1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Partitioned Fluid-Structure Interaction Techniques Applied to theMixed-Elastohydrodynamic Solution of Dynamically Loaded

Connecting-Rod Big-End Bearings

F.J. Profitoa,b,∗, D.C. Zachariadisa, D. Dinib,∗

aDepartment of Mechanical Engineering, Polytechnic School of the University of Sao Paulo, Sao Paulo, BrazilbDepartment of Mechanical Engineering, Imperial College London, South Kensington Campus, London, UK

Abstract

The present contribution proposes different partitioned techniques, which are commonly used in

fluid-structure interaction (FSI) applications, in the context of tribological simulations of the tran-

sient mixed-elastohydrodynamic problem of dynamically loaded connecting-rod bearings. With the

premise that the FSI framework developed is general, in the current work the fluid flow effects have

been considered through the averaged Reynolds equation by Patir & Cheng and the mass-conserving

p − θ Elrod-Adams cavitation model. The multiphysics simulation framework developed has been

used to simulate the connecting-rod big-end bearings of both heavy-duty diesel and high-speed

motorcycle engines. In the latter case, the influence of polymer concentration in VM-containing

oils with similar HTHS150 values on the bearing power loss is investigated and discussed in details.

Keywords: Conformal EHL, Partitioned FSI techniques, Connecting-rod bearings, Numerical

simulation

2010 MSC: 00-01, 99-00

1. Introduction

The solution of the fluid-structure interaction (FSI) problem established between the hydrody-

namic pressures and solid deformations is crucial for determining the tribological behaviour of EHL

contacts. For conformal contacts often encountered in journal and sliding bearing applications, such

FSI calculations are traditionally addressed by using either the nodal or modal approach [1]. The5

∗Corresponding authorsEmail addresses: fprofito@usp.br (F.J. Profito), dczachar@usp.br (D.C. Zachariadis),

d.dini@imperial.ac.uk (D. Dini)

Preprint submitted to Journal of XXXXX April 27, 2019

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modal (or mode-based) approach was introduced by [2] and is less widespread in the literature. Ac-

cording to this method, the nodal displacements are computed by adopting a linear combination of

particular mode shapes determined from the linear elastic solution of the bearing structure. On the

other hand, the nodal (or node-based) methodology is most predominant in publications involving

journal bearing EHL problems. In the latter case, the nodal displacements are determined directly10

from the linear matrix-vector relationship between the nodal load vector and the compliance ma-

trix of the elastic structure (quasi -static analysis). The computation of the compliance matrix is

usually carried out by applying some model reduction (or condensation) technique to the complete

FEM bearing model [3], from which only the degrees of freedom of the nodes placed on the internal

bearing surface are retained for the EHL solutions.15

Two subclasses of methods, namely indirect or direct methods, can be identified for the node-

based approach. The nodal indirect (or monolithic) methods often employ sophisticated implicit

Newton-Raphson schemes, where the solution step of each Newton iteration is computed by solving

a system of residual equations defined both in terms of the applied external loads and a Taylor series

expansion of Reynolds equation, which thus allows the ready evaluation of the Jacobian matrix [4–20

15]. In other words, in the indirect methods all the equations involved in the EHL modelling are

solved simultaneously. In contrast, the nodal direct (or partitioned) methods are established by

means of direct iterative schemes [16, 17], in which the hydrodynamic and structural problems

are solved separately. In that case, a coupling algorithm is required to incorporate the interaction

between the fluid and structural solvers. The numerical convergence of the direct schemes commonly25

used to solve flexible bearing systems is often difficult, especially when large deformations take

place, hence justifying the predominance of the indirect methods for EHL solutions of highly loaded

bearings.

In this scenario, three partitioned methods [18, 19], namely (i) Fixed Point Gauss-Seidel Method

(PGMF), (ii) Point Gauss-Seidel Method with Aitken Acceleration (PGMA) and (iii) Interface30

Quasi -Newton Method with an approximation for the Inverse of the Jacobian from a Least-Squares

model (IQN-ILS), are analysed in this contribution. The aim is to evaluate and improve the use

of these techniques to provide more robust and stable direct (nodal) solutions particularly applied

to conformal EHL contacts under different operational conditions. Partitioned methods are used

in a wide range of applications of fluid-structure interaction problems, such as in the analysis of35

flutter in wings of aircrafts and blades of turbo-machines, in the complex fluid-structure interaction

2

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of life-saving equipment such as parachutes and air bags, and even in the investigation of blood

flow through arteries and heart valves [19]. The main advantage of the partitioned approach is

the possibility of using optimized codes to solve hydrodynamic equations and structural equations

separately as “black-box” solvers. In other words, each component of the EHL problem can be solved40

with different techniques specialized for each type of equations. This flexibility is not available in the

implicit solutions, where all equations are generally solved in the same solution framework. Thus,

keeping these advantages in mind, the introduction of the aforementioned partitioned coupling

techniques for simulating dynamically loaded tribological systems constitute an important goal of

the present contribution.45

We now turn to recent developments in the area of coupled algorithms to solve lubrication prob-

lems to provide a general overview and place our contribution in the context of existing solutions.

The growth of computer processing power along with the efficient combination of multilevel and

multigrid algorithms and parallel computing have recently allowed the consideration of complex

multiscale tribological phenomena in multiphysics and multibody simulation analysis [20]. From50

an engineering point of view, it should be advantageous to have specialized libraries of models and

dedicated packages, or at least efficient co-simulation platforms, available for commercial multi-

physics software, which take into account the tribological phenomena arising in dry and lubricated

contacts. Commercial multiphysics software usually provide high level interface to describe the

problem, as well as a wide range of pre- and post-processing tools, customized functions and solvers55

that can be used without needing to understand the complexities of numerical implementations.

For journal bearing applications, for instance, various attempts have recently been made to develop

FSI solutions. Ref. [21] has proposed an thermohydrodynamic model with a new mass conserving

cavitation algorithm and implemented this into the general purpose commercial software COMSOL

Multiphysics [22]. Similarly, Ref. [23] has studied the influence of different mass-conserving cav-60

itation modelling approaches on the dynamic stability of a flexible multibody turbocharge rotor

through an explicit co-simulation approach, in which the lubrication problem based on the com-

pressible Reynolds equation was also solved in the software COMSOL Multiphysics [22] and the

turbocharger rotor dynamics was calculated with the commercial multibody dynamic simulation

software MSC ADAMS [24]. Considering conformal EHL problems, Ref. [25] used the COMSOL65

software to solve the time dependent Reynolds equation, equations of motion and structural defor-

mations simultaneously in order to investigate the influence of pad compliance on nonlinear dynamic

3

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characteristics of tilting pad journal bearings. Moreover, Ref. [26] used the same commercial multi-

physics software to build a elastohydrodynamic journal bearing design tool. Furthermore, Ref. [27]

compared two different approaches to calculate the asperity contact pressure for an elastohydrody-70

namic analysis of a conrod small-end/piston pin coupling, and a model validation was performed

considering the commercial software AVL Excite Power Unit [28]. The numerical solution procedure

employed in most of the works mentioned above for solving the FSI problem in case of EHL journal

bearings was based on relatively simple partitioned underrelaxation iterative processes equivalent

to the PGMF method.75

With regard to EHL applications involving concentrated contacts (e.g. line and point con-

tacts), Ref. [29] proposed a multiscale framework for EHL and micro-EHL and Ref. [30] evalu-

ated different computational approaches for modelling elastohydrodynamic lubrication using mul-

tiphysics software. Recognizing the absence of a commercial software package for solving thermo-

elastohydrodynamic lubrication (TEHL) problems embedded in larger multiphysics software, Ref.80

[31] provided detailed guidelines on how to implement a TEHL simulation model in commercial

multiphysics software with particular focus on gear contact applications; this was followed by the

work performed in Ref. [32] that utilised a similar approach to conceive a more generalised mod-

elling framework for lubricated and dry line contacts. The solution of the FSI problem in these

works was essentially based on the development of co-simulation frameworks structured considering85

implementation of a full-system monolithic approach proposed in Refs. [33–35] in the software

COMSOL Multiphysics [22]. Recent attempts have also been made to develop coupled strategies to

solve EHL lubrication problems using finite-volume CFD solvers, which provide the opportunity to

go beyond Reynolds’ approximations and allow tackling particularly complex lubrication problems

(see e.g. Refs. [36, 37]).90

Although only journal bearing conformal contacts are investigated in this work, the coupling

techniques under consideration can be readily extended to non-conformal EHL line and point con-

tacts, as well as being used as alternative strategies for implementing dedicated lubrication solvers

to multiphysics software. The introduction of such methods for solving EHL connecting-rod big-end

bearing problems under static and particular transient conditions has already been published by95

the authors’ in [36, 38] and recently applied to stationary flexible bearings in [39]. Thus, besides

a detailed introduction of the aforementioned partitioned techniques for solving a thorough mixed-

elastohydrodynamic lubrication modelling also proposed in this contribution, the novelty of the

4

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present work is the further development and extension of such FSI techniques for the solution of

problems characterised by transient loading conditions, here evaluated through the EHL simulation100

of connecting-rod big-end bearings of both heavy-duty diesel and high-speed motorcycle engines;

for such class of problems, the abrupt variation of the load magnitude and direction, together with

the time variation of the pressure and film fraction (cavitation) fields and time discretization, all

require a generalization of the solution framework originally proposed in [36]. Furthermore, for the

high-speed motorcycle engine case, the influence of polymer concentration in VM-containing engine105

oils with similar HTHS150 value subject to high shear rate conditions on the bearing power loss is

also investigated and discussed in details.

2. Mathematical Modelling

The complete mathematical modelling adopted in this contribution for describing the mixed-

elastohydrodynamic lubrication regime of conformal journal bearing systems under transient loading110

conditions is outlined in the following sub-sections.

2.1. Fluid Film Lubrication

The fluid film lubrication phenomenon taking place on the bearing interface are mathematically

described through the modified Reynolds equation [40] based on the Patir & Cheng’s average flow

model for mixed lubrication [41, 42]. The choice of this average flow model is simply based on its

popularity; however, incorporation of other average flow models, such as those proposed by [43–48] is

straightforward. Such modified Reynolds equation is derived by locally averaging the lubricant flows

at the microscopic scale for a representative rough bearing cell, thus providing specific flow factors

coefficients that allow the incorporation of the roughness induced flow perturbations effects directly

on the macroscopic scale. Furthermore, the fluid film cavitation is taking into account through

the p − θ Elrod-Adams mass-conserving cavitation model [49, 50], which automatically satisfies

the complementary JFO conditions for mass conservation throughtout the lubricated domain [51–

53]. Accordingly, the transient modified Reynolds equation for a general misaligned journal bearing

system with axial movement, including geometric deviations around the bearing/journal cilindricity

due to superficial defects and/or elastic deformations (EHL), is expressed as follows (see Fig. 1):

5

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Cylindrical geometry

Actual geometry with

superficial defects

𝒀𝑌𝑗

𝒁

𝑍𝑗

𝒔

𝒔

𝑪𝒃′

𝑪𝒋′𝑪𝒃

𝑪𝒋

𝑷𝒃

𝑷𝒋

Cross

Section (s)

𝒀

𝑿

𝑌𝑏𝑟

𝑋𝑏𝑟

𝑷𝒃

𝑷𝒋

𝜽𝒃𝝎𝒃

𝜶𝒃

𝒆

𝜹

𝒙

𝒚 𝑥𝑗𝑙

𝑦𝑗𝑙

𝑪𝒋

𝑪𝒃

Centre Line

𝝎𝒋

𝑿𝒓

𝒀𝒓

𝑨𝒓

Figure 1: Main coordinate systems and the geometric and kinematic characteristics of the journal bearing modelproposed in the present work.

