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On the propagation of nonlinear water waves in a three-dimensional numerical wave flume using the generalized finite difference method. (English) Zbl 1464.76123
Summary: Nonlinear water waves are common physical phenomena in the field of coastal and ocean engineering, which plays a critical role in the investigation of hydrodynamics regarding offshore and deep-water structures. In the present study, a three-dimensional (3D) numerical wave flume (NWF) is constructed to simulate the propagation of nonlinear water waves. On the basis of potential flow theory, the second-order Runge-Kutta method (RKM2) combining with a semi-Lagrangian approach is carried out to discretize the temporal variable of the 3D Laplace’s equation. For the spatial variables, the generalized finite difference method (GFDM) is adopted to solve the governing equations for the deformable computational domain at each time step. The upstream condition is considered as a wave-making boundary with imposing horizontal velocity while the downstream condition as a wave-absorbing boundary with a pre-defined sponge layer to deal with the phenomenon of wave reflection. Three numerical examples are investigated and discussed in detail to validate the accuracy and stability of the developed 3D GFDM-based NWF. The results show that the newly-proposed numerical method has good performance in the prediction of the dynamic evolution of nonlinear water waves, and suggests that the novel 3D “RKM2-GFDM” meshless scheme can be employed to further investigate more complicated hydrodynamic problems in practical applications.
MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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[1] Akylas, T., David J. Benney: nonlinear wave and instability processes in fluid flows, Annu Rev Fluid Mech, 52, 1, 21-36 (2020) · Zbl 1439.76013
[2] Benito, J.; Garca, A.; Gavete, L.; Negreanu, M.; Urea, F.; Vargas, A., On the numerical solution to a parabolic-elliptic system with chemotactic and periodic terms using generalized finite differences, Eng Anal Bound Elem, 113, 181-190 (2020) · Zbl 1464.92004
[3] Benito, J.; Urea, F.; Gavete, L., Influence of several factors in the generalized finite difference method, Appl Math Model, 25, 12, 1039-1053 (2001) · Zbl 0994.65111
[4] Benito, J.; Urea, F.; Gavete, L., Solving parabolic and hyperbolic equations by the generalized finite difference method, J Comput Appl Math, 209, 2, 208-233 (2007) · Zbl 1139.35007
[5] Benito, J.; Urea, F.; Gavete, L.; Salete, E.; Urea, M., Implementations with generalized finite differences of the displacements and velocity-stress formulations of seismic wave propagation problem, Appl Math Model, 52, 1-14 (2017) · Zbl 07166309
[6] Chan, H.-F.; Fan, C.-M.; Kuo, C.-W., Generalized finite difference method for solving two-dimensional non-linear obstacle problems, Eng Anal Bound Elem, 37, 9, 1189-1196 (2013) · Zbl 1287.74056
[7] Chen, C. S.; Cho, H. A.; Golberg, M. A., Some comments on the ill-conditioning of the method of fundamental solutions, Eng Anal Bound Elem, 30, 5, 405-410 (2006) · Zbl 1187.65136
[8] Chen, J. T.; Wu, C. S.; Lee, Y. T.; Chen, K. H., On the equivalence of the Trefftz method and method of fundamental solutions for laplace and biharmonic equations, Comput Math Appl, 53, 6, 851-879 (2007) · Zbl 1125.65110
[9] Dalrymple, R. A.; Rogers, B. D., Numerical modeling of water waves with the SPH method, Coastal Eng, 53, 2-3, 141-147 (2006)
[10] Dong, C-M; Huang, C.-J., Generation and propagation of water waves in a two-dimensional numerical viscous wave flume, J Waterw Port Coastal Ocean Eng, 130, 3, 143-153 (2004)
[11] Improved Localized and Hybrid Meshless Methods - Part 1. · Zbl 1464.74266
[12] Fu, Z.-J.; Xie, Z.-Y.; Ji, S.-Y.; Tsai, C.-C.; Li, A.-L., Meshless generalized finite difference method for water wave interactions with multiple-bottom-seated-cylinder-array structures, Ocean Eng, 195, 106736 (2020)
[13] Gavete, L.; Gavete, M.; Benito, J., Improvements of generalized finite difference method and comparison with other meshless method, Appl Math Model, 27, 10, 831-847 (2003) · Zbl 1046.65085
[14] Gu, Y.; Lei, J.; Fan, C.-M.; He, X.-Q., The generalized finite difference method for an inverse time-dependent source problem associated with three-dimensional heat equation, Eng Anal Bound Elem, 91, 73-81 (2018) · Zbl 1403.65039
