Two Dimensional Simulation of Incompressible Fluid Flow Using Lattice Boltzmann Method

Abstract

We present the mains of the new method of modeling and simulation lattice Boltzmann methods for fluid flows. Showing some comparisons between classical computations in fluid mechanics based on the resolution of the Navier-Stokes equations and this method. The two dimensional square lattice model with 9 velocities (d2q9) and a simple relaxation time are presented and applied to simulate a laminar flow over backward facing step. The results obtained are in good agreement with those published in the literature.

Full text

Two Dimensional Simulation of Incompressible Fluid Flow Using
Lattice Boltzmann Method
S. Houat1, A.Youcefi2
Abstract – We present the mains of the new method of modeling and simulation lattice
Boltzmann methods for fluid flows. Showing some comparisons between classical computations in
fluid mechanics based on the resolution of the Navier-Stokes equations and this method. The two
dimensional square lattice model with 9 velocities (d2q9) and a simple relaxation time are
presented and applied to simulate a laminar flow over backward facing step. The results
obtained are in good agreement with those published in the literature
Keywords: CFD, Lattice Boltzmann Method, Incompressible Fluid Flow.
Nomenclature

 .
f single particle mass distribution function
m masse
v particle velocity vector
r spatial position vector

 massdensity
u

fluid velocity
P pressure
F body force
eq
f equilibrium distribution function
Ω collision term
kB Boltzmann constant.
T temperature
ei discrete particle velocity in LBE method
cs speed of sound
 constant specific to each set
 weighting factor
 relaxation time
ƒi distribution function in discretized particle
velocity space
eq
i
f equilibrium distribution function in
discretized particle
velocity space
 energy
xspace step
ttime step
i
f
~
post-collision distribution function.
ν kinematic viscosity
h height of the inlet duct
Re Reynolds number
I. Introduction
The development of novel numerical methods used
in computational fluid dynamics (CFD) has made rapid
progress in recent years, especially that of lattice gas
cellular automata (LGCA) introduced by U.Frisch, B
Hasslacher and Y. Pomeau [1]-[4]. This was followed
by that of the lattice Boltzmann Methods (LBM) [5]-
[9]. This latter is gaining gradually more ground, and
subsequently, it is challenging the traditional (CFD)
ones in all domains. These include all numerical
schemes that aim at solving the Navier-Stokes equations
by some direct discretization. Indeed, the macroscopic
variables of interest, such as velocity u and pressure p
are usually obtained by solving the Navier–Stokes (N-
S) equations [10],[11]. However, the LBM is based on a
more rigorous description of the transport phenomena.
In this approach we are able to solve the kinetic
equation of the particle velocity distribution
function ),v,( trff

where v is the particle velocity
vector, r is the spatial position vector and t is the time.
The macroscopic quantities (such as mass density  and
momentum density u) can then be obtained by
evaluating the hydrodynamic moments of the
distribution function f [8]. This method was first
proposed by McNamara and Zanetti [5], Qian et al. [6],
Higuera and Jimenez [12], Koelman [13].
The lattice Boltzmann methods make use of several
significant physically motivated simplifications that
allow not only an efficient and competitive
computational codes, but also superior ones if compared
to the classical approach. These advantages have urged
a lot of researchers in the field of fluid dynamics to rely
more heavily on this method which has, in fact,
emerged as an attractive alternative in many domains:
chemically reacting flow [14], porous media [15],
combustions [16], crystallization [17]-[18], magneto
hydrodynamics [19] and many others [20].

S. Houat, A. Youcefi
In this paper, we present the mains of the lattice
Boltzmann methods and give some comparisons
between the classical computation in fluid mechanics
based on the resolution of the Navier-Stokes equation
and the Boltzman ones.
The two dimensional square lattice model with 9
velocities (d2q9) [4] and simple relaxation time are
presented and applied to simulate a laminar flow over
backward facing step. The results obtained are
compared with those published in the literature.
II. Classical Governing Equations
The macroscopic behaviour of fluid flow is modelled
by means of the Navier-Stokes equations [10],[11]. The
main assumption is that at a macroscopic scale, the
behaviour of fluid motion is not directly influenced by
the detailed molecular interactions, but depends on the
average motion of many microscopic particles, the so-
called continuum limit assumption. Fluid motion can
therefore be described by average physical quantities
like velocity, pressure, temperature and density. The
flow of a single component phase, incompressible,
isothermal and Newtonian fluid, in a given geometry
can be described by the classical Navier-Stokes
equation.
Fupu).u(
u
u.




