We present three algorithms for robust and efficient geometric calculations in the context of immersed boundary method (IBM), including classification of mesh cells as inside/outside of a closed surface, projection of points onto a surface, and accurate calculation of the solid volume fraction field created by a closed surface overlapping with a background Cartesian mesh. The algorithms use the signed distance field (SDF) to represent the surface and remove the intersection tests, which are usually required by other algorithms developed before, no matter the surface is described in analytic or discrete form. The errors of the algorithms are analyzed. We also develop an approximate method on efficient SDF field calculation for complex geometries. We demonstrate how the algorithms can be implemented within the framework of IBM with a volume-average discrete-forcing scheme and applied to simulate fluid–structure interaction problems.

References

References
1.
Akiki
,
G.
,
Jackson
,
T. L.
, and
Balachandar
,
S.
,
2017
, “
Pairwise Interaction Extended Point-Particle Model for a Random Array of Monodisperse  Spheres
,”
J. Fluid Mech.
,
813
, pp.
882
928
.
2.
Beetstra
,
R.
,
Van Der Hoef
,
M. A.
, and
Kuipers
,
J. A. M.
,
2007
, “
Drag Force of Intermediate Reynolds Number Flow past Mono- and Bidisperse Arrays of Spheres
,”
AIChE J.
,
53
(
2
), pp.
489
501
.
3.
Diaz-Goano
,
C.
,
Minev
,
P. D.
, and
Nandakumar
,
K.
,
2003
, “
A Fictitious Domain/Finite Element Method for Particulate Flows
,”
J. Comput. Phys.
,
192
(
1
), pp.
105
123
.
4.
Veeramani
,
C.
,
Minev
,
P. D.
, and
Nandakumar
,
K.
,
2007
, “
A Fictitious Domain Formulation for Flows With Rigid Particles: A Non-Lagrange Multiplier Version
,”
J. Comput. Phys.
,
224
(
2
), pp.
867
879
.
5.
Peskin
,
C.
,
1977
, “
Numerical Analysis of Blood Flow in the Heart
,”
J. Comput. Phys.
,
25
(
3
), pp.
220
252
.
6.
Uhlmann
,
M.
,
2005
, “
An Immersed Boundary Method With Direct Forcing for the Simulation of Particulate Flows
,”
J. Comput. Phys.
,
209
(
2
), pp.
448
476
.
7.
Monaghan
,
J. J.
,
2012
, “
Smoothed Particle Hydrodynamics and Its Diverse Applications
,”
Annu. Rev. Fluid Mech.
,
44
(
1
), pp.
323
346
.
8.
Ye
,
T.
,
Mittal
,
R.
,
Udaykumar
,
H. S.
, and
Shyy
,
W.
,
1999
, “
An Accurate Cartesian Grid Method for Viscous Incompressible Flows With Complex Immersed Boundaries
,”
J. Comput. Phys.
,
156
(
2
), pp.
209
240
.
9.
Mittal
,
R.
, and
Iaccarino
,
G.
,
2005
, “
Immersed Boundary Methods
,”
Annu. Rev. Fluid Mech.
,
37
(
1
), pp.
239
261
.
10.
Fogelson
,
A. L.
, and
Peskin
,
C. S.
,
1988
, “
A Fast Numerical Method for Solving the Three-Dimensional Stokes' Equations in the Presence of Suspended Particles
,”
J. Comput. Phys.
,
79
(
1
), pp.
50
69
.
11.
Yang
,
J.
, and
Stern
,
F.
,
2014
, “
A Sharp Interface Direct Forcing Immersed Boundary Approach for Fully Resolved Simulations of Particulate Flows
,”
ASME J. Fluids Eng.
,
136
(
4
), p.
40904
.
12.
Zhu
,
X.
,
He
,
G.
, and
Zhang
,
X.
,
2014
, “
An Improved Direct-Forcing Immersed Boundary Method for Fluid-Structure Interaction Simulations
,”
ASME J. Fluids Eng.
,
136
(
4
), p.
40903
.
13.
Lai
,
M.-C.
, and
Peskin
,
C. S.
,
2000
, “
An Immersed Boundary Method With Formal Second-Order Accuracy and Reduced Numerical Viscosity
,”
J. Comput. Phys.
,
160
(
2
), pp.
