A review on approaches to solving Poisson’s equation in projection-based meshless methods for modelling strongly nonlinear water waves

Three meshless methods, including incompressible smooth particle hydrodynamic (ISPH), moving particle semi-implicit (MPS) and meshless local Petrov–Galerkin method based on Rankine source solution (MLPG_R) methods, are often employed to model nonlinear or violent water waves and their interaction with marine structures. They are all based on the projection procedure, in which solving Poisson’s equation about pressure at each time step is a major task. There are three different approaches to solving Poisson’s equation, i.e. (1) discretizing Laplacian directly by approximating the second-order derivatives, (2) transferring Poisson’s equation into a weak form containing only gradient of pressure and (3) transferring Poisson’s equation into a weak form that does not contain any derivatives of functions to be solved. The first approach is often adopted in ISPH and MPS, while the third one is implemented by the MLPG_R method. This paper attempts to review the most popular, though not all, approaches available in literature for solving the equation.

[1]  Abbas Khayyer,et al.  A higher order Laplacian model for enhancement and stabilization of pressure calculation by the MPS method , 2010 .

[2]  Hitoshi Gotoh,et al.  A 3D higher order Laplacian model for enhancement and stabilization of pressure calculation in 3D MPS-based simulations , 2012 .

[3]  Deborah Greaves,et al.  APPLICATION OF THE FINITE VOLUME METHOD TO THE SIMULATION OF NONLINEAR WATER WAVES , 2010 .

[4]  J. Morris,et al.  Modeling Low Reynolds Number Incompressible Flows Using SPH , 1997 .

[5]  Javier Bonet,et al.  A simplified approach to enhance the performance of smooth particle hydrodynamics methods , 2002, Appl. Math. Comput..

[6]  S. Shao,et al.  Corrected Incompressible SPH method for accurate water-surface tracking in breaking waves , 2008 .

[7]  S. Cummins,et al.  An SPH Projection Method , 1999 .

[8]  J. K. Chen,et al.  Completeness of corrective smoothed particle method for linear elastodynamics , 1999 .

[9]  M. Lastiwka,et al.  Truncation error in mesh‐free particle methods , 2006 .

[10]  Taro Arikawa,et al.  On enhancement of Incompressible SPH method for simulation of violent sloshing flows , 2014 .

[11]  Sivakumar Kulasegaram,et al.  Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations , 2000 .

[12]  Qingwei Ma.  MLPG Method Based on Rankine Source Solution for Simulating Nonlinear Water Waves , 2005 .

[13]  A. Chorin.  Numerical Solution of the Navier-Stokes Equations* , 1989 .

[14]  Q. W. Ma,et al.  Incompressible SPH method based on Rankine source solution for violent water wave simulation , 2014, J. Comput. Phys..

[15]  Benedict D. Rogers,et al.  Numerical Modeling of Water Waves with the SPH Method , 2006 .

[16]  Edmond Y.M. Lo,et al.  Simulation of near-shore solitary wave mechanics by an incompressible SPH method , 2002 .

[17]  Abbas Khayyer,et al.  Corrected higher order Laplacian for enhancement of pressure calculation by projection-based particle methods with applications in ocean engineering , 2015 .

[18]  Nikolaus A. Adams,et al.  An incompressible multi-phase SPH method , 2007, J. Comput. Phys..

[19]  Seiichi Koshizuka,et al.  Least squares moving particle semi-implicit method , 2014, Computational Particle Mechanics.

[20]  Qingwei Ma,et al.  Meshless local Petrov-Galerkin method for two-dimensional nonlinear water wave problems , 2005 .

[21]  Juntao Zhou,et al.  MLPG Method Based on Rankine Source Solution for Modelling 3D Breaking Waves , 2010 .

[22]  Behzad Ataie-Ashtiani,et al.  Numerical simulation of landslide impulsive waves by incompressible smoothed particle hydrodynamics , 2008 .

[23]  Ping Dong,et al.  Wave Impact Simulations by an Improved ISPH Model , 2014 .

[24]  H. Schwaiger.  An implicit corrected SPH formulation for thermal diffusion with linear free surface boundary conditions , 2008 .

[25]  D. Graham,et al.  Simulation of wave overtopping by an incompressible SPH model , 2006 .

[26]  Seiichi Koshizuka,et al.  Improvement of stability in moving particle semi‐implicit method , 2011 .

[27]  Bertrand Alessandrini,et al.  An improved SPH method: Towards higher order convergence , 2007, J. Comput. Phys..

[28]  Rui Xu,et al.  Comparisons of weakly compressible and truly incompressible algorithms for the SPH mesh free particle method , 2008, J. Comput. Phys..

[29]  Q. W. Ma,et al.  A new meshless interpolation scheme for MLPG_R method , 2008 .

[30]  Nan-Jing Wu,et al.  Meshless numerical simulation for fully nonlinear water waves , 2006 .