2
1
0
v
t

Where

is the fluid density, u is the velocity, p is the
pressure, v is the kinematics viscosity and F is a body
force, e.g. the gravity force. The velocity and the
pressure are functions of space and time. The first
equation (1) expresses the conservation of mass. It
states that the in-and outflow of mass in a fluid element
are balanced. Equation (2) expresses the conservation of
momentum. This equation describes the velocity
changes in time, due to convection (u.)u spatial
variation in pressure p and viscous forces 2u In the
stationary incompressible flow the term
t

u=0.
The next step in CFD is to discretized the continuum
equations [11],[21]. There are many ways to accomplish
this goal, namely by means of finite difference (FDM),
finite volume (FVM), finite element (FEM) or spectral
methods. Each of these strategies has their own
philosophy for obtaining a set of discrete algebraic
equations. Finally, the evolution of the flow field is
computed by solving the discrete equations constrained
by well-defined initial and boundary conditions via an
appropriate numerical algorithm.
III. Lattice Boltzmann Method
III.1. Basic theory
The basic kinetic model is given by the Boltzmann
equation:
)(.
F
.v vff
m
f
t
f

 (3)
Where ),v,( trff

is the single particle distribution
function (supposed identical of fluid particles with
masse m, v is the particle velocity and F=F(r,t) is the
local force acting on fluid particles. The collision term
Ω (ƒ) is usually linearized using the Bhatnagar-Gross-
Krook [22] (BGK) approximation after introducing a
relaxation time 


),(),(.
1
)( tftff eq vr,vr, 

(4)
The equilibrium distribution function ),( tff eqeq vr, is
the Maxwell-Boltzmann distribution function:






 2
2/1 ),(.
2
exp.)
..2
).(,(),( t
Tk
m
Tk
m
ttf
BB
eq ruvrvr,


(5)
Here kB is the Boltzmann constant, T is the absolute
temperature of the system.
The particle number density is given by:

v),(),( dtft eq vr,r

(6)
The local fluid velocity and energy respectively are
expressed by [7]:

v),(.
),(
1
), dtf
t
teq vr,v
r
u(r

(7)

v),(.)(
2
1
),(
1
),( 2dtf
t
teq vr,u-v
r
rε

(8)
Investigation [2],[3] and [7],[8] resulted in a general
procedure to construct lattice Boltzmann models for
single component fluids. After discretization of the
phase space [1]-[4], the distribution functions are
defined only in the nodes r of a discrete lattice in the
one, two or three dimensional space, while the velocity
is reduced to a discrete set {ei}, i=0,1,....N[...]. This
velocity is currently used literature LB in the [4, 5] are
expressed using the speed mTkc Bs ./.

, where
is a constant specific to each sets. The discretization
procedure is presented by neglecting the outside forces
for a simplified Boltzmann equation. The equation (3) is
replaced by the set of N equations:
)],(),([
1
),(.
),( tftftf
t
tf eq
iii
ixxxe
x
i



i=0,1,....N (9)
(1)
(2)

S. Houat, A. Youcefi
Where the distribution function ƒi (x, t) express the
probability of finding at node x belonging a lattice, a
particle having the velocity ei . To discrete the equation
(9) in time and space, we get the Boltzmann equation
discretized:
)],(),([
1
),(),.( tftftftttf eq
iiiii xxxex 


(10)
The particle number density  the local velocity u and
the energy are now expressed as:
 




Ni
i
eq
i
Ni
i
itf
t
tf
t
t
00
),(
),(
1
),(
),(
1
),( xe
x
xe
x
xuu ii

(11)
 



 Ni
i
eq
i
Ni
i
itf
t
tf
t
t
0
2
0
2),()(
),(
1
),()(
),(
1
),( xue
x
xue
x
xii


(12)
While the equilibrium distribution function are given as
series expansion in the local velocity:







