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Journal ArticleDOI

About an inverse problem for a free boundary compressible problem in hydrodynamic lubrication

01 Jan 2016-Journal of Inverse and Ill-posed Problems (Walter de Gruyter GmbH)-Vol. 24, Iss: 5, pp 599-623
TL;DR: In this article, an inverse problem is considered for a non-cooperative partial di erential equation, issued from a mass conservation cavitation model for a slightly compressible finite domain.
Abstract: In this paper an inverse problem is considered for a non coercive partial di erential equation, issued from a mass conservation cavitation model for a slightly compressible uid. The cavitation phenomenon and compressibility take place and are described by the Elrod model. The existence of an optimal solution is proven. Optimality conditions are derived and some numerical results are given.

Summary (2 min read)

1 Introduction

  • The present work comes within the scope of inverse problems in hydrodynamic lubrication.
  • The interest of the more physical compressible Elrod-Adams model is that there is only one unknown (the density of lubricant) instead of two (P and θ) in the JFO model.
  • In [30] the identification procedure for a gap with known pressure is studied.
  • The optimality system for the initial problem is derived in the one-dimensional case.

3 Approached state equation

  • The two parameters and η have different meanings.
  • The first one is a regularization parameter and the second one a penalization parameter.
  • This condition will be used in the sequel.
  • The proof is based upon the implicit function theorem for the equation Fη (vη (h), h) = 0. (12) Applying the Fredholm’s alternative [11] allows us to conclude.

4 Approximate cost function and necessary optimality conditions

  • To cope with the possible non uniqueness of the solution for the control problem, the authors introduce as in [5, 10] a modified cost function which forces the solution of the approximate problem to converge towards the solution (h∗, u∗) deduced from theorem (2).
  • The strong convergence will be proved for each of the two steps, so inducing the result.
  • The first term of the right hand side in (32) tends to 0 and from the compact inclusion of H1(Ω) in C0(Ω) the third term also tends to 0.

5 Numerical realization

  • The theory in the previous sections has been mainly developped in the context of the hydrodynamic lubrication(The Reynolds state is the governing equation for thin film flow).
  • Equation (15) is solved using the P1 finite elements and fixed-point procedure.
  • The value chosen for β is a usual value for the bulk modulus of a lubricant [32].
  • The value Θ0 = 0.675 for the input mass flow is chosen so that two cavitation regions exist.
  • Fig. 3 and Fig. 4 depict a comparaison between the computed pressure and thickness profiles and the exact ones for m = 401 after various number of iterations.

6 Conclusion

  • The present work shows how is it possible to get some identification process for thin film flow taking a correct mass preserving condition into account.
  • It opens the way to optimization problems in the same mechanical area, as finding the best shape (the function h) to minimize friction or oil consumption.
  • An interesting feature is that recent physical studies of the gaseous cavitation in lubrication tends to give a physical meaning to some mathematical problems close to the approximate one (6)[7] used in the present paper.
  • This induces an additional reason to introduce more efficient numerical methods for solving both direct and inverse problems.
  • This work was partially supported by the French-Morocco cooperation CNRS-CNRST ( SPM02/12 ) and the CMIMF action intégrée (MA/04/94).

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About an inverse problem for a free boundary
compressible problem in hydrodynamic lubrication
K. Ait Hadi, Guy Bayada, M. El Alaoui Talibi
To cite this version:
K. Ait Hadi, Guy Bayada, M. El Alaoui Talibi. About an inverse problem for a free boundary
compressible problem in hydrodynamic lubrication. Journal of Inverse and Ill-posed Problems, De
Gruyter, 2015. �hal-00822388�

