Hamiltonian moving-particle semi-implicit (HMPS) method for incompressible fluid flows

Particle methods are meshless simulation techniques in which motion of continua is approximated by discrete dynamics of a finite number of particles, and thus have a great degree of flexibility, for instance, in dealing with the complex motion of surfaces or boundaries. In this paper, a particle method is developed from a direct discretization of the Lagrangian for inviscid incompressible fluid flows. The discretized Lagrangian is transformed into a Hamiltonian with holonomic constraints that stem from the incompressibility condition. The RATTLE algorithm that is a symplectic scheme for holonomically constrained Hamiltonian systems is adopted so that the resulting particle method inherits structures possessed by the partial differential equations that govern inviscid incompressible flows. The algorithm of the particle method is semi-implicit, so we call it the Hamiltonian moving-particle semi-implicit (HMPS) method. Some numerical tests indicate the excellence of the HMPS method in conservation of mechanical energy (as well as linear and angular momenta).

[1]  Oscar Gonzalez,et al.  Exact energy and momentum conserving algorithms for general models in nonlinear elasticity , 2000 .

[2]  Frederic A. Rasio.  Particle Methods in Astrophysical Fluid Dynamics , 2000 .

[3]  J. C. Simo,et al.  The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics , 1992 .

[4]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

[5]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multi-dimensional advective-diffusive systems , 1987 .

[6]  M. Ortiz,et al.  Lagrangian finite element analysis of Newtonian fluid flows , 1998 .

[7]  M. Lesieur,et al.  New Trends in Large-Eddy Simulations of Turbulence , 1996 .

[8]  S. Reich,et al.  Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity , 2001 .

[9]  L. Lucy.  A numerical approach to the testing of the fission hypothesis. , 1977 .

[10]  J. Monaghan,et al.  Smoothed particle hydrodynamics: Theory and application to non-spherical stars , 1977 .

[11]  P. Morrison,et al.  Hamiltonian description of the ideal fluid , 1998 .

[12]  Wing Kam Liu,et al.  Reproducing kernel particle methods , 1995 .

[13]  Nobuatsu Tanaka.  Hamiltonian Particle Dynamics, CIVA-Particle Method and Symplectic Upwind Scheme , 2001 .

[14]  Y. Chikazawa,et al.  Numerical Analysis of Three-dimensional Sloshing in an Elastic Cylindrical Tank using Moving Particle Semi-implicit Method , 2001 .

[15]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier—Stokes equations and the second law of thermodynamics , 1986 .

[16]  Eugenio Oñate,et al.  A Lagrangian meshless finite element method applied to fluid-structure interaction problems , 2003 .

[17]  J. C. Simo,et al.  Unconditional stability and long-term behavior of transient algorithms for the incompressible Navier-Stokes and Euler equations , 1994 .

[18]  C. W. Hirt,et al.  Volume of fluid (VOF) method for the dynamics of free boundaries , 1981 .

[19]  J. Marsden,et al.  Discrete mechanics and variational integrators , 2001, Acta Numerica.

[20]  R. Salmon,et al.  Geophysical Fluid Dynamics , 2019, Classical Mechanics in Geophysical Fluid Dynamics.

[21]  Sivakumar Kulasegaram,et al.  Variational formulation for the smooth particle hydrodynamics (SPH) simulation of fluid and solid problems , 2004 .

[22]  P. Moin,et al.  Fully Conservative Higher Order Finite Difference Schemes for Incompressible Flow , 1998 .

[23]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[24]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[25]  J. Monaghan,et al.  Kernel estimates as a basis for general particle methods in hydrodynamics , 1982 .

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

[27]  R. Eatock Taylor,et al.  Numerical wave tank based on a σ‐transformed finite element inviscid flow solver , 2003 .

[28]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[29]  Jason Frank,et al.  A Hamiltonian Particle-Mesh Method for the Rotating Shallow Water Equations , 2003 .

[30]  Alistair G.L. Borthwick,et al.  A PSEUDOSPECTRAL σ -TRANSFORMATION MODEL OF 2-D NONLINEAR WAVES , 1999 .

[31]  P. Moin,et al.  DIRECT NUMERICAL SIMULATION: A Tool in Turbulence Research , 1998 .

[32]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.

[33]  R. Eatock Taylor,et al.  Finite element analysis of two-dimensional non-linear transient water waves , 1994 .

[34]  Oden,et al.  An h-p adaptive method using clouds , 1996 .

[35]  Yoshiaki Oka,et al.  Numerical Analysis of Droplet Breakup Behavior using Particle Method , 2001 .

[36]  Wing Kam Liu,et al.  Lagrangian-Eulerian finite element formulation for incompressible viscous flows☆ , 1981 .

[37]  Daniel J. Price,et al.  Variational principles for relativistic smoothed particle hydrodynamics , 2001 .

[38]  S. Koshizuka,et al.  GENERALIZATION OF PHYSICAL COMPONENT BOUNDARY FITTED CO-ORDINATE (PCBFC) METHOD FOR THE ANALYSIS OF FREE-SURFACE FLOW , 1992 .

[39]  Francis H. Harlow,et al.  Numerical Study of Large‐Amplitude Free‐Surface Motions , 1966 .

[40]  Eugenio Oñate,et al.  The meshless finite element method , 2003 .

[41]  B. Leimkuhler,et al.  Symplectic Numerical Integrators in Constrained Hamiltonian Systems , 1994 .

[42]  J. M. Sanz-Serna,et al.  Numerical Hamiltonian Problems , 1994 .

[43]  S. Koshizuka,et al.  International Journal for Numerical Methods in Fluids Numerical Analysis of Breaking Waves Using the Moving Particle Semi-implicit Method , 2022 .

[44]  Particle method for fluid and solid dynamics , 2001 .

[45]  E. Oñate,et al.  The particle finite element method. An overview , 2004 .

[46]  Li,et al.  Moving least-square reproducing kernel methods (I) Methodology and convergence , 1997 .

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

[48]  J. Monaghan.  Smoothed particle hydrodynamics , 2005 .

[49]  J. Marsden,et al.  Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs , 1998, math/9807080.

[50]  Harry Yserentant,et al.  The Finite Mass Method , 2000, SIAM J. Numer. Anal..

[51]  Huafeng Liu,et al.  Meshfree Particle Methods , 2004 .

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

[53]  Yoshiaki Oka,et al.  A particle-gridless hybrid method for incompressible flows , 1999 .