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# An EHL Extension of the Unsteady FBNS Algorithm

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## Abstract and Figures

Many engineering applications rely on lubricated gaps where the hydrodynamic pressure distribution is influenced by cavitation phenomena and elastic deformations. To obtain details about the conditions within the lubricated gap, solvers are required that can model cavitation and elastic deformation effects efficiently when a large amount of discretization cells is employed. The presented unsteady EHL-FBNS solver can compute the solution of such large problems that require the consideration of both mass-conserving cavitation and elastic deformation. The execution time of the presented algorithm scales almost with Nlog(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\log (N)$$\end{document} where N is the number of computational grid points. A detailed description of the algorithm and the discretized equations is presented. The MATLAB©\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\copyright }$$\end{document} code is provided in the supplements along with a maintained version on GitHub to encourage its usage and further development. The output of the solver is compared to and validated with analytical, simulated, and experimental results from the literature to provide a detailed comparison of different discretization schemes of the Couette term in presence of gap height discontinuities. As a final result, the most favorable scheme is identified for the unsteady study of surface textures in ball-on-disc tribometers under EHL conditions.
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Tribology Letters (2022) 70:80
https://doi.org/10.1007/s11249-022-01615-1
ORIGINAL PAPER
An EHL Extension oftheUnsteady FBNS Algorithm
ErikHansen1 · AltayKacan1· BettinaFrohnapfel1· AndreaCodrignani2
Received: 23 November 2021 / Accepted: 14 May 2022
© The Author(s) 2022
Abstract
Many engineering applications rely on lubricated gaps where the hydrodynamic pressure distribution is inﬂuenced by cavi-
tation phenomena and elastic deformations. To obtain details about the conditions within the lubricated gap, solvers are
required that can model cavitation and elastic deformation eﬀects eﬃciently when a large amount of discretization cells is
employed. The presented unsteady EHL-FBNS solver can compute the solution of such large problems that require the con-
sideration of both mass-conserving cavitation and elastic deformation. The execution time of the presented algorithm scales
almost with
Nlog(N)
where N is the number of computational grid points. A detailed description of the algorithm and the
discretized equations is presented. The MATLAB
code is provided in the supplements along with a maintained version on
GitHub to encourage its usage and further development. The output of the solver is compared to and validated with analytical,
simulated, and experimental results from the literature to provide a detailed comparison of diﬀerent discretization schemes
of the Couette term in presence of gap height discontinuities. As a ﬁnal result, the most favorable scheme is identiﬁed for
the unsteady study of surface textures in ball-on-disc tribometers under EHL conditions.
Keywords Elasto-Hydrodynamic Lubrication (EHL)· Fischer-Burmeister-Newton-Schur (FBNS)· Jakobsson-Floberg-
Olsson (JFO)· Mass-conserving cavitation· Elastic half-space
1 Introduction
In the case of lubrication ﬂows in narrow gaps, the Reyn-
olds equation [1] is a handy tool to determine the hydrody-
namic pressure distribution in a simpler way than by using
the full Navier-Stokes Equations (NSE) [2, Ch. 7]. Since
cavitation commonly occurs in lubrication ﬂows, various
models have been developed to describe this phenomenon
[3]. Especially when it occurs within surface textures, mass-
conserving properties of the cavitation model are required
to properly describe the ﬂow’s transition from the cavita-
tion region to the full-ﬁlm region, because this full-ﬁlm
reformulation interface has a great eﬀect on the extension
of the cavitated area and the subsequent downstream rise
in pressure within the full-ﬁlm region [4]. The required
mass-conserving properties can be taken into account with
the Jakobsson-Floberg-Olsson (JFO) [5, 6] cavitation model
[3]. Starting from the cavitation algorithm of Elrod [7], Gia-
copini etal. [8] developed a one-dimensional ﬁnite element
method (FEM) solver that couples the Reynolds equation
with the mass-conserving JFO cavitation model through a
complementarity formulation. This work was extended by
Bertocchi etal. [9] to consider two-dimensional problems
with compressible, piezoviscous, and shear-thinning ﬂuid
behavior. The arising complementarity problem was refor-
mulated to be expressed by an unconstrained equation sys-
tem by Woloszynski etal. [10], resulting in the Fischer-Bur-
meister-Newton-Schur (FBNS) algorithm. As demonstrated
by Woloszynski etal., the FBNS algorithm is of remarkable
computational eﬃciency also for high spatial resolutions.
