Computer modeling of complex fluid flows usually presents great challenges for conventional grid-based numerical methods. Smoothed particle hydrodynamics (SPH) is a meshfree Lagrangian particle method and has special advantages in modeling complex fluid flows, especially those with large fluid deformations, fluid-structure interactions, and multi-scale physics. In this paper, we review the recent developments of SPH in methodology and applications for modeling complex fluid flows. Specifically, in methodology, some important issues including modified SPH particle approximation schemes for improving discretization accuracy, different particle regularization techniques, and various boundary treatment algorithms for solid boundary, free surface, or multiphase interface are described. More importantly, the SPH method with ideas from the dissipative particle dynamics for complex fluids in macro- or meso-scales is discussed. In applications, different complex fluid flows, including biological flows, microfluidics and droplet dynamics, non-Newtonian fluid flows, free surface flows, multiphase flows, and flows with fluid-structure interaction, are reviewed. Some concluding remarks in SPH modeling of complex fluid flows are provided.
REFERENCES
1.
M. B.
Liu
, G. R.
Liu
, L. W.
Zhou
, and J. Z.
Chang
, “Dissipative particle dynamics (DPD): An overview and recent developments
,” Arch. Comput. Methods Eng.
22
, 529
(2015
).2.
3.
G. R.
Liu
and M. B.
Liu
, Smoothed Particle Hydrodynamics: A Meshfree Particle Method
(World Scientific Publishing Co. Pte. Ltd.
, Singapore
, 2003
).4.
C. W.
Hirt
and B. D.
Nichols
, “Volume of fluid (VOF) method for the dynamics of free boundaries
,” J. Comput. Phys.
39
, 201
(1981
).5.
M.
Sussman
, P.
Smereka
, and S.
Osher
, “A level set approach for computing solutions to incompressible two-phase flow
,” J. Comput. Phys.
114
, 146
(1994
).6.
S.
Koshizuka
and Y.
Oka
, “Moving-particle semi-implicit method for fragmentation of incompressible fluid
,” Nucl. Sci. Eng.
123
, 421
(1996
).7.
G. T.
Duan
and B.
Chen
, “Large eddy simulation by particle method coupled with sub-particle-scale model and application to mixing layer flow
,” Appl. Math. Modell.
39
, 3135
(2015
).8.
M. B.
Liu
and G. R.
Liu
, “Smoothed particle hydrodynamics (SPH): An overview and recent developments
,” Arch. Comput. Methods Eng.
17
, 25
(2010
).9.
A.
Di Mascio
, M.
Antuono
, A.
Colagrossi
, and S.
Marrone
, “Smoothed particle hydrodynamics method from a large eddy simulation perspective
,” Phys. Fluids
29
, 035102
(2017
).10.
L. B.
Lucy
, “Numerical approach to testing of fission hypothesis
,” Astron. J.
82
, 1013
(1977
).11.
R. A.
Gingold
and J. J.
Monaghan
, “Smoothed particle hydrodynamics: Theory and application to non-spherical stars
,” Mon. Not. R. Astron. Soc.
181
, 375
(1977
).12.
J. J.
Monaghan
, “Smoothed particle hydrodynamics and its diverse applications
,” Annu. Rev. Fluid Mech.
44
, 323
(2012
).13.
F. R.
Ming
, P. N.
Sun
, and A. M.
Zhang
, “Numerical investigation of rising bubbles bursting at a free surface through a multiphase SPH model
,” Meccanica
52
, 2665
(2017
).14.
J. J.
Monaghan
, “Smoothed particle hydrodynamics
,” Annu. Rev. Astron. Astrophys.
30
, 543
(1992
).15.
J. J.
Monaghan
, “Smoothed particle hydrodynamics
,” Rep. Prog. Phys.
68
, 1703
(2005
).16.
D.
Violeau
and B. D.
Rogers
, “Smoothed particle hydrodynamics (SPH) for free-surface flows: Past, present and future
,” J. Hydraul. Res.
54
, 1
(2016
).17.
H.
Gotoh
and A.
Khayyer
, “On the state-of-the-art of particle methods for coastal and ocean engineering
,” Coastal Eng. J.
60
, 79
(2018
).18.
A. M.
Zhang
, P.
Sun
, F.
Ming
, and A.
Colagrossi
, “Smoothed particle hydrodynamics and its applications in fluid-structure interactions
,” J. Hydrodyn.
29
, 187
(2017
).19.
M. S.
Shadloo
, G.
Oger
, and D.
Le Touze
, “Smoothed particle hydrodynamics method for fluid flows, towards industrial applications: Motivations, current state, and challenges
,” Comput. Fluids
136
, 11
(2016
).20.
Z. B.
Wang
, R.
Chen
, H.
Wang
, Q.
Liao
, X.
Zhu
, and S.-Z.
Li
, “An overview of smoothed particle hydrodynamics for simulating multiphase flow
,” Appl. Math. Modell.
40
, 9625
(2016
).21.
M. P.
Allen
and D. J.
Tildesley
, Computer Simulation of Liquids
(Oxford University Press
, 1987
).22.
P. J.
Hoogerbrugge
and J.
Koelman
, “Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics
,” Europhys. Lett.
19
, 155
(1992
).23.
M. B.
Liu
, P.
Meakin
, and H.
Huang
, “Dissipative particle dynamics with attractive and repulsive particle-particle interactions
,” Phys. Fluids
18
, 017101
(2006
).24.
P.
Espanol
, “Fluid particle dynamics: A synthesis of dissipative particle dynamics and smoothed particle dynamics
,” Europhys. Lett.
39
, 605
(1997
).25.
P.
Español
and M.
Revenga
, “Smoothed dissipative particle dynamics
,” Phys. Rev. E
67
, 026705
(2003
).26.
P. W.
Randles
and L. D.
Libersky
, “Smoothed particle hydrodynamics: Some recent improvements and applications
,” Comput. Methods Appl. Mech. Eng.
139
, 375
(1996
).27.
J. K.
Chen
and J. E.
Beraun
, “A generalized smoothed particle hydrodynamics method for nonlinear dynamic problems
,” Comput. Methods Appl. Mech. Eng.
190
, 225
(2000
).28.
M. B.
Liu
, W. P.
Xie
, and G. R.
Liu
, “Modeling incompressible flows using a finite particle method
,” Appl. Math. Modell.
29
, 1252
(2005
).29.
M. B.
Liu
and G. R.
Liu
, “Restoring particle consistency in smoothed particle hydrodynamics
,” Appl. Numer. Math.
56
, 19
(2006
).30.
J.
Bonet
and T.-S. L.
Lok
, “Variational and momentum preservation aspects of smooth particle hydrodynamic formulations
,” Comput. Methods Appl. Mech. Eng.
180
, 97
(1999
).31.
M. B.
Liu
, G. R.
Liu
, and K. Y.
Lam
, “Constructing smoothing functions in smoothed particle hydrodynamics with applications
,” J. Comput. Appl. Math.