∂

∂x

(φpx

ρH3

12µ

∂ph∂x

)+

∂

∂z

(φpz

ρH3

12µ

∂ph∂z

)︸ ︷︷ ︸

Poiseuille Flow

=∂

∂x

[ρθ(UHφc − ∆Uσφsx

)]+

∂

∂z

[ρθ(WHφc − ∆Wσφsz

)]︸ ︷︷ ︸

Couette Flow

+

ρθ

[Rωj

(φcj

∂δnj∂x

)−Rωb

(φcb

∂δnb∂x

)]+

[Wjφcj

∂(δnj − h

)∂z

−Wb

(φcb

∂δnb∂z

)]︸ ︷︷ ︸

Translation Squeeze

+

ρθ

[R(ωjδ

tj − ωbδ

tb

)+ φc

H

∂H

∂t

]︸ ︷︷ ︸

Normal Squeeze

+ (Hφc)∂ (ρθ)

∂t︸ ︷︷ ︸Expansion

(1)

with the complementary boundary conditions for cavitation:

(ph − pcav) (1− θ) = 0→

ph > pcav → θ = 1 (pressured regions)

ph = pcav → 0 ≤ θ < 1 (cavitation regions)

(2)

In the above equations, ph(x, z, t) is the hydrodynamic pressure, θ(x, z, t) the lubricant film fraction

(cavitation), pcav the limit cavitation pressure, H(x, z, t) the oil film thickness, and µ(x, z, t) and

ρ(x, z, t) the lubricant dynamic viscosity and density, respectively. Moreover, U = R(ωj+ωb

2

),

6

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∆U = R(ωj−ωb

2

), W =

(Wj+Wb

2

)and ∆W =

(Wj−Wb

2

)are, correspondingly, the mean and

relative velocities along the bearing circumferential (x = Rθb) and axial (z) directions. Furthermore,

R is the nominal bearing radius, and (ωj ,Wj) and (ωb,Wb) are the respective rotational speeds

and axial velocities of the journal and bearing components. The coefficients φpx,z and φsx,z are

the respective pressure and shear flow factors, σ the combined standard deviation of the surface

roughness, and φc = HTH , φc

H= ∂HT

∂H and φc(j,b) =∂δn(j,b)T∂δn

(j,b)are the contact factors defined according

to [54, 55], respectively; the quantities with overbars appearing in the definitions of the contact

factors are the corresponding average heights. Eq. 1 is a general form of Reynolds equation for

journal bearing applications, in which shape deviations and surface deformations, mass-conserving

cavitation and mixed lubrication are contemplated, along with the rotational and axial movements

of both journal and bearing parts. Notice that the translation squeeze term usually cancelled out

from Reynolds equation in typical journal bearing models may modify the load carrying capacity

and frictional torque in the presence of local geometric deviations [56, 57]. Moreover, if the shaft is

assumed rigid and the bearing stationary, Eq. 1 is reduced to the conventional Reynolds equation

commonly used to solve EHL journal bearing problems. According to Fig. 1, the expressions for the

geometry and kinematics of the lubricant film thickness described in the local bearing coordinate

system xyz are given as:

H (x, z, t) = c− (Yr −Arz) cos θb + (Xr +Brz) sin θb︸ ︷︷ ︸h

+(δnb − δnj

)(3a)

∂H (x, z, t)

∂t=(−Yr + Arz

)cos θb +

(Xr + Brz

)sin θb︸ ︷︷ ︸

∂h∂t

+(δnb − δnj

)(3b)

where c is the bearing nominal clearance, (Xr, Yr) and (Ar, Br) the rigid body displacements and

misalignments of the journal relative to the bearing along the X and Y axes, respectively, δnj,b are

the normal geometric deviations of the journal/bearing surfaces in the normal (radial) direction115

and h is the lubricant film thickness for a perfectly cylindrical bearing in the absence of geometric

deviations. Note that in the above equations the subscripts j and b denote journal and bearing,

and the over-dots designates time derivatives.

The expressions of the lubricant shear stresses acting on the journal surface needed to calculate

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the hydrodynamic friction forces can be written as:

τhx =H

2

∂ph

∂xφfpx + µθ

(ωj − ωbH

)R (φf + φfsx) (4a)

τhz =H

2

∂ph

∂zφfpz + µθ

(Wj −Wb

H

)(φf + φfsz ) (4b)

where φfpx,z and φfsx,z are the shear pressure and shear stress flow factors, respectively, and φfs

friction factor defined in the Patir & Cheng’s theory [41, 42].120

2.2. Lubricant Rheology

The lubricant rheological properties (µ and ρ) are strongly affected by the temperature, pressure

and shear rate conditions of the contact interface. As the thermal effects have been neglected in this

work, only the isothermal density-pressure, viscosity-pressure and viscosity-shear-thinning effects

will be considered for the corrections of the lubricant properties. Here some of the mainstream

rheological constitutive relationships will be used, with the caveat that other laws can be readily

implemented. Accordingly, the viscous-pressure relationship is calculated through the traditional

isothermal Roelands equation [58]:

µp = µR

e

[ln(µ0µR

)(1+

phpR

)Z](5)

where µR

and pR

are constants having the values of 6.31× 10=5 Pa s and 196 MPa, respectively, and

µ0 is the low pressure dynamic viscosity at a given temperature [58]. The exponent Z is associated

with the specific lubricant, typically around 0.6.

Under high shear rate conditions not rarely found in lubricated contacts, the linear relationship

between shear stress and shear rate (Newtonian fluid) is not totally valid so that the lubricant

starts to behave as a non-Newtonian fluid. The non-Newtonian characteristic typically observed

in lubricant oils is denominated shear-thinning. The multi-viscous lubricants widely used in the

lubrication of internal combustion engines are particularly susceptible to experience shear-thinning

effects, especially due to the considerable amount of polymeric additives existent in their compo-

sitions. In the present contribution, the following power-law based Carreau-Yasuda equation is

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adopted to describe shear-thinning fluids [58]:

µ = µ∞ +(µp− µ∞

)[1 + (λγe)

a]n−1a (6)

where a and n (n < 1) are constants, λ the characteristic relaxation time of the polymer, µp

the low

shear Newtonian viscosity at a given temperature and pressure, and µ∞Pis the lubricant dynamic

viscosity at infinite shear rate. Such rheological parameters strongly depend on the lubricant

formulation and are obtained from measurements conducted with precise high-shear viscometers.

The equivalent shear rate γe corresponds to the scalar magnitude of the second invariant of the

shear rate tensor, which for fluid film lubrication assumes the following simplified expression [59]:

γe =

√(γx)

2+ (γz)

2(7)

Regarding the density-pressure correction, the well-known Dowson-Higginson equation is adopted

[60]:

ρ = ρ0

(C1 + C2 phC1 + ph

)(8)

where ρ0 is the reference lubricant density at atmospheric pressure and C1 and C2 are constants125

with typical values of 0.59 GPa and 1.34, respectively. Notice that in this case the fluid pressure

has to be in GPa.

2.3. Bearing Structure

As the contact arrangement established in journal bearing systems is typically conformal, the

surfaces deformation induced by the hydrodynamic pressures under EHL conditions are affected

essentially by the overall bearing flexibility [1]. Since only the solid displacements of the contact

interface are important for the fluid pressure calculations, it is convenient to use FEM substructure

(or superelement/condensation) techniques [61] to reduce the entire structural FEM model of the

bearing system to an equivalent one that retains only the degrees of freedom of the nodes located

on the bearing surfaces. In this case, the reduced FEM model of the bearing system, including its

structural dynamics and distributed inertia, can be written as:

[Mr] ~δ + [Br] ~δ + [Kr]~δ = [A] ~ph (9)

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where [Mr], [Br] and [Kr] are the respective reduced mass, damping and stiffness matrices, and

the vectors ~δ, ~δ and ~δ are the nodal surfaces displacements, velocities and accelerations represented

in the local bearing coordinates system (tangential, normal and axial directions). Furthermore, ~ph

is the vector of nodal hydrodynamic pressures and [A] is the area matrix that converts the fluid

pressures into distributed loads on the contact surfaces. For the particular case of quasi -static

solutions, the structural dynamics and distributed inertia are neglected, thus Eq. 9 is simplified as

follows:

~δ = [Lr] ~pH (10)

where Lr is the so-called flexibility matrix of the reduced system.

2.4. Asperity Contact130

A plethora of statistical [62–67] and deterministic [68–73] methods are available to describe the

interactions at asperity level and all of them can be considered for implementation in the proposed

FSI framework. In the present work, the asperity contact pressures that arise under boundary and

mixed lubrication conditions are calculated through the statistical-based Greenwood & Tripp model

for rough contacts [62]:

pc(H)

=

16π√

215 E∗

(η2sβ

3/2s σ

5/2s

)F5/2

(H)

, pc ≤ HV

HV , pc > HV

(11)

where pc (x, z, t) is the rough contact pressure, H (x, z, t) = H−Zsσs

the dimensionless separation

distance of the surfaces, E∗ =(

1−υ21

E1+

1−υ22

E2

)−1

the combined elastic modulus and HV the Vickers’

hardness of the softer material that allows the extension of the originally elastic Greenwood &

Tripp model to elasto-perfectly plastic contacts. The asperity contact pressures mainly depend

upon the statistical distribution and shape of the asperity heights, which are represented in Eq.135

11 in terms of the contact parameters Zs (mean asperity height), σs (standard deviation of the

asperity heights), βs (mean asperity radius od curvature) and ηs (asperity density). The function

F5/2

(H), which represents the Gaussian distribution of the asperity heights, is approximated by a

polynomial function whose coefficients can be found in [74].

The friction forces produced by the asperity interactions will be here calculated based on the

Coulomb-Amontons’ Laws applied to dry solid contacts. Therefore, the contact shear stresses can

10

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be defined mathematically as:

τcx,z(H)

= µbpc(H)

(12)

where µb is the boundary coefficient of friction often obtained from experimental results. Similarly140

to the hydrodynamic shear stresses, once the contact shear stress is determined the total associated

friction forces can be easily calculated by numerical integration over the contact domain.