Noname manuscript No.
(will be inserted by the editor)
About an inverse problem for a free boundary
compressible problem in hydrodynamic lubrication
K. Ait Hadi · G. Bayada ·
M. El Alaoui Talibi
Received: date / Accepted: date
Abstract In this paper an inverse problem is considered for a non coercive
partial differential equation, issued from a mass conservation cavitation model
for a slightly compressible fluid. The cavitation phenomenon and compress-
ibility take place and are described by the Elrod model. The existence of an
optimal solution is proven. Optimality conditions are derived and some nu-
merical results are given.
Keywords Inverse problem · Elrod model · optimality system · numerical
results.
Mathematics Subject Classification (2000) MCS 35J70 · MCS 35Q35 ·
MCS 49K20 · MCS 65N21 · MCS 65N30.
1 Introduction
The present work comes within the scope of inverse problems in hydrodynamic
lubrication. Numerous works are based upon the computation of the solution
of various forms of the Reynolds partial differential equation established in
1886. This equation enables one to compute the hydrodynamic pressure (P )
Khalid Ait Hadi
Universit´e Cadi Ayyad, Laboratoire LIBMA de Math´ematiques Appliqu´ees Facult´e des Sci-
ences Semlalia, epartement de Math´ematiques, BP 2390, Marrakech Morocco.
E-mail: k.aithadi@ucam.ac.ma
Guy Bayada
Universit´e de Lyon, INSA , Institut Camille Jordan , UMR CNRS 5208 , Bat . Leonard de
Vinci Math 69621 Villeurbanne Cedex France.
E-mail: guy.bayada@insa-lyon.fr
Corresponding author: Mohamed El Alaoui Talibi
Universit´e Cadi Ayyad, Laboratoire LIBMA de Math´ematiques Appliqu´ees Facult´e des Sci-
ences Semlalia, epartement de Math´ematiques, BP 2390, Marrakech Morocco.
E-mail: elalaoui@uca.ma

2 K. Ait Hadi et al.
in a lubricated device, from data like the velocities distribution on the surfaces
surrounding the thin film flow, the gap between these surfaces(or equivalently
the shape of one of these surfaces) and the rheological characteristics of the
fluids. However, due to severe operational conditions(the gap between the sur-
faces can be of some micrometers only and the relative velocity of the surfaces
some meters/second), some of these data are not really well known.
This is the case for example of the surfaces which are deformed from an initial
known shape by the hydrodynamic pressure inside the fluid. Experimentally,
the knowledge of this pressure (P ) can now be obtained with a good precision.
It becomes then possible to find the real shape (h) of the surfaces by solving an
inverse problem with Reynolds equation as state equation. Such information
is important as if the related gap is too small, it means that some contact is
possible between the two surfaces thus inducing wear and possible failure of
the device.
Two other very similar problems in the lubrication field can be mentioned.
The first one is to find the gap (h) such that the pressure (P ) is the greatest
possible. In some specific situation the solution is the Rayleigh-step bearing
[31] in which the optimum gap h is a discontinuous function. More recently,
generalized Reynolds equation for heterogeneous slip/no slip engineered sur-
faces have been proposed [9]. As a consequence, a no-slip condition for the
velocity is valid on some part of the fluid boundary and slip occurs on the
other part. Due to this non homogeneity, the resulting Reynolds equation con-
tains discontinuous coefficients. Finding the best location of the slip/no-slip
regions is mathematically close to the previous optimization problem.
Another difficulty must be considered in these aforementioned identification-
optimization problems. It is due to cavitation, the well-known phenomenon in
fluid mechanics when fluid is no longer homogeneous and takes some diphasic
aspect with the appearance of air bubbles. This phenomenon occurs very often
in lubrication and cannot be ignored, especially as the gap (h) is not constant.
Taking cavitation into account implies considering a new non linear operator
as state equation instead of the classical Reynolds equation. Various models
exist in the mechanical literature. The most common one is based upon a vari-
ational inequality for the pressure [27]. However, as it is not a mass preserving
model, it is often replaced by the Jacobsson Floberg Olssen (JFO) model, in
which a new variable θ is introduced. This variable describes the local pro-
portion of fluid (or saturation) considering the presence of air bubbles in the
flow [25]. In a widely referenced paper [20] , it has been pointed out that the
(JFO) model is equivalent to a free boundary complementary problem with
two unknowns, pressure P and saturation θ with
P 0, 0 θ 1, P (1 θ) = 0
Existence and uniqueness aspects of such models have been mathematically
studied in a lot of papers [2] . However another aspect of the Elrod-Adams pa-
per has been less studied in the mathematical literature: authors try to recover
the somewhat heuristic JFO model by introducing a small compressibility in
the classical (non cavitated) Reynolds equation. They show that the JFO