In many cases, the hydrodynamic pressure can deform the
lubricated surfaces notably leading to the regime of Elasto-
Hydrodynamic Lubrication (EHL) [11]. Various solvers
have been developed to tackle EHL problems, some of the
most prominent ones are the ﬁnite diﬀerence method (FDM)
Multigrid solver of Venner and Lubrecht [12] and the FEM
solver of Habchi [13]. Some algorithms are also capable of
simulating surface contact along with the Reynolds equation
* Erik Hansen
erik.hansen@kit.edu
1 Institute ofFluid Mechanics (ISTM), Karlsruhe Institute
ofTechnology (KIT), Kaiserstr. 10, 76131Karlsruhe,
Germany
2 Fraunhofer-Institut für Werkstoﬀmechanik (IWM),
Wöhlerstraße 11, 79108Freiburg, Germany
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Tribology Letters (2022) 70:80
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80 Page 2 of 25
[1418]. Since the full-ﬁlm reformulation interface is often
not of relevance in EHL problems, many EHL solvers do
not employ mass-conserving cavitation models. However,
in some cases - such as starved lubrication—mass-conserv-
ing cavitation is crucial and has been considered in several
works [19, 20]. Among them, the coupling of pressure,
mass-conserving cavitation, elastic deformation, a rough-
ness asperity contact model and the FBNS algorithm was
achieved by Ferretti [21, 22]. In contrast to Ferretti’s work,
the FBNS algorithm is coupled with the elastic deforma-
tion of an elastic half-space within this paper, thus present-
ing the new EHL-FBNS algorithm. Due to the half-space
assumption, the elastic deformation is a linear convolu-
tion of a kernel function with the hydrodynamic pressure
ﬁeld. This allows exploiting the fast Fourier transformation
(FFT) to speed up the computation of the elastic deforma-
tion [23]. Furthermore, a proportional integral derivative
(PID) controller is employed to meet the load balance equa-
tion through adjustment of the rigid body displacement
as already introduced by Wang etal. [24]. Eventually, the
EHL-FBNS algorithm is capable of eﬃciently computing
the solution of large problems that require the consideration
of both mass-conserving cavitation and elastic deformation
at the same time.
In the beginning of this paper, the basic equations and
the general procedure of the EHL-FBNS algorithm are
summarized. The algorithm is implemented in MATLAB
with the ﬁnite volume method (FVM) and a generic order
spatial discretization scheme for the Couette term of the
Reynolds equation. The discretized equations are supplied
in Appendix. Then, the performance of the steady EHL-
FBNS implementation is compared to the original FBNS
algorithm of Woloszynski etal. [10]. Afterward, one- and
two-dimensional literature reference cases [9, 25] of a con-
vergent slider with rectangular pocket are used to compare
the EHL-FBNS output to analytical and simulated reference
results, assess the inﬂuence of the discretization order of the
Couette term in the presence of gap height discontinuities
through grid convergence studies and give an example case
where both mass-conserving cavitation and elastic proper-
ties of the solver are required. Moreover, unsteady EHL-
FBNS simulations are performed for the set-up of a single
texture that passes through the EHL contact of a ball-on-disc
tribometer. The results are compared to experimental and
simulated data of Mourier etal. [26]. This allows to demon-
strate the stability of the EHL-FBNS algorithm under EHL
operating conditions with discontinuous surface textures and
to provide recommendations about the most suitable discre-
tization scheme. Eventually, the EHL-FBNS algorithm is
validated for surface texture investigations with ball-on-disc
tribometers under unsteady EHL operating conditions.
The MATLAB
code, set-up, and visualization scripts
are provided in the supplements. The MATLAB
scripts are
thoroughly commented to encourage their usage and further
development. A maintained and publicly available version of
the code can also be found on GitHub: https:// github. com/
ErikH ansen Git/ EHL.
2 Numerical Methods
2.1 Governing Equations
The lubrication ﬂow in a narrow gap (schematically depicted
in Fig.1) is governed by the Reynolds equation considering
mass-conserving cavitation with the JFO model [5, 6] at any
set of spatial coordinates
x1
and
x2
and time t [9]:
In this equation, h denotes the gap height,
𝜌l
is the density
of the liquid phase and
𝜇l
describes the dynamic viscosity of
(1)
𝜕
𝜕x1
l
12𝜇l
𝜕x1
+𝜕
𝜕x2
l
12𝜇l
𝜕x2
𝜕
𝜌lhum(1𝜃)
𝜕
𝜌lh(1𝜃)
=
h
Uup
Ulow
x3
x2
x
1
Fig. 1 Schematic sketch of the lubricated gap
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Tribology Letters (2022) 70:80
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the liquid phase. All of them can vary in space and time. The
mean velocity
u
m=
U
up
+Ulow
2
is composed of
Uup
as the veloc-
ity of the upper surface and
Ulow
as the velocity of the lower
surface in
x1
-direction. The relative pressure
p=phd pcav
and the cavity fraction
𝜃
=1
𝜌
𝜌l
are the solution variables,
where
𝜌
is the mixture density of the ﬂow. The hydrody-
namic pressure
phd
is prevented from falling below the cavi-
tation pressure
pcav
by adding the following complementary
constraints [9]:
Depending on whether the liquid phase is modeled as iso-
or piezoviscous through the Barus or Roelands model, its
dynamic viscosity reads [12, Ch. 1.3.3], [27]:
where
z
R=
𝛼
R
p
0,R
ln
(
𝜇
0
+9.67
)
[26] is the pressure viscosity index,
𝛼B
and
𝛼R
denote the pressure viscosity coeﬃcient and
p0,R
is a
constant in the Roelands equation. The dynamic viscosity of
the liquid phase at ambient pressure is
𝜇0
. Moreover,
depending on whether the liquid phase is assumed to be of
constant density or to be compressible according to the
Dowson-Higginson model, the liquid phase density is given
by [12, Ch. 1.3.4], [9, 27]:
where
𝜌0
is the density of the liquid phase at ambient pres-
sure and
C1
and
C2
are constants.