155
, 263
(2003
).32.
C.
Huang
, J. M.
Lei
, M. B.
Liu
, and X. Y.
Peng
, “A kernel gradient free (KGF) SPH method
,” Int. J. Numer. Methods Fluids
78
, 691
(2015
).33.
C.
Huang
, J. M.
Lei
, M. B.
Liu
, and X. Y.
Peng
, “An improved KGF-SPH with a novel discrete scheme of Laplacian operator for viscous incompressible fluid flows
,” Int. J. Numer. Methods Fluids
81
, 377
(2016
).34.
G. M.
Zhang
and R. C.
Batra
, “Symmetric smoothed particle hydrodynamics (SSPH) method and its application to elastic problems
,” Comput. Mech.
43
, 321
(2009
).35.
T.
Jiang
, Z. C.
Chen
, W. G.
Lu
, J. Y.
Yuan
, and D. S.
Wang
, “An efficient split-step and implicit pure mesh-free method for the 2D/3D nonlinear Gross–Pitaevskii equations
,” Comput. Phys. Commun.
231
, 19
(2018
).36.
Z. L.
Zhang
and M. B.
Liu
, “A decoupled finite particle method for modeling incompressible flows with free surfaces
,” Appl. Math. Modell.
60
, 606
(2018
).37.
Z. L.
Zhang
, K.
Walayat
, J. Z.
Chang
, and M. B.
Liu
, “Meshfree modeling of a fluid-particle two-phase flow with an improved SPH method
,” Int. J. Numer. Methods Eng.
116
, 530
(2018
).38.
J. P.
Morris
, P. J.
Fox
, and Y.
Zhu
, “Modeling low Reynolds number incompressible flows using SPH
,” J. Comput. Phys.
136
, 214
(1997
).39.
J. J.
Monaghan
, “Simulating free surface flows with SPH
,” J. Comput. Phys.
110
, 399
(1994
).40.
S. J.
Cummins
and M.
Rudman
, “An SPH projection method
,” J. Comput. Phys.
152
, 584
(1999
).41.
A.
Rafiee
and K. P.
Thiagarajan
, “An SPH projection method for simulating fluid-hypoelastic structure interaction
,” Comput. Methods Appl. Mech. Eng.
198
, 2785
(2009
).42.
M.
Antuono
, A.
Colagrossi
, S.
Marrone
, and D.
Molteni
, “Free-surface flows solved by means of SPH schemes with numerical diffusive terms
,” Comput. Phys. Commun.
181
, 532
(2010
).43.
S.
Marrone
, M.
Antuono
, A.
Colagrossi
, G.
Colicchio
, D.
Le Touze
, and G.
Graziani
, “δ-SPH model for simulating violent impact flows
,” Comput. Methods Appl. Mech. Eng.
200
, 1526
(2011
).44.
D.
Le Touze
, A.
Colagrossi
, G.
Colicchio
, and M.
Greco
, “A critical investigation of smoothed particle hydrodynamics applied to problems with free-surfaces
,” Int. J. Numer. Methods Fluids
73
, 660
(2013
).45.
M.
Antuono
, B.
Bouscasse
, A.
Colagrossi
, and S.
Marrone
, “A measure of spatial disorder in particle methods
,” Comput. Phys. Commun.
185
, 2609
(2014
).46.
J. J.
Monaghan
, “Smoothed particle hydrodynamics
,” J. Comput. Phys.
82
, 1
(1989
).47.
S.
Shahriari
, I. G.
Hassan
, and L.
Kadem
, “Modeling unsteady flow characteristics using smoothed particle hydrodynamics
,” Appl. Math. Modell.
37
, 1431
(2013
).48.
J. J.
Monaghan
, “SPH without a tensile instability
,” J. Comput. Phys.
159
, 290
(2000
).49.
T.
Jiang
, L. G.
Lu
, and W. G.
Lu
, “The numerical investigation of spreading process of two viscoelastic droplets impact problem by using an improved SPH scheme
,” Comput. Mech.
53
, 977
(2014
).50.
Y.
Melean
, L. D.
Sigalotti
, and A.
Hasmy
, “On the SPH tensile instability in forming viscous liquid drops
,” Comput. Phys. Commun.
157
, 191
(2004
).51.
X. F.
Yang
, M. B.
Liu
, and S. L.
Peng
, “Smoothed particle hydrodynamics modeling of viscous liquid drop without tensile instability
,” Comput. Fluids
92
, 199
(2014
).52.
N.
Tsuruta
, A.
Khayyer
, and H.
Gotoh
, “A short note on dynamic stabilization of moving particle semi-implicit method
,” Comput. Fluids
82
, 158
(2013
).53.
R.
Xu
, P.
Stansby
, and D.
Laurence
, “Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach
,” J. Comput. Phys.
228
, 6703
(2009
).54.
S. J.
Lind
, R.
Xu
, P.
Stansby
, and B.
Rogers
, “Incompressible smoothed particle hydrodynamics for free-surface flows: A generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves
,” J. Comput. Phys.
231
, 1499
(2012
).55.
A.
Colagrossi
, B.
Bouscasse
, M.
Antuono
, and S.
Marrone
, “Particle packing algorithm for SPH schemes
,” Comput. Phys. Commun.
183
, 1641
(2012
).56.
A.
Khayyer
, H.
Gotoh
, and Y.
Shimizu
, “Comparative study on accuracy and conservation properties of two particle regularization schemes and proposal of an optimized particle shifting scheme in ISPH context
,” J. Comput. Phys.
332
, 236
(2017
).57.
C.
Huang
, D. H.
Zhang
, Y. X.
Shi
, Y. L.
Si
, and B.
Huang
, “Coupled finite particle method with a modified particle shifting technology
,” Int. J. Numer. Methods Eng.
113
, 179
(2018
).58.
G.
Oger
, S.
Marrone
, D. L.
Touze
, and M. D.
Leffe
, “SPH accuracy improvement through the combination of a quasi-Lagrangian shifting transport velocity and consistent ALE formalisms
,” J. Comput. Phys.
313
, 76
(2016
).59.
D. H.
Zhang
, Y. X.
Shi
, C.
Huang
, Y. L.
Si
, and W.
Li
, “A mixed characteristic boundary condition for simulating viscous incompressible fluid flows around a hydrofoil
,” J. Marine Sci. Technol.
(published online).60.
C.
Huang
, J. M.
Lei
, and M. B.
Liu
, “An iterative method for homogenizing particles
,” in IVth International Conference on Particle-based Methods
, Barcelona, Spain
, 2015
.61.
R.
Vacondio
and B. D.
Rogers
, “Consistent iterative shifting for SPH methods
,” in Proceedings of 12th International SPHERIC Workshop
, Ourense, Spain
, 2017
.62.
M. B.
Liu
, J. R.
Shao
, and J. Z.
Chang
, “On the treatment of solid boundary in smoothed particle hydrodynamics
,” Sci. China Technol. Sci.
