K2_SPH method and its application for 2-D water wave simulation

Abstract

Smoothed Particle Hydrodynamics (SPH) is a Lagrangian meshless particle method. However, its low accuracy of kernel approximation when particles are distributed disorderly or located near the boundary is an obstacle standing in the way of its wide application. Adopting the Taylor series expansion method and solving the integral equation matrix, the second order kernel approximation method can be obtained, namely K2_SPH, which is discussed in this paper. This method is similar to the Finite Particle Method. With the improvement of kernel approximation, some numerical techniques should be adopted for different types of boundaries, such as a free surface boundary and solid boundary, which are two key numerical techniques of K2_SPH for water wave simulation. This paper gives some numerical results of two dimensional water wave simulations involving standing wave and sloshing tank problems by using K2_SPH. From the comparison of simulation results, the K2_SPH method is more reliable than standard SPH.

Keywords

meshless method / SPH / K2_SPH / water wave simulation

References

[1] Atluri S.N., Shen S. The Meshless Local Petrove-Galerkin (MLPG) method: a simple less-costly alternative to the finite element and boundary element methods. Computer Modeling in Engineering & Sciences, 2002, 3(1): 11-52

[2] Atluri S.N., Zhu T. A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput. Mech., 1998, 22: 117-127

[3] Belytschko T., Lu Y.Y., Gu L. Element free Galerkin method. Int. J. Num. Meth. Engng., 1994, 37: 229-256

[4] Benz W., Asphaug E. Simulations of brittle solids using smooth particle hydrodynamics. Comput. Phys. Commun., 1995, 87: 253-265

[5] Chen J.K., Beraun J.E., Jih C.J. An improvement for tensile instability in smoothed particle hydrodynamics. Computational Mechanics, 1999, 23: 279-287

[6] Chen J.K., Beraun J.E., Jih C.J. A corrective smoothed particle method for boundary value problems in heat conduction. Int. J. Numer. Meth. Engng., 1999, 46: 231-252

[7] Cleary P.W., Monaghan J.J. Conduction modeling using smoothed particle hydrodynamics. J. Comput. Phys., 1999, 148: 227-264

[8] Colagrossi A., Landrini M. Numerical simulation of interfacial flow by Smoothed Particle Hydrodynamics. J. Computational Physic., 2003, 191: 448-475

[9] Cummins S.J., Rudman M. An SPH projection method. J. Comput. Phys., 1999, 152: 584-607

[10] Ellero M., Kroger M., Hess S. Viscoelastic flows studied by smoothed particle dynamics. J. Non-Newtonian Fluid Mech., 2002, 105: 35-51

[11] Ellero M., Tanner R.I. SPH simulations of transient viscoelastic flows at low Reynolds number. J. Non-Newtonian Fluid Mech., 2005, 132: 61-72

[12] Faltinsen O.M. A numerical non-linear method for sloshing in tanks with two dimensional flow. J. Ship Research, 1976, 18(4): 224-241

[13] Fang J.N., Owens R.G., Tacher L. A numerical study of the SPH method for simulating transient viscoelastic free surface flows. J. Non-Newtonian Fluid Mech., 2006, 139: 68-84

[14] Fang J.N., Parriaux A. A regularized lagrangian finite point method for the simulation of incompressible viscous flows. J. Comput. Phys., 2008, 227: 8894-8908

[15] Fang J.N., Parriaux A., Rentschler M. Improved SPH methods for simulating free surface flows of viscous fluids. Appl. Numer. Math., 2009, 59(2): 251-271

[16] Gingold R.A., Monaghan J.J. Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon. Not. Roy. Astrou. Soc., 1977, 181: 375-389

[17] Gotoh H., Sakai T. Key issues in the particle method for computation of wave breaking. Coastal Engineering, 2006, 53: 171-179

[18] Johnson G.R., Beissel S.R. Normalized smoothed functions for sph impact computations. Int. J. Numer. Meth. Eng., 1996, 39: 2725-2741

[19] Koshizuka S., Oka Y. Moving Particle Semi-implicit method for fragmentation of incompressible fluid. Nuclear Science and Engineering, 1996, 123: 421-434

[20] Libersky L. D., Petschek A. G., Carney T. C., et al. High strain lagrangian hydrodynamics. J. Comput. Phys., 1993, 109: 67-75

[21] Liu GR, Liu MB (2003). Smoothed Particle Hydrodynamics-a Meshfree particle method. World Scientific

[22] Liu G.R., Xu X.G. A gradient smoothing method (GSM) for fluid dynamics problems. Int. J. Numer. Methods Fluids, 2008, 58: 1101-1133

[23] Liu M.B., Liu G.R. Restoring particle consistency in smoothed particle hydrodynamics. Applied Numerical Mathematics, 2006, 56: 9-36

[24] Liu M.B., Liu G.R. Smoothed particle hydrodynamics (SPH): an overview and recent developments. Arch. Comput. Methods Eng., 2010, 17: 25-76

[25] Liu M.B., Liu G.R., Lam K.Y. Investigations into water mitigations using a meshless particle method. Shock waves, 2002, 12(3): 181-195

[26] Liu M.B., Liu G.R., Lam K.Y. Smoothed particle hydrodynamics for numerical simulation of underwater explosion. Comput. Mech., 2003, 30(2): 106-118

[27] Liu M.B., Liu G.R., Zong Z. Numerical simulation of underwater explosion by SPH. Advances in Computational Engineering & Science, 2000, 1: 1475-1480