The gap height h can be constructed as a superposition
of the rigid body displacement of the two surfaces
hd
, the
variation of the gap height due to the rigid geometry of the
surfaces
hg
and the elastic deformation of the gap height
hel
due to the hydrodynamic pressure [2, Ch. 19.2]:
(2)
p
𝜃=0, p0, 𝜃0.
(3)
𝜇
l=
𝜇0,
𝜇0exp 𝛼Bphd pcav 
𝜇0exp
ln
𝜇0
+9.67
1+
1+(phdpcav )
p0,R
zR
,
(4)
𝜌
l=
{
𝜌0,
𝜌0
C1+C2(phdpcav )
C
1
+
(
p
hd
p
cav ),
Depending on whether the upper and lower surfaces are
assumed to be rigid or elastic half-spaces, the elastic defor-
mation of the gap height can be expressed as [12, Ch. 1.3.5]:
where the reduced elastic modulus is stated as [12, Ch.
1.3.5]:
In this equation, E denotes the corresponding Young’s mod-
ulus and
𝜈
the Poisson ratio of the upper and lower surface.
If a constant rigid body displacement
hd
is prescribed, the
provided set of equations is suﬃcient to describe the EHL
problem. If, however, a constant imposed normal load force
FN,imp
is prescribed, it needs to be satisﬁed by the normal
force
FN
resulting from the hydrodynamic pressure proﬁle
[28]:
where
pamb
is the ambient pressure. The rigid body displace-
ment
hd
needs to be set such that the load balance Equation
(8) is fulﬁlled.
2.2 EHL‑FBNS Algorithm
The set of equations described above can be solved numeri-
cally with the EHL-FBNS algorithm presented in the fol-
lowing. This new algorithm is based on the FBNS algorithm
developed by Woloszynski etal. [10] and extends it by tak-
ing elastic surface deformation and the load balance equa-
tion into account. First of all, the dimensionless Reynolds
equation considering mass-conserving cavitation is deﬁned
as:
(5)
h=hd+hg+hel.
(6)
h
el
x1,x2
=
0,
2
𝜋E+∞
−∞ +∞
−∞
phd(x
1,x
2)
(x1x
1)2+(x2x
2)2dx
1dx
2
,
(7)
E
=
2
1𝜈2
low
E
low
+1𝜈2
up
E
up
.
(8)
F
N,imp =FN=
+∞
−∞ +∞
−∞
phd pambdx1dx2
,
(9)
G
=
x
1
Po
p
x
1
+
x1,ref
x2,ref
2
x
2
Po
p
x
2

Poiseuille ter m
x
1
Co(1)

Couett e term
t
Ti(1)

.
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80 Page 4 of 25
It can be derived by inserting the following non-dimensional
quantities (indicated by
) and reference quantities (denoted
by the index
ref
) into and reformulating Equation (1):
Similar to Venner and Lubrecht [12, Ch. 6.3], the coeﬃ-
cients of the Poiseuille, Couette, and unsteady term within
the dimensionless Reynolds equation can be consolidated as:
The complimentary constraints (2) are replaced by the Fis-
cher-Burmeister equation in non-dimensional form [10]:
The EHL-FBNS algorithm uses the Newton-Raphson
method to determine the values of
p
and
𝜃
such that G and
F get suﬃciently close to 0, thus solving the dimensionless
Reynolds Equation (9) and the dimensionless Fischer-Bur-
meister Equation (12). By evaluating the discretized form
of G and F at each discrete position, an equation system is
created. The discretized equations are obtained though the
FVM, where the second-order midpoint rule is applied to
evaluate surface and volume integrals. The required values
and derivatives of the Poiseuille term are discretized with a
second-order central scheme, of the Couette term by either
the ﬁrst-order upwind interpolation (UI) or the third-order
quadratic upwind interpolation (QUICK) and the unsteady
term with the ﬁrst-order Euler implicit scheme [29, Chs.
3.3, 4, 6.3.2]. The discretized expressions of Equations (9)
and (12) are provided in Appendix A.1. The set of discrete
dimensionless Reynolds equations
G
can be expressed in
matrix-vector notation through the pressure coeﬃcients con-
tributed by the Poiseuille term
APo
, the discretized dimen-
sionless relative pressures
p
, the cavity fraction coeﬃcients
contributed by the Couette and unsteady term B, the discre-
tized cavity fractions
𝜃
, and the remaining constants from
the Couette and unsteady term
c
[10]:
The set of discrete dimensionless Fischer-Burmeister Equa-
tions (12) are denoted by
F(
p
,
𝜃)
. The non-dimensional
properties
𝜇
and
𝜌
at each discrete point are computed
according to the respective Eqs. (3, 4 and 10).
The gap height h at each discrete point is computed
according to Eqs. (5 and 6). It is prevented from becom-
ing lower than 1
nm
by using truncation at this instant.
(10)
x
1
=x
1
x
1,ref
,x
2
=x
2
x
2,ref
,t=tt
ref
,𝜌
l
=𝜌𝜌
ref
,
𝜇
l
=𝜇𝜇
ref
,h=hh
ref
,u
m
=u
m
u
m,ref
,p=pp
ref .
(11)
𝜉
Po =
𝜌h3
𝜇,𝜉
Co =12 x1,ref um,ref 𝜇ref
h2
ref
pref
𝜌hu
m,𝜉
Ti =12
x
2
1,ref 𝜇ref
tref h2
ref
pref
𝜌h
.