[31]  James J. Feng,et al.  Pressure boundary conditions for computing incompressible flows with SPH , 2011, J. Comput. Phys..

[32]  B. Nayroles,et al.  Generalizing the finite element method: Diffuse approximation and diffuse elements , 1992 .

[33]  Christophe Kassiotis,et al.  Unified semi-analytical wall boundary conditions applied to 2-D incompressible SPH , 2014, J. Comput. Phys..

[34]  Nikolaus A. Adams,et al.  A constant-density approach for incompressible multi-phase SPH , 2009, J. Comput. Phys..

[35]  Stéphane Zaleski,et al.  TWO-DIMENSIONAL NAVIER-STOKES SIMULATION OF BREAKING WAVES , 1996, comp-gas/9605002.

[36]  Ronald Fedkiw,et al.  A new grid structure for domain extension , 2013, ACM Trans. Graph..

[37]  S. J. Lind,et al.  Incompressible smoothed particle hydrodynamics for free-surface flows: A generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves , 2012, J. Comput. Phys..

[38]  M. Manzari,et al.  An incompressible SPH method for simulation of unsteady viscoelastic free-surface flows , 2007 .

[39]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[40]  Songdong Shao,et al.  Numerical study of PPE source term errors in the incompressible SPH models , 2015 .

[41]  Yoshimi Sonoda,et al.  A Stabilized Incompressible SPH Method by Relaxing the Density Invariance Condition , 2012, J. Appl. Math..

[42]  Songdong Shao,et al.  Incompressible smoothed particle hydrodynamics simulation of multifluid flows , 2012 .

[43]  J. P. Hughes,et al.  Accuracy of SPH viscous flow models , 2008 .

[44]  S. Koshizuka,et al.  Moving-Particle Semi-Implicit Method for Fragmentation of Incompressible Fluid , 1996 .

[45]  Hitoshi Gotoh,et al.  Enhancement of stability and accuracy of the moving particle semi-implicit method , 2011, J. Comput. Phys..

[46]  Juntao Zhou,et al.  MLPG_R Method for Numerical Simulation of 2D Breaking Waves , 2009 .

[47]  Hitoshi Gotoh,et al.  Enhancement of performance and stability of MPS mesh-free particle method for multiphase flows characterized by high density ratios , 2013, J. Comput. Phys..

[48]  Behzad Ataie-Ashtiani,et al.  Modified incompressible SPH method for simulating free surface problems , 2008, Journal of the Brazilian Society of Mechanical Sciences and Engineering.

[49]  Hitoshi Gotoh,et al.  ENHANCED PREDICTIONS OF WAVE IMPACT PRESSURE BY IMPROVED INCOMPRESSIBLE SPH METHODS , 2009 .

[50]  O M Valtinsen.  A Nonlinear Theory of Sloshing in Rectangular Tanks , 1974 .

[51]  E. Oñate,et al.  A stabilized finite point method for analysis of fluid mechanics problems , 1996 .

[52]  Masashi Kashiwagi,et al.  Numerical simulation of violent sloshing by a CIP-based method , 2006 .

[53]  Jose L. Cercos-Pita,et al.  On the consistency of MPS , 2013, Comput. Phys. Commun..

[54]  Ling Qian,et al.  DEVELOPMENTS IN MULTI-FLUID FINITE VOLUME FREE SURFACE CAPTURING METHODS , 2010 .

[55]  Rui Xu,et al.  Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach , 2009, J. Comput. Phys..

[56]  L. Brookshaw,et al.  A Method of Calculating Radiative Heat Diffusion in Particle Simulations , 1985, Publications of the Astronomical Society of Australia.

[57]  Hitoshi Gotoh,et al.  Key issues in the particle method for computation of wave breaking , 2006 .

[58]  R. Fatehi,et al.  Error estimation in smoothed particle hydrodynamics and a new scheme for second derivatives , 2011, Comput. Math. Appl..

[59]  J. K. Chen,et al.  A corrective smoothed particle method for transient elastoplastic dynamics , 2001 .

[60]  J. Monaghan.  Simulating Free Surface Flows with SPH , 1994 .

[61]  B. Rogers,et al.  State-of-the-art of classical SPH for free-surface flows , 2010 .

[62]  Seiichi Koshizuka,et al.  On the consistency and convergence of particle-based meshfree discretization schemes for the Laplace operator , 2017 .

[63]  Hitoshi Gotoh,et al.  Current achievements and future perspectives for projection-based particle methods with applications in ocean engineering , 2016 .

[64]  D. Violeau,et al.  Smoothed particle hydrodynamics (SPH) for free-surface flows: past, present and future , 2016 .

[65]  V. Springel,et al.  Thermal conduction in cosmological SPH simulations , 2004, astro-ph/0401456.

[66]  S. Koshizuka.  A particle method for incompressible viscous flow with fluid fragmentation , 1995 .