SPH Study on Mitigating the Incident Wave Effect on Seawalls Using the Bragg Scattering Phenomenon in the Presence of Wavy-shaped Submerged Plates: Case Study of Chabahar Port

Abstract

This study aims to investigate the effectiveness of solid and perforated wavy plates in mitigating water level fluctuations near the Chabahar impermeable seawall. This arrangement creates a gap between the wavy plates and the seawall, inducing Bragg resonance, which can be exploited to diminish water level fluctuations near the seawall. Numerical simulations were conducted using the DualSPHysics software (V 5.2.0), which is based on the smoothed particle hydrodynamics (SPH) method. Initially, the numerical model was validated against experimental results. Subsequently, the effects of wave height, period, and seawall slope on the wave reflection coefficient and water level variation were examined. The results indicate that the wave-mitigating performance of wavy plates decreases with increasing submersion depth. Furthermore, these plates are more effective in mitigating the effects of waves with greater heights and shorter periods. Higher reflection coefficients correspond to lower water level fluctuations within the gap. Solid plates can effectively mitigate wave action on seawalls with low to moderate slopes. Additionally, water level fluctuations outside the gap decrease as the porosity of the wavy plate increases. This study offers valuable insights for designing resilient, environmentally sensitive coastal defenses; however, further experimental validation is recommended to support broader applications.

Keywords

Reflection coefficient / Bragg scattering / SPH / Chabahar port / Coastal protection

References

[1] Aghaei B, Arasteh AM, Lari K, Torabi Azad M (2021) Numerical modeling of wave reflection from sloped impermeable seawalls using the SPH method: Case study of Chabahar port. Advances in Civil Engineering 2021:Article ID 7382416. https://doi.org/10.1155/2021/7382416

[2] Altomare C, Crespo AJC, Domínguez JM, Gómez-Gesteira M, Suzuki T, Verwaest T. Applicability of smoothed particle hydrodynamics for estimation of sea wave impact on coastal structures. Coastal Engineering, 2015, 96: 1-12

[3] Antuono M, Colagrossi A, Marrone S. Numerical diffusive terms in weakly-compressible SPH schemes. Computer Physics Communications, 2012, 183(12): 2570-2580

[4] Badarch A, Fenton JD, Hosoyamada T. Application of free-surface immersed-boundary lattice boltzmann method to waves acting on coastal structures. Journal of Hydraulic Engineering, 2020, 146(2): 04019062

[5] Bautista E, Bahena-Jimenez S, Quesada-Torres A, Méndez F, Arcos E. Interaction between long water waves and two fixed submerged breakwaters of wavy surfaces. Wave Motion, 2022, 112: 102926

[6] Bora SN, Martha SC. Scattering of surface waves over an uneven sea-bed. Applied Mathematics Letters, 2008, 21(101082-1089

[7] Bouscasse B, Colagrossi A, Marrone S, Antuono M. Nonlinear water wave interaction with floating bodies in SPH. Journal of Fluids and Structures, 2013, 42: 112-119

[8] Bragg WH, Bragg WL. The reflection of X-rays by crystals. Proceedings of the Royal Society of London Series A, Containing Papers of a Mathematical and Physical Character. London:Royal Society, 1913, 88(605428-438

[9] Cao XY, Ming FR, Zhang AM. Sloshing in a rectangular tank based on SPH simulation. Applied Ocean Research, 2014, 47: 241-254

[10] Carmigniani R, Leroy A, Violeau D. A simple SPH model of a free surface water wave pump: Waves above a submerged plate. Coastal Engineering Journal, 2019, 61(1): 96-108

[11] Chen H Bin, Tsai CP, Jeng CC (2007) Wave transformation between submerged breakwater and seawall. Journal of Coastal Research (SI 50): 1069–1074

[12] Chen W, Warmink JJ, van Gent MRA, Hulscher SJMH. Numerical modelling of wave overtopping at dikes using OpenFOAM®. Coastal Engineering, 2021, 166: 103890

[13] Chen YK, Meringolo DD, Liu Y, Liang JM. Energy balance during Bragg wave resonance by submerged porous breakwaters through a mixture theory-based (5-LES-SPH model. Coastal Engineering, 2025, 196: 104652

[14] Chen Z, Zong Z, Li HT, Li J. An investigation into the pressure on solid walls in 2D sloshing using SPH method. Ocean Engineering, 2013, 59: 129-141