(12)
F
=p+
𝜃
p2+
𝜃
2
.
(13)
G=APo
p
+B
𝜃+c.
Afterward, the non-dimensional gap height
h
is determined
through Equation (10). If the surfaces are chosen to be elas-
tic, Equation (6) is discretized by assuming a constant pres-
sure over the rectangular discretization cell [30, Ch. 3.3],
[31, Ch. 3.1], the discretized equation is also provided in
Appendix A.2. Since the resulting equation is a linear con-
volution of a kernel function with the hydrodynamic pres-
sure ﬁeld, it is computed in Fourier space by means of FFT
to speed up the computation. Attention is paid to double
the size of the kernel in each direction and to zero pad the
hydrodynamic pressure ﬁeld such that a linear instead of a
circular convolution is obtained. After the convolution, the
deformation and pressure ﬁelds are resized to their original
size [23, 32, 33].
After computing
G
and
F
, the Newton-Raphson method
is used to determine the updates of non-dimensional relative
pressure
𝛿p
and cavity fraction
𝛿𝜃
[10]:
The most important extension of the EHL-FBNS algorithm
compared to the original FBNS algorithm is the approxi-
mation of the pressure Jacobian
JG,p
of the dimensionless
Reynolds equation when elastic deformation is taken into
account. The idea is to consider the dependence
h(p)
by
inserting it into
c
, thus creating the matrix
Ah
. Due to the
kernel function, this would result in
Ah
being a full matrix
which is prone to lose its diagonal dominance and there-
fore being unfeasible to invert and likely to cause unsta-
ble behavior in the iteration process. This is rectiﬁed by
approximating
Ah
only by some of its diagonals as already
done in the literature for other EHL algorithms: for example,
Venner and Lubrecht [12, Chs. C.1, C.3.2] who combine it
with distributive relaxation and multigrid methods or Wang
etal. [15] who employ the semi-system method. In case of
the EHL-FBNS algorithm,
Ah
is reduced to the 5 diagonals
that correspond to the South, West, Center, East, and North
cells. Eventually, the Jacobians of
G
JG,p
=APo +Ah
and
JG,𝜃=B
. The boundary conditions of
p
are considered
in
APo
and
c
and the boundary conditions of
𝜃
in
F
and
JF,𝜃
.
If Neumann boundary conditions are used for
𝜃
, the Jaco-
bian
JF,𝜃
would contain several diagonals. In this case, it is
approximated only by its main diagonal. It is worthwhile to
note that this approximation of the Jacobians eventually only
aﬀects the updates of non-dimensional relative pressure
𝛿p
and cavity fraction
𝛿𝜃
, but never the computation of
G
and
F
.
(14)
J
𝛿=
JF,pJF,𝜃
JG,pJG,𝜃
𝛿p
𝛿
𝜃
=−
F
G
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Tribology Letters (2022) 70:80
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The discrete formulations of the Jacobians
JG,p
=APo +Ah
and
JG,𝜃=B
are provided in Appendix A.1 and A.3. The
center entries of the Jacobians of the dimensionless Fischer-
Burmeister equation
JF,p
,C
and
JF,𝜃,C
for each discrete point
are [10]:
Here,
p
C,aux
and
𝜃
C,aux
are the auxiliary dimensionless pres-
sure and cavity fraction which are adjusted such that
JF,p
and
JF,𝜃
do not become singular [10]. To prevent them from
having center entries close to zero within the range
(−𝜀,𝜀)
,
they are adjusted as:
where machine epsilon is given by
𝜀2.2204 1016
. As
already done in the original FBNS algorithm, the corre-
sponding columns of the Jacobian J and rows of the updates
𝛿
are swapped if
JF,p
,C<JF,𝜃,C
to obtain a reordered system
[10]:
Due to the swapping,
AF
is better conditioned than
JF,p
which is exploited when Equation system (19) is solved [10]:
After obtaining
𝛿a
and
𝛿b
, the earlier performed row swap-
ping is reversed to get the updates of non-dimensional
(15)
J
F,p,C=1
p
C,aux
p2
C,aux +𝜃2
C,aux
,
(16)
J
F,𝜃,C=1
𝜃
C,aux
p2
C,aux +𝜃2
C
,aux
(17)
p
C,aux =
p
Cif p
C
𝜀or p
C
𝜀
,
𝜀if 0 p
C<𝜀,
𝜀if 𝜀<p
C<0,
(18)
𝜃
C,aux =
𝜃Cif 𝜃C
𝜀or 𝜃C
𝜀
,
𝜀if 0 𝜃C<𝜀,
𝜀if 𝜀<𝜃
C<0,
(19)
AFBF
AGBG
𝛿a
𝛿b
=−
F
G
.
(20)
𝛿
b=
BGAG
A1
FBF

1
G+AG
A1
F
F
,
(21)
𝛿
a=A1
F
(
FBF
𝛿b
).
pressure
𝛿
n
p
and cavity fraction
𝛿n
𝜃
[10]. The new values
p,n
and
𝜃n
at iteration n are obtained by means of relaxation:
where
𝛼p
,
p,n1
,
𝛼𝜃
and
𝜃
n
1
are the relaxation factors and
previous solutions of non-dimensional relative pressure and
cavity fraction. Depending on the simulated case, relaxa-
tion coeﬃcients between 0.05 and 1 resulted in good trade-
oﬀs between convergence speed and stability. Preventing
p,n
from having values below 0 and
𝜃n
from having values
below 0 or above 1 through truncation furthermore enhances
favorable convergence properties.