2.5. Rigid Body Motion of the Bearing System

By applying Newton’s second law of motion and assuming that only the relative rigid body

displacements and misalignments of small magnitude are considered in the modelling, i.e. the

eventual gross motion of the system is ignored in the dynamic equations and ultimately incorporated

into the analysis as (external) inertial loads, the rigid body dynamic equations can be expressed

with respect to the bearing coordinate system (XY Z):

[M ] ~q (t) + ~F(~q, ~q, t

)= ~Fext (t) (13)

where [M ] is the inertia matrix, ~F is the vector of internal loads associated with the hydrodynamic

and asperity contact pressures and ~Fext the vector of the external loads acting upon the system.

Furthermore, ~q, ~q and ~q are the vectors of the rigid body displacements, velocities and accelerations

of the journal relative to the bearing, respectively. The components of such matrix and vector

quantities are defined as follows:

[M ] =

m 0 0 0

0 m 0 0

0 0 JX 0

0 0 0 JY

~q =

Xr

Yr

Ar

Br

~q =

Xr

Yr

Ar

Br

~q =

Xr

Yr

Ar

Br

(14a)

~F =

WXh +WX

c

WYh +WY

c

MXh +MX

c

MYh +MY

c

~Fext =

FXext

FYext

MXext

MYext

(14b)

11

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where m is the equivalent journal mass and JX and JY are the correspondent moments of inertia

of the journal described in the bearing coordinate system. Moreover, W and M denote the result-145

ing forces and moments obtained by integrating the hydrodynamic and asperity contact pressures

throughout the lubricated domain. For the particular case of quasi -static solutions, the inertia

effects associated with the gross rigid body motions are assumed to be implicitly accounted into

the external loads, thus the inertia term of Eq. 13 can be neglected and the rigid body equations

are reduced to ~F(~q, ~q, t

)− ~Fext (t) = ~0.150

2.6. Fully Coupled System of Nonlinear Equations

In this section, the fully coupled system of nonlinear equations that comprises the entire math-

ematical modelling described in the previous sections for journal bearing systems operating under

mixed-EHL conditions is summarized in terms of the following solver operators:

[~q, ~q, ph, θ, γe, pc

]= E

(~δ, ~δ, ρ, µ, ~F ext, t

)(15a)

[ρ] = Lρ (ph) (15b)

[µ] = Lµ (ph, γe) (15c)[~δ, ~δ

]= S (ph) (15d)

The operators defined in the above expressions are associated with the governing equations of

each particular physical phenomenon of the system and must be solved simultaneously during the

problem solution. The operator E represents the complete solution of the system “instantaneous

equilibrium” for the external load condition ~Fext (t). This solver is composed of three intermediate155

solvers that embrace the calculations of the hydrodynamic (Eq. 1) and asperity contact (Eq. 11)

pressures, as well as the solution of the rigid body dynamic equations (Eq. 13). During the solution

of E, the rheological (ρ, µ) and structural (~δ, ~δ) variables are hold fixed, so that only the instanta-

neous rigid body kinematics (~q, ~q) and the associated hydrodynamic pressure (ph) and film fraction

(θ) distributions are determined by solving the rigid body dynamic equations (Eq. 13). Afterwards,160

from the existing results, both the lubricant rheological variables and structural displacements are

updated through the solvers Lρ (density-pressure operator), Lµ (viscosity-pressure-shear-thinning

operator) and S (structural operator), respectively.

12

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3. Solution Framework

In this section, the solution framework developed for solving the fully coupled system of nonlinear165

equations summarized in the preceding section will be exposed in details. Special attention is

dispensed to describe the partitioned FSI methods adopted to solve the EHL problem. It is worth

mentioning that the use of such coupling techniques to calculate the highly nonlinear fluid-structure

interaction problem established on journal bearing systems subjected to transient, high loading

conditions represents the major contribution of the present work. Furthermore, regarding the170

solution of the hydrodynamic problem, the reader is referred to Ref. [75] for an extensive description

of the general finite volume discretization scheme adopted here to solve the Reynolds equation

(Eq. 1). In this same reference, the hydrodynamic solver was also fully validated against other

codes and semi-analytical solutions for simulations of stationary loaded bearings. Moreover, the

asperity contact and hydrodynamic parts of the current solution framework, including the cavitation175

and rheological constitutive models, have been used and validated by the authors in previous

publications [74, 76, 77]. Finally, the algorithm and the deformation solver has also been validated

using full finite element deformations calculations and comparisons with the respective reduced order

FEM model. More details about the structural solution and interpolations on the fluid-structure

interface can be found in Ref. [36].180

3.1. Time Discretization

The time discretization of the solver operators defined in Section 2.6 is based on an second-

order accuracy, four-step linear BDF-like scheme proposed by [78]. The scheme is fully implicit and

derived in terms of parameters that control the stability and dissipative properties of the temporal

solution. Its convergence characteristics have been demonstrated to be considerably stable for

nonlinear dynamic applications where the responses are controlled by a relatively small number of

low frequency modes, as those encountered in the EHL problems considered in the present work.

Furthermore, due to the multistep aspect of the algorithm, it is also possible to determine more

accurate extrapolation formulas for the initial guesses for each time step that considerably speed-up

the convergence rates. To the best of the authors’ knowledge, that is the first time that such a four-

order implicit scheme has been employed in the context of dynamic EHL solutions. Accordingly,

13

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

the time discretization of each solver operator can be expressed as:

[~q n, pnh , γne ] = E

(~δ n, ρn, µn, ~F

n

ext

)(16a)[

~δ n]

= S (pnh ) (16b)

[ρn] = Lρ (pnh ) (16c)

[µn] = Lµ (pnh , γne ) (16d)

where the superscript n denotes time iteration and tn = n∆t. Notice that as the adopted time

discretization scheme is fully implicit, the rigid body and structural velocities and accelerations can

be determined in terms of the quantities calculated in the previous time steps. Hence, the variables

~qn, ~q

n, ~δ

n

and ~δn

can be omitted in the above equations without loss of generality, retaining185

explicitly only the variables directly involved in the coupled EHL solution.

3.2. Solution of the Rigid Body Dynamic Equations: Advanced Newton-Raphson Method

The solution of the rigid body dynamic equations (Eq. 13) is performed during the calculation

of the solver operator E (see Section 2.6) and is based on the same implicit time discretization

scheme described above. Thus, at each time step n the discrete form of Eq. 13 can be expressed in

terms of ~q n as:

D (~q n) = M~q n + ~Fc

(~q n, ~q n, tn

)− ~Fext (tn) = ~0 (17)

where the auxiliary operator D represents the set of nonlinear equations that has to be solved

at each time step to determine the “instantaneous equilibrium” of the bearing system. This set

of equations is solved by using the advanced Newton-Raphson method with Armijo’s line search

technique to improve the solution step size at each iteration [79]. Accordingly, the Newton sequence

derived from the local linear model of D (~q n) is

~q n,w = ~q n,w−1 + λ′w~sw D′ (~q n,w−1)~sw = −D

(~q n,w−1

)(18)

where D′ is the Jacobian matrix, ~sw the Newton step and λ′w

the step length. For each Newton

iteration w, the Jacobian matrix is approximated by finite differences, while λ′w

is selected to

guarantee the decrease of ‖D‖ in accordance with the Armijo’s line search method. The iterative190

process terminates whenever the norm of D is less than a relative error tolerance εNR, which

14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

depends on the magnitude of the external loads, i.e. ‖D (~q n,w)‖ ≤ εNR

∥∥∥~Fext (tn)∥∥∥. The initial

iterate (w = 0) for a given time step n can be estimated from the previous converged solutions

by extrapolating the expressions of the BDF scheme. As already mentioned, as a high-order time

discretization scheme is being considered, such initial iterate tends to provide favourable starting195

guesses which contribute to the speeding-up of the iterative calculations.

3.3. Solution of the Coupled Mixed-EHL Equations: Partitioned FSI Methods

In this section, the partitioned methods considered in the present contribution is described in

details. Although only conformal contacts are here investigated, the proposed coupling methods

may well have their use extended to EHL non-conformal contacts, such as in applications involving200

line and point contact configurations. More in-depth explanations concerning the convergence of

discussed partitioned techniques can be found in the comprehensive references [18, 19]. It should

be noted that the superscript n associated with the time discretization is omitted in the following

derivations for the sake of notation clarity. Furthermore, the superscript k is introduced to indicate

the coupling iteration within time step tn and all the coupled field variables (ph, γe, ρ, µ) will205

be denoted in terms of vectors (~ph, ~γe, ~ρ, ~µ) representing their respective nodal values on the

hydrodynamic mesh.

3.3.1. Fixed Point Gauss-Seidel Method (PGMF)

The Fixed Point Gauss-Seidel Method (PGMF) is the simplest iterative technique employed

for the direct solution of EHL conformal contacts. In this method, at each coupling iteration k,

the vectors of the structural displacements and rheological properties are updated according to the

residual vectors (~rδ,ρ,µk) computed in the previous iteration:

~δ k = ~δ k−1 + ωδ~rδk (19a)

~ρ k = ~ρ k−1 + ωρ~rρk (19b)

~µ k = ~µ k−1 + ωµ~rµk (19c)

The fixed under-relaxation parameters ωδ,ρ,µ, with 0 < ωδ,ρ,µ ≤ 1, are introduced in order to

accommodate abrupt changes in the variables that might cause difficulties in the numerical solution

convergence. The optimum values of ωδ,ρ,µ are problem dependent and have to be determined

15

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empirically for each particular simulation case. The computation of the residual vectors is based

on the intermediate solutions obtained in the previous iteration. Mathematically:

~rδk = ~δ k − ~δ k−1 ~δ k = S E

(~δ k−1, ~ρ k−1, ~µ k−1, ~q k

)(20a)

~rρk = ~ρ k − ~ρ k−1 ~ρ k = Lρ E

(~δ k−1, ~ρ k−1, ~µ k−1, ~q k

)(20b)

~rµk = ~µ k − ~µ k−1 ~µ k = Lµ E

(~δ k−1, ~ρ k−1, ~µ k−1, ~q k

)(20c)

where the over-tilde represents intermediate solutions and the notation ‘’ indicates function com-

position, in which the results from the E solver are given as input to the other operators. Notice210

that E is shown in terms of ~q k, the converged rigid body position obtained by solving the “in-

stantaneous equilibrium” equations (see Section 3.2) for ~F ext and with the rheological properties

and structural displacements evaluated in the previous iteration (k − 1). The solution of the equi-

librium equations also provides the values of the hydrodynamic pressures (~phk) and shear rates

( ~γek) needed to compute the intermediate structural displacements (

~δ k), lubricant density (~ρ k)215

and viscosity (~µ k).

