Special Issue Paper
Received 24 June 2016 Published online 12 December 2016 in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/mma.4252
MOS subject classiﬁcation: 68U20; 65M99
Smoothed particle hydrodynamics method
with partially deﬁned fluid particles
and S. Nakata
Communicated by J. Vigo-Aguiar
This paper presents a method for nested ﬂuid simulation based on smoothed particle hydrodynamics. Given suitable
background ﬂow information of an “external ﬂow” as it evolves in time, our method simulates the motion of particles only
within a local material region. In order to perform the simulation, the background physical quantities need to be trans-
ferred to the local ﬂuid particles. We employ ghost particles to carry the given physical quantities to the nested ﬂuid.
We also solve the problem of density computation appropriately for the ghost particles. Our numerical tests show
that accurate local ﬂuid motion can be obtained in such a nested volume of ﬂuid particles. Copyright © 2016 John Wiley &
Keywords: smoothed particle hydrodynamics; ﬂuid simulation
This paper proposes a method for simulating ﬂuid ﬂow in a nested Lagrangian domain by smoothed particle hydrodynamics (SPH)  as
exempliﬁed by the labelled region in Figure 1. For geophysical and environmental ﬂows, nesting of ﬁne resolution domains into larger
domains is the dominant technique to affordably resolve ﬂow in a local domain that is inﬂuenced by complex large-scale phenomena
in the surrounding environment.
Our method applies SPH only within a limited material volume and speciﬁes an algorithm to transfer velocity and density from the
“outer computation” to the bounding material surface of that volume.
Similarly, the particle tracking method [2,3] tracks ﬂuid particle motion in a given “external” ﬂow ﬁeld. Generally, the velocities applied
in particle tracking method to move particles are simple interpolations that do not satisfy momentum conservation, nor commonly
local mass conservation and travelling time. The proposed method considers the physical laws at every time step with SPH and thereby
conserves local mass and momentum within the interpolation limits of SPH.
In particle-based simulation of liquid ﬂow, enforcing incompressibility is very important. During initial development, SPH in com-
puter graphics has been successfully applied to simulate liquid ﬂow with interactive speed  but does not enforce incompressibility.
For incompressible ﬂuid, moving particle semi-implicit  solves a pressure Poisson equation. Weakly compressible SPH (WCSPH) 
uses Tait’s equation to satisfy a weak compressibility condition while reducing the computational cost. Later, predictive-corrective
incompressible SPH , implicit incompressible SPH  and divergence-free SPH  decrease the computational cost by applying a
longer time step. This work employs WCSPH for weak incompressibility of ﬂuid ﬂow.
As mentioned earlier, nested models are typically applied with ﬁxed-grid solvers. For example in , a grid-based nested model has
been developed to offer a cost-effective accurate ﬂuid simulation with high-spatial resolution in limited areas. Recently, a hybrid model
combining a Boussinesq model and a local SPH model has been also developed to study coastal wave propagation efﬁciently . In
the future, the method proposed here should work to nest a high-resolution particle simulation into either a Lagrangian or Eulerian
computation of the outer domain. One can anticipate the need for both one-way and two-way coupling, but the current work considers
only the simpler one-way coupling.
In the proposed method, the background information needs to be transferred to the partially deﬁned ﬂuid particles. We solve this
problem by generating ghost particles around the ﬂuid and interpolating the background information to the ghost particles. We apply
Graduate School of Information Science and Engineering, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu-shi, Shiga-ken, Japan
College of Science and Engineering, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu-shi, Shiga-ken, Japan
College of Information Science and Engineering, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu-shi, Shiga-ken, Japan
Correspondence to: Y. Kanetsuki, Graduate School of Information Science and Engineering, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu-shi, Shiga-ken, Japan.
Copyright © 2016 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2018, 41 2299–2306