[15] Hu, X. Y.; Adams, N. A., A multi-phase SPH method for macroscopic and mesoscopic flows, J Comput Phys, 213, 2, 844-861 (2006) · Zbl 1136.76419
[16] Hu, X. Y.; Adams, N. A., An incompressible multi-phase SPH method, J Comput Phys, 227, 1, 264-278 (2007) · Zbl 1126.76045
[17] Huang, J.; Lyu, H.; Fan, C.-M.; Chen, J.-H.; Chu, C.-N.; Gu, J., Wave-structure interaction for a stationary surface-piercing body based on a novel meshless scheme with the generalized finite difference method, Mathematics, 8, 7 (2020)
[18] Lee, E.. S.; Moulinec, C.; Xu, R.; Violeau, D.; Laurence, D.; Stansby, P., Comparisons of weakly compressible and truly incompressible algorithms for the SPH mesh free particle method, J Comput Phys, 227, 18, 8417-8436 (2008) · Zbl 1256.76054
[19] Lee, W.-M.; Chen, J.-T., Scattering of flexural wave in a thin plate with multiple circular holes by using the multipole Trefftz method, Int J Solids Struct, 47, 9, 1118-1129 (2010) · Zbl 1193.74069
[20] Lei, J.; Xu, Y.; Gu, Y.; Fan, C.-M., The generalized finite difference method for in-plane crack problems, Eng Anal Bound Elem, 98, 147-156 (2019) · Zbl 1404.74194
[21] Li, P.-W.; Fan, C.-M., Generalized finite difference method for two-dimensional shallow water equations, Eng Anal Bound Elem, 80, 58-71 (2017) · Zbl 1403.76133
[22] Li, Z.-C.; Lu, T.-T.; Huang, H.-T.; Cheng, A. H.. D., Trefftz, collocation, and other boundary methods - a comparison, Numer Methods Partial Differ Equ, 23, 1, 93-144 (2007) · Zbl 1223.65093
[23] Liu, C.-S., An effectively modified direct Trefftz method for 2Dpotential problems considering the domain’s characteristic length, Eng Anal Bound Elem, 31, 12, 983-993 (2007) · Zbl 1259.65183
[24] Liu, C.-S., A modified Trefftz method for two-dimensional laplace equation considering the domain’s characteristic length, Cmes-Comput Model Eng Sci, 21, 1, 53-65 (2007) · Zbl 1232.65157
[25] Liu, C.-S., A modified collocation Trefftz method for the inverse cauchy problem of laplace equation, Eng Anal Bound Elem, 32, 9, 778-785 (2008) · Zbl 1244.65188
[26] Liu, C.-S.; Atluri, S. N., Numerical solution of the Laplacian cauchy problem by using a better postconditioning collocation Trefftz method, Eng Anal Bound Elem, 37, 1, 74-83 (2013) · Zbl 1352.65569
[27] Liu, M. B.; Liu, G. R., Smoothed particle hydrodynamics (SPH): an overview and recent developments, Arch Comput Methods Eng, 17, 1, 25-76 (2010) · Zbl 1348.76117
[28] Madsen, O. S., On the generation of long waves, J Geophys Res (1896-1977), 76, 36, 8672-8683 (1971) · Zbl 1245.65070
[29] Ohyama, T.; Beji, S.; Nadaoka, K.; Battjes, J., Experimental verification of numerical model for nonlinear wave evolutions, J Waterw Port Coastal Ocean Eng, 120 (1994)
[30] Sarler, B., Solution of potential flow problems by the modified method of fundamental solutions: formulations with the single layer and the double layer fundamental solutions, Eng Anal Bound Elem, 33, 12, 1374-1382 (2009) · Zbl 1244.76084
[31] Wei, T.; Hon, Y. C.; Ling, L., Method of fundamental solutions with regularization techniques for cauchy problems of elliptic operators, Eng Anal Bound Elem, 31, 4, 373-385 (2007) · Zbl 1195.65206
[32] Xu, R.; Stansby, P.; Laurence, D., Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach, J Comput Phys, 228, 18, 6703-6725 (2009) · Zbl 1261.76047
[33] Yan, L.; Fu, C.-L.; Yang, F.-L., The method of fundamental solutions for the inverse heat source problem, Eng Anal Bound Elem, 32, 3, 216-222 (2008) · Zbl 1244.80026
[34] Young, D. L.; Jane, S. J.; Fan, C. M.; Murugesan, K.; Tsai, C. C., The method of fundamental solutions for 2D and 3Dstokes problems, J Comput Phys, 211, 1, 1-8 (2006) · Zbl 1160.76332
[35] Zhang, T.; Lin, Z.-H.; Huang, G.-Y.; Fan, C.-M.; Li, P.-W., Solving Boussinesq equations with a meshless finite difference method, Ocean Eng, 198, 106957 (2020)
[36] Zhang, T.; Ren, Y.-F.; Fan, C.-M.; Li, P.-W., Simulation of two-dimensional sloshing phenomenon by generalized finite difference method, Eng Anal Bound Elem, 63, 82-91 (2016) · Zbl 1403.65044
[37] Zhang, T.; Ren, Y.-F.; Yang, Z.-Q.; Fan, C.-M.; Li, P.-W., Application of generalized finite difference method to propagation of nonlinear water waves in numerical wave flume, Ocean Eng, 123, 278-290 (2016)
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