 242
2
2c.2
)(
.2
)(
.
)(
1),(


u.u.ue.ue
xii
cc
tff i
eq
i
eq
i (13)
Proceeding with the Chapman–Enskog analysis [7]-[8],
the NS equations are recovered in the near
incompressible limit (i.e., the Mach number
M=|u|/cs<<1).
III.2. Two Dimensional (D2Q9) Model
The nine-velocity square lattice model, which is often
referred to as the 2-D 9-velocity (D2Q9) model (Fig. 1),
has been widely and successfully used for simulating
two-dimensional (2D) flows. In the D2Q9 model [6],
and N=8, ei denotes the discrete velocity. In
what follows all quantities are given in non-dimensional
unit’s x. In the advection step of the LBE, particles
move from one node of the grid to one of its
neighbours.
Fig.1: A Two Dimensions 9-Velocities Lattice (D2Q9) model.



















































5,6,7,8( .c2.
2
)5(
4
,
2
)5(
4
1,2,3,4
)
( .c
2
)1(
,
2
)1(
0)( 0
i
i
Sin
i
Cos
i
i
Sin
i
Cos
i
i


e
(14)
While the weight factors are









)8,7,6,5( 36/1
)4,3,2,1( 9/1
)0( 9/4
i
i
i
i

(15)
Where c=x/t, x and t are the lattice constants and
the time step size, respectively. The speed of sound in
this model is 3/cc s [6] and the equation of state
is that of an ideal gas and the viscosity is expressed by
[6]:
tc s

..
2
12





 (16)
The viscosity’s value is positive only if


>1/2.
The lattice Boltzmann scheme consists of two
computational steps [5], [9]:
- Collision step:
)],(),([
1
),(),(
~tftftftf eq
iiii


xxxx  (17a)
- Streaming step:
),(
~
),.( tftttf iii


xex  (17b)
Where fi and i
f
~
denote the pre- and post-collision state
of the distribution function, respectively. The
advantages of solving the lattice Boltzmann equation
over the NS equations can now be seen. In the kinetic
equation for fi given by (11), the advection operator is
linear in the phase space whereas the convection term is
nonlinear in the NS equation. In traditional CFD
methods, the pressure is typically obtained by solving
the Poisson or Poisson-like equation derived from the
incompressible NS equations that can be time
consuming. In the LBE method, the pressure is obtained
through an extremely simple equation of state
2
ps
c.

[9]. This is an appealing feature of the LBE
method. The discretized in (17a-b) for fi is explicit in
form, easy to implement, and natural to parallelize. The
collision step is completely local. The streaming step
takes very little computational effort at every time step.
III.3. Boundary Conditions Treatments
In LBM the implementation of boundary conditions
is very important in simulation. Two classes of
boundaries are frequently encountered in CFD: open
boundaries include inlet and outlet flow and the solid
wall. A difficulty of the LBM is that the boundary