If a constant load force is prescribed, the dimensionless
rigid body displacement
h
d=
h
d
h
ref
a PID controller to meet the load balance Equation (8) as
already done by Wang etal. [24]. This is done by ﬁrst deter-
mining the resulting normal load force
Fn
N
through the dis-
cretized load balance equation:
where
N
x
1
,
Δx1
,
N
x
2
, and
Δx2
are the amount and spacing
of the discretization cells in
x1
- and
x2
-direction and
pn
hd,C
is the hydrodynamic pressure at the center of each discrete
cell. The residual of the load balance equation is deﬁned as:
where
FN,ref
is a reference normal force that is usually just
set equal to the imposed normal force
FN,imp
. Note that
rn
FN
can be either positive or negative, depending on whether
Fn
N
is larger or smaller than
FN,imp
. This is required for the PID
controller to work properly. Finally,
rn
FN
is fed into the PID
controller to determine
h,n+1
d
of the next iteration step [24]:
Note that
KI
is only multiplied with the sum up until
rn1
FN
,
since
rn
F
N
is already considered by
KP
. For all of the later
considered simulations with imposed normal load force,
KP=0.001
,
KI=0.02
, and
KD=0.001
worked well. At last,
the following residuals are computed:
(22)
p
,n=p,n1+𝛼p
𝛿n
p
,
(23)
𝜃
n=
𝜃n1+𝛼
𝜃
𝛿n
𝜃,
(24)
F
n
N=
N
x2
N
x1
(
pn
hd,Cpamb
)
Δx1Δx2
,
(25)
r
n
FN
=
Fn
N
F
N,imp
F
N,ref
,
(26)
h
,n+1
d=KPrn
FN
+KI
n1
i
ri
FN
+KD
(
rn
FN
rn1
FN
).
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Tribology Letters (2022) 70:80
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80 Page 6 of 25
(27)
r
n
max,𝛿p=max
(
abs
(
𝛿n
p
))
,
rn
max,𝛿𝜃 =max (abs(
𝛿n
𝜃)),
rn
max,𝛿G=max(abs(
Gn
Gn1))
,
rn
max,G=max (abs(
Gn)),
rn
max,𝛿F=max(abs(
Fn
Fn1)),
rn
max,F=max
(
abs
(
Fn
))
,
Note that the residuals
rn
max,𝛿G
and
rn
max,𝛿F
are directly aﬀected
by the relaxation factors and
Gn
and
Fn
are computed through
the solutions
p,n1
and
𝜃
n
1
. The EHL-FBNS algorithm is
repeated as long as
rn
EHL-FBNS
and in case of an imposed nor-
abs(rn
FN
)
are above the tolerance
tol =106
.
(28)
r
n
EHL-FBNS =max
(
rn
max,𝛿p,rn
max,𝛿𝜃,rn
max,𝛿G,rn
max,G,rn
max,𝛿F,rn
max,F
).
Fig. 2 Flowchart of the most
relevant steps of the EHL-
FBNS algorithm
Initialize: p0
hd(x1,x
2), θ0(x1,x
2), h0
d,n=1,t=0
Compute h
el(pn1
hd ) and h(x1,x
2,t), truncate hif h<1nm
Compute µ
l(pn1
hd ), ρ
l(pn1
hd ), ξ
Po,ξ
Co,ξ
Ti
Construct APo,B,c,J
Compute
Gn(p
,n1,
θn1)and
Fn(p
,n1,
θn1)
Swap columns of Jand rows of
δ
Compute
δaand
δb
Reswap rows of
δaand
δbto obtain
δn
pand
δn
θ
Compute p
,n and
θn,truncate p
,n below0
and
θnoutside of 0 and 1
Compute pn
hd,Fn
Nand resiuduals rn,t
d(rn,t
FN,r
n1,t
FN,...,r
n,tt
FN, ...) withPID controller
rn,t
EHLFBNS and
abs(rn,t
FN)tol?
t=tfinal?
Export pn
hd(x1,x
2,t), θn(x1,x
2,t)and h(x1,x
2,t)
No
nn+1
No
tt+∆t
p0
hd pn
hd,θ0θn,h0
dhn
d
n=1
Yes
Yes
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Tribology Letters (2022) 70:80
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Page 7 of 25 80
The most important steps of the algorithm structure are also
visualized in Fig.2. The initial guess is always a zero cavity
fraction ﬁeld and a pressure ﬁeld at ambient pressure. If an
unsteady simulation is performed, the solution at
t=0
is
obtained through the steady problem caused by the geometry
at
t=0
. Furthermore, the PID controller also takes the resid-
uals of the load balance equation of the previous time steps
into account if
t>0
.