705
719
.
14.
Liu
,
X.
,
Yang
,
B.
,
Ji
,
C.
,
Chen
,
Q.
, and
Song
,
M.
,
2018
, “
Research on the Turbine Blade Vibration Base on the Immersed Boundary Method
,”
ASME J. Fluids Eng.
,
140
(
6
), p.
61402
.
15.
González
,
F. A.
,
Cruchaga
,
M. A.
, and
Celentano
,
D. J.
,
2017
, “
Analysis of Flow past Oscillatory Cylinders Using a Finite Element Fixed Mesh Formulation
,”
ASME J. Fluids Eng.
,
139
(
8
), p.
81202
.
16.
Alan Wei
,
Z.
, and
Charlie Zheng
,
Z.
,
2017
, “
Fluid–Structure Interaction Simulation on Energy Harvesting From Vortical Flows by a Passive Heaving Foil
,”
ASME J. Fluids Eng.
,
140
(
1
), p.
11105
.
17.
Fadlun
,
E. A.
,
Verzicco
,
R.
,
Orlandi
,
P.
, and
Mohd-Yusof
,
J.
,
2000
, “
Combined Immersed-Boundary Finite Difference Methods for Three-Dimensional Complex Flow Simulations
,”
J. Comput. Phys.
,
161
(
1
), pp.
35
60
.
18.
Yang
,
J.
, and
Balaras
,
E.
,
2006
, “
An Embedded-Boundary Formulation for Large-Eddy Simulation of Turbulent Flows Interacting With Moving Boundaries
,”
J. Comput. Phys.
,
215
(
1
), pp.
12
40
.
19.
Yang
,
J.
, and
Stern
,
F.
,
2009
, “
Sharp Interface Immersed-Boundary/level-Set Method for Wave-Body Interactions
,”
J. Comput. Phys.
,
228
(
17
), pp.
6590
6616
.
20.
Kim
,
J.
,
Kim
,
D.
, and
Choi
,
H.
,
2001
, “
An Immersed-Boundary Finite Volume Method for Simulations of Flow in Complex Geometries
,”
J. Comput. Phys.
,
171
(
1
), pp.
132
150
.
21.
Choi
,
J.-I.
,
Oberoi
,
R. C.
,
Edwards
,
J. R.
, and
Rosati
,
J. A.
,
2007
, “
An Immersed Boundary Method for Complex Incompressible Flows
,”
J. Comput. Phys.
,
224
(
2
), pp.
757
784
.
22.
Kajishima
,
T.
, and
Takiguchi
,
S.
,
2002
, “
Interaction Between Particle Clusters and Particle-Induced Turbulence
,”
Int. J. Heat Fluid Flow
,
23
(
5
), pp.
639
646
.
23.
Yuki
,
Y.
,
Takeuchi
,
S.
, and
Kajishima
,
T.
,
2007
, “
Efficient Immersed Boundary Method for Strong Interaction Problem of Arbitrary Shape Object With the Self-Induced Flow
,”
J. Fluid Sci. Technol.
,
2
(
1
), pp.
1
11
.
24.
Tseng
,
Y.-H.
, and
Ferziger
,
J. H.
,
2003
, “
A Ghost-Cell Immersed Boundary Method for Flow in Complex Geometry
,”
J. Comput. Phys.
,
192
(
2
), pp.
593
623
.
25.
Bigot
,
B.
,
Bonometti
,
T.
,
Lacaze
,
L.
, and
Thual
,
O.
,
2014
, “
A Simple Immersed-Boundary Method for Solid-Fluid Interaction in Constant- and Stratified-Density Flows
,”
Comput. Fluids
,
97
, pp.
126
142
.
26.
Gsell
,
S.
,
Bonometti
,
T.
, and
Astruc
,
D.
,
2016
, “
A Coupled Volume-of-Fluid/Immersed-Boundary Method for the Study of Propagating Waves Over Complex-Shaped Bottom: Application to the Solitary Wave
,”
Comput. Fluids
,
131
, pp.
56
65
.
27.
Nakagawa
,
N.
,
Yabu
,
T.
,
Otomo
,
R.