3.3.2. Point Gauss-Seidel Method with Aitken Acceleration (PGMA)

The convergence of the PGMF method can be improved by introducing the so-called Aitken

relaxation scheme [80–83]. This technique is based on “dynamically” varying the scalar under-

relaxation parameters ωδ,ρ,µ (which were previously assumed constant in the fixed-point method)

within a time step. In this case, the Gauss-Seidel iteration can be rewritten as follows:

~δ k = ~δ k−1 + ωkδ~rδk (21a)

~ρ k = ~ρ k−1 + ωkρ~rρk (21b)

~µ k = ~µ k−1 + ωkµ~rµk (21c)

Notice that iteration index k also appears in the relaxation parameters, since their values are

allowed to change in every coupling iteration. The computation of the residual vectors remains the

same as those defined for the PGMF method (Eqs. 20). As for the values of the relaxation factors,

16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

they are updated according to the residues calculated in the previous iteration as:

ωδ,ρ,µk = ωδ,ρ,µ

k−1

(~rδ,ρ,µ

k−1)T·(~rδ,ρ,µ

k−1 − ~rδ,ρ,µk)

(~rδ,ρ,µ

k − ~rδ,ρ,µk−1)T·(~rδ,ρ,µ

k − ~rδ,ρ,µk−1) (22)

where superscript T denotes transpose vector and the symbol ‘·’ means dot product. In order to

ensure that 0 < ωδ,ρ,µ ≤ 1 (under-relaxation) and to avoid either negative or very small values, the

following restrictions are imposed on the magnitude of the relaxation parameters:220

ωδ,ρ,µk ← min

[max

(ωδ,ρ,µ

k, ωmin), ωmax

](23)

The limit values of ωmin = 0.001 and ωmax = 1 are adopted throughout the thesis. Moreover,

the first relaxation in a time step is executed with the relaxation factor from the end of the previous

time step.

3.3.3. Interface Quasi-Newton Method (IQN-ILS)

The Gauss-Seidel based methods previously described are prone to numerical instabilities or low225

convergence rates, especially when large elastic deformations occur [6]. In this section, a method

expected to be more robust is proposed, in which the inverse of the Jacobian matrices of the complete

nonlinear system of equations are successively approximated during the iterative procedure. The

present derivation evolves from the method detailed in reference [19].

Initially, the full EHL problem (Eqs. 16) is reformulated as a set of nonlinear equations, as

follows:

E(~δ, ~ρ, ~µ, ~F ext

)−[~q, ~ph, ~γe

]= ~0 (24a)

S (~ph)− ~δ = ~0 (24b)

Lρ (~ph)− ~ρ = ~0 (24c)

Lµ

(~ph, ~γe

)− ~µ = ~0 (24d)

Considering the same function composition used in Eqs. 20, where the outputs of the E solver

17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

are given as input to the other operators, the above equations can be restated as:

S E(~δ, ~ρ, ~µ, ~q

)− ~δ = ~0 (25a)

Lρ E(~δ, ~ρ, ~µ, ~q

)− ~ρ = ~0 (25b)

Lµ E(~δ, ~ρ, ~µ, ~q

)− ~µ = ~0 (25c)

Furthermore, the introduction of the residual operators

<δ(~δ)

= S E(~δ, ~ρ, ~µ, ~q

)− ~δ = ~0 (26a)

<ρ (~ρ) = Lρ E(~δ, ~ρ, ~µ, ~q

)− ~ρ = ~0 (26b)

<µ (~µ) = Lµ E(~δ, ~ρ, ~µ, ~q

)− ~µ = ~0 (26c)

yields compact equations for three unknown variables, namely the structural displacements (~δ)230

and the lubricant density (~ρ) and viscosity (~µ); such unknown quantities, in turn, are implicitly

dependent on the hydrodynamic pressure (~ph) and shear rate ( ~γe) fields.

The above defined residuals can be solved by using the well-known Newton iterative scheme:

~δ k = ~δ k−1 + ∆~δ k[<′δk−1]

∆~δ k = −~rδk (27a)

~ρ k = ~ρ k−1 + ∆~ρ k[<′ρk−1]

∆~ρ k = −~rρk (27b)

~µ k = ~µ k−1 + ∆~µ k[<′µk−1]

∆~µ k = −~rµk (27c)

with residuals computed as:

~rδk = <δ

(~δ k−1

)= ~δ k − ~δ k−1 ~δ k = S E

(~δ k−1, ~ρ k−1, ~µ k−1, ~q k

)(28a)

~rρk = <ρ

(~ρ k−1

)= ~ρ k − ~ρ k−1 ~ρ k = Lρ E

(~δ k−1, ~ρ k−1, ~µ k−1, ~q k

)(28b)

~rµk = <µ

(~µ k−1

)= ~µ k − ~µ k−1 ~µ k = Lµ E

(~δ k−1, ~ρ k−1, ~µ k−1, ~q k

)(28c)

In the above expressions,[<′δ,ρ,µ

k−1]

are the Jacobian matrices of <δ,ρ,µ evaluated at ~δ k−1,

~ρ k−1 and ~µ k−1, respectively, all at time level n. Again, notice that E is shown in terms of ~q k, the

converged rigid body position obtained by solving the “instantaneous equilibrium” equations (see235

18

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Section 3.2) for ~Fext and with the rheological properties and structural displacements evaluated in

the previous iteration (k − 1).

In partitioned solutions, the exact Jacobian matrix is unknown a priori since all the operators

involved are assumed to be “black-box” solvers. Although the number of degrees of freedom on the

contact interface is smaller than in the entire bearing structure (e.g. substructure techniques, see240

Section 2.3), Jacobian matrices <′δ,ρ,µ are normally dense. Consequently, the solution of the linear

systems in Eqs. 27 at each iteration would still represent a drawback in terms of computational

efforts, particularly in problems with a large number of retained nodes on the bearing surfaces.

A possible way to circumvent the issue of the inexistence of the exact Jacobian matrix is the use

of some approximation technique. Quasi -Newton iterations could be used in such cases, but the

direct solution of the linear systems would still have to be computed. However, one notices that

the approximation of the Jacobian matrix can in fact be suppressed, in the sense that it is more

advantageous to approximate the inverse of the Jacobian directly. In this case, the Quasi -Newton

iterations can be rewritten as:

~δ k = ~δ k−1 +[

<′δk−1]−1 (

−~rδk)

(29a)

~ρ k = ~ρ k−1 +[

<′ρk−1]−1 (

−~rρk)

(29b)

~µ k = ~µ k−1 +[

<′µk−1]−1 (

−~rµk)

(29c)

where[

<′δ,ρ,µ

k−1]−1

indicates the approximation for the inverse of the Jacobian matrices. Addi-

tionally, it can also be observed in Eqs. 29 that such approximation does not need to be performed245

explicitly; only a procedure to calculate the matrix-vector products[

<′δ,ρ,µ

k−1]−1 (

~rδ,ρ,µk)

is

sufficient.

The main purpose of the coupling method presented in this section is essentially to conceive an

ingenious and efficient approximation solution for the matrix-vector products of Eqs. 29. This intent

is accomplished with a formulation that progressively reconstructs and improves the solution of the

matrix-vector products by using the results calculated in the previous Quasi -Newton iterations.

These pieces of information are consecutively stored as the columns in the following auxiliary

19

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

matrices:

[V kδ

]=[

∆~rδk−1 ∆~rδ

k−2 · · · ∆~rδ0] [

W kδ

]=[

∆~δ k−1 ∆~δ k−2 · · · ∆~δ 0]

(30a)[V kρ

]=[

∆~rρk−1 ∆~rρ

k−2 · · · ∆~rρ0] [

W kρ

]=[

∆~ρ k−1 ∆~ρ k−2 · · · ∆~ρ 0]

(30b)[V kµ

]=[

∆~rµk−1 ∆~rµ

k−2 · · · ∆~rµ0] [

W kµ

]=[

∆~µ k−1 ∆~µ k−2 · · · ∆~µ 0]

(30c)

The information from previous time steps can be reused due to the similarity among their

results. The matrices V δ,ρ,µk

and W δ,ρ,µk

are then combined with those from the previous time

steps, q, (if, of course, at least q time steps have already been computed), providing the following

full auxiliary matrices (attention should be paid to the different superscripts for implementation):

[Vδ

k]

=[V kδ Vδ

n−1 · · · Vδn−q

] [Wδ

k]

=[W kδ Wδ

n−1 · · · Wδn−q

](31a)[

Vρk]

=[V kρ Vρ

n−1 · · · Vρn−q

] [Wρ

k]

=[W kρ Wρ

n−1 · · · Wρn−q

](31b)[

Vµk]

=[V kµ Vµ

n−1 · · · Vµn−q

] [Wµ

k]

=[W kµ Wµ

n−1 · · · Wµn−q

](31c)

The consideration of information from previous time steps substantially accelerates the conver-

gence of the coupling iterations. However, if results from too many times steps are reused, the

convergence can decay as the information from the time step (n − q) might no longer be relevant250

in time step n. The optimum value of q is problem-dependent but the convergence does not vary

significantly near the optimum; the value of q = 5 has shown to be effective for the EHL solutions

presented later in this article. Notice that, since the matrices[Vδ,ρ,µ

k]

and[Wδ,ρ,µ

k]

contain at

least one column, simple relaxation has to be performed only in the beginning of the first time step

if information from the previous time steps is reused (q > 0); if this is not the case (q = 0), such255

relaxation has to be done at the beginning of every time step.

In Eqs. 30, the differences among the vectors from the coupling iterations k and (k − 1) are

defined as (see Eqs. 28):

∆~rδk−1 = ~rδ

k − ~rδk−1 ∆~δ k−1 = ~δ k − ~δ k−1 (32a)

∆~rρk−1 = ~rρ

k − ~rρk−1 ∆~ρ k−1 = ~ρ k − ~ρ k−1 (32b)

∆~rµk−1 = ~rµ

k − ~rµk−1 ∆~µ k−1 = ~µ k − ~µ k−1 (32c)

20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Moreover, the difference between the desired (converged) and current residuals, i.e. ∆~rδ,ρ,µ =

~0− ~rδ,ρ,µk, can be approximated as a linear combination of the previous differences:

∆~rδ ≈[Vδ

k]~cδk →

[Vδ

k]~cδk = −~rδk (33a)

∆~rρ ≈[Vρ

k]~cρ

k →[Vρ

k]~cρ

k = −~rρk (33b)

∆~rµ ≈[Vµ

k]~cµ

k →[Vµ

k]~cµ

k= −~rµk (33c)

where ~cδ,ρ,µk are vectors with the coefficients of the linear approximation. As the number of

columns of matrices[Vδ,ρ,µ

k]

is k ≤ NP , where NP is the number of nodes of the hydrodynamic

mesh, the system is overdetermined for ~cδ,ρ,µk and hence has to be calculated in least-squares sense.