3 Results andDiscussion
In order to assess the performance of the presented EHL-
FBNS algorithm, it is ﬁrstly employed in a numerical litera-
ture test case of a textured parallel slider. Then, the results
of the EHL-FBNS algorithm are compared to the analytical
solution of a rigid one-dimensional convergent slider with
rectangular pocket to show the eﬀect of the discretization
order of the Couette term on the accuracy of the simulation
result when gap height discontinuities are present. Subse-
quently, the slider is extended to a two-dimensional geome-
try and an elastic model is employed to give an example case
where both mass-conserving cavitation and elasticity show
relevant eﬀects. Afterward, another experimental–numerical
literature case is simulated with the EHL-FBNS algorithm
to validate the code for textured ball-on-disc investigations,
evaluate the code’s stability in unsteady EHL conditions,
and compare diﬀerent spatial discretization schemes. For
all considered cases, the second-order midpoint rule is used
for the evaluation of integrals arising from the FVM and the
Poiseuille term is always discretized with a second-order
central scheme. Consequently, the resulting order of the
dimensionless Reynolds equation discretized with the FVM
in the steady case is ﬁrst order with the UI and second order
with the QUICK scheme. In the unsteady case, only ﬁrst
order is achievable for both UI and QUICK since the ﬁrst-
order Euler implicit scheme is employed. All of the EHL-
FBNS simulations are performed with MATLAB
R2020a.
Fig. 3 Exemplary cell array
resulting in the case of
Kx1
=
Kx2
=
2
hg=hmin,p =pamb
=0
hg=hmin
hg=hmin +hp/2
hg=hmin +hp
Lx1
x2
x
1
L
x2
Table 1 Summary of the parameters and values used in the EHL-
FBNS simulations of the parallel slider with a various amount of
trapezoidal pockets
Param. Value Param. Value Param. Value
Uup
5
ms−1
𝜇0
3102
Pas
Ulow
0
𝛼B
Nx1
2+30
Kx1
um
2.5
ms−1
𝛼R
N
x
2
2+30
K
x
2
p0,R
Lx1
8102
m
pamb
105
Pa
Lx2
Lx1
pcav
3104
Pa
𝜌0
850
kg m−3
hd
0
phd,SB
pamb
C1
hp
12 106
m
phd,WB
pamb
C2
hmin
15
10
6
m
phd,EB
pamb
Eup
–,
5109
Pa
phd,NB
pamb
Elow
–,
Eup
K
x
2
K
x
1
𝜃SB
0
𝜈up
–, 0.3
Δx1
Lx1∕(
N
x1
1
)
𝜃WB
0
𝜈low
–,
𝜈up
Δx2
L
x
2∕(N
x
21)
𝜃EB
0
E
–,
5.49 109
Pa
𝜃NB
0
FN,imp
href
hmin
x1,ref
Lx1
𝛼p
1, 0.5
x2,ref
L
x
2
𝛼𝜃
1, 0.5
pref
106
Pa
𝜌ref
𝜌0
tref
𝜇ref
𝜇0
FN,ref
uref
um
Param. Value
N1024, 3844, 14,884, 58,564, 91,204, 3,62,404,
14,44,804, 57,69,604
Kx1
1, 2, 4, 8, 10, 20, 40, 80
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Tribology Letters (2022) 70:80
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80 Page 8 of 25
3.1 Parallel Slider withaVarying Amount
ofTrapezoidal Pockets
The parallel slider with a varying amount of trapezoidal
pockets used by Woloszynski etal. [10] serves as ﬁrst test
case. This set-up is chosen because it can cause a generic
amount of distinctive cavitation regions. This is to demon-
strate the good performance and stability properties of the
EHL-FBNS algorithm since the simulation of such cases
showed to be diﬃcult or ineﬃcient with other codes from
the literature. Furthermore, the solid properties are chosen
such that noticeable eﬀects due to elastic deformations on
the pressure proﬁle occur. Thereby, it is shown that the con-
sideration of elasticity does not alter the performance scal-
ing. While being of little physical interest, this numerical
set-up allows a comparison of the performance scaling to the
original FBNS algorithm of Woloszynski etal.
The variation of the gap height due to the rigid geom-
etry of the surfaces
hg
is constructed by assembling several
unit geometries. Each unit geometry is composed of
20 20
cells with
hg=hmin +hp
. This square is surrounded by a one
cell thick layer with
hg=hmin +hp2
, which is in turn sur-
rounded by a layer of four cells with
hg=hmin
, resulting in a
square of
30 30
cells. Depending on the desired size of the
computational domain, a certain amount
K
x
1
K
x
2
of these
unit cells is attached to each other and ﬁnally surrounded
by a layer of cells for the Dirichlet boundary conditions of
hydrodynamic pressure at
pamb
and zero cavity fraction. At
last, the coordinates of each cell center are set such they
are in the range of
[0Lx1][0Lx2]
. The resulting array of
cells in the exemplary case of
K
x
1
=K
x
2
=2
is visualized in
Fig.3. The rigid body displacement
hd
is set to 0 since it is
hg
.