About an inverse problem for a free boundary 3
model can be obtained as the limit of this compressible model and propose
some numerical procedures to solve it. Although being somewhat heuristic,
this result has been supported by recent rigorous mathematic results [14] for
the asymptotic thin film compressible Navier Stokes system. The interest of
the more physical compressible Elrod-Adams model is that there is only one
unknown (the density of lubricant) instead of two (P and θ) in the JFO model.
The price to pay is that the state equation is a non linear degenerate partial
differential equation. The importance of the choice of a model of cavitation in
the simulation of lubrication devices has often been mentioned [9, 4].
Although these inverse-optimization problems are of permanent interest in the
mechanical literature [3, 12, 15, 17, 19, 21, 29], few mathematical works ex-
ist. One of the first attempts [29] is devoted to the optimum Rayleight step
bearing. In [30] the identification procedure for a gap with known pressure
is studied. However, in these works cavitation is not considered. In [8], varia-
tional inequality is chosen to model the cavitation and optimality problems are
studied by way of penalization. The Elrod-Adams model is considered in [19]
using some regularizations which enables one to gain optimality conditions.
The present study addresses the identification process of the gap for a given
pressure and takes the cavitation into account by way of the slightly compress-
ible model proposed by Elrod-Adams. This model has also the advantage of
preserving the mass flow.
This problem presents many mathematical difficulties essentially bound to the
complexity of the criterion considered as a function of the thickness. It is not
even a locally Lipschitz function. We are lead to consider a more general func-
tional framework which is the space of the functions of bounded variation. To
deal with these difficulties a double regularization is introduced by approach-
ing the degenerate part of the state equation with a particular sequence of
monotonous and continuously differentiable functions. Thus a sequence of reg-
ularized control problems can be obtained.
The paper is organized as follows. Section 2 is devoted to the physical
background, the statement of the problem and the proof of the existence of a
control. In section 3, we formulate the regularized control problem for which
an existence result is proved. Optimality conditions are given in section 4,
and some a priori estimates are obtained, so allowing us to pass to the limit.
The optimality system for the initial problem is derived in the one-dimensional
case. In the last section some numerical results show that the optimality system
associated to an inverse algorithm is efficient.
Finally let us mention that from a mathematical aspect, the problem here
studied is very close to the one of the ”dam problem” [13] with unsaturated
porous media or multiple fluid saturated porous media. In this case, the un-
knowns to be determined are the value of the porosity parameters of the soil
[18] and the measured data is the hydraulic pressure.

4 K. Ait Hadi et al.
2 Formulation of the problem and existence of control
2.1 Description of the physical problem (Elrod-Adams model)
Let the square ]0, 1[×]0, 1[ of the (x, y) plane, Γ
0
= {(x, y) , x = 0},
and Γ = \Γ
0
. The lubricant is assumed to be contained in a three dimen-
sional volume between the lower flat surface with horizontal velocity
U and
an upper one describe by z = h(x, y) at rest. If h is small, the pressure is
known to be independent from z and to obey the following Reynolds equation
defined on [23].
div
ρ
12η
h
3
P
= div
h
U ρ
2
!
on , (1)
where P (x, y) is the pressure of the lubricant, h(x, y) is the film thickness, ρ
is the density of lubricant, η is the viscosity,
U is the relative velocity of the
surfaces of the mechanism in which lubricant takes place.
For a slightly compressible fluid like water or oil, the law linking P and ρ is
usually described by introducing the bulk number β with: see [25]
P = β log(ρ), ρ > 1, P > 0, (2)
Taking into account the possible existence of a cavitation area where P is zero
and ρ 1 lead to generalize (2) in
P = β log((ρ 1)
+
+ 1), ρ 0, β > 0.
in which f
+
= sup(f, 0). Rewriting (1) in term of ρ, we gain by differentiating
in the distributional sense
div
β
h
3
12η
(ρ 1)
+
ρ
h
U
2
!
= 0. (3)
Introducing u = β(ρ 1) as the primary unknown, equation (3) becomes
div
h
3
12η
u
+
1 +
1
β
u
h
U
2
!
= 0.
In the sequel, we will consider more precisely the case of a journal bear-
ing (see Figure 1) with a supply line located at x = 0. It is a usual lu-
brication device in which a known quantity of fluid Θ
0
is supplied through
Γ
0
= {(x, y) , x = 0, 0 < y < 1} while the pressure is assumed to be
known on Γ = \Γ
0
and equal to zero (the atmospheric pressure).
For simplicity sake we assume η =
1
6
,
U ==
1
0
so that the weak formula-
tion of the state problem reads :
(P
h
)
Find u L
2
() verifying u
+
V,
1 +
1
β
u
0
R
h
3
u
+
· φ dxdy
R
h
1 +
1
β
u
φ
x
dxdy =
R
Γ
0
Θ
0
φ dy φ V
Θ
0
L
(Γ
0
), Θ
0
0