The solution of the posed least-squares problem can be accomplished by using the simplest normal260

decomposition of[Vδ,ρ,µ

k]. However, for cases involving a high number of iterations, a more sta-

ble economy-size QR-decomposition method is recommended. Thus, for instance, by adopting the

Householder transformation [19], such matrices are decomposed in[Vδ,ρ,µ

k]

=[Qδ,ρ,µ

k] [Rδ,ρ,µ

k],

with[Qδ,ρ,µ

k]

being orthogonal matrices and[Rδ,ρ,µ

k]

upper triangular matrices. In this man-

ner, Eqs. 33 may be calculated for the coefficients vectors by solving the triangular systems265 [Rδ,ρ,µ

k]~cδ,ρ,µ

k =[Qδ,ρ,µ

k]T (−~rδ,ρ,µk

).

Analogously, the differences of the intermediate variables that correspond to ∆~rδ,ρ,µ can be also

approximated as a linear combination, giving:

∆~δ ≈

[Wδ

k]~cδk (34a)

∆~ρ ≈[Wρ

k]~cρ

k (34b)

∆~µ ≈[Wµ

k]~cµ

k (34c)

From Eqs. 28, the following combination of approximations can be established:

∆~rδ ≈ ∆~δ −∆~δ (35a)

∆~rρ ≈ ∆~ρ−∆~ρ (35b)

∆~rµ ≈ ∆~µ−∆~µ (35c)

21

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Substituting Eqs. 34 into Eqs. 35 and assuming again that ∆~rδ,ρ,µ = ~0− ~rδ,ρ,µk, one has:

∆~δ =[Wδ

k]~cδk + ∆~rδ

k (36a)

∆~ρ =[Wρ

k]~cρ

k + ∆~rρk (36b)

∆~µ =[Wµ

k]~cµ

k + ∆~rµk (36c)

Finally, since ~cδ,ρ,µk are functions of ~rδ,ρ,µ

k (see Eqs. 33), Eqs. 36 can be interpreted as an

approximation of the products between the inverse of the Jacobian matrices and the associated

residual vectors (see Eqs. 29). Mathematically:

∆~δ =[

<′δk−1]−1 (

−~rδk)

=[Wδ

k]~cδk + ~rδ

k ≡ ∆~δ k (37a)

∆~ρ =[

<′ρk−1]−1 (

−~rρk)

=[Wρ

k]~cρ

k + ~rρk ≡ ∆~ρ k (37b)

∆~µ =[

<′µk−1]−1 (

−~rµk)

=[Wµ

k]~cµ

k + ~rµk ≡ ∆~µ k (37c)

3.3.4. Implementation Details of the Partitioned FSI Methods

The iterative process established by each partitioned method within a given time step (Eqs. 19,

Eqs. 21 or Eqs. 27) has to be computed until a convergence criterion is reached. The criterion

adopted in this thesis is based on the definition of a relative error (eEHLk) that embraces all the

solution variables, so that the calculation process is halted whenever the combined error is less than

a limit value (εEHL). Mathematically:

[~ph

k,~δ k, ~ρ k, ~µ k]→[~ph,~δ, ~ρ, ~µ

]nif eEHL

k ≤ εEHL (38)

with

eEHLk =

∥∥∥~pH

k − ~phk−1∥∥∥∥∥∥~phk∥∥∥+

∥∥∥~δ k − ~δ k−1

∥∥∥∥∥∥~δ k∥∥∥+

(∥∥~ρ k − ~ρ k−1∥∥

‖~ρ k‖

)+

∥∥~µ k − ~µ k−1∥∥∥∥∥~µ k

∥∥∥ (39)

22

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For the calculation of the residual vectors at each coupling iteration (Eqs. 20 and 28), the

bearing “instantaneous equilibrium” position ~q k has to be determined for the current external load

~Fext and the rheological properties and structural displacements obtained in the previous iteration.270

This calculation is undertaken by solving operator E, which in turn also provides the updated

hydrodynamic pressures ( ~phk) and shear rates ( ~γe

k) values needed for the next computations of

the coupled variables (~δ k, ~ρ k, ~µ k). This also explains the reason why ~q k appears within E in

the function compositions defined for the residual equations. Furthermore, all coupling algorithms

begin each time step with suitable extrapolations for both structural displacements and rigid body275

positions; all the other coupled variables are not extrapolated at all, thus only their values at the

previous time step are used to initiate the EHL iterations.

4. Applications

In this section, the complete mathematical modelling and numerical solution framework pro-

posed in this work are evaluated through two case studies involving dynamically loaded connecting-280

rod big-end bearings of a heavy-duty diesel and a high-speed motorcycle internal combustion en-

gines.

4.1. Dynamically Loaded Connecting-Rod Big-End Bearing of a Heavy-Duty Diesel Engine

The present application case is devoted to evaluate the tribological performance of the connecting-

rod big-end bearing of a heavy-duty diesel (HDD) engine subjected to extremely high combustion285

loads. Furthermore, the numerical simulations of such bearing application is prone to present con-

vergence issues due to the high concentrated fluid pressures and relatively large solid deformations

involved, thus providing appropriate challenges for the evaluation of the robustness of the new

partitioned coupling techniques contemplated in this work.

4.1.1. Input Data290

The main data of the connecting-rod part, as well as the general properties and operational

conditions associated with the HDD big-end bearing are listed in Table 1. The bearing bore and

the conrod cap are assumed to have the same material properties of the entire rod structure. Any

design details and shape variations provided by the system assemblage have been ignored in the

analysis. Moreover, Table 2 displays the rheological properties of the SAE 15W40 engine oil at295

23

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Structural Reduced Mesh

Cross section plane for clamped boundary condition

(a) 3D FEM model (b) Load chart of the big-end bearing

Figure 2: Geometry and loading conditions of the HDD connecting-rod system.

the operational temperature of 95 C. Notice that the viscosity correction due to lubricant shear-

thinning effect has not been considered in this analysis.

Similarly to the connecting-rod of the high-speed engine that will be studied in Section 4.2,

the 3D FEM model of the current conrod part was condensed to the set of nodes located on the

big-end bearing surface using the Abaqus® software [87]. However, an alternative approach widely300

adopted in the literature [6–8, 88–94] was chosen for the boundary condition of the solid structure.

In this case, instead of the free-free condition used in the high-speed conrod case, the clamped

boundary condition was imposed on an orthogonal cross section plane situated above the big-end

bore, as illustrated in Fig. 2a. This condition automatically restrains the free-free rigid body motion

of the structure, and is here considered for evaluating the appropriateness of the FSI partitioned305

techniques under different structural boundary conditions. In particular, the corresponding clamped

constraint tend to yield large, unrealistic bending deformations in the rod, which in turn have to

be removed to properly consider the local influence of the superficial distortions on the lubricant

film thickness. The usual approach to deal with this unrealistic effect, and also employed, consists

in subtracting the mean bending displacement of the structure from the calculated global bearing310

distortions [91]. Regarding the journal flexibility, it was assumed as rigid due to the absence of a

24

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Table 1: General properties and operational conditions of the HDD connecting-rod system.

Parameter Value Unit

Connecting-Rod Part

Material Steel −Young modulus 210 GPaPoisson’s ratio 0.3 −Density 7800 kg/m3

Rod length 192 mm

Big-End Bearing

Radius 36.50 mmWidth 30.14 mmRadial clearance 37.50 µmAmbient pressure (bearing sides) 105 PaCavitation pressure 0 PaEngine speed 3000 RPM

Big-End Bearing Shells

Bearing material* AlSnSi alloy −Overlay material* PbSnCu alloy −E (bearing material)* 75 GPaυ (bearing material)* 0.34 −HV (overlay material)* 18 HV

Sa 0.65 µmSq 0.77 µmγ 9 -ZS 0.9 µmσS 0.49 µmβS 26.17 µmηS 2.68× 109 m−2

µBL 0.12 −*See [84, 85]

25

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(a) Original topography as measured (b) Roughness topography after form and wavinessremoval (cut-off of 0.8 mm)

Figure 3: Representative surface topography of the HDD big-end bearing shells.

Table 2: Rheological properties of the SAE 15W40 engine oil at 95 C for the HDD connecting-rod rearing simulations.

Parameter Value Unit

Reference Values

η0 13.55 mPa sρ0 850 kg/m3

Piezo-Viscosity Effect (Roelands equation)*

α 1.10× 10−8 Pa−1

Piezo-Density Effect (Dowson-Higgison equation)*

C1 5.9× 108 PaC2 1.34 −*See [60, 86]

26

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hollow pin-journal linking the conrod to the crankshaft. Furthermore, the effects of the distributed

inertia were neglected, so that a quasi -static solution approach was considered. The external

load forces acting on the big-end bearing over a full engine cycle and expressed in the bearing

coordinate system (see Fig. 2a) are depicted in Fig. 2b. These external load forces were calculated315

from prior multibody dynamic simulations performed for the entire connecting-rod mechanism with

ideal joints. In these earlier multibody dynamic simulations, the differential-algebraic equations of

motion and constraints of the whole connecting-rod system formed by the crankshaft, conrod and

piston rigid parts were solved numerically considering the continuous variation of the combustion

pressure acting on the piston head over a full engine cycle. Additionally, the joints connecting (i)320

the crankshaft pin to the conrod big-end bearing, and (ii) the conrod small-end bearing to the piston

pin, were assumed perfectly rigid with no relative motion between the corresponding parts. Thus,

the reaction forces acting on the conrod big-end bearing joint associated with the gross motion of

the entire connecting-rod system calculated from these prior multibody dynamic simulations were

used as external load forces for the local quasi -static EHL analysis.325

With regard to the inputs for the statistical mixed lubrication models (GW/GT models for

asperity contact and Patir & Cheng’s formulation for the average flow model), Table 1 also lists the

roughness and contact parameters obtained from surface measurements of a bearing shell topog-

raphy used in a similar HDD engine application (see Fig. 3). Notice that the shell topography in

consideration is clearly designed with a circumferential microgroove pattern (250µm pitch), whose330

influence on the lubricant flow under mixed lubrication conditions is accounted for through the

Patir and Cheng’s flow factors for longitudinally oriented surface roughness, i.e. γ →∞. The Patir

& Cheng average flow model has already been used to simulate the lubrication performance of

microgrooved journal bearings operating under mixed lubrication regime in numerous publications

(e.g. see Ref. [54]). Nonetheless, explicit expressions of the flow factors for microgrooved surfaces335

could be derived by considering the microgrooves as a deterministic longitudinal parallel pattern

and assuming the original roughness of the bearing and journal surfaces small compared with the

microgroove depth, as proposed in [84]. Furthermore, an alternative approach would be the use of

the more general homogenization method, whose comparisons with the Patir & Cheng flow factor

formulation are reported in [43]. Indeed, more accurate simulation results for microgroove patterns340

with high wavelengths, whose scale is like that of textures, could be obtained through fully deter-

ministic calculations using very fine meshes. However, the computational efforts to resolve these

27

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patterns would be prohibitive in practical applications involving full engine cycle simulations. Thus,

the use of an averaged flow model is a good trade-off for such component-scale analysis. In this

case, deterministic simulations can be used to calculate the flow factors (or homogenization factors)345

needed for more accurate predictions with the averaged models, as adopted in several publications

involving textures [95, 96], honed valleys [97, 98] and general roughness patterns [99–102]. A more

in-depth analysis regarding the definition and calculation of flow factors is outside the scope of the

present contribution and would not affect the use of the FSI methods proposed in this work. The

steel journal surface has been assumed smooth due to the considerably lower roughness measured350

when compared to the bearing one. Finally, the hydrodynamic problem was solved on a numerical

mesh with 120x30 elements and the structural problem (reduced FEM model) on a mesh with 984

nodes.