The values of the parameters used in the EHL-FBNS
simulations are summarized in Table1. Piezoviscosity and
compressibility of the liquid phase are not considered. The
Fig. 4 Performance of EHL-FBNS and FBNS algorithm along with
reference scalings. The performance of the FBNS algorithm was
taken from Woloszynski etal. [10]
Fig. 5 Performance of combinations of rigid, elastic, UI, and QUICK
EHL-FBNS simulations
Table 2 Summary of the code execution times of the FBNS algorithm given by Woloszynski etal. [10] and the ones of the EHL-FBNS algo-
rithm with UI or QUICK discretization and rigid or elastic geometry to simulate the parallel slider with a various amount of trapezoidal pockets
The FBNS simulations of Woloszynski etal. were performed on a workstation with 32
GB
RAM and an Intel Xeon 3.3
GHz
processor while the
computations of the EHL-FBNS algorithm were conducted on a workstation with a 64
GB
RAM and an AMD Ryzen 9 3900X 12-Core 3.8
GHz
processor
Resolution Execution time
tex[s]
N[−]
FBNS [10] Ri, UI El, UI Ri, QUICK El, QUICK
1024 0.1 0.0285 0.1133 0.0456 0.1958
3844 0.3 0.0828 0.2591 0.3294 0.2884
14,884 1.3 0.3665 1.0631 0.9767 1.3695
58,564 5.7 2.0455 5.3531 4.1428 6.0378
91,204 11.3 3.5748 16.1899 6.7058 10.1669
362,404 47.1 17.6511 41.6014 31.8607 54.4950
1,444,804 103.5338 225.9930 179.1673 287.9650
5,769,604 632.0061 1,776.8957 1,042.3062 2,096.1574
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Tribology Letters (2022) 70:80
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Page 9 of 25 80
amount of unit geometries in
x1
-direction
Kx1
is varied while
Kx2
=Kx1
is always enforced such that the resulting mesh is
always quadratic. By increasing the amount of unit geom-
etries, the total amount of discretization cells N is increased.
Each resulting geometry is simulated with UI and QUICK
for the rigid and elastic case. In the elastic case, the solid
bodies’ Young’s modulus E and Poisson ratio
𝜈
are set such
that the elastic deformation shows notable effects even
though the overall pressures are low. At the same time, this
set-up produces many distinctive cavitation regions. The
rigid simulations use relaxation factors of
𝛼=1
. Since the
elastic simulations are generally more unstable, the relaxa-
tion factors have to be reduced to 0.5.
To quantify the algorithm’s performance independently
of the hardware’s computational power, the non-dimensional
code execution time is deﬁned as:
where
tex
is the code execution time for a certain total
amount of discretization cells N. The non-dimensional code
execution time
t
ex
of the EHL-FBNS algorithm in the rigid
UI case is compared with the one of the original FBNS algo-
rithm of Woloszynski etal. [10] in Fig.4. Their code was
also implemented in MATLAB
, considered rigid geome-
tries and was discretized with the FVM, where the Poiseuille
term was discretized with second-order central diﬀerences
and a ﬁrst-order upwind scheme was employed for the cavity
fraction. While Woloszynski etal. [10] use diﬀerent toler-
ances for diﬀerent residuals (
103
for
rn
max,G
and
106
for
rn
max,𝛿G
), the EHL-FBNS algorithm employs an even stricter
convergence criterion (
106
for
rn
EHL-FBNS
given by Equation
(28)). Furthermore, three reference curves for the scaling of
the non-dimensional code execution time are displayed: lin-
ear
t
ex,lin =M
1
N
, logarithmic
t
ex,log =M
2
Nlog(N)
(29)
t
ex =
t
ex
(N)
t
ex
(N=1024)
,
Fig. 6 Hydrodynamic pressure
proﬁles
phd
for a UI simulation
with
Kx1
=
8
: a rigid and b
elastic case
Fig. 7 Hydrodynamic pres-
sure proﬁles
phd
for a QUICK
simulation with
Kx1
=
8
: a rigid
and b elastic case
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Tribology Letters (2022) 70:80
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80 Page 10 of 25
ratic
t
=M3N
2
. The coeﬃcients
M1
=
9.7656
104
,
M2
=1.4089 10
4
and
M3
=9.5367 10
7
are chosen such
that
t
ex(N=1024)=1
in all cases. It can be seen that the
original FBNS algorithm has a performance scaling close to
the linear reference while the EHL-FBNS algorithm per-
forms a little bit slower than the
Nlog(N)
reference for large
N but is always much faster than the quadratic reference. The
diﬀerence between the FBNS and EHL-FBNS performances
might be due to the fact that the EHL-FBNS algorithm con-
structs the matrices
APo
and B and vector
c
at each iteration
step, while the FBNS algorithm might exploit that they are
constant in a rigid and isoviscous simulation with an incom-
pressible liquid phase. However, the exact details of the
implementation of the original FBNS algorithm are
unknown and the diﬀerence in time scaling cannot be pinned
down rigorously.