[15] Chen Z, Zong Z, Liu MB, Zou L, Li HT, Shu C. An SPH model for multiphase flows with complex interfaces and large density differences. Journal of Computational Physics, 2015, 283: 169-188

[16] Chen Z, Zong Z, Liu MB, Li HT. A comparative study of truly incompressible and weakly compressible SPH methods for free surface incompressible flows. Int J Numer Methods Fluids, 2013, 73(9): 813-829

[17] Cho IH, Kim MH. Wave absorbing system using inclined perforated plates. Journal of Fluid Mechanics, 2008, 608: 1-20

[18] Crespo AJC, Gómez-Gesteira M, Dalrymple RA. Boundary conditions generated by dynamic particles in SPH methods. Computers, Materials and Continua, 2007, 5(3): 173-184

[19] Dai Z, Xu K, Wang F, Yang H, Qin S. Numerical investigation on the kinetic characteristics of the Yigong debris flow in Tibet, China. Water (Switzerland), 2021, 13(8): 1076

[20] Dang BL, Nguyen-Xuan H, Abdel Wahab M. Numerical study on wave forces and overtopping over various seawall structures using advanced SPH-based method. Engineering Structures, 2021, 226: 111349

[21] Davies AG, Heathershaw AD. Surface-wave propagation over sinusoidally varying topography. Journal of Fluid Mechanics, 1984, 144: 419-443

[22] Dean RG, Dalrymple RA. Water wave mechanics for engineers and scientists. world scientific publishing company. Advanced Series on Ocean Engineering, 1991, 2: 368

[23] Dehnen W, Aly H. Improving convergence in smoothed particle hydrodynamics simulations without pairing instability. Monthly Notices of the Royal Astronomical Society, 2012, 425(21068-1082

[24] Delorme L, Colagrossi A, Souto-Iglesias A, Zamora-Rodríguez R, Botía-Vera E. A set of canonical problems in sloshing, Part I: Pressure field in forced roll-comparison between experimental results and SPH. Ocean Engineering, 2009, 36(2168-178

[25] Delorme L, Iglesias AS, Perez SA. Sloshing Loads Simulation in Lng Tankers With SPH. International Conference on Computational Methods in Marine Engineering, 2005, Barcelona, CIMNE110

[26] Demir A. Hydro-elastic analysis of standing submerged structures under seismic excitations with sph-fem approach. Latin American Journal of Solids and Structures, 2020, 17(9): 1-14

[27] Dentale F, Donnarumma G, Carratelli EP. Rubble mound breakwater: Run-up, reflection and overtopping by numerical 3D simulation. Coasts, Marine Structures and Breakwaters 2013: From Sea to Shore-Meeting the Challenges of the Sea, 2014

[28] Domínguez JM, Altomare C, Gonzalez-Cao J, Lomonaco P. Towards a more complete tool for coastal engineering: solitary wave generation, propagation and breaking in an SPH-based model. Coastal Engineering Journal, 2019, 61: 15-40

[29] Domínguez JM, Fourtakas G, Altomare C, Canelas RB, Fafuni A, García-Feal O, Martínez-Estévez I, Mokos A, Vacondio R, Crespo AJC, Rogers BD, Stansby PK, Gómez-Gesteira M. DualSPHysics: from fluid dynamics to multiphysics problems. Computational Particle Mechanics, 2021, 9: 867-895

[30] Dong J, Wang B, Liu H. Wave forces on a submerged horizontal plate over a sloping beach due to a solitary wave. Proceedings of the 12th ISOPE Pacific-Asia Offshore Mechanics Symposium, PACOMS 2016: ISOPE-P-16-079, 2016

[31] Duan X, Kazemi E, Gui Q, Fang K. Smoothed particle hydrodynamics for enhanced coastal protection: A study on wave dynamics and Bragg resonance over different topographic conditions. Physics of Fluids, 2025, 37(4): 047111

[32] English A, Domínguez JM, Vacondio R, Crespo AJC, Stamsby PK, Lind SJ, Chiapponi L, Gómez-Gesteira M. Modified dynamic boundary conditions (mDBC) for general-purpose smoothed particle hydrodynamics (SPH): application to tank sloshing, dam break and fish pass problems. Computational Particle Mechanics, 2022, 9: 1-15

[33] Fang Q, Yang C, Guo A. Hydrodynamic performance of submerged plates during focused waves. Journal of Marine Science and Engineering, 2019, 7(11389