4.1.2. Results

In the following, the tribological behaviour and the performance of the partitioned coupling355

methods described in Section 3 are evaluated for the HDD conrod big-end bearing case. Similar

analysis have been conducted for the same bearing under static loading conditions and reported in

[36].

The overall scalar results of the conrod big-end bearing considered in the present analysis for

the dynamic loading conditions depicted in Fig. 2b are summarized in Fig. 4. The plots illustrate360

the variation of the most important operational variables of the system over a full engine cycle.

The results for both rigid (blue lines) and flexible (red lines) bearing cases are shown. As can be

seen in Fig. 4e, the journal eccentricity considerably increased when the bearing flexibility was

taken into account in comparison to the rigid condition; this situation can also be observed in

Fig. 4c, where the journal trajectory for the EHL case clearly extrapolated the unit eccentricity365

circle. This situation typically occurs in conrod applications and is explained by the significant solid

deformations that allow the larger rigid body displacements of the journal within the bearing bore.

In particular, two important localized differences can be further highlighted with respect to the

bearing eccentricity: (1) at CA ≈ 360°, which corresponds to the situation of maximum structural

distortions (see Fig. 9a) induced by the alignment of the external loads to the more compliant370

direction of the lower bearing shell, the journal rigid body displacements were significantly higher

for the flexible bearing case in comparison with the rigid one; and (2) at CA ≈ 450°, where the

28

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Crank angle (deg)0 90 180 270 360 450 540 630 720

Forc

es (

kN

)

-50

0

50

100

150

200Load Chart

FX

FY

(a) External load forces

Crank angle (deg)0 90 180 270 360 450 540 630 720

MO

FT

(µ

m)

0

4

8

12

16Minimum Oil Film Thickness

Rigid

Flexible

(b) Minimum oil film thickness

Xr (µm)

-60 -40 -20 0 20 40 60

Yr (µ

m)

-60

-40

-20

0

20

40

60Trajectory

Rigid

Flexible

(c) Journal trajectory

Crank angle (deg)0 90 180 270 360 450 540 630 720

PP

(M

Pa)

0

200

400

600

800

1000Hydrodynamic Peak Pressure

Rigid

Flexible

(d) Hydrodynamic peak pressure

Crank angle (deg)0 90 180 270 360 450 540 630 720

ǫ (µ

m)

20

30

40

50

60Eccentricity

Rigid

Flexible

(e) Journal eccentricity

Crank angle (deg)0 90 180 270 360 450 540 630 720

PL

(W

)

0

1000

2000

3000

4000Power Loss

Rigid

Flexible

(f) Power loss

Figure 4: Scalar results of the dynamically loaded HDD big-end bearing. (a) External load forces. (b) Minimum oilfilm thickness. (c) Journal trajectory. (d) Hydrodynamic peak pressure. (e) Journal eccentricity. (f) Power loss.

29

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magnitude of the external loads is relatively small, the rapid eccentricity drop observed for the rigid

bearing was attenuated when the system flexibility is considered.

The general behaviour of the hydrodynamic peak pressures (PP) and the minimum oil film375

thicknesses (MOFT) were essentially the same, but the magnitudes of both tended to be lower when

the bearing flexibility was taken into account. For the peak pressures (see Fig. 4d), very significant

reductions from almost 1 GPa to 300 MPa were observed during combustion for 0° < CA < 45°

and 675° < CA < 720°. This decrease in PP is associated with the beneficial “spreading” of the

hydrodynamic pressures over the deformed bearing surface, i.e. the bearing distortions are prone to380

“accommodate” the fluid pressures throughout the contact (increase of the contact wet area), thus

attenuating eventual localized “spikes” in the pressure distribution. Regarding the minimum oil

film thicknesses (see Fig. 4b), a distinguished effect can also be visualized for the EHL case during

combustion, where the MOFT reached very small values (≈ 1.5µm) that led the system to operate

under mixed-EHL conditions (see Fig. 5b). Such critical lubrication situation occurred only locally385

at the bearing sides and was caused by deformations of the contact geometry surrounding those

regions; this effect was not observed for the rigid bearing case, and even for the flexible high-speed

big-end bearing to be investigated in Section 4.2. This is noteworthy as accurate predictions of the

transition between full-film and mixed-EHL regimes, along with the identification of the regions and

positions at which asperity contact takes place, are crucial for designing reliable bearing components390

for applications with high loading conditions and abrupt transients. Finally, the curves shown in

Fig. 4f indicate that the changes in the power friction losses were not so pronounced for the analyzed

cases. Once again, the major differences were detected during combustion and around CA = 360°.

In the former situation, the friction losses were reduced for the flexible bearing due to the lower

lubricant viscosities produced by the moderate influence of the piezoviscous effect (see results for395

PP in Fig. 4d) in comparison with the rigid case. In contrast, for the crank angle positions near

360°, the smaller film thicknesses obtained for the deformable bearing (see results for the MOFT

in Fig. 4b) contributed to raising the overall viscous dissipations.

The surface plots depicted in Figs. 5-10 illustrate the hydrodynamic pressure, hydrodynamic

film fraction (cavitation), asperity contact pressure and bearing displacement fields for 8 crank angle400

positions covering the whole engine cycle (only results for the flexible bearing case are shown). The

intention is to provide a global picture on how such field variables change along the engine cycle;

the reader is referred to the video animations provided as a supplementary material in the online

30

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version of this article, which show the time evolution of the bearing operational variables. Hence,

one can emphasize the EHL effects under the high loading conditions around CA = 27°, where405

the structural distortions were mainly verified close to the regions of high pressure concentrated

on the top bearing shell. In this case, however, despite the high intensity of the external loads,

the magnitudes of the displacements were smaller than those obtained for the high deformation

situations in the vicinity of CA = 360°. As already pointed out, such discrepancy is due to the

limited flexibility of the bearing structure along the direction of the conrod shank in which the410

high loads are applied. Furthermore, for the same high deformation conditions, the local bearing

displacements also yielded significant perturbations in the hydrodynamic pressure fields, which in

turn generated the well-known bimodal pressure distributions.

31

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(a) CA = 0° (b) CA = 27°

Figure 5: Field results of the HDD big-end bearing. Left: CA = 0°. Right: CA = 27°. From top to bottom: hydrodynamicpressure, film fraction (cavitation), asperity contact pressure and radial bearing displacement.

32

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(a) CA = 90° (b) CA = 180°

Figure 6: Field results of the HDD big-end bearing. Left: CA = 90°. Right: CA = 180°. From top to bottom: hydrodynamicpressure, film fraction (cavitation) and radial bearing displacement. Asperity contact pressure was null for both crank angles.

(a) CA = 240° (b) CA = 270°

Figure 7: Field results of the HDD big-end bearing. Left: CA = 240°. Right: CA = 270°. From top to bottom: hydrodynamicpressure, film fraction (cavitation) and radial bearing displacement. Asperity contact pressure was null for both crank angles.

33

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(a) CA = 300° (b) CA = 330°

Figure 8: Field results of the HDD big-end bearing. Left: CA = 300°. Right: CA = 330°. From top to bottom: hydrodynamicpressure, film fraction (cavitation) and radial bearing displacement. Asperity contact pressure was null for both crank angles.

(a) CA = 360° (b) CA = 450°

Figure 9: Field results of the HDD big-end bearing. Left: CA = 360°. Right: CA = 450°. From top to bottom: hydrodynamicpressure, film fraction (cavitation) and radial bearing displacement. Asperity contact pressure was null for both crank angles.

34

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(a) CA = 540° (b) CA = 630°

Figure 10: Field results of the HDD big-end bearing. Left: CA = 540°. Right: CA = 630°. From top to bottom: hydrodynamicpressure, film fraction (cavitation) and radial bearing displacement. Asperity contact pressure was null for both crank angles.

35

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Table 3: Performance of the partitioned FSI techniques for the HDD big-end bearing.

Partitioned Method EHL Iterations* Time Elapsed*

PGMF N/C N/CPGMA 6 (5) 26.28 s (16.57 s)IQN-ILS 8 (6) 37.74 s (23.82 s)*Values between brackets correspond to the statistical medians

N/C: not converged

The connecting-rod system in study also showed an interesting effect regarding the collapse of

the cavitation zones under situations of abrupt changes in the orientation of the external loads [103–415

106]. This phenomenon was first observed for the crank angle interval 240° < CA < 330°, as can

be seen by following the evolution of the film fraction field in Figs. 5-10. Similar trends were also

encountered for the stroke ranges 350° < CA < 450° and 620° < CA < 720°. Under these conditions,

the fast change in the journal movement, along with the relative high structural deformations and

the associated lubricant squeeze effect, yielded the generation of a large depressurization zone in420

the contact, which in turn contributed to hindering the local convergence ratio of the EHL solution.

4.1.3. Performance of the Partitioned FSI Techniques

The overall performance of the partitioned coupling algorithms for the present big-end bearing

case under dynamic loading conditions is summarized in Table 3. The simple average and the

statistical median of the number of EHL iterations and the associated elapsed time for the solu-425

tion convergence are listed in the table. Both performance parameters are determined from their

respective values required for the convergence of each crank angle of the entire dynamic solution.

All simulations were carried out in a computer with processor Intel Core i7-3630 CPU 2.40 GHz

and 8 GB memory. According to the average values, the PGMA method has shown to be more

efficient, stable and more robust than the IQN-ILS method, in contrast to the static simulations430

reported in [36]. The main reason for such a difference is that the high transients experienced by

the bearing tended to weaken the robustness of the IQN-ILS method, especially due to the accen-

tuated squeeze film effects induced both by the rapid changes in the journal rigid body movement

and the “vibrational” velocity of the bearing structure. Such effects added further challenges to

the convergence of the EHL problem that were overcome more efficiently by the PGMA technique.435

The performances of the PGMF method were extremely poor, showing no convergence (within the

36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(a) Assembly of the main engine parts. (b) Details of the connecting-rod system.

Figure 11: Illustration of the high-speed motorcycle (HSM) engine.

maximum number of iterations allowed) for practically all time steps. Therefore, one can conclude

that the proposed PGMA method is preferable and recommended for partitioned EHL solutions of

bearings operating under severe lubrication conditions.