Next, it is investigated how combinations of UI or
QUICK discretization and rigid or elastic geometry aﬀect
the performance. The code execution times
tex
are provided
in Table2. The corresponding non-dimensional code execu-
tion times
t
ex
are displayed in Fig.5. Because all curves
have almost the same inclinations in the double logarithmic
diagram, it can be deduced that the performance scaling
of the EHL-FBNS algorithm stays similar in all cases. To
prove that the operating conditions were chosen such that
discretization scheme and elastic model have a noticeable
impact on the results while the performance scaling remains
unchanged, exemplary results are considered in the follow-
ing for the geometry with
Kx1
=8
. The pressure proﬁles of
the UI simulations are visualized in Fig.6 for the rigid (a)
hmin
Uup
x3
x1
hp
pamb pamb
b
a
Lx1
hmax
Uup
x3
x2
x1
Lx2
w
hmin
hmax pamb
(a)
(b)
Fig. 8 Schematic sketch of the convergent slider with rectangular
pocket: a one-dimensional conﬁguration, b two-dimensional geom-
etry with one-dimensional conﬁguration along center line. Adapted
from Bertocchi etal. [9]
Table 3 Summary of the parameters and values used in the EHL-
FBNS simulations of the one-dimensional convergent slider with rec-
tangular pocket
Param. Value Param. Value Param. Value
Uup
1
ms−1
𝜇0
102
Pas
Ulow
0
𝛼B
um
0.5
ms−1
𝛼R
N
x
2
3
ur
1
ms−1
p0,R
Lx1
102
m
pamb
105
Pa
Lx2
(Nx21x2
pcav
0
𝜌0
850
kg m−3
hd
0
phd,SB
pamb
C1
hmax
1.05 ×106
m
phd,WB
pamb
C2
hmin
106
m
phd,EB
pamb
Eup
a
2103
m
phd,NB
pamb
Elow
b
3103
m
𝜃SB
Neumann
𝜈up
Δx1
Lx1∕(Nx11)
𝜃WB
0
𝜈low
Δx2
Δx1
𝜃EB
0
E
hp
106
m
𝜃NB
Neumann
FN,imp
href
hmin
x1,ref
Lx1
𝛼p
0.5
x2,ref
Lx2
𝛼𝜃
0.5
pref
106
Pa
𝜌ref
𝜌0
tref
-
𝜇ref
𝜇0
FN,ref
-
uref
um
Param. Value
N243, 483, 963, 1923
Nx1
81, 161, 321, 641
Fig. 9 Hydrodynamic pressure proﬁle
phd
for UI simulations with dif-
ferent resolutions
N
x
2
N
x
1
in comparison to the analytical solution
derived by Fowell etal. [25]
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Tribology Letters (2022) 70:80
1 3
Page 11 of 25 80
and elastic (b) case. The contour of the regions where the
hydrodynamic pressure
phd
reaches the cavitation pressure
pcav
is visualized by orange lines. The elastic deformation
drastically reduces the resulting pressure proﬁle in compari-
son to the rigid simulation. The results of the same cases but
with the QUICK scheme are shown in Fig.7 for the rigid (a)
and elastic (b) case for comparison. Diﬀerences between the
UI and the QUICK scheme are noticeable.
3.2 Convergent Slider withRectangular Pocket
The next test case is a convergent slider with a single rec-
tangular pocket that introduces discontinuities in the gap
height. For this set-up, a mass-conserving cavitation model
is essential to predict the full-ﬁlm reformulation properly.
The aim of the simulations is to show the eﬀect of the spa-
cial discretization order on the pressure distribution when
gap height discontinuities are present. For this steady case,
the UI scheme eventually results in ﬁrst and the QUICK
scheme in second-order accuracy. The investigation is ﬁrstly
done for a rigid one-dimensional geometry because it has
the analytical solution of Fowell etal. [25] for comparison.
Next, the two-dimensional set-up of Bertocchi etal. [9] is
used on the one hand to demonstrate that the algorithm of
Bertocchi etal. and the EHL-FBNS algorithm give consist-
ent results in the rigid case and on the other hand to show
that the additional consideration of elastic deformations even
at traditional hydrodynamic operating conditions is of great
relevance. A sketch of the one-dimensional conﬁguration
is depicted in Fig.8a while its extension to the two-dimen-
sional geometry is described in Fig.8b.
The analytical solution of a rigid one-dimensional con-
verging slider with rectangular pocket and incompressible
isoviscous liquid phase was derived by Fowell etal. [25] and
was also used for code veriﬁcation by Giacopini etal. [8].
The parameters used in the current study are summarized
in Table3. The one-dimensional geometry was replicated
Fig. 10 Hydrodynamic pressure proﬁle
phd
for QUICK simulations
with diﬀerent resolutions
Nx2
Nx1
in comparison to the analytical
solution derived by Fowell etal. [25]
Table 4 Summary of the parameters and values used in the EHL-
FBNS simulations of the two-dimensional convergent slider with rec-
tangular pocket
Param. Value Param. Value Param. Value
Uup
1
ms−1
𝜇0
102
Pas
N8320
Ulow
0
𝛼B
1.2 108
Pa−1
Nx1
128
um
0.5
ms−1
𝛼R
Nx2
65
ur
1
ms−1
p0,R
L
x
1
2102
m
pamb
105
Pa
pcav
0
𝜌0
850
kg m−3
hd
0
phd,SB
pamb
C1
2.22 109
Pa
hmax
1.1 106
m
phd,WB
pamb
C2
1.66
hmin
106
m
phd,EB
pamb
a
4103
m
phd,NB
pamb
Elow
Eup
b
6
103
m
𝜃SB
0
Δx1
L
x
1∕(N
x
11)
𝜃WB
0
𝜈low
𝜈up
Δx2
Lx2∕(
N
x2
1
)
𝜃EB
0
hp
0.4 10
6
m
𝜃NB
0
FN,imp
href
hmin
x1,ref
L
x
1
𝛼p
0.05
x2,ref