55
, 244
(2012
).63.
B. D.
Rogers
and R. A.
Dalrymple
, “SPH modeling of tsunami waves
,” in 3rd International Workshop on Long-wave Runup Models
(Catalina
, 2008
).64.
J. J.
Monaghan
and J. B.
Kajtar
, “SPH particle boundary forces for arbitrary boundaries
,” Comput. Phys. Commun.
180
, 1811
(2009
).65.
Y. W.
Han
, H. F.
Qiang
, J. L.
Zhao
, and W. R.
Gao
, “A new repulsive model for solid boundary condition in smoothed particle hydrodynamics
,” Acta Phys. Sin.
62
, 044702
(2013
).66.
D. H.
Zhang
, Y. X.
Shi
, C.
Huang
, Y. L.
Si
, B.
Huang
, and W.
Li
, “SPH method with applications of oscillating wave surge converter
,” Ocean Eng.
152
, 273
(2018
).67.
P. N.
Sun
, A.
Colagrossi
, S.
Marrone
, M.
Antuono
, and A. M.
Zhang
, “Multi-resolution Delta-plus-SPH with tensile instability control: Towards high Reynolds number flows
,” Comput. Phys. Commun.
224
, 63
(2018
).68.
M. S.
Shadloo
, A.
Zainali
, M.
Yildiz
, and A.
Suleman
, “A robust weakly compressible SPH method and its comparison with an incompressible SPH
,” Int. J. Numer. Methods Eng.
89
, 939
(2012
).69.
T.
Long
, D.
Hu
, D.
Wan
, C.
Zhuang
, and G.
Yang
, “An arbitrary boundary with ghost particles incorporated in coupled FEM-SPH model for FSI problems
,” J. Comput. Phys.
350
, 166
(2017
).70.
S. D.
Shao
and E. Y. M.
Lo
, “Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface
,” Adv. Water Resour.
26
, 787
(2003
).71.
E. S.
Lee
, C.
Moulinec
, R.
Xu
, D.
Violeau
, D.
Laurence
, and P.
Stansby
, “Comparisons of weakly compressible and truly incompressible algorithms for the SPH mesh free particle method
,” J. Comput. Phys.
227
, 8417
(2008
).72.
S.
Marrone
, A.
Colagrossi
, D.
Le Touze
, and G.
Graziani
, “Fast free-surface detection and level-set function definition in SPH solvers
,” J. Comput. Phys.
229
, 3652
(2010
).73.
A.
Colagrossi
and M.
Landrini
, “Numerical simulation of interfacial flows by smoothed particle hydrodynamics
,” J. Comput. Phys.
191
, 448
(2003
).74.
Q. W.
Ma
and J. T.
Zhou
, “MLPG_R method for numerical simulation of 2D breaking waves
,” Cmes-Comput. Model. Eng. Sci.
43
, 277
(2009
).75.
C.
Huang
, D. H.
Zhang
, Y. L.
Si
, Y. X.
Shi
, and Y. G.
Lin
, “Coupled finite particle method for simulations of wave and structure interaction
,” Coastal Eng.
140
, 147
(2018
).76.
A.
Zhang
, P.
Sun
, and F.
Ming
, “An SPH modeling of bubble rising and coalescing in three dimensions
,” Comput. Methods Appl. Mech. Eng.
294
, 189
(2015
).77.
K.
Szewc
, J.
Pozorski
, and J. P.
Minier
, “Spurious interface fragmentation in multiphase SPH
,” Int. J. Numer. Methods Eng.
103
, 625
(2015
).78.
A.
Krimi
, M.
Rezoug
, S.
Khelladi
, X.
Nogueira
, M.
Deligant
, and L.
Ramírez
, “Smoothed particle hydrodynamics: A consistent model for interfacial multiphase fluid flow simulations
,” J. Comput. Phys.
358
, 53
(2018
).79.
N.
Grenier
, M.
Antuono
, A.
Colagrossi
, T.
Le
, and B.
Alessandrini
, “An Hamiltonian interface SPH formulation for multi-fluid and free surface flows
,” J. Comput. Phys.
228
, 8380
(2009
).80.
S.
Adami
, X. Y.
Hu
, and N. A.
Adams
, “A new surface-tension formulation for multi-phase SPH using a reproducing divergence approximation
,” J. Comput. Phys.
229
, 5011
(2010
).81.
X. Y.
Hu
and N. A.
Adams
, “A constant-density approach for incompressible multi-phase SPH
,” J. Comput. Phys.
228
, 2082
(2009
).82.
X. Y.
Hu
and N. A.
Adams
, “A multi-phase SPH method for macroscopic and mesoscopic flows
,” J. Comput. Phys.
213
, 844
(2006
).83.
X. Y.
Hu
and N. A.
Adams
, “An incompressible multi-phase SPH method
,” J. Comput. Phys.
227
, 264
(2007
).84.
G. X.
Zhu
, L.
Zou
, Z.
Chen
, A. M.
Wang
, and M. B.
Liu
, “An improved SPH model for multiphase flows with large density ratios
,” Int. J. Numer. Methods Fluids
86
, 167
(2017
).85.
Z.
Chen
, Z.
Zong
, M. B.
Liu
, L.
Zou
, H. T.
Li
, and C.
Shu
, “An SPH model for multiphase flows with complex interfaces and large density differences
,” J. Comput. Phys.
283
, 169
(2015
).86.
M.
Zhang
, “Simulation of surface tension in 2D and 3D with smoothed particle hydrodynamics method
,” J. Comput. Phys.
229
, 7238
(2010
).87.
M.
Zhang
and X.-L.
Deng
, “A sharp interface method for SPH
,” J. Comput. Phys.
302
, 469
(2015
).88.
T.
Breinlinger
, P.
Polfer
, A.
Hashibon
, and T.
Kraft
, “Surface tension and wetting effects with smoothed particle hydrodynamics
,” J. Comput. Phys.
243
, 14
(2013
).89.
G. T.
Duan
, S.
Koshizuka
, and B.
Chen
, “A contoured continuum surface force model for particle methods
,” J. Comput. Phys.
298
, 280
(2015
).90.
A. M.
Tartakovsky
and A.
Panchenko
, “Pairwise force smoothed particle hydrodynamics model for multiphase flow: Surface tension and contact line dynamics
,” J. Comput. Phys.
305
, 1119
(2016
).91.
J. U.
Brackbill
, D. B.
Kothe
, and C.
Zemach
, A Continuum Method for Modeling Surface Tension
(Academic Press Professional, Inc.
, 1992
).92.
B.
Lafaurie
, C.
Nardone
, R.
Scardovelli
, and G.
Zanetti
, “Modelling merging and fragmentation in multiphase flows with SURFER
,” J. Comput. Phys.
113
, 134
(1994
).93.
H.
Takeda
, S. M.
Miyama
, and M.
Sekiya
, “Numerical simulation of viscous flow by smoothed particle hydrodynamics
,” Prog. Theor. Phys.
