momentum cannot be conserved. For this reason, some modifications are made for pressure gradient and Laplacian models for particles whose interaction radius is different from that of its neighboring particles. In order to derive the modifications in pressure gradient and Laplacian models, an imaginary calculation point k is introduced between the position vector of particle i and its neighboring particles j, whichisthesameas Khayyer and Gotoh s treatment [2009]. The gradient terms now can be expressed as Eqs. (14) and(15) considering the imaginary point k and the corresponding position vector r ik. P i = D P k P i n 0 r k r i 2 (r k r i ) W ( r k r i ), (14) P j = D n 0 i j P k P j r k r j 2 (r k r j ) W ( r k r j ). (15) Here, it is assumed that the distance r ik and r kj are proportional to the diameter of particle i and j, respectively. Accordingly, corresponding influence distance r e ik and r e kj are also in proportion to their diameters. Then, the following equations can be given: r ik = r ij, r e ik = r e ij, (16) + + r kj = r ij, r e kj = r e ij, (17) + + where s the particle s diameter. Following Khayyer and Gotoh s [2009] treatment, the weight function W ik between i and k is equal to the one in the initial influence domain W ij. 1 W ik = 0.85 r ik r e ik +0.15 1= 1 0.85 rij r e ij +0.15 1=W ij. (18) 1641018-6

Multi-Resolution MPS Method for Free Surface Flows On the other hand, the force due to pressure on particle i owing to k can be expressed as [Khayyer and Gotoh (2008)]: F P k i = D P k P i ρ n 0 r k r i 2 (r k r i ) W ( r k r i ), (19) while the pressure force on particle j owing to k would be [Khayyer and Gotoh (2008)]: F P k j = m j D P k P j ρ n 0 r k r j 2 (r k r j ) W ( r k r j ). (20) To ensure the continuity of the force between two adjacent particles i and j, we require F P k i = F P k j. (21) Substitute equations (16) (20) into(21), the inner-particle pressure at point k can be given as following: P k = P i + mj P j + mj. (22) According to Eqs. (14) and(22), the pressure gradient at particle i can be discretized as: P i = D P k P i n 0 r k r i 2 (r k r i ) W ( r k r i ) where A j i = = D n 0 m j L ( i+) L + m j i P j P i r j r i 2 (r j r i )W ( r j r i )A j i, (23) can be considered as the modification for the kernel function in pressure gradient. Similar to Tanaka et al. [2009], this modification is also employed in PND and velocity divergence models. In 2D case, the particle mass is equal to its volume multiplying fluid density and = ρl 2 i, m j = ρl 2 j can be obtained. Then, this modification A j i can further simply to A j i = Lj for 2D case, which coincides with the modified term by Tanaka et al. [2009]. In the present work, following gradient model is adopted to conserve momentum: P i = D n 0 P j + P i r j r i 2 (r j r i )W ( r j r i )A j i. (24) 2.6. Derivation of Laplacian model To derive the Laplacian model, we split the diffusion amount into two parts: firstly, part of quantity φ i of particle i is distributed to the imaginary point k; then, part 1641018-7

Z.Tang,Y.Zhang&D.Wan of quantity φ k of imaginary particle k is distributed to particle j. Therefore, the diffusion amount per unit volume in each part can be given as following: φ ik = 2Dν t λ i n 0 (φ k φ i )W ik = 2Dν t m j λ i n 0 φ kj = 2Dν t λ j n 0 (φ j φ k )W kj = 2Dν t λ j n 0 (φ j φ i ) + mj W ij, (25) (φ j φ i ) + mj W ij, (26) where φ k is the quantity in the imaginary point k, which can be expressed as L φ φ k = i i+ m j L φ j j + m j. We use the summation of diffusion amount in the above two parts to calculate the diffusion amount from particle i to its neighboring particles j approximatively, ignoring the nonlinear effect: φ ij = φ ik + m j φ kj = 2Dν t ( / (φ j φ i ) mi m j n 0 λ + mj j + m ) im j /λ i W ij, (27) which gives φ ij = 2Dν t m ( j L n 0 (φ j φ i )W i /λ j + mj /λ i ) ij + mj. (28) Finally, the quantity transfer can be superposed, and the Laplacian model can be represented by: 2 φ i = 2D n 0 (φ j φ i )W ij ( mj /λ j + mj /λ i ). (29) + mj 3. Numerical Examples 3.1. The hydrostatic case In this section, the hydrostatic problem similar to Antuono et al. s case [2010] is carried out by the employment of the single resolution and multi-resolution MPS. The computational domain for this test is shown in Fig. 3, where both the width of the water tank and the height of fluid are H =1.0m. The computational parameters are summarized in Table 1. In Case A1, the entire computational domain is discretized by high resolution particles and the corresponding initial particle space is d =0.01m. In Case A2, two kinds of particle size are employed where the yellow and blue regions are presented by high resolution (H/d = 100) and low resolution (H/d = 50) particles, respectively. The height of the high resolution region in Case A2 is 0.4 m as the yellow region in Fig. 3. Figure 4 shows the pressure field after a long time evolution of the hydrostatic test. The contour of pressure field in the fine region by multi-resolution MPS is 1641018-8