S. Houat, A. Youcefi
conditions for the distribution function are not known.
One must construct suitable conditions for ƒi(x, t) based
on the macroscopic flow variables.
In inlet and Outlet flow, the velocity or pressure is
usually specified in the macroscopic description of fluid
flows [23],[24].
Solid wall: The most commonly used method to apply a
no-slip boundary condition is the particle bounce-back
scheme Proposed by Zou and He [23] the particles
arriving at the stationary wall are reflected back in the
direction it came from. Referring to Inamuro and al
[25], the errors produced by the conditions of border are
negligible if a time relaxation is around the 0.5.
IV. Lattice Boltzmann Method and
Navier-Stokes Solver
The differences between the Lattice Boltzmann
Equation (LBE) method and the Navier-Stokes (N-S)
equations solver are as follows [9]:
1. The N-S equations are second-order parabolic
differential equations (PDEs) applied in macroscopic
study as in (2); the discrete velocity model from which
the LBE model is derived consists of a set of first-order
PDEs (kinetic equations (3)).
2. The N-S solver must deal with nonlinear convective
terms; in the LBE model the convection terms are linear
and handled by simple advection (uniform data
shifting).
3. For incompressible flow, the N-S solver needs to
solve the Poisson equation for the pressure, which
involves global data communication. In the LBE
method, pressure is obtained through an equation of
state and data communication is always local.
4. Usually in the LBE method, the grid Courant-
Friedrichs-Lewy (CFL) number is equal to 1, based on
the lattice units of δx = δt = 1. Also, the coupling
between the discretized momentum space and physical
space leads to regular square grids.
5. Due to the kinetic nature of the Boltzmann equation,
the physics associated with the molecular level
interaction can be incorporated more easily in the LBE
model.
6. For boundary conditions (BCs), in the LBE model
there is no counterpart of the BCs found in a continuum
framework, e.g. no-slip at the wall. Thus the BCs in
LBE model need to be developed.
7. The N-S solver usually employs iterative procedures
to obtain a converged solution; the LBE models are
usually explicit and don’t need iterative procedures.
V. Numerical Results and Discussions
In this section, we applied the lattice Boltzmann
method for a problem of incompressible laminar flow
over backward facing step. This geometry produces
flow separation at the step and subsequent reattachment
which is recognized as important within industrial
situation.
The flow over backward facing step was analyzed by
many researchers, initially by the experimental and
numerical study of Armaly and al. [26], and Morgan
and al [27]. A common configuration is adopted for the
numerical study of the flow in streamline flow with a
ratio of expansion equal to 1/2 (Fig.2). The numerical
results obtained by Armaly and al. are not in good
agreement with the experimental results for medium and
great Reynolds numbers. This difference must probably
be with the inadequate grid in the resolution associated
with the first order differential diagram used in the
study of Armaly and al [26].
Later, by the predictions by digital simulation (Kim
and Moin [28], Orlando and al [29], Patankar and al
[30], Guj and Stella [31], Shon [32], Thangam and
Knight [33]-[34] overcome these problems and obtained
results in good agreement with the experimental one for
all Reynolds numbers, except for large ones where this
difference is due to the three-dimensional aspect of the
flow as it was mentioned in [26].
The flow through a back facing step with an
extension ratio equal ER=1/2 was always a case test
where a parabolic velocity profile u is adopted at the
inlet of the flow.
In the present study, the geometry of the flow and the
boundary conditions are presented on figure 2. The
expansion ratio is ER=1/2 and the length of the field for
calculation is 20h, where h is the height of the inlet
duct. The entrance channel is 3h. It prevents thus
downstream effects of the inlet boundary conditions.
Along with the entrance channel, there is an inlet
velocity with a fully developed parabolic profile.
Fig.2. Sketch of the geometry for the back step facing flow
The geometrical flow is simple. After a study of the
grid, we adopted a uniform grid for calculation with
299×26 nodes. The Reynolds numbers considered are:
50, 100, 150, 200, 250, 300, 400, 500 and 600 where
Re=2h u0/ν, u0 is initial velocity fixed 0.05 and ν is the
kinematic viscosity of the fluid defined by (16). The
velocity was chosen to be lower than 10% of the speed
of sound for the LBM simulation to avoid significant
compressibility effects which are known to increase
with square of the Mach number. The relaxation time is
varied from =0.5065 for Re=600 to =0.5780 for
Re=50. The convergence criterion of the numerical