92
, 939
(1994
).94.
O.
Kum
, W. G.
Hoover
, and H. A.
Posch
, “Viscous conducting flows with smooth-particle applied mechanics
,” Phys. Rev. E
52
, 4899
(1995
).95.
S. J.
Watkins
, A. S.
Bhattal
, N.
Francis
, J. A.
Turner
, and A. P.
Whitworth
, “A new prescription for viscosity in smoothed particle hydrodynamics
,” Astron. Astrophys. Suppl.
119
, 177
(1996
).96.
P. W.
Cleary
and J. J.
Monaghan
, “Conduction modelling using smoothed particle hydrodynamics
,” J. Comput. Phys.
148
, 227
(1999
).97.
P.
Espanol
and P.
Warren
, “Statistical mechanics of dissipative particle dynamics
,” Europhys. Lett.
30
, 191
(1995
).98.
R. D.
Groot
and P. B.
Warren
, “Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation
,” J. Chem. Phys.
107
, 4423
(1997
).99.
M. B.
Liu
, P.
Meakin
, and H.
Huang
, “Dissipative particle dynamics simulation of pore-scale multiphase fluid flow
,” Water Resour. Res.
43
, W04411
, https://doi.org/10.1029/2006wr004856 (2007
).100.
M. B.
Liu
, P.
Meakin
, and H.
Huang
, “Dissipative particle dynamics simulation of multiphase fluid flow in microchannels and microchannel networks
,” Phys. Fluids
19
, 033302
(2007
).101.
I.
Pagonabarraga
and D.
Frenkel
, “Dissipative particle dynamics for interacting systems
,” J. Chem. Phys.
115
, 5015
(2001
).102.
P. B.
Warren
, “Hydrodynamic bubble coarsening in off-critical vapor-liquid phase separation
,” Phys. Rev. Lett.
87
, 225702
(2001
).103.
S. Y.
Trofimov
, E. L. F.
Nies
, and M. A. J.
Michels
, “Thermodynamic consistency in dissipative particle dynamics simulations of strongly nonideal liquids and liquid mixtures
,” J. Chem. Phys.
117
, 9383
(2002
).104.
P. B.
Warren
, “Vapor-liquid coexistence in many-body dissipative particle dynamics
,” Phys. Rev. E
68
, 066702
(2003
).105.
M.
Arienti
, W.
Pan
, X.
Li
, and G.
Karniadakis
, “Many-body dissipative particle dynamics simulation of liquid/vapor and liquid/solid interactions
,” J. Chem. Phys.
134
, 204114
(2011
).106.
A.
Ghoufi
, J.
Emile
, and P.
Malfreyt
, “Recent advances in many body dissipative particles dynamics simulations of liquid-vapor interfaces
,” Eur. Phys. J. E
36
, 10
(2013
).107.
R. M.
Füchslin
, H.
Fellermann
, A.
Eriksson
, and H. J.
Ziock
, “Coarse graining and scaling in dissipative particle dynamics
,” J. Chem. Phys.
130
, 214102
(2009
).108.
P.
Español
and P. B.
Warren
, “Perspective: Dissipative particle dynamics
,” J. Chem. Phys.
146
, 150901
(2017
).109.
I. V.
Pivkin
and G. E.
Karniadakis
, “Coarse-graining limits in open and wall-bounded dissipative particle dynamics systems
,” J. Chem. Phys.
124
, 184101
(2006
).110.
Z.
Li
, Y. H.
Tang
, H.
Lei
, B.
Caswell
, and G. E.
Karniadakis
, “Energy-conserving dissipative particle dynamics with temperature-dependent properties
,” J. Comput. Phys.
265
, 113
(2014
).111.
M.
Ellero
and P.
Español
, “Everything you always wanted to know about SDPD (but were afraid to ask)
,” Appl. Math. Mech. Engl. Ed.
39
, 103
(2018
).112.
W. M.
Gelbart
and A.
Benshaul
, “The ‘new’ science of ‘complex fluids
,’” J. Phys. Chem.
100
, 13169
(1996
).113.
S. E.
Spagnolie
, Complex Fluids in Biological Systems
(Springer
, New York
, 2015
).114.
J.
Li
and M.
Kwauk
, “Exploring complex systems in chemical engineering—The multi-scale methodology
,” Chem. Eng. Sci.
58
, 521
(2003
).115.
W.
Ge
, J.
Ma
, J.
Zhang
, D.
Tang
, F.
Chen
, X.
Wang
, L.
Guo
, and J.
Li
, “Particle methods for multi- scale simulation of complex flows
,” Chin. Sci. Bull.
50
, 1057
(2005
).116.
X.
Chen
, “Numerical modeling of fluid-structure interaction with rheologically complex fluids
,” Ph.D. thesis, Technische Universität Darmstadt
, 2014
.117.
J. G.
Oldroyd
, “On the formulation of rheological equations of state
,” Proc. R. Soc. London
200
, 523
(1950
).118.
A.
Peterlin
, “Hydrodynamics of macromolecules in a velocity field with longitudinal gradient
,” J. Polym. Sci. Part B: Polym. Lett.
4
, 287
(1996
).119.
N. P.
Thien
and R. I.
Tanner
, “A new constitutive equation derived from network theory
,” J. Non-Newtonian Fluid Mech.
2
, 353
(1977
).120.
J. N.
Fang
, R. G.
Owens
, L.
Tacher
, and A.
Parriaux
, “A numerical study of the SPH method for simulating transient viscoelastic free surface flows
,” J. Non-Newtonian Fluid Mech.
139
, 68
(2006
).121.
A.
Rafiee
, M. T.
Manzari
, and M.
Hosseini
, “An incompressible SPH method for simulation of unsteady viscoelastic free-surface flows
,” Int. J. Non-Linear Mech.
42
, 1210
(2007
).122.
M.
Ellero
, M.
Kröger
, and S.
Hess
, “Viscoelastic flows studied by smoothed particle dynamics
,” J. Non-Newtonian Fluid Mech.
105
, 35
(2002
).123.
A.
Vázquez-Quesada
, M.
Ellero
, and P.
Español
, “Smoothed particle hydrodynamic model for viscoelastic fluids with thermal fluctuations
,” Phys. Rev. E
79
, 056707
(2009
).124.
A.
Vázquez-Quesada
, M.
Ellero
, and P.
Español
, “A SPH-based particle model for computational microrheology
,” Microfluidics Nanofluidics
13
, 249
(2012
).125.
S. B.
Dan
, G. S.
Grest
, J. B.
Lechman
, F.
Pierce
, S. J.
Plimpton
, and P. R.
Schunk
, “Particle dynamics modeling methods for colloid suspensions
,” Comput. Particle Mech.
1
, 321
(2014
).126.
T.
Ye
, N.
Phan-Thien
, and C. T.
Lim
, “Particle-based simulations of red blood cells—A review
,” J. Biomech.