4.2. Dynamically Loaded Connecting-Rod Big-End Bearing of a High-Speed Motorcycle Engine440

In this second case study, a set of dynamic simulations were performed for the connecting-rod

big-end bearing of a high-speed motorcycle (HSM) engine will be presented. Figure 11 illustrates an

assemblage of the main engine parts with particular emphasis on the connecting-rod system. This

analysis is aimed to evaluate the suitability and robustness of the EHL coupling techniques, as well

as the effectiveness of the high-order time discretization scheme described in Section 3 for simulating445

journal bearing systems under dynamic loading conditions. Furthermore, the influence of engine

oils with different shear-thinning behaviour on the tribological performance of the bearing system

is also evaluated to provide further information on the potential use of such oils for lubricating real

engine components. Part of the results presented in this section have been published in reference

[38].450

37

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4.2.1. Input Data

The main input data of the connecting-rod part, as well as the principal properties and working

conditions of the HSM big-end bearing are listed in Table 4. The entire 3D FEM model of the

connecting-rod part created in the Abaqus® software [87] is depicted in Fig. 12a. The bearing sur-

face is highlighted to illustrate the region containing the nodes retained in the reduced FEM model455

and effectively used for EHL calculations (see Section 2.3). Moreover, the system was admitted with

no physical constraints, i.e. free-free boundary condition was assumed such that the singularities of

the reduced matrices associated with the rigid body modes were suppressed through modal trunca-

tion [107, 108]. The effects of the distributed inertia were neglected, so that a quasi -static solution

approach is considered. The external load forces acting on the big-end bearing and represented in460

the bearing coordinate system shown in Fig. 12a are illustrated in the plots of Fig. 12b. Such

loads correspond to the reaction forces obtained from multibody dynamic simulations of the whole

connecting-rod system, assuming the respective bearings as ideal rigid joints; see Section 4.1.1 for

more details about the calculation of these reaction forces. Furthermore, by considering the ab-

sence of a hollow pin-journal, the journal flexibility tends to be considerably lower than the bearing465

structure one, so that the journal may be admitted as rigid and only the bearing deformations are

significant for the analysis.

Table 5 summarizes the rheological properties of the engine oils here evaluated for potential

friction reduction at the operational temperature of 80 C. Oil0 is the standard lubricant commonly

used in the specified engine, and as such will be assumed as the reference for comparisons. On470

the other hand, OilA, OilB and OilC are VM-containing engine oils formulated with different

polymer concentrations. OilA was formulated with the higher base oil low shear viscosity. OilB

and OilC were formulated with similar HTHS150 values (High Temperature High Shear viscosity

measured at 150 C and a shear rate of 106 s−1), however OilB was blended with a lower base oil

viscosity and a higher polymer content than OilC. Nearly all engine lubricants contain polymer-475

based Viscosity Index Improvers (VIIs), which are used to attenuate the temperature dependence

of the blend. These polymers may undergo both temporary and permanent shear-thinning at high

shear stresses. Furthermore, it is now commonly recognised that the temporary shear-thinning effect

yields to local temporary viscosity loss, hence contributing to reducing the hydrodynamic friction at

high sliding velocities [110–112], as those encountered in the particular high-speed engine studied480

in this section. Figure 14 illustrates the variation of the dynamic viscosity with shear rate for

38

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(a) 3D FEM model (b) Load chart of the big-end bearing

Figure 12: Geometry and loading conditions of the HSM connecting-rod system.

(a) Original topography as measured. (b) Roughness topography after form and wavinessremoval (cut-off of 0.8 mm).

Figure 13: Representative surface topography of the HSM big-end bearing shells.

39

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Table 4: General properties and operational conditions of the HSM connecting-rod system.

Parameter Value Unit

Connecting-Rod Part

Material Steel −Young modulus 210 GPaPoisson’s ratio 0.3 −Density 8050 kg/m3

Rod length 150 mm

Big-End Bearing

Radius 21.5 mmWidth 20.0 mmRadial clearance 50.0 µmAmbient pressure (bearing sides) 105 PaCavitation pressure 0 PaEngine speed 9000 RPM

Big-End Bearing Shells

Bearing material* AlSnSi alloy −Overlay material* PbSnCu alloy −E (bearing material)* 75 GPaυ (bearing material)* 0.34 −HV (overlay material)* 18 HV

Sa 0.24 µmSq 0.32 µmγ 0.1 -ZS 0.28 µmσS 0.23 µmβS 61.42 µmηS 6.50× 109 m−2

µBL 0.12 −*See [84, 85]

40

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Table 5: Rheological properties of the lubricants for the HSM connecting-rod simulations.

Parameter Value Unit

OIL 0 OIL A OIL B OIL C(Standard) (−) (−) (−)

Piezo-viscosity effect (Roelands equation)*

α 1.10× 10−8 Pa−1

Shear-thinning effect (Carreau-Yasuda equation)**

η0 1.90× 10−2 1.90× 10−2 1.12× 10−2 9.12× 10−3 Pa sη∞ − 3.87× 10−3 1.85× 10−3 3.87× 10−3 Pa sA − 5.00× 10−5 9.00× 10−5 3.00× 10−5 −n − 0.79 0.79 0.85 −a − 1.9 1.9 2 −

Piezo-density effect (Dowson-Higgison equation)*

ρ0 850 kg/m3

C1 5.90× 108 PaC2 1.34 −*See [60, 86] **See [109]

Shear Rate (s-1

)

100

102

104

106

108

1010

Dy

nam

ic v

isco

sity

(P

a.s)

0

0.005

0.01

0.015

0.02Shear-Thinning Oils

Oil A

Oil B

Oil C

Figure 14: Variation of the dynamic viscosity with shear rate for the different VM-containing engine oils investigatedfor friction reduction.

41

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the three VM-containing engine oils under consideration. The permanent shear thinning effect is

considered in this analysis through the rheological parameters listed in Table 5, which were obtained

from measurements with an ultrashear viscometer (USV). The magnitude of the permanent shear

thinning expressed in terms of the rheological parameters was established in accordance with the485

permanent shear stability index (PSSI) for viscosities measurements at 106 s−1 as defined in Ref.

[110].

As for the inputs of the statistical mixed lubrication models (Greenwood & Tripp model for

asperity contact and Patir & Cheng’s formulation for the average fluid flow), Table 4 also shows the

contact and roughness parameters obtained for the surface topography depicted in Fig. 13, which490

was measured from a bearing shell used in a similar high-speed engine. Notice that the surface

roughness is assumed longitudinally-oriented, so that γ →∞.

4.2.2. Results: Standard Engine Lubricant

The overall results of the HSM conrod big-end bearing for lubricant Oil0 are summarized in Fig.

15 for both rigid (blue lines) and flexible (red lines) bearing cases. The curves depict the evolution495

of the main operational lubrication variables over a full engine cycle subjected to the working

conditions described in Section 4.2.1. The general distortion behaviour observed for the present

high-speed bearing is similar to that of the heavy-duty bearing shown in Section 4.1.2. As expected,

under EHL conditions the journal orbit deviated around the unit eccentricity circle due to the large

solid deformations of the bearing bore. Furthermore, the maximum structural displacements also500

occurred under equivalent loading conditions aligned to the more compliant direction of the lower

shell (CA ≈ 180° and CA ≈ 540°).

The variation tendencies of the minimum oil film thickness (MOFT) and the hydrodynamic peak

pressures (PP) showed to be similar along the complete engine cycle, except for the magnitudes of

PP that were clearly lower for the flexible bearing case. In particular, significant decrease of PP is505

observed during combustion, which can be explained by the “spreading” effect of the fluid pressure

over the distorted bearing surface. Analogously, distinguishable reduction in the hydrodynamic

peak pressures also took place around the positions of larger structural deformations (CA ≈ 180°

and CA ≈ 540°), which equally contributed to the “accommodation” of the lubricant pressure

distributions on the interface. Regarding the MOFT, slightly lower values were obtained under EHL510

conditions, but in contrast to the heavy-duty diesel engine analysis they were not small enough to

42

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Crank angle (deg)

0 90 180 270 360 450 540 630 720

Fo

rces

(k

N)

-40

-20

0

20

40

60External Load Forces

FX

FY

(a) External load forces

Crank angle (deg)

0 90 180 270 360 450 540 630 720

MO

FT

(µ

m)

0

10

20

30

40

50MOFT

Rigid

Flexible

(b) Minimum oil film thickness

Xr (µm)

-90 -60 -30 0 30 60 90

Yr (µ

m)

-90

-60

-30

0

30

60

90Trajectory

Rigid

Flexible

(c) Journal trajectory

Crank angle (deg)

0 90 180 270 360 450 540 630 720

PP

(M

Pa)

0

100

200

300

400

500

600Hydrodynamic Peak Pressure

Rigid

Flexible

(d) Hydrodynamic peak pressure

Crank angle (deg)0 90 180 270 360 450 540 630 720

ǫ (µ

m)

0

20

40

60

80Eccentricity

Rigid

Flexible

(e) Journal eccentricity

Crank angle (deg)

0 90 180 270 360 450 540 630 720

PL

(W

)

0

500

1000

1500

2000

2500Power Loss

Rigid

Flexible

(f) Power loss

Figure 15: Scalar results of the dynamically loaded HSM big-end bearing for the lubricant ‘Oil0’. (a) Externalload forces. (b) Minimum oil film thickness. (c) Journal trajectory. (d) Hydrodynamic peak pressure. (e) Journaleccentricity. (f) Power loss.

produce asperity contact interactions. Likewise, the main discrepancies in the power friction losses

(PL) were noticed during combustion and for the situations of large structural displacements. In

the former condition, the differences in the viscous dissipation are linked to the varying intensity of

the piezoviscous effect, while in the latter situation the deviations are associated with the combined515

drop in the hydrodynamic pressures and film thickness for the flexible bearing case.

43

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The surface plots illustrated in Figs. 16-22 aim to outline the progression in time of the hydro-

dynamic pressure, hydrodynamic film fraction (cavitation), asperity contact pressure and bearing

displacement fields over the entire engine cycle; the reader is referred to the video animations

provided as a supplementary material in the online version of this article, which show the time520

evolution of the bearing operational variables. Equivalently to the heavy-duty diesel engine case,

for the situations of maximum structural deformations (CA ≈ 180° and CA ≈ 540°), the local geo-

metric perturbations on the bearing surface yielded the hydrodynamic pressure distributions to the

same bimodal shape pattern. Furthermore, a collapse of the cavitation zones comparable with that

previously described in Section 4.1.2 was detected as well, but now only for the crank angle interval525

330° < CA < 390°, as can be visualized by following the evolution of the film fraction field through

Figs. 18-20. Physically, such sudden disruption onto the cavitation regions is closely related to the

abrupt change of orientation and magnitude of the external loads acting on the big-end bearing

[103–106]. This can be verified in the load chart of Fig. 12b, with special attention to the almost

null loading situation experienced by the system at CA ≈ 360°. This instantaneous “unloading”530

condition explains the “spike” in the eccentricity and MOFT curves of Fig. 15. Furthermore, this

extremely rapid transient behaviour was the only bottleneck for the convergence of the solution of

the EHL problem.