S. Houat, A. Youcefi
simulations natural steady state flow is defined by
velocities calculation as follow [7]:
6
1001 




x
tx
txttx
ji
ji
ji
jiji
.
),(u
),(u),(u
,
,
,
,,

(18)
In (fig. 3) the visualisation of the flow is illustrated
by field velocity obtained between Re=50 after 8000
iterations and Re=600 after 100000 iterations. It is
shown clearly the reattachment flow phenomena after
the back step, where its length on the lower wall
increased and the point of flow reattachment moves
towards the outlet section with the increase in the
Reynolds number.
Re=50
Re=100
Re=150
Re=200
Re=250
Re=300
Re=400
Re=500
Re=600
Fig.3.Velocity field contour
The second separation and reattachment flows is
showing on upper wall after the back step when Re>200
(see Fig.3) this phenomena is confirmed experimentally
in [26].
Quantitative results of the present simulation are
compared to the data found in literature (experimental
and CFD results [26], [28], [35]) in (Fig.4) which
indicates the reattachment distance, according to
Reynolds numbers, starting from the step to the lower
wall. The comparison shows a good agreement across
the whole range of investigation.
0
1
2
3
4
5
6
7
8
9
10
11
12
50 100 150 200 250 300 400 500 600 650
Re
r/h
J. Kim et Moin [28]
TH Le et Al. [35 ]
Armaly et al. [26]
Present result s (LBM)
Fig. 4. Variation of the ratio r/h with Reynolds number
VI. Conclusion
In this study, the bases of the lattice Boltzmann
method to simulate an incompressible fluid flow are
presented. The two dimensional models with nine
velocities and simple relaxation time is detailed. We
also showed the difference in use of this method with
the traditional CFD ones which are based on the
resolution of Navier-Stokes Equations.
The application of this method for the flow
simulation over backward facing step (in order to
measure the reattachment distance flow from the step
according to the Reynolds number), give satisfying
results compared to experimental results and those
using traditional CFD methods.
In this study, we wish to emphasize the kinetic nature
of the LBM. Because it is a mesoscopic and dynamic
description of the physics of fluids, it can model
problems wherein both macroscopic hydrodynamics
and microscopic statistics are important. The LBM can
be considered as an efficient numerical method for
computational fluid dynamics with an advantage of the
simpler code implement. It is also a powerful tool for
modeling new physical phenomena that are not yet
easily described by macroscopic equations. This would
open many new scopes of application. From a
computational point of view, the LBM equation is
hyperbolic and can be solved locally, explicitly, and
efficiently on parallel computers.
References
[1] U. Frisch, B Hasslacher and Y. Pomeau , Lattice gas automata
for the Navier-Stokes Equation. Phys.Rev. Lett., 56(14), pp.
1505-1508, 1986.
[2] U. Frisch, D. d’Humi`eres, B. Hasslacher, et al., Lattice gas
hydrodynamics in two and three dimensions, Complex Syst. 1,
649, 1987.