[34] Fathi A, Ketabdari MJ (2018) Modeling of emerged semi-circular breakwater performance against solitary waves using SPH method. Journal of the Brazilian Society of Mechanical Sciences and Engineering 40: article number 290. https://doi.org/10.1007/s40430-018-1179-4

[35] Fourtakas G, Dominguez JM, Vacondio R, Rogers BD. Local uniform stencil (LUST) boundary condition for arbitrary 3-D boundaries in parallel smoothed particle hydrodynamics (SPH) models. Computers Fluids, 2019, 190: 346-361

[36] Fu D, Zhao X, Wang S, Yan D. Numerical study on the wave dissipating performance of a submerged heaving plate breakwater. Ocean Engineering, 2021, 219: 108310

[37] Gayathri R, Benny C, Behera H. Wave attenuation by a submerged flexible permeable membrane. AIP Conference Proceedings, 2020, 2277: 210003

[38] Gharechae A, Ketabdari MJ. Analytical solution based on BEM to oblique waves scattering by thin arc-shaped permeable barrier applied for array of aquaculture cages. Engineering Analysis with Boundary Elements, 2022, 134: 241-258

[39] Goda Y, Fukumori T. Laboratory investigation of wave pressures exerted upon vertical and composite walls. Coastal Engineering in Japan, 1972, 15: 81-90

[40] Goda Y, Suzuki T. Estimation of incident and reflected waves in random wave experiments. Coastal Engineering Proceedings, 1976, 1(1547

[41] Google Maps. Chabahar Port, 2025Accessed 10 Oct 2025

[42] Gotoh H, Khayyer A, Ikari H, et al.. On enhancement of Incompressible SPH method for simulation of violent sloshing flows. Applied Ocean Research, 2014, 46: 104-115

[43] Gotoh H, Shao S, Memita T. SPH-LES model for numerical investigation of wave interaction with partially immersed breakwater. Coastal Engineering Journal, 2004, 46(139-63

[44] Guo YC, Mohapatra SC, Guedes Soares C. Wave energy dissipation of a submerged horizontal flexible porous membrane under oblique wave interaction. Applied Ocean Research, 2020, 94: 101948

[45] He M, Gao X, Xu W, Ren B, Wang H. Potential application of submerged horizontal plate as a wave energy breakwater: A 2D study using the WCSPH method. Ocean Engineering, 2019, 185: 27-46

[46] He M, Xu W, Gao X, Ren B. SPH simulation of wave scattering by a heaving submerged horizontal plate. International Journal of Ocean and Coastal Engineering, 2018, 1(2): 1840004

[47] He SY, Liu Y, Zhao Y, Li HJ. New analytical solutions of oblique wave scattering by submerged horizontal perforated plates using quadratic pressure drop condition. Ocean Engineering, 2021, 220: 108444

[48] Hirt CW, Nichols BD. Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics, 1981, 39(1201-225

[49] Jiang C, Liu X, Yao Y, Deng B, Chen J. Numerical investigation of Tsunami-like solitary wave interaction with a seawall. Journal of Earthquake and Tsunami, 2017, 11(11740006

[50] Ketabdari MJ, Soleimani K. Numerical study of motion response of a floating body using SPH method. AIP Conference Proceedings, 2015, 1648(1770007

[51] Ketabdari MJ, Soleimani K, Khorasani F. Vasel-Be-Hagh A, Ting DK. Modeling of multipurpose fixed breakwater wave energy converter using SPH method. Complementary Resources for Tomorrow, 2020, Cham, Springer149-162

[52] Kumaran V, Neelamani S, Vijay KG, Al-Anjari N, Al-Ragum A. Wave attenuation by multiple slotted barriers with a zigzag arrangement -A physical and numerical approach. Journal of Hydro-environment Research, 2022, 41: 25-37

[53] Lee S, Ko K, Hong JW. Comparative study on the breaking waves by a piston-type wavemaker in experiments and SPH simulations. Coastal Engineering Journal, 2020, 62: 267-284

[54] Ling Z, Li Y, Zhang Y. Investigation of Bragg resonant reflection of linear and nonlinear water waves by multiple fixed floating structures based on viscous meshfree method. Ocean Engineering, 2023, 278: 114270

[55] Liu BJ, Cheng D, Sun ZC, Zhao XZ, Chen Y, Lin WD. Experimental and numerical study of regular waves past a submerged breakwater. Journal of Hydrodynamics, 2019, 31: 641-653