Multi-Resolution MPS Method for Free Surface Flows Fig. 3. (Color online) A schematic sketch of the computational domain for hydrostatic problem. Table 1. Computational parameters in the simulations. Cases Initial particle space (H/d) Description A1 100 Single resolution A2 50, 100 Multi-resolution (a) Single resolution (b) Multiresolution Fig. 4. The pressure field predicted in Cases A1 and A2. quite similar to that by single resolution MPS with fine particles. Furthermore, in the vicinity of the interface between the high and low resolution particles, a relative smooth pressure field can also be predicted by multi-resolution MPS. Figure 5 shows the comparison between the kinetic energy predicted by multiresolution MPS and single resolution MPS, where the entire domain is represented 1641018-9

Z.Tang,Y.Zhang&D.Wan Fig. 5. The comparison between the kinetic energy predicted by single resolution and multiresolution MPS. by fine particles in the later and the fine particles are only distributed in the yellow region in the former. In Fig. 5, both the kinetic energy by multi-resolution MPS and single resolution MPS first increase and then decrease. In addition, the kinetic energy by multi-resolution MPS quickly decreases to the same level as that by single resolution MPS. This means that the disturbance produced in the interface between different particle sizes is not large, and can be reduced quickly as the initial disturbance in the uniform particle size simulation in this case. 3.2. Dam break flow Dam breaking probles a typical violent free surface flow. In the present work, a 2D dam-break flow is numerically simulated by the multi-resolution MPS method and Fig. 6. A schematic view of the computation domain for dam breaking. 1641018-10

Multi-Resolution MPS Method for Free Surface Flows Fig. 7. Initial particle mass distribution for 2D dam breaking problem. Table 2. Computational parameters in dam breaking cases. Cases Initial particle space (H/d) Description B1 120 Single resolution B2 30, 60, 120 Multi-resolution B3 30 Single resolution the uniform resolution MPS method, respectively. The computational domain and the dimensions of water column are shown in Fig. 6. A wave height probe is placed at 2.725 m from the left boundary and two pressure probes are placed on the right wall. The initial particle mass distribution for multi-resolution simulation is depicted in Fig. 7, where three kinds of particle size are selected and listed in Table 2, including H/d =30, 60, 120, and corresponding masses are 0.4, 0.1 and 0.025, respectively. The wave front propagation along the downstream horizontal dry bed after the dam door release are shown in Fig. 8. The multi-resolution result is quite agreement with that of single resolution MPS (H/d = 120) and is also quite similar to that of BEM [Colagrossi and Landrini (2003)]. However, the speeds of the leading edge by these numerical results are quick than that of the experiment [Buchner (2002)]. Similar results can also be reported in literatures [Koshizuka and Oka (1996); Rogers et al. (2010); Abdolmaleki et al. (2004)]. In Figs. 9 and 10, the pressure fields by single resolution and multi-resolution MPS are depicted for comparison. It can be seen that the computed pressure fields are both relative smooth throughout the time of flow propagation, free surface overturning and impacting the underline water. The contour of the pressure field by multi-resolution MPS is also similar to that by single resolution MPS. In addition, the left pressure field by single resolution (H/d = 120) is slightly smoother than that by multi-resolution MPS and single resolution with H/d = 30, while the right pressure field by multi-resolution MPS and high resolution case are both smoother than that by low resolution case. Generally speaking, multi-resolution MPS can produce the fine local pressure field due to the high resolution in this region. 1641018-11

Z.Tang,Y.Zhang&D.Wan Fig. 8. Propagation of the surge front after dam gate removal compared to literature data (Experimental data [Buchner (2002)]; BME result [Colagrossi and Landrini (2003)]). (a) Single resolution (H/d = 120) (b) Multiresolution (c) Single resolution (H/d = 30) Fig. 9. Comparisons of dam-break flows using single resolution and multi-resolution MPS at t p g/h =1.45. The detailed comparisons of time variations of recorded pressure at the bottom of the probe P1 [Ozbulut et al. (2014); Colicchio et al. (2002)] are shown in Fig. 11, it can be seen that the overall tendency of pressure variation by both the single resolution (H/d = 120) and multi-resolution MPS is quite in agreement with experimental data [Buchner (2002)] except a clear discrepancy between the position of second pressure peak by the numerical results and experimental data [Buchner (2002)], which is also reported by many researchers employing the single phase model [Marrone et al. (2011); Khayyer et al. (2009)]. However, the pressure variation by multi-resolution MPS is quite close to that of single resolution MPS 1641018-12