,
Kase
,
A.
,
Makino
,
M.
,
Itano
,
T.
, and
Sugihara-Seki
,
M.
,
2015
, “
Inertial Migration of a Spherical Particle in Laminar Square Channel Flows From Low to High Reynolds Numbers
,”
J. Fluid Mech.
,
779
, pp. 776–793.
28.
Kempe
,
T.
, and
Fröhlich
,
J.
,
2012
, “
An Improved Immersed Boundary Method With Direct Forcing for the Simulation of Particle Laden Flows
,”
J. Comput. Phys.
,
231
(
9
), pp.
3663
3684
.
29.
Blais
,
B.
,
Lassaigne
,
M.
,
Goniva
,
C.
,
Fradette
,
L.
, and
Bertrand
,
F.
,
2016
, “
A Semi-Implicit Immersed Boundary Method and Its Application to Viscous Mixing
,”
Comput. Chem. Eng.
,
85
, pp.
136
146
.
30.
Ghosh
,
S.
,
Choi
,
J.-I.
, and
Edwards
,
J. R.
,
2010
, “
Numerical Simulations of Effects of Micro Vortex Generators Using Immersed-Boundary Methods
,”
AIAA J.
,
48
(
1
), pp.
92
103
.
31.
Liu
,
X.-D.
,
Fedkiw
,
R. P.
, and
Kang
,
M.
,
2000
, “
A Boundary Condition Capturing Method for Poisson's Equation on Irregular Domains
,”
J. Comput. Phys.
,
160
(
1
), pp.
151
178
.
32.
Gibou
,
F.
,
Fedkiw
,
R. P.
,
Cheng
,
L.-T.
, and
Kang
,
M.
,
2002
, “
A Second-Order-Accurate Symmetric Discretization of the Poisson Equation on Irregular Domains
,”
J. Comput. Phys.
,
176
(
1
), pp.
205
227
.
33.
Marella
,
S.
,
Krishnan
,
S.
,
Liu
,
H.
, and
Udaykumar
,
H. S.
,
2005
, “
Sharp Interface Cartesian Grid Method—I: An Easily Implemented Technique for 3D Moving Boundary Computations
,”
J. Comput. Phys.
,
210
(
1
), pp.
1
31
.
34.
Cheny
,
Y.
, and
Botella
,
O.
,
2010
, “
The LS-STAG Method: A New Immersed Boundary/Level-Set Method for the Computation of Incompressible Viscous Flows in Complex Moving Geometries With Good Conservation Properties
,”
J. Comput. Phys.
,
229
(
4
), pp.
1043
1076
.
35.
Jasak
,
H.
,
1996
, “
Error Analysis and Estimation for the Finite Volume Method With Applications to Fluid Flows
,” Doctoral Dissertation, Imperial College London, Kensington, London.
36.
Kajishima
,
T.
,
Takiguchi
,
S.
,
Hamasaki
,
H.
, and
Miyake
,
Y.
,
2001
, “
Turbulence Structure of Particle-Laden Flow in a Vertical Plane Channel Due to Vortex Shedding
,”
JSME Int. J. Ser. B
,
44
(
4
), pp.
526
535
.
37.
Liao
,
C.-C.
,
Chang
,
Y.-W.
,
Lin
,
C.-A.
, and
McDonough
,
J. M.
,
2010
, “
Simulating Flows With Moving Rigid Boundary Using Immersed-Boundary Method
,”
Comput. Fluids
,
39
(
1
), pp.
152
167
.
38.
Sharma
,
N.
, and
Patankar
,
N. A.
,
2005
, “
A Fast Computation Technique for the Direct Numerical Simulation of Rigid Particulate Flows
,”
J. Comput. Phys.
,
205
(
2
), pp.
439
457
.
39.
Hager
,
A.
,
Kloss
,
C.
,
Pirker
,
S.
, and
Goniva
,
C.
,
2014
, “
Parallel Resolved Open Source CFD-DEM: Method, Validation and Application
,”
J. Comput. Multiphase Flows
,
6
(
1
), pp.
13
27
.
40.
Jain
,
A. K.
,
1989
,
Fundamentals of Digital Image Processing
,
Prentice Hall
, Englewood Cliffs, NJ.