[56] Liu HW, Shi YP, Cao DQ. Optimization of parabolic bars for maximum Bragg resonant reflection of long waves. Journal of Hydrodynamics, 2015, 27: 373-382

[57] Liu Y, Li A J, Fang ZB. Oblique wave scattering by porous breakwaters/seawalls: Novel analytical solutions based on contour integral without finding complex roots. Applied Ocean Research, 2020, 101: 102258

[58] Liu Y, Li HJ. Iterative multi-domain BEM solution for water wave reflection by perforated caisson breakwaters. Engineering Analysis with Boundary Elements, 2017, 77: 70-80

[59] Liu Y, Li HJ, Li YC. A new analytical solution for wave scattering by a submerged horizontal porous plate with finite thickness. Ocean Engineering, 2012, 42: 83-92

[60] Liu Y, Li YC, Teng B, Dong S. Wave motion over a submerged breakwater with an upper horizontal porous plate and a lower horizontal solid plate. Ocean Engineering, 2008, 35(16): 1588-1596

[61] Liu Z, Wang Y, Wang W, Hua X. Numerical modeling and optimization of a winged box-type floating breakwater by Smoothed Particle Hydrodynamics. Ocean Engineering, 2019, 188: 106246

[62] Luthfi M, Syamsidik, Fauzi A, Fatimah E. Numerical simulations of Tsunami wave properties on coastal slopes using one piston-wavemaker method. IOP Conference Series: Earth and Environmental Science, the 11th Aceh International Workshop and Expo on Sustainable Tsunami Disaster Recovery. Banda Aceh, 2019, 273: 012011

[63] Lyu HG, Sun PN. Establishment and validation of a versatile SPH-based numerical tank for generating wave-alone, current-alone, and wave-current-combined fields. Coastal Engineering, 2025, 197: 104663

[64] Mallayachari V, Sundar V. Reflection characteristics of permeable seawalls. Coastal Engineering, 1994, 23(1–2): 135-150

[65] Manenti S, Panizzo A, Ruol P, Martinelli L. SPH simulation of a floating body forced by regular waves. 3rd ERCOFTAC SPHERIC Workshop on SPH Applications, 200836-41

[66] Marsh A, Prakash M, Semercigil E, Turan F. A study of sloshing absorber geometry for structural control with SPH. Journal of Fluids Structures, 2011, 27(8): 1165-1181

[67] Md N N, Greenhow M. Wave impacts on structures with rectangular geometries: Part 1. Seawalls. Applied Ocean Research, 2015, 53: 132-141

[68] Medina-Rodríguez A, Bautista E, Méndez F. Asymptotic analysis of the interaction between linear long waves and a submerged floating breakwater of wavy surfaces. Applied Ocean Research, 2016, 59: 345-365

[69] Mei CC. Resonant reflection of surface water waves by periodic sandbars. Journal of Fluid Mechanics, 1985, 152: 315-335

[70] Mohapatra AK, Vijay KG, Sahoo T. Bragg scattering of surface gravity waves by a submerged wavy porous plate. Ocean Engineering, 2021, 219: 108273

[71] Monaghan J. Smoothed particle hydrodynamics. Annual Review of Astronomy and Astrophysics, 1992, 30: 543-574

[72] Muttray M, Oumeraci H, ten Oever E. Wave reflection and wave run-up at rubble mound breakwaters. 30th ICCE, 2007, Los Angeles, ICCE43134324

[73] Neelamani S, Reddy MS. Wave transmission and reflection characteristics of a rigid surface and submerged horizontal plate. Ocean Engineering, 1992, 19(4): 327-341

[74] Neelamani S, Sandhya N. Surface roughness effect of vertical and sloped seawalls in incident random wave fields. Ocean Engineering, 2005, 32(3–4): 395-416

[75] Neelamani S, Schüttrumpf H, Muttray M, Oumeraci H. Prediction of wave pressures on smooth impermeable seawalls. Ocean Engineering, 1999, 26(8): 739-765

[76] Ning D, Li Q, Lin H, Teng B. Numerical investigation of nonlinear wave scattering by a horizontal submerged plate. Procedia Engineering, 2015, 116: 237-244

[77] Ning D, Li X, Zhang C. Nonlinear simulation of wave train impact on a vertical seawall. Water, 2018, 10(8): 986

[78] Omidvar P, Stansby PK, Rogers BD. SPH for 3D floating bodies using variable mass particle distribution. International Journal for Numerical Methods in Fluids, 2013, 72(4427-452