49
, 2255
(2016
).127.
D. Y.
Pan
, N.
Phan-Thien
, and B. C.
Khoo
, “Dissipative particle dynamics simulation of droplet suspension in shear flow at low Capillary number
,” J. Non-Newtonian Fluid Mech.
212
, 63
(2014
).128.
J. T.
Padding
and W. J.
Briels
, “Systematic coarse-graining of the dynamics of entangled polymer melts: The road from chemistry to rheology
,” J. Phys.: Condens. Matter
23
, 233101
(2011
).129.
S. M.
Hosseini
and J. J.
Feng
, “A particle-based model for the transport of erythrocyte in capillaries
,” Chem. Eng. Sci.
64
, 4488
(2009
).130.
H. N.
Polwaththe-Gallage
, S. C.
Saha
, E.
Sauret
, R.
Flower
, and Y.
Gu
, “A coupled SPH-DEM approach to model the interactions between multiple red blood cells in motion in capillaries
,” Int. J. Mech. Mater. Design
12
, 477
(2016
).131.
G. W.
Schmid-Schönbein
, R.
Skalak
, S.
Usami
, and S.
Chien
, “Cell distribution in capillary networks
,” Microvasc. Res.
19
, 18
(1980
).132.
I. H.
Sarelius
and B. R.
Duling
, “Direct measurement of microvessel hematocrit, red cell flux, velocity, and transit time
,” Am. J. Physiol.
243
, H1018
(1982
).133.
A. R.
Pries
, T. W.
Secomb
, P.
Gaehtgens
, and J. F.
Gross
, “Blood flow in microvascular networks. Experiments and simulation
,” Circ. Res.
67
, 826
(1990
).134.
B. M.
Fenton
, R. T.
Carr
, and G. R.
Cokelet
, “Nonuniform red cell distribution in 20 to 100 micrometers bifurcations
,” Microvasc. Res.
29
, 103
(1985
).135.
S. P.
Sutera
and R.
Skalak
, “The history of Poiseuille’s law
,” Annu. Rev. Fluid Mech.
25
, 1
(1993
).136.
S.
Chien
, C. D.
Tvetenstrand
, M. A.
Epstein
, and G. W.
Schmid-Schönbein
, “Model studies on distributions of blood cells at microvascular bifurcations
,” Am. J. Physiol.
248
, H568
(1985
).137.
A. W.
El-Kareh
and T. W.
Secomb
, “A model for red blood cell motion in bifurcating microvessels
,” Int. J. Multiphase Flow
26
, 1545
(2000
).138.
T.
Hyakutake
, T.
Matsumoto
, and S.
Yanase
, “Lattice Boltzmann simulation of blood cell behavior at microvascular bifurcations
,” Math. Comput. Simul.
72
, 134
(2006
).139.
J. B.
Freund
, “Numerical simulation of flowing blood cells
,” Annu. Rev. Fluid Mech.
46
, 67
(2014
).140.
Y.
Imai
, T.
Omori
, Y.
Shimogonya
, T.
Yamaguchi
, and T.
Ishikawa
, “Numerical methods for simulating blood flow at macro, micro, and multi scales
,” J. Biomech.
49
, 2221
(2016
).141.
T. W.
Secomb
, B.
Styp-Rekowska
, and A. R.
Pries
, “Two-dimensional simulation of red blood cell deformation and lateral migration in microvessels
,” Ann. Biomed. Eng.
35
, 755
(2007
).142.
J. O.
Barber
, J. P.
Alberding
, J. M.
Restrepo
, and T. W.
Secomb
, “Simulated two-dimensional red blood cell motion, deformation, and partitioning in microvessel bifurcations
,” Ann. Biomed. Eng.
36
, 1690
(2008
).143.
T. W.
Secomb
, J. O.
Barber
, J. P.
Alberding
, and J. M.
Restrepo
, “Motion of red blood cells in microvessels and bifurcations: Computational simulations
,” Biorheology
45
, 35
(2008
).144.
T. W.
Secomb
, “Mechanics and computational simulation of blood flow in microvessels
,” Med. Eng. Phys.
33
, 800
(2011
).145.
W. K.
Liu
, Y.
Liu
, D.
Farrell
, L.
Zhang
, X. S.
Wang
, Y.
Fukui
, N.
Patankar
, Y.
Zhang
, C.
Bajaj
, and J.
Lee
, “Immersed finite element method and its applications to biological systems
,” Comput. Methods Appl. Mech. Eng.
195
, 1722
(2006
).146.
Y.
Liu
and W. K.
Liu
, “Rheology of red blood cell aggregation by computer simulation
,” J. Comput. Phys.
220
, 139
(2006
).147.
Y.
Sui
, Y. T.
Chew
, P.
Roy
, Y. P.
Cheng
, and H. T.
Low
, “Dynamic motion of red blood cells in simple shear flow
,” Phys. Fluids
20
, 112106
(2008
).148.
J.
Zhang
, “Lattice Boltzmann method for microfluidics: Models and applications
,” Microfluidics Nanofluidics
10
, 1
(2011
).149.
S. M.
Hosseini
and J. J.
Feng
, “How malaria parasites reduce the deformability of infected red blood cells
,” Biophys. J.
103
, 1
(2012
).150.
H. N.
Polwaththe-Gallage
, S. C.
Saha
, E.
Sauret
, R.
Flower
, W.
Senadeera
, and Y. T.
Gu
, “SPH-DEM approach to numerically simulate the deformation of three-dimensional RBCs in non-uniform capillaries
,” Biomed. Eng. Online
15
, 161
(2016
).151.
D. A.
Fedosov
, B.
Caswell
, and G. E.
Karniadakis
, “Systematic coarse-graining of spectrin-level red blood cell models
,” Comput. Methods Appl. Mech. Eng.
199
, 1937
(2010
).152.
L.
Xiao
, Y.
Liu
, S.
Chen
, and B.
Fu
, “Simulation of deformation and aggregation of two red blood cells in a stenosed microvessel by dissipative particle dynamics
,” Cell Biochem. Biophys.
74
, 513
(2016
).153.
P.
Balogh
and P.
Bagchi
, “A computational approach to modeling cellular-scale blood flow in complex geometry
,” J. Comput. Phys.
334
, 280
(2017
).154.
T.
Ye
, N.
Phan-Thien
, B. C.
Khoo
, and C. T.
Lim
, “Stretching and relaxation of malaria-infected red blood cells
,” Biophys. J.
105
, 1103
(2013
).155.
T.
Ye
, N.
Phan-Thien
, B. C.
Khoo
, and C. T.
Lim
, “A file of red blood cells in tube flow: A three-dimensional numerical study
,” J. Appl. Phys.
116
, 124703
(2014
).156.
T.
Ye
, N.
Phan-Thien
, B. C.
Khoo
, and C. T.
Lim
, “Numerical modelling of a healthy/malaria-infected erythrocyte in shear flow using dissipative particle dynamics method
,” J. Appl. Phys.