41.
Green
,
C.
,
2007
, “
Improved Alpha-Tested Magnification for Vector Textures and Special Effects
,”
ACM SIGGRAPH 2007 Courses
(
SIGGRAPH '07
), San Diego, CA, Aug. 5–9, pp.
9
18
.
42.
Frisken
,
S. F.
,
Perry
,
R. N.
,
Rockwood
,
A. P.
, and
Jones
,
T. R.
,
2000
, “
Adaptively Sampled Distance Fields
,”
27th Annual Conference on Computer Graphics and Interactive Techniques—SIGGRAPH
, pp.
249
254
.
43.
Teschner
,
M.
,
Kimmerle
,
S.
,
Heidelberger
,
B.
,
Zachmann
,
G.
,
Raghupathi
,
L.
,
Fuhrmann
,
A.
,
Cani
,
M. P.
,
Faure
,
F.
,
Magnenat-Thalmann
,
N.
,
Strasser
,
W.
, and
Volino
,
P.
,
2005
, “
Collision Detection for Deformable Objects
,”
Comput. Graph. Forum
,
24
(
1
), pp.
61
81
.
44.
Osher
,
S.
, and
Fedkiw
,
R.
,
2003
,
Level Set Methods and Dynamic Implicit Surfaces
, Springer Verlag, Berlin.
45.
Osher
,
S.
, and
Sethian
,
J. A.
,
1988
, “
Fronts Propagating With Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations
,”
J. Comput. Phys.
,
79
(
1
), pp.
12
49
.
46.
van der Pijl
,
S. P.
,
Segal
,
A.
,
Vuik
,
C.
, and
Wesseling
,
P.
,
2005
, “
A Mass-Conserving Level-Set Method for Modelling of Multi-Phase Flows
,”
Int. J. Numer. Methods Fluids
,
47
(
4
), pp.
339
361
.
47.
Hua
,
H.
,
Shin
,
J.
, and
Kim
,
J.
,
2013
, “
Level Set, Phase-Field, and Immersed Boundary Methods for Two-Phase Fluid Flows
,”
ASME J. Fluids Eng.
,
136
(
2
), p.
21301
.
48.
Sussman
,
M.
,
Smereka
,
P.
, and
Osher
,
S.
,
1994
, “
A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow
,”
J. Comput. Phys.
,
114
(
1
), pp.
146
159
.
49.
Lorensen
,
W. E.
, and
Cline
,
H. E.
,
1987
, “
Marching Cubes: A High Resolution 3D Surface Construction Algorithm
,”
14th Annual Conference on Computer Graphics and Interactive Techniques
(
SIGGRAPH '87
), Anaheim, CA, July 27–31, pp.
163
169
.
50.
Koschier
,
D.
,
Deul
,
C.
, and
Bender
,
J.
,
2016
, “
Hierarchical Hp-Adaptive Signed Distance Fields
,”
Symposium on Computer Animation
, Zurich, Switzerland, July 11–13, pp.
189
198
.
51.
ten Cate
,
A.
,
Nieuwstad
,
C. H.
,
Derksen
,
J. J.
, and
Van den Akker
,
H. E. A.
,
2002
, “
Particle Imaging Velocimetry Experiments and Lattice-Botlzmann Simulations on a Single Sphere Settling Under Gravity
,”
Phys. Fluids
,
14
(
11
), pp.
4012
4025
.
52.
Lacis
,
U.
,
Brosse
,
N.
,
Ingremeau
,
F.
,
Mazzino
,
A.
,
Lundell
,
F.
,
Kellay
,
H.
, and
Bagheri
,
S.
,
2014
, “
Passive Appendages Generate Drift Through Symmetry Breaking
,”
Nat. Commun.
,
5
, p.
5310
.
53.
Glowinski
,
R.
,
Pan
,
T. W.
,
Hesla
,
T. I.
,
Joseph
,
D. D.
, and
Périaux
,
J.
,
2001
, “
A Fictitious Domain Approach to the Direct Numerical Simulation of Incompressible Viscous Flow Past Moving Rigid Bodies: Application to Particulate Flow
,”
J. Comput. Phys.
,
169
(
2
), pp.
363
426
.
You do not currently have access to this content.