[79] Patarapanich M. Maximum and zero reflection from submerged plate. Journal of Waterway, Port, Coastal, and Ocean Engineering, 1984, 110(2): 171-181

[80] Peng C, Xu G, Wu W, Yu H, Wang C. Multiphase SPH modeling of free surface flow in porous media with variable porosity. Computers and Geotechnics, 2017, 81: 239-248

[81] Rao S, Shirlal KG, Varghese RV, Govindaraja KR. Physical model studies on wave transmission of a submerged inclined plate breakwater. Ocean Engineering, 2009, 36(15–16): 1199-1207

[82] Reeve DE, Soliman A, Lin PZ. Numerical study of combined overflow and wave overtopping over a smooth impermeable seawall. Coastal Engineering, 2008, 55(2): 155-166

[83] Ren B, He M, Dong P, Wen H. Nonlinear simulations of wave-induced motions of a freely floating body using WCSPH method. Applied Ocean Research, 2015, 50: 1-12

[84] Ren B, Jin Z, Gao R, Wang YX, Xu ZL. SPH-DEM Modeling of the Hydraulic Stability of 2D Blocks on a Slope. Journal of Waterway, Port, Coastal, and Ocean Engineering, 2014, 140(6): 04014022-1-04014022-12

[85] Ren XF, Sun ZC, Wang XG, Liang SX. SPH numerical modeling for the wave - thin structure interaction. China Ocean Engineering, 2018, 32: 157-168

[86] Rostami Varnousfaaderani M, Ketabdari MJ. Numerical simulation of solitary wave breaking and impact on seawall using a modified turbulence SPH method with Riemann solvers. Journal of Marine Science and Technology, 2015, 20: 344-356

[87] Sarfaraz M, Pak A. SPH numerical simulation of tsunami wave forces impinged on bridge superstructures. Coastal Engineering, 2017, 121: 145-157

[88] Shao JR, Li HQ, Liu GR, Liu MB. An improved SPH method for modeling liquid sloshing dynamics. Computers Structures, 2012, 100–101: 18-26

[89] Shao S, Gotoh H. Simulating coupled motion of progressive wave and floating curtain wall by SPH-LES model. Coastal Engineering Journal, 2004, 46: 171-202

[90] Sharma M, Kaligatla RB, Sahoo T. Wave interaction with a submerged floating tunnel in the presence of a bottom mounted submerged porous breakwater. Applied Ocean Research, 2020, 96: 102069

[91] Shi Y, Li S, Peng S, Chen H, He M. Numerical modeling of flexible floating boom using a coupled SPH-FEM model. Coastal Engineering Journal, 2018, 60(2): 140-158

[92] Shimizu Y, Gotoh H. Volume-conserved wavy interface boundary for (5-SPH-based numerical wave flume. Coastal Engineering Journal, 2025, 67(2214-231

[93] Shimizu Y, Khayyer A, Gotoh H. An enhanced incompressible SPH method for simulation of fluid flow interactions with saturated/unsaturated porous media of variable porosity. Ocean Systems Engineering, 2022, 12: 63-86

[94] Sigalotti LDG, Klapp J, Gesteira MG. The mathematics of smoothed particle hydrodynamics (SPH) consistency. Frontiers in Applied Mathematics and Statistics, 2021, 7: 797455

[95] Soleimani K, Ketabdari MJ. Meshfree modeling of near field two-liquid mixing process in the presence of different obstacles. Ocean Engineering, 2020, 213: 107625

[96] Soleimani K, Ketabdari MJ. Performance analysis of a tuned point absorber using SPH calm water and wave tank simulations. Journal of Ocean Engineering and Science, 2022, 9(5): 409-436

[97] Soleimani K, Ketabdari MJ. SPH modeling of heave motion response of pile-supported floating breakwater. Journal of Coastal and Marine Engineering, 2018, 1(139-45

[98] Soleimani K, Ketabdari MJ, Gharechae A. Smoothed particle hydrodynamics study of a heaving point absorber in various waves using wave tank and calm-water models. Physics of Fluids, 2023, 35: 033116

[99] Subramaniam SP, Scheres B, Schilling M, Liebisch S, Kerpen NB, Schlurmann T, Altomare C, Schüttrumpf H. Influence of convex and concave curvatures in a coastal dike line on wave run-up. Water, 2019, 11(7): 1333

[100] Tsurudome C, Liang D, Shimizu Y, Khayyer A, Gotoh H. Study of beach permeability’s influence on solitary wave runup with ISPH method. Applied Ocean Research, 2021, 117: 102957