115
, 224701
(2014
).157.
T.
Ye
, N.
Phan-Thien
, B. C.
Khoo
, and C. T.
Lim
, “Dissipative particle dynamics simulations of deformation and aggregation of healthy and diseased red blood cells in a tube flow
,” Phys. Fluids
26
, 111902
(2014
).158.
T.
Ye
, H.
Shi
, L.
Peng
, and Y.
Li
, “Numerical studies of a red blood cell in rectangular microchannels
,” J. Appl. Phys.
122
, 084701
(2017
).159.
T.
Ye
, N.
Phan-Thien
, C. T.
Lim
, L.
Peng
, and H.
Shi
, “Hybrid smoothed dissipative particle dynamics and immersed boundary method for simulation of red blood cells in flows
,” Phys. Rev. E
95
, 063314
(2017
).160.
T.
Ye
, N.
Phan-Thien
, C. T.
Lim
, and Y.
Li
, “Red blood cell motion and deformation in a curved microvessel
,” J. Biomech.
65
, 12
(2017
).161.
T.
Ye
, L.
Peng
, and Y.
Li
, “Three-dimensional motion and deformation of a red blood cell in bifurcated microvessels
,” J. Appl. Phys.
123
, 064701
(2018
).162.
T.
Ye
, H.
Shi
, N.
Phan-Thien
, C. T.
Lim
, and Y.
Li
, “Relationship between transit time and mechanical properties of a cell through a stenosed microchannel
,” Soft Matter
14
, 533
(2018
).163.
C. S.
Peskin
, “The immersed boundary method
,” Acta Numer.
11
, 479
(2002
).164.
P.
Balogh
and P.
Bagchi
, “Direct numerical simulation of cellular-scale blood flow in 3D microvascular networks
,” Biophys. J.
113
, 2815
(2017
).165.
X. F.
Yang
, M.
Ray
, S.-C.
Kong
, and C.-B. M.
Kweon
, “SPH simulation of fuel drop impact on heated surfaces
,” Proc. Combust. Inst.
(in press).166.
X. F.
Yang
and S.-C.
Kong
, “3D simulation of drop impact on dry surface using SPH method
,” Int. J. Comput. Methods
15
, 1850011
(2018
).167.
J.
Kordilla
, A. M.
Tartakovsky
, and T.
Geyer
, “A smoothed particle hydrodynamics model for droplet and film flow on smooth and rough fracture surfaces
,” Adv. Water Resour.
59
, 1
(2013
).168.
K.
Ferrara
, R.
Pollard
, and M.
Borden
, “Ultrasound microbubble contrast agents: Fundamentals and application to gene and drug delivery
,” Annu. Rev. Biomed. Eng.
9
, 415
(2007
).169.
C. C.
Coussios
and R. A.
Roy
, “Applications of acoustics and cavitation to noninvasive therapy and drug delivery
,” Annu. Rev. Fluid Mech.
40
, 395
(2008
).170.
S. L.
Anna
, “Droplets and bubbles in microfluidic devices
,” Annu. Rev. Fluid Mech.
48
, 285
(2016
).171.
A. K.
Das
and P. K.
Das
, “Incorporation of diffuse interface in smoothed particle hydrodynamics: Implementation of the scheme and case studies
,” Int. J. Numer. Methods Fluids
67
, 671
(2011
).172.
X.
Bian
, S.
Litvinov
, M.
Ellero
, and N. J.
Wagner
, “Hydrodynamic shear thickening of particulate suspension under confinement
,” J. Non-Newtonian Fluid Mech.
213
, 39
(2014
).173.
A.
Rahmat
and M.
Yildiz
, “A multiphase ISPH method for simulation of droplet coalescence and electro-coalescence
,” Int J Multiphas Flow
105
, 32
(2018
).174.
L. G.
Leal
, Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes
(Cambridge University Press
, 2007
).175.
D.
Bonn
, J.
Eggers
, J.
Indekeu
, J.
Meunier
, and E.
Rolley
, “Wetting and spreading
,” Rev. Mod. Phys.
81
, 739
(2009
).176.
J. H.
Snoeijer
and B.
Andreotti
, “Moving contact lines: Scales, regimes, and dynamical transitions
,” Annu. Rev. Fluid Mech.
45
, 269
(2013
).177.
Y.
Sui
, H.
Ding
, and P. D. M.
Spelt
, “Numerical simulations of flows with moving contact lines
,” Annu. Rev. Fluid Mech.
46
, 97
(2014
).178.
M.
Huber
, F.
Keller
, W.
Sackel
, M.
Hirschler
, P.
Kunz
, S. M.
Hassanizadeh
, and U.
Nieken
, “On the physically based modeling of surface tension and moving contact lines with dynamic contact angles on the continuum scale
,” J. Comput. Phys.
310
, 459
(2016
).179.
F.
Yeganehdoust
, M.
Yaghoubi
, H.
Emdad
, and M.
Ordoubadi
, “Numerical study of multiphase droplet dynamics and contact angles by smoothed particle hydrodynamics
,” Appl. Math. Modell.
40
, 8493
(2016
).180.
E.
Shigorina
, J.
Kordilla
, and A. M.
Tartakovsky
, “Smoothed particle hydrodynamics study of the roughness effect on contact angle and droplet flow
,” Phys. Rev. E
96
, 033115
(2017
).181.
L.
Li
, L.
Shen
, G. D.
Nguyen
, A.
El-Zein
, and F.
Maggi
, “A smoothed particle hydrodynamics framework for modelling multiphase interactions at meso-scale
,” Comput. Mech.
62
, 1071
(2018
).182.
D. Y.
Pan
, J. X.
Hu
, and X. M.
Shao
, “Lees-Edwards boundary condition for simulation of polymer suspension with dissipative particle dynamics method
,” Mol. Simul.
42
, 328
(2016
).183.
H. H.
Bui
, R.
Fukagawa
, K.
Sako
, and S.
Ohno
, “Lagrangian meshfree particles method (SPH) for large deformation and failure flows of geomaterial using elastic–plastic soil constitutive model
,” Int. J. Numer. Anal. Methods Geomech.
32
, 1537
(2008
).184.
M.
Ellero
and R. I.
Tanner
, “SPH simulations of transient viscoelastic flows at low Reynolds number
,” J. Non-Newtonian Fluid Mech.
132
, 61
(2005
).185.
J.
Ren
, J.
Ouyang
, and T.
Jiang
, “An improved particle method for simulation of the non-isothermal viscoelastic fluid mold filling process
,” Int. J. Heat Mass Transfer
85
, 543
(2015
).186.
M.
Vakilha
and M. T.
Manzari
, “Modelling of power-law fluid flow through porous media using smoothed particle hydrodynamics
,” Transp. Porous Media
74
, 331
(2008
).187.
X. J.
Fan
, R. I.
Tanner
, and R.
Zheng
, “Smoothed particle hydrodynamics simulation of non-Newtonian moulding flow
,” J. Non-Newtonian Fluid Mech.