S. Houat, A. Youcefi
[3] G. Zenetti, The hydrodynamics of lattice gas automata. Phys.
Rev. A40, pp. 1539-1548, 1989.
[4] P. Clavin, P. Lallemand, Y. Pomeau and G. Searby,
Simulation of free boundaries in flow systems by lattice-gas
models. J. Fluid Mech. vol.188, pp. 437-464, 1988
[5] McNamara G, Zanetti GUse of the Boltzmann equation to
simulate lattice-gas automata. Phys Rev Lett;61 pp. 2332–2335.
1988.
[6] Y. H. Qian, D. D'Humieres, & P. Lallemand, Lattice BGK
models for Navier-Stokes equation. Europhys. Lett. 17, pp. 479-
484, 1992.
[7] X. He & L. Luo, Lattice Boltzmann model the incompressible
Navier Stokes equation, journal of statistical physics 88, pp.
927-944., 1997.
[8] X. He & L. Luo, Theory of the lattice Boltzmann method:
from the Boltzmann equation to the lattice Boltzmann equation,
Phys. Rev. E 56, pp.6811-6817, 1997.
[9] S. Chen and G. D. Doolen, Lattice Boltzmann method for fluid
flows, Annu. Rev. Fluid Mech. 30, pp.329-364, 1998.
[10] CAJ. Fletcher Computational techniques for fluid dynamics,
vols.I and II. ( New York: Springer 1988).
[11] P Wesseling Principles of Computational Fluid Dynamics
(Springer, 2001).
[12] FJ. Higuera, J. Jimienez Boltzmann Approach to lattice gas
simulations. Europhys Lett 9, pp. 663–668. . 1989.
[13] Koelman JMVA, A simple lattice Boltzmann scheme for
Navier–Stokes fluid flow. Europhys Lett1, 5, pp. 603–607, 1991.
[14] S. Chen, SP. Dawson, GD Doolen, DR. Janecky, A.
Lawniczak. Lattice metods and their applications to reacting
systems. Comput Chem Eng ,19, PP. 617-646, 1995.
[15] NS. Martys, H. Chen, Simulation of multi-component fluids in
complex three-dimensional geometries by lattice Boltzmann
method. Phys. Rev. E 53, pp. 743-750,1996
[16] H.Yu, L-S. Luo, SS. Girimaji. Scalar mixing and chemical
reaction simulations using lattice Boltzmann method. Int. J.
Computat Eng Sci 3, pp. 73-87, 2002.
[17] W. Miller, S. Succi, D. Mansutti, A lattice Boltzmann model
for anisotropic liquid/solid phase transition. Phys. Rev. Lett 86,
pp. 3578-3581, 2001.
[18] W. Miller, S. Succi, D. A lattice Boltzmann model for
anisotropic crystal growth from melt. J Stat Phys 107, pp. 173-
186, 2002.
[19] S. Chen, H. Chen D. Martinez, WH. Matthaeus. Lattice
Boltzmann model for simulation of magneto hydrodynamics,
phys. Rev. Lett 67, pp.3776-3779, 1991.
[20] S. Succi, The lattice Boltzmann equation for fluid dynamics
and beyond. (New York: Oxford University press, 2001).
[21] John D Anderson Computational Fluid Dynamics The Basics
With Applications (McGraw Hill, Inc NeyYork 1995).
[22] P.L. Bhatnagar, E.P. Gross and M. Krook A model for
collision processes in gases. I. Small amplitude processes in
charged and neutral one-component system, Phys. Rev. 94, pp.
511-525, 1954.
[23] Q. Zou, X. He, On pressure and velocity boundary conditions
for the lattice Boltzmann BGK model, Phys. Fluids 9, pp. 1591-
1598,1997
[24] J. Bernsdorf, Stromungsbereshnungen mit Lattice Boltzmann
automaten. Kunrzlehrgang NUMET, LSTM, Universitat
Erlangen-Nrnberg, 2000.
[25] T. Inamuro, M. Yoshineand F. Ogine, A non-slip boundary
condition for lattice Boltzmann simulation. Phys. Fluids, Vol.7,
pp.2928-2930, 1995.
[26] B.F. Armaly, F. Durst, J. C.F. Pereira and B.Schonung.
Experimental and theorical investigation of backward-facing
step flow, Journ. Fluid. Mech., Vol. 127, pp. 473-4961, 1983.
[27] K. Morgan, J. Periaux, F. Thomasset, Analysis of laminar
flow over a backward step. In notes on numerical fluid
mechanics, (Vol. 9, Vieweg 1984).
[28] J. Kim, P. Moin, Application of a fractional-step method to
incompressible Navier-Stokes equations. J. Comp. Phys. 59, pp.
308-323, 1985.
[29] P. Orlandi, Vorticity-Velocity formulation for high Re flows,
Computers & Fluids, 15, pp. 137-149, 1987.
[30] S. Patankar, K. C. Karki, H. C. Monia, Development and
evaluation of improved numerical schemes for recirculating
flows. Proc. AIAA 25th Aerospace Sciences Meeting (Reno,
NV), AIAA, New York, 1987, pp. 87-91.
[31] G. Guj, F. Stella, Numerical solutions of high-Re recirculating
flows in vorticity-velocity formulation. Int. J. Numer. Meth.
Fluids, 8, pp. 405-416, 1988.
[32] J. L. Shon, Evaluation of FIDAP on some classical laminar
and turbulent benchmarks. Int. J. Numer. Meth. Fluids, 8,
pp.1469-1490, 1988.
[33] S. Thangam, D. D, Knight, Effect of step height on the
separated flow past a backward-facing. Phys. Fluids, 1, pp. 604-
606, 1989.
[34] S. Thangam, D. D. Knight, A computational scheme in
generalized coordinates for viscous incompressible flows.
Computers & Fluids, 18 pp. 317-327, 1990.
[35] T.H. Le, B. Troff, P. Sagault, K. Dang-Tra, L. Ta Phuoc,
PEGASE: A Navier-Stokes solver for direct numerical
simulation of incompressible flows, Int. Jour. Num. Meth.
Fluids, Vol. 24, 833-861, 1997.
AUTHORS’ INFORMATION
1Department of Mechanical Engineering, University Ibn Badis of
Mostaganem, B.P.188, Mostaganem 27000 Algeria
E_mail: sa_houat@yahoo.fr
2Appleid Mechanic Laboratory, University of Sciences and
Technology of Oran, USTO 31000 Algeria
Samir Houat born in Algeria in 1965, is
Senior lecturer since 1991 at the Department
of mechanics engineering in Mostaganem,
Algeria. He graduated from the University of
Science and Technology, in Oran, Algeria in
1988 with a BEng Degree and in 1991 with a
Magister degree. Currently, he is Head of Department of mechanics
engineering in Mostaganem. His current research interests include the
use of the Lattice Boltzmann method to analyse the incompressible
fluid flow through obstacles.
Abdelakader Youcefi is a Professor at department of mechanics
engineering in Oran, Algeria. In 1993, he received his Ph.D. in fluid
mechanics from “institut national Polytechique de Toulouse” in
France. He has refereed many conference papers.