2nd Reading April 1, 2016 15:52 WSPC/0219-8762 196-IJCM 1641018 Multi-Resolution MPS Method for Free Surface Flows (a) Single resolution (H/d = 120) (b) Multiresolution (c) Single resolution (H/d = 30) Fig. p 10. Comparisons of dam-break flows using single resolution and multi-resolution MPS at t g/h = 5.7. Fig. 11. Time variations of dimensionless pressure at the bottom of the probe P1 (Experimental date [Buchner (2002)]). (H/d = 120), including the first impact time and the position of the second pressure peak. Specially note that different particle methods may obtain different pressure peaks [Colagrossi and Landrini (2003); Chen et al. (2015)] due to various boundary treatments, one or two phase model and so on. Here, the effects of these factors are not discussed since this is not our target in this paper. Figure 12 depicts the number of particles in the two numerical simulations and the corresponding CPU time by the multi-resolution MPS method and the uniform resolution MPS method. In Fig. 12, single resolution represents the case with fine particles in the whole domain. Both of these two cases are carried out on personal computer with Intel i7-3770. From Fig. 12, the number of particles by multi-resolution MPS is nearly half of that by single resolution MPS, while the 1641018-13

Z.Tang,Y.Zhang&D.Wan Fig. 12. The number of particles and CPU time for flowing 3 s. consuming CPU time in the later is about two times and a half than that in the former. In the other words, the number of particles in multi-resolution MPS is less than that in single resolution MPS, which means that multi-resolution MPS can decrease the required CPU time through reducing the number of particles based on the relationship between the consuming CPU time for solving PPE and number of particles [Koshizuka et al. (1998)]. Therefore, the multi-resolution method can be an alternative way to reduce the required computational time by decreasing the necessary number of particles if one only concerns the local region. Furthermore, considering the pressure variation and the contour of pressure field between the multi-resolution MPS and the uniform resolution MPS with fine particles in the whole domain, the multi-resolution MPS can reduce the CPU time without sacrificing the accuracy. 4. Conclusions In this paper, the multi-resolution MPS method is applied into 2D free surface flows based on in-house particle solver MLParticle-SJTU. In the multi-resolution MPS method, the entire computation domain is discretized with both the low resolution and high resolution particles. Considering the effect of different size particles, both the influence radiuses of two adjacent particles are replaced by the arithmetic mean of their interaction radiuses. Then, the modifications for kernel function of differential operator models are derived respectively. To verify the availability and efficiency of the present multi-resolution MPS, two cases are carried out. Firstly, a hydrostatic case is performed. The results show that the contour of pressure field 1641018-14

Multi-Resolution MPS Method for Free Surface Flows by multi-resolution MPS is nearly the same as that of single resolution MPS. Especially, the multi-resolution MPS can still provide a relative smooth pressure in the vicinity of the interface between the high resolution and low resolution particles. For a long time simulation, the kinetic energy of particles by multi-resolution MPS can decrease quickly to the same level as that of single resolution MPS with fine particles distributed in the entire domain. In addition, a 2D dam-break flow is also carried out and the multi-resolution case can run stably during the whole simulation. Both the pressure variation at the measuring position and the contour of the pressure field at different times by multi-resolution MPS are quite in agreement with that of single resolution MPS. Considering the required CPU time of these two methods, multi-resolution MPS can reduce the computational time without sacrificing its accuracy. Acknowledgments This work is supported by the National Natural Science Foundation of China (51379125, 51490675, 11432009, 51579145, 11272120), Chang Jiang Scholars Program (T2014099), Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning (2013022), Innovative Special Project of Numerical Tank of Ministry of Industry and Information Technology of China (2016-23) and Foundation of State Ley Laboratory of Ocean Engineering (GKZD010065), to which the authors are most grateful. References Abdolmaleki, K., Thiagarajian, K. P. and Morris-Thomas, M. T. [2004] Simulation of the dam break problem and impact flows using a Navier Stokes solver, Proc. 15th Australasian Fluid Mechanics Conf., Sydney. Antuono, M., Colagrossi, A. and Molteni, D. [2010] Free-surface flows solved by means of SPH schemes with numerical diffusive terms, Computer Physics Communications 181, 532 549. Barcarolo, D. A., Le Touzé, D., Oger, G. and de Vuyst, F. [2014] Adaptive particle refinement and derefinement applied to the smoothed particle hydrodynamics method, J. Comput. Phys. 273, 640 657. Buchner, B. [2002] Green water on ship-type offshore structures, Delft University of Technology, PhD thesis. Chen, Z., Zong, Z., Liu, M. B., Zou, L., Li, H. T. and Shu, C. [2015] An SPH model for multiphase flows with complex interfaces and large density differences, J. Comput. Phys. 283, 169 188. Colagrossi, A. and Landrini, M. [2003] Numerical simulation of interfacial flows by smoothed particle hydrodynamics, J. Comput. Phys. 191, 448 475. Colicchio, G., Colagrossi, A., Greco, M. and Landrini, M. [2002] Free surface flow after a dam break a comparative study, Ship Technol. Res. 49, 95 104. Feldman, J. and Bonet, J. [2007] Dynamic refinement and boundary contact forces in SPH with applications in fluid flow problems, Int. J. Numer. Methods Eng. 72, 295 324. Gotoh, H. and Sakai, T. [2006] Key issues in the particle method for computation of wave breaking, Coastal Eng. 53, 171 179. 1641018-15

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