165
, 219
(2010
).188.
D.
Laigle
, P.
Lachamp
, and M.
Naaim
, “SPH-based numerical investigation of mudflow and other complex fluid flow interactions with structures
,” Comput. Geosci.
11
, 297
(2007
).189.
H. N.
Zhu
, N. S.
Martys
, C.
Ferraris
, and D.
De Kee
, “A numerical study of the flow of Bingham-like fluids in two-dimensional vane and cylinder rheometers using a smoothed particle hydrodynamics (SPH) based method
,” J. Non-Newtonian Fluid Mech.
165
, 362
(2010
).190.
X. Y.
Xu
, J.
Ouyang
, T.
Jiang
, and Q.
Li
, “Numerical simulation of 3D-unsteady viscoelastic free surface flows by improved smoothed particle hydrodynamics method
,” J. Non-Newtonian Fluid Mech.
177
, 109
(2012
).191.
J.
Ren
, T.
Jiang
, W.
Lu
, and G.
Li
, “An improved parallel SPH approach to solve 3D transient generalized Newtonian free surface flows
,” Comput. Phys. Commun.
205
, 87
(2016
).192.
W.
Wang
, G. Q.
Chen
, Z.
Han
, S. H.
Zhou
, H.
Zhang
, and P. D.
Jing
, “3D numerical simulation of debris-flow motion using SPH method incorporating non-Newtonian fluid behavior
,” Nat Hazards
81
, 1981
(2016
).193.
M. R.
Hashemi
, R.
Fatehi
, and M. T.
Manzari
, “SPH simulation of interacting solid bodies suspended in a shear flow of an Oldroyd-B fluid
,” J. Non-Newtonian Fluid Mech.
166
, 1239
(2011
).194.
X. Y.
Xu
and J.
Ouyang
, “A SPH-based particle method for simulating 3D transient free surface flows of branched polymer melts
,” J. Non-Newtonian Fluid Mech.
202
, 54
(2013
).195.
E.
Bertevas
, J.
Ferec
, B. C.
Khoo
, G.
Ausias
, and P.-T.
Nhan
, “Smoothed particle hydrodynamics (SPH) modeling of fiber orientation in a 3D printing process
,” Phys. Fluids
30
, 103103
(2018
).196.
W.
Pan
, A. M.
Tartakovsky
, and J. J.
Monaghan
, “Smoothed particle hydrodynamics non-Newtonian model for ice-sheet and ice-shelf dynamics
,” J. Comput. Phys.
242
, 828
(2013
).197.
J. W.
Glen
, “The creep of polycrystalline ice
,” Proc. R. Soc. London Seri. A
228
, 519
(1955
).198.
J. R.
Shao
, H. Q.
Li
, G. R.
Liu
, and M. B.
Liu
, “An improved SPH method for modeling liquid sloshing dynamics
,” Comput. Struct.
100
, 18
(2012
).199.
M.
Antuono
, A.
Colagrossi
, S.
Marrone
, and C.
Lugni
, “Propagation of gravity waves through an SPH scheme with numerical diffusive terms
,” Comput. Phys. Commun.
182
, 866
(2011
).200.
C.
Lugni
, “ A study about the free-surface waves and the freely floating structure interactions
,” Ph.D. thesis, University of Rome La Sapienza
, 1999
.201.
X. F.
Yang
and M. B.
Liu
, “Numerical study of Rayleigh-Taylor instability by using smoothed particle hydrodynamics
,” Acta Phys. Sin.
66
, 164701
(2017
).202.
X. F.
Yang
and S.-C.
Kong
, “Smoothed particle hydrodynamics method for evaporating multiphase flows
,” Phys. Rev. E
96
, 033309
(2017
).203.
A. M.
Tartakovsky
and P.
Meakin
, “A smoothed particle hydrodynamics model for miscible flow in three-dimensional fractures and the two-dimensional Rayleigh Taylor instability
,” J. Comput. Phys.
207
, 610
(2005
).204.
G. T.
Duan
, B.
Chen
, X. M.
Zhang
, and Y. C.
Wang
, “A multiphase MPS solver for modeling multi-fluid interaction with free surface and its application in oil spill
,” Comput. Methods Appl. Mech. Eng.
320
, 133
(2017
).205.
X. F.
Yang
and M. B.
Liu
, “Numerical modeling of oil spill containment by boom using SPH
,” Sci. China Phys., Mech. Astron.
56
, 315
(2013
).206.
F. R.
Ming
, A. M.
Zhang
, H.
Cheng
, and P. N.
Sun
, “Numerical simulation of a damaged ship cabin flooding in transversal waves with smoothed particle hydrodynamics method
,” Ocean Eng.
165
, 336
(2018
).207.
F. R.
Ming
, A. M.
Zhang
, Y. Z.
Xue
, and S. P.
Wang
, “Damage characteristics of ship structures subjected to shockwaves of underwater contact explosions
,” Ocean Eng.
117
, 359
(2016
).208.
Z. L.
Zhang
, D. L.
Feng
, and M. B.
Liu
, “Investigation of explosive welding through whole process modeling using a density adaptive SPH method
,” J. Manufact. Processes
35
, 169
(2018
).209.
C.
Antoci
, M.
Gallati
, and S.
Sibilla
, “Numerical simulation of fluid-structure interaction by SPH
,” Comput. Struct.
85
, 879
(2007
).210.
J. B.
Kajtar
and J. J.
Monaghan
, “On the swimming of fish like bodies near free and fixed boundaries
,” Eur. J. Mech. B:
33
, 1
(2012
).211.
X. F.
Yang
, M. B.
Liu
, S. L.
Peng
, and C. G.
Huang
, “Numerical modeling of dam-break flow impacting on flexible structures using an improved SPH-EBG method
,” Coastal Eng.
108
, 56
(2016
).212.
X. F.
Yang
, M. B.
Liu
, and S. L.
Peng
, “Smoothed particle hydrodynamics and element bending group modeling of flexible fibers interacting with viscous fluids
,” Phys. Rev. E
90
, 063011
(2014
).213.
S. S. P.
Kumar
, B. S. V.
Patnaik
, and K.
Ramamurthi
, “Prediction of air blast mitigation in an array of rigid obstacles using smoothed particle hydrodynamics
,” Phys. Fluids
30
, 046105
(2018
).214.
P. D.
Thomas
and C. K.
Lombard
, “Geometric conservation law and its application to flow computations on moving grids
,” AIAA J.
17
, 1030
(1979
).215.
M. B.
Liu
, J. R.
Shao
, and H. Q.
Li
, “An SPH model for free surface flows with moving rigid objects
,” Int. J. Numer. Methods Fluids
74
, 684
(2014
).216.
M.
Greenhow
and S.
Moyo
, “Water entry and exit of horizontal circular cylinders
,” Philosoph. Trans. R. Soc. A
355
, 551
(1997
).217.
P.