[101] Venkateswarlu V, Suresh Kumar D, Srinivasula Reddy I, Martha SC. Effect of seabed geometry on the hydrodynamic performance of a thick wavy porous barrier. Journal of Marine Science and Application, 2025, 24: 998-1018

[102] Venkateswarlu V, Vijay KG, Nishad CS. Iterative dual boundary element analysis of a wavy porous plate near an inclined seawall. Ocean Engineering, 2021, 235: 109242

[103] Verduzco-Zapata MG, Javier Ocampo-Torres F, Mendoza E, Silva R, Liñán-Cabello M, Torres-Orozco E. Optimal submergence of horizontal plates for maximum wave energy dissipation. Ocean Engineering, 2017, 142: 78-86

[104] Vicinanza D, Lauro E D, Contestabile P, Gisonni C, Lara JL, Losada IJ (2019) Review of innovative harbor breakwaters for wave-energy conversion. Journal of Waterway, Port, Coastal, and Ocean Engineering 145(4). https://doi.org/10.1061/(asce)ww.1943-5460.0000519

[105] Vignjevic R, Djordjevic N, Gemkow S, De Vuyst T, Campbell J. SPH as a nonlocal regularisation method: Solution for instabilities due to strain-softening. Computer Methods Applied Mechanics and Engineering, 2014, 277: 281-304

[106] Vijay KG, Neelamani S, AlYousif A. Hydrodynamic analyses of multiple porous plates attached to the front of a vertical composite breakwater. Ocean Engineering, 2022, 266(Part3112964

[107] Vijay KG, Neelamani S, Sahoo T. Wave interaction with multiple slotted barriers inside harbour: Physical and numerical modelling. Ocean Engineering, 2019, 193: 106623

[108] Violeau D, Rogers BD. Smoothed particle hydrodynamics (SPH) for free-surface flows: Past, present and future. Journal of Hydraulic Research, 2016, 54: 1-26

[109] Wen H, Ren B, Dong P, Wang Y. A SPH numerical wave basin for modeling wave-structure interactions. Applied Ocean Research, 2016, 59: 366-377

[110] Xu H, Lin P. A new two-step projection method in an ISPH model for free surface flow computations. Coastal Engineering, 2017, 127: 68-79

[111] Xu W, Chen C, Htet MH, Sarkar MSI, Tao A, Wang Z, Fan J, Jiang D. Experimental investigation on bragg resonant reflection of waves by porous submerged breakwaters on a horizontal seabed. Water, 2022, 14(172682

[112] Yeganeh-Bakhtiary A, Houshangi H, Hajivalie F, Abolfathi S. A numerical study on hydrodynamics of standing waves in front of caisson breakwaters with WCSPH model. Coastal Engineering Journal, 2017, 59: 1750005-1-1750005-31

[113] Yip TL, Chwang AT. Perforated wall breakwater with internal horizontal plate. Journal of Engineering Mechanics, 2000, 126(5533

[114] Yueh BCY, Tsaur DH. Wave scattering by submerged vertical plate-type breakwater using composite. Coastal Engineering Journal, 1999, 41(165-83

[115] Yueh CY, Chuang SH, Wen CC. Wave scattering by submerged composite wavy plate breakwaters using a dual BEM. Ocean Engineering, 2016, 124: 192-203

[116] Yueh CY, Chuang SH, Wen CC. Bragg reflection of water waves due to submerged wavy plate breakwater. Journal of Hydro-Environment Research, 2018, 21: 52-59

[117] Zeng J, Chen G, Pan CH, Zhang ZY. Effect of dike line adjustment on the tidal bore in the Qiantang Estuary, China. Journal of Hydrodynamics, 2017, 29: 452-459

[118] Zhang A, Sun P, Ming F. An SPH modeling of bubble rising and coalescing in three dimensions. Computer Methods in Applied Mechanics and Engineering, 2015, 294: 189-209

[119] Zhang W, Ji J, Gao Y, Li X, Zhang C. Spatial variability effect of internal friction angle on the post-failure behavior of landslides using a random and non-Newtonian fluid based SPH method. Geoscience Frontiers, 2020, 11(4): 1107-1121

[120] Zhao Y, Liu Y, Li H. Wave interaction with a partially reflecting vertical wall protected by a submerged porous bar. Journal of Ocean University of China, 2016, 15: 619-626

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