44

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(a) CA = 0° (b) CA = 90°

Figure 16: Field results of the HSM big-end bearing for lubricant ‘Oil0’. Left: CA = 0°. Right: CA = 90°. From top to bottom:hydrodynamic pressure, film fraction (cavitation) and radial bearing displacement. Asperity contact pressure was null for bothcrank angles.

(a) CA = 180° (b) CA = 270°

Figure 17: Field results of the HSM big-end bearing for lubricant ‘Oil0’. Left: CA = 180°. Right: CA = 270°. From top tobottom: hydrodynamic pressure, film fraction (cavitation) and radial bearing displacement. Asperity contact pressure was nullfor both crank angles.

45

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(a) CA = 330° (b) CA = 351°

Figure 18: Field results of the HSM big-end bearing for lubricant ‘Oil0’. Left: CA = 330°. Right: CA = 351°. From top tobottom: hydrodynamic pressure, film fraction (cavitation) and radial bearing displacement. Asperity contact pressure was nullfor both crank angles.

(a) CA = 355° (b) CA = 357°

Figure 19: Field results of the HSM big-end bearing for lubricant ‘Oil0’. Left: CA = 355°. Right: CA = 357°. From top tobottom: hydrodynamic pressure, film fraction (cavitation) and radial bearing displacement. Asperity contact pressure was nullfor both crank angles.

46

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(a) CA = 360° (b) CA = 390°

Figure 20: Field results of the HSM big-end bearing for lubricant ‘Oil0’. Left: CA = 360°. Right: CA = 390°. From top tobottom: hydrodynamic pressure, film fraction (cavitation) and radial bearing displacement. Asperity contact pressure was nullfor both crank angles.

(a) CA = 420° (b) CA = 450°

Figure 21: Field results of the HSM big-end bearing for lubricant ‘Oil0’. Left: CA = 420°. Right: CA = 450°. From top tobottom: hydrodynamic pressure, film fraction (cavitation) and radial bearing displacement. Asperity contact pressure was nullfor both crank angles.

47

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(a) CA = 540° (b) CA = 630°

Figure 22: Field results of the HSM big-end bearing for lubricant ‘Oil0’. Left: CA = 540°. Right: CA = 630°. From top tobottom: hydrodynamic pressure, film fraction (cavitation) and radial bearing displacement. Asperity contact pressure was nullfor both crank angles.

48

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Crank angle (deg)

0 90 180 270 360 450 540 630 720

Fo

rces

(k

N)

-40

-20

0

20

40

60External Load Forces

FX

FY

(a) External load forces

Crank angle (deg)

0 90 180 270 360 450 540 630 720

MO

FT

(µ

m)

0

10

20

30

40

50MOFT

Oil A

Oil B

Oil C

(b) Minimum oil film thickness

Xr (µm)

-90 -60 -30 0 30 60 90

Yr (µ

m)

-90

-60

-30

0

30

60

90Trajectory

Oil A

Oil B

Oil C

(c) Journal trajectory

Crank angle (deg)

0 90 180 270 360 450 540 630 720

PP

(M

Pa)

0

20

40

60

80

100

120Hydrodynamic Peak Pressure

Oil A

Oil B

Oil C

(d) Hydrodynamic peak pressure

Crank angle (deg)

0 90 180 270 360 450 540 630 720

ǫ (µ

m)

0

20

40

60

80

100Eccentricity

Oil A

Oil B

Oil C

(e) Journal eccentricity

Crank angle (deg)0 90 180 270 360 450 540 630 720

PL

(W

)

0

200

400

600

800

1000

1200Power Loss

Oil A

Oil B

Oil C

(f) Power loss

Figure 23: Scalar results for the dynamically loaded HSM big-end bearing for the three VM-containing engine oils.(a) External load forces. (b) Minimum oil film thickness. (c) Journal trajectory. (d) Hydrodynamic peak pressure.(e) Journal eccentricity. (f) Power loss.

4.2.3. Results: VM-Containing Engine Oils

Figure 23 outlines the main operational lubrication variables obtained after full EHL simulations535

for the three VM-containing engine oils listed in Table 5. Only the flexible bearing case subjected to

the same dynamic loading conditions displayed in Fig. 12b was considered at this point. As can be

seen in Figs. 23b-23c, the journal movement is hardly affected by the shear-thinning properties of

49

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the lubricant. In fact, the journal eccentricity tended to be slightly higher for OilB and progressively

lower for OilC and OilA. This is explained by the predominant higher viscosities of OilA in contrast540

to the reduced values of OilB within the range of operational shear rate conditions, i.e. 106 s=1 <

γ < 107 s=1 (see Fig. 14). Such aspects are also visualized in Fig. 23b, where the MOFT is in general

subtly superior for OilA and, to the same extent, lower for Oil B. As for the hydrodynamic peak

pressures (Fig. 23d), no meaningful difference was found among the investigated oils. Regarding

the field results, e.g. hydrodynamic pressures, film fraction, structural displacements, etc., they545

were very similar to those calculated for the lubricant Oil0 and illustrated in Figs. 16-22.

The influence of the VM-containing engine oils on the bearing performance was clearly observed

through the power loss curves (Fig. 23f). In this case, the lubricant (OilB) with lower effective

dynamic viscosity for the operating shear rate range, i.e. 106 s=1 < γ < 107 s=1 yielded the lowest

overall viscous dissipations, while OilC and OilA produced higher frictional losses, with OilA being550

the one producing the largest losses. Such results indicate, at least from the point of view of

connecting-rod bearing applications, the potential benefits of using lubricants with similar shear-

thinning properties of OilB for engine lubrication.

Furthermore, although OilB and OilC had similar HTHS150 values and the low shear dynamic

viscosity of OilB was higher than OilC, the overall frictional power dissipation computed for OilB555

was lower. Our investigation suggests that the HTHS150 parameter, which has traditionally been

used as an indicator of bearing durability, may not necessarily be the best indicator for estimating

the fuel economy performance of an oil. This is particularly important for connecting-rod engine

bearings, since the shear rates in such components may be greater than 106 s=1; in such conditions

the shear thinning behaviour of lubricants may differ significantly depending on their formulation.560

Similar remarks were reported in Ref. [113], which also highlighted the importance of considering

the coupled temperature and shear rate effects on the lubricant rheology for a more accurate

assessment of the fuel economy potential of a lubricant. However, detailed TEHL analysis, including

heat transfer in the components, should be conducted to provide more accurate results about the

combined effect of permanent shear thinning, temperature distribution and high share rates within565

the conjunction to support any further discussion regarding the use of HTHS150 parameter for the

ranking of lubricants. The effect of local temperature rises on the contacting surfaces is especially

important for engine bearings applications, which can reach values as higher as 150 C, as reported

in numerous publications (see e.g. Refs. [91, 114–117]).

50

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Table 6: Performance of the partitioned algorithms for the HSM big-end bearing.

Partitioned Method EHL Iterations* Time Elapsed*

Oil 0

PGMF N/C N/CPGMA 6 (5) 13.98 s (10.56 s)

IQN-ILS 7 (6) 20.20 s (12.66 s)

Oil A

PGMF N/C N/CPGMA 6 (4) 14.94 s (10.96 s)

IQN-ILS 7 (5) 21.35 s (14.26 s)

Oil B

PGMF N/C N/CPGMA 6 (5) 18.45 s (13.33 s)

IQN-ILS 8 (6) 24.05 s (15.82 s)

Oil C

PGMF N/C N/CPGMA 6 (5) 15.92 s (11.87 s)

IQN-ILS 8 (6) 25.14 s (16.89 s)*Values between brackets correspond to the statistical medians

N/C: not converged

4.2.4. Performance of the Partitioned FSI Techniques570

The performance of the coupling algorithms for all the simulation cases undertaken for the

present connecting-rod big-end bearing is summarized in Table 6. All simulations were carried out

in a computer with processor Intel Core i7-3630 CPU 2.40 GHz and 8 GB memory. As can be seen,

the PGMA method has shown the best performance, both in terms of the average number of EHL

iterations and the computational time for convergence. No convergence has been obtained with575

the PGMF method within the maximum number of iteration assumed for the problem solution.

The values listed in Table 6 are the average results per crank angle calculated from their respective

values of each time step of the full engine cycle. It should be noted that all the converged scalar

and field results were identical regardless of the coupling method considered.

5. Conclusions580

In the present work an extensive mathematical modelling and a computational platform for

simulating the mixed-elastohydrodynaimc lubrication regime of journal bearing systems operating

under varying working conditions have been proposed. The application of different partitioned FSI

methods, namely (i) Fixed Point Gauss-Seidel Method (PGMF), (ii) Point Gauss-Seidel Method

with Aitken Acceleration (PGMA) and (iii) Interface Quasi-Newton Method with an approximation585

51

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for the Inverse of the Jacobian from a Least-Squares model (IQN-ILS), has been considered for

solving the EHL problem of dynamically loaded connecting-rod big-end bearings of both heavy-

duty diesel and high-speed motorcycle engines. The following conclusions can be drawn from the

simulation analysis:

In contrast to previous results [36], which showed that the IQN-ILS method was advantageous590

in terms of computational efforts for statically loaded situations, the PGMA method was

shown here to be more efficient for bearing under dynamic loading conditions. The PGMF

method produced the worst results with no convergence for either conrod bearing considered.

The frictional performance of different VM-containing engine oils for the high-speed big-end

bearing was investigated, showing clearly the influence of the polymeric structure and con-595

centration of lubricant viscosity modifiers on the bearing power dissipation, and how crucial

is the consideration of the actual operating shear rate range conditions to properly assess

the lubrication performance of engine oils. Particularly, a discussion on the need of more

in-depth TEHL analyses to assess the effectiveness of the HTHS150 parameter to estimate

fuel economy performance of an oil for engine bearings was provided.600

The practical engineering application cases illustrate examples of how the solution framework

developed herein can be used to support parametric and/or optimization analyses during the

engineering design process.

Finally, it is worth emphasizing that the use of the proposed partitioned methods can be read-

ily extended to other tribological contacts, e.g. line and point contacts, and further developed605

to perform thermo-elastohydrodynamic (TEHL) calculations. The main advantage of such parti-

tioned approaches is the possibility of using optimized codes to solve hydrodynamic equations and

structural equations separately as “black-box” solvers and is ideal for the development of modular

coupled schemes. Furthermore, the FSI techniques evaluated in this contribution can also be used

as alternative strategies for coupling specialized lubrication solvers to general purpose Multiphysics610

software. Additionally, besides the extension of the proposed simulation framework to TEHL anal-

ysis, the same simulation platform can also be used to predict the new worn shape of bearings,

especially for bearing systems operating under intermittent conditions, such as in modern start-

stop engines and wind turbines. Moreover, comparisons of the proposed FSI techniques with other

52

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monolithic approaches for connecting-rod EHL simulations, which would require benchmarking the615

speed and accuracy of the codes using the same computer architecture and resources, as for example

reported in Ref. [27], are further aspects that the authors endeavour to consider in forthcoming

studies.

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