Lin
, “A fixed-grid model for simulation of a moving body in free surface flows
,” Comput. Fluids
36
, 549
(2007
).218.
P. A.
Tyvand
and T.
Miloh
, “Free-surface flow due to impulsive motion of a submerged circular cylinder
,” J. Fluid Mech.
286
, 67
(1995
).219.
M. B.
Liu
, J. R.
Shao
, and H. Q.
Li
, “Numerical simulation of hydro-elastic problems with smoothed particle hydro-dynamics method
,” J. Hydrodyn.
25
, 673
(2013
).220.
G.
Zhou
, Z.
Chen
, W.
Ge
, and J.
Li
, “SPH simulation of oil displacement in cavity-fracture structures
,” Chem. Eng. Sci.
65
, 3363
(2010
).221.
B.
Seiffert
, M.
Hayatdavoodi
, and R. C.
Ertekin
, “Experiments and computations of solitary-wave forces on a coastal-bridge deck. Part I: Flat Plate
,” Coastal Eng.
88
, 194
(2014
).222.
Y.
Wei
, T.
Abadie
, A.
Henry
, and F.
Dias
, “Wave interaction with an oscillating wave surge converter. Part II: Slamming
,” Ocean Eng.
113
, 319
(2016
).223.
F.
Dias
, E.
Renzi
, S.
Gallagher
, D.
Sarkar
, Y.
Wei
, T.
Abadie
, C.
Cummins
, and A.
Rafiee
, “Analytical and computational modelling for wave energy systems: The example of oscillating wave surge converters
,” Acta Mech. Sin.
33
, 647
(2017
).224.
F.
Dias
and J.-M.
Ghidaglia
, “Slamming: Recent progress in the evaluation of impact pressures
,” Annu. Rev. Fluid Mech.
50
, 243
(2018
).225.
A.
Shakeri
, K.-W.
Lee
, and T.
Poeschel
, “Limitation of stochastic rotation dynamics to represent hydrodynamic interaction between colloidal particles
,” Phys. Fluids
30
, 013603
(2018
).226.
J. W.
Swan
and G.
Wang
, “Rapid calculation of hydrodynamic and transport properties in concentrated solutions of colloidal particles and macromolecules
,” Phys. Fluids
28
, 011902
(2016
).227.
T.-D.
Thien
, P.-T.
Nhan
, and B. C.
Khoo
, “A smoothed particle hydrodynamics (SPH) study on polydisperse sediment from technical activities on seabed
,” Phys. Fluids
30
, 023302
(2018
).228.
T.-D.
Thien
, P.-T.
Nhan
, and B. C.
Khoo
, “A smoothed particle hydrodynamics (SPH) study of sediment dispersion on the seafloor
,” Phys. Fluids
29
, 083302
(2017
).229.
E.
Chaparian
, A.
Wachs
, and I. A.
Frigaard
, “Inline motion and hydrodynamic interaction of 2D particles in a viscoplastic fluid
,” Phys. Fluids
30
, 033101
(2018
).230.
Z.
Ouyang
, J.-Z.
Lin
, and X.
Ku
, “Hydrodynamic interactions between a self-rotation rotator and passive particles
,” Phys. Fluids
29
, 103301
(2017
).231.
S.
Turek
and C.
Berker
, FEATFLOW: Finite Element Software for the Incompressible Navier Stokes Equations
(IWR
, 1996
).232.
Z. L.
Zhang
and M. B.
Liu
, “Smoothed particle hydrodynamics with kernel gradient correction for modeling high velocity impact in two- and three-dimensional spaces
,” Eng. Anal. Boundary Elem.
83
, 141
(2017
).233.
F.
Plassard
, J.
Mespoulet
, and P.
Hereil
, “Hypervelocity impact of aluminium sphere against aluminium plate : Experiment and LS-DYNA correlation
,” in 8th European LS-DYNA Users Conference
, Strasbourg, France
, 2011
.234.
G. R.
Cowan
, O. R.
Bergmann
, and A. H.
Holtzman
, “Mechanism of bond zone wave formation in explosion-clad metals
,” Metall. Mater. Trans. B
2
, 3145
(1971
).235.
M. B.
Liu
, G. R.
Liu
, and K. Y.
Lam
, “A one-dimensional meshfree particle formulation for simulating shock waves
,” Shock Waves
13
, 201
(2003
).236.
D. A.
Hu
, T.
Long
, Y.
Xiao
, X.
Han
, and Y.
Gu
, “Fluid–structure interaction analysis by coupled FE–SPH model based on a novel searching algorithm
,” Comput. Methods Appl. Mech. Eng.
276
, 266
(2014
).237.
A. M.
Zhang
, F. R.
Ming
, and S. P.
Wang
, “Coupled SPHS–BEM method for transient fluid–structure interaction and applications in underwater impacts
,” Appl. Ocean Res.
43
, 223
(2013
).238.
G. R.
Johnson
, R. A.
Stryk
, and S. R.
Beissel
, “SPH for high velocity impact computations
,” Comput. Methods Appl. Mech. Eng.
139
, 347
(1996
).239.
M. B.
Liu
and G. R.
Liu
, Particle Methods for Multi-Scale and Multi-Physics
(World Scientific
, 2015
).240.
P. A.
Cundall
and O. D. L.
Strack
, “A discrete numerical model for granular assemblies
,” Geotechnique
29
, 47
(1979
).241.
U. E.
Shamy
and M.
Zeghal
, “Coupled continuum-discrete model for saturated granular soils
,” J. Eng. Mech.
131
, 413
(2005
).242.
P. W.
Cleary
and M.
Prakash
, “Discrete–element modelling and smoothed particle hydrodynamics: Potential in the environmental sciences
,” Philosoph. Trans.: Math., Phys. Eng. Sci.
362
, 2003
(2004
).243.
J. U.
Brackbill
and H. M.
Ruppel
, “Flip: A method for adaptively zoned, particle-in-cell calculations of fluid flows in two dimensions
,” J. Comput. Phys.
65
, 314
(1986
).244.
D.
Sulsky
, Z.
Chen
, and H. L.
Schreyer
, “A particle method for history-dependent materials
,” Comput. Methods Appl. Mech. Eng.
118
, 179
(1994
).245.
P. M.
Kulkarni
, C. C.
Fu
, M. S.
Shell
, and L. G.
Leal
, “Multiscale modeling with smoothed dissipative particle dynamics
,” J. Chem. Phys.
138
, 234105
(2013
).246.
N. D.
Petsev
, L. G.
Leal
, and M. S.
Shell
, “Hybrid molecular-continuum simulations using smoothed dissipative particle dynamics
,” J. Chem. Phys.
142
, 044101
(2015
).247.
N. D.
Petsev
, L. G.
Leal
, and M. S.
Shell
, “Multiscale simulation of ideal mixtures using smoothed dissipative particle dynamics
,” J. Chem. Phys.
144
, 084115
(2016
).© 2019 Author(s).
2019
Author(s)
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