[28] Liu M.B., Xie W.P., Liu G.R. Modeling incompressible flows using a finite particle method. Appl. Math. Model., 2005, 29(12): 1252-1270

[29] Liu W.K., Chen Y., Jun S., et al. Overview and application of the reproducing kernel particle methods. Arch. Comput. Methods Eng.: State of the Art Rev., 1996, 3: 3-80

[30] Liu W.K., Chen Y., Uras R.A. Generalized multiple scale reproducing kernel particle methods. Computer Methods in Applied Mechanics and Engineering, 1996, 139: 91-157

[31] Liu W.K., Jun S., Li S.F. Reproducing kernel particle methods for structural dynamics. Int. J. Numer. Meth. Engng., 1995, 38: 1655-1679

[32] Liu W.K., Jun S., Sihling D.T. Multiresolution reproducing kernel particle method for computational fluid dynamics. Int. J. Numer. Meth. Fluids., 1997, 24: 1391-1415

[33] Liu W.K., Jun S., Zhang Y.F. Reproducing kernel particle methods. Int J. Numer Methods Eng., 1995, 20: 1081-1106

[34] Lo Edmond Y.M., Shao S.D. Simulation of near-shore solitary wave mechanics by an incompressible SPH method. Applied Ocean research, 2002, 24: 275-286

[35] Lucy L.B. A numerical approach to the testing of the fission hypothesis. The Astron. J., 1977, 82(12): 1013-1024

[36] Ma Q.W. Meshless local Petrov-Galerin method for two dimensional nonlinear water wave problems. J. Compu. Phys., 2005, 205(2): 611-625

[37] Ma Q.W. MLPG method based on rankine source solution for simulating nonlinear water waves. Computer Modelling in Engineering & Sciences, 2005, 9(2): 193-209

[38] Ma Q.W. A new Meshless interpolation scheme for MLPG_R method. Computer Modeling in Engineering & Sciences, 2008, 23(2): 75-89

[39] Ma Q.W., Zhou J. MLPG_R method for numerical simulation of 2D breaking waves. Computer Modeling in Engineering & Sciences, 2009, 43(3): 277-304

[40] Maveyraud C., Benz W., Sornette A. Solid friction at high sliding velocities: an explicit three-dimensional dynamical smoothed particle hydrodynamics approach. J. Geophys. Res., 1999, 104: 28769-28788

[41] Monaghan J.J. An introduction to SPH. Comput. Phys. Comm., 1988, 48: 89-96

[42] Monaghan J.J. Simulation Free Surface Flows with SPH. J. Compu. Phys., 1994, 110(4): 399-406

[43] Monaghan J.J., Kocharyan A. SPH simulation of multi-phase flow. Comput. Phys. Commun., 1995, 87: 225-235

[44] Morris J.P., Fox P.J., Zhu Y. Modeling low Reynolds number incompressible flows using SPH. J. Comput. Phys., 1997, 136: 214-226

[45] Nayroles B., Touzot G., Vilon P. Generalizing the finite element method: diffuse approximation and diffuse elements. Comput. Mech., 1992, 10: 307-318

[46] Onate E., Idelsohn S., Zinnkinewicz O.C. A finite point method in computational mechanics: Applications to convective transport and fluid flow. Int. J. Num. Meth. Engng., 1996, 39: 3839-3866

[47] Shao S.D. Incompressible SPH simulation of water entry of a free-falling object. Int. J. Numer Methods Fluids, 2009, 59(1): 91-115

[48] Shao S.D., Ji C.M., Graham D.I. Simulation of wave overtopping by an incompressible SPH model. Coast Eng., 2006, 53(9): 723-735

[49] Shao S., Lo Edmond Y.M. Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Adv. Water Resour., 2003, 26(7): 787-800

[50] Takeda H., Miyama S.M., Sekiya M. Numerical simulation of viscous flow by smoothed particle hydrodynamics. Prog. Theor. Phys., 1994, 92(5): 939-959

[51] Wu G.X. Second order resonance of sloshing in a tank. Ocean Engineering, 2007, 34: 2345-2349

[52] Xu HH, Ha DM, Soon CE (2010). A SPH model with C1 particle consistency. 5th International SPHERIC Workshop, Manchester, UK, 194–200.

[53] Zhang G.M., Batra R.C. Wave propagation in functionally graded materials by modified smoothed particle hydrodynamics (MSPH) method. J. Comput. Phys., 2007, 222: 374-390

[54] Zhang G.M., Batra R.C. Modified smoothed particle hydrodynamics (MSPH) basis functions for meshless method, and their application to axisymmetric Taylor impact test. J. Comput. Phys., 2008, 227: 1962-1981

[55] Zheng X., Duan W.Y. Study on the precision of second order algorithm for smoothed particle hydrodynamics. Advances in Water Science, 2008, 19(6): 86-92

[56] Zheng X., Duan W.Y. Numerical simulation of dam breaking using smoothed particle hydrodynamics and viscosity behavior. J. Marine Sci. Appl., 2010, 9(1): 34-41

[57] Zheng X., Duan W.Y., Ma Q.W. Comparison of improved meshless interpolation schemes for SPH method and accuracy analysis. J. Marine Sci. Appli., 2010, 9(3): 223-230

[58] Zou Z.L. Water wave theories and their applications, 2005, Beijing: Science Press