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TL;DR: In this paper, a class of partial differential equations that generalize and are represented by Laplace's equation was studied. And the authors used the notation D i u, D ij u for partial derivatives with respect to x i and x i, x j and the summation convention on repeated indices.
Abstract: We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equations of second order. A linear partial differential operator L defined by $$ Lu{\text{: = }}{a_{ij}}\left( x \right){D_{ij}}u + {b_i}\left( x \right){D_i}u + c\left( x \right)u $$ is elliptic on Ω ⊂ ℝ n if the symmetric matrix [a ij ] is positive definite for each x ∈ Ω. We have used the notation D i u, D ij u for partial derivatives with respect to x i and x i , x j and the summation convention on repeated indices is used. A nonlinear operator Q, $$ Q\left( u \right): = {a_{ij}}\left( {x,u,Du} \right){D_{ij}}u + b\left( {x,u,Du} \right) $$ [D u = (D 1 u, ..., D n u)], is elliptic on a subset of ℝ n × ℝ × ℝ n ] if [a ij (x, u, p)] is positive definite for all (x, u, p) in this set. Operators of this form are called quasilinear. In all of our examples the domain of the coefficients of the operator Q will be Ω × ℝ × ℝ n for Ω a domain in ℝ n . The function u will be in C 2(Ω) unless explicitly stated otherwise.

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TL;DR: In this article, the authors define and define elementary properties of BV functions, including the following: Sobolev Inequalities Compactness Capacity Quasicontinuity Precise Representations of Soboleve Functions Differentiability on Lines BV Function Differentiability and Structure Theorem Approximation and Compactness Traces Extensions Coarea Formula for BV Functions isoperimetric inequalities The Reduced Boundary The Measure Theoretic Boundary Gauss-Green Theorem Pointwise Properties this article.
Abstract: GENERAL MEASURE THEORY Measures and Measurable Functions Lusin's and Egoroff's Theorems Integrals and Limit Theorems Product Measures, Fubini's Theorem, Lebesgue Measure Covering Theorems Differentiation of Radon Measures Lebesgue Points Approximate continuity Riesz Representation Theorem Weak Convergence and Compactness for Radon Measures HAUSDORFF MEASURE Definitions and Elementary Properties Hausdorff Dimension Isodiametric Inequality Densities Hausdorff Measure and Elementary Properties of Functions AREA AND COAREA FORMULAS Lipschitz Functions, Rademacher's Theorem Linear Maps and Jacobians The Area Formula The Coarea Formula SOBOLEV FUNCTIONS Definitions And Elementary Properties Approximation Traces Extensions Sobolev Inequalities Compactness Capacity Quasicontinuity Precise Representations of Sobolev Functions Differentiability on Lines BV FUNCTIONS AND SETS OF FINITE PERIMETER Definitions and Structure Theorem Approximation and Compactness Traces Extensions Coarea Formula for BV Functions Isoperimetric Inequalities The Reduced Boundary The Measure Theoretic Boundary Gauss-Green Theorem Pointwise Properties of BV Functions Essential Variation on Lines A Criterion for Finite Perimeter DIFFERENTIABILITY AND APPROXIMATION BY C1 FUNCTIONS Lp Differentiability ae Approximate Differentiability Differentiability AE for W1,P (P > N) Convex Functions Second Derivatives ae for convex functions Whitney's Extension Theorem Approximation by C1 Functions NOTATION REFERENCES

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Abstract: This book presents a carefully selected group of methods for unconstrained and bound constrained optimization problems and analyzes them in depth both theoretically and algorithmically. It focuses on clarity in algorithmic description and analysis rather than generality, and while it provides pointers to the literature for the most general theoretical results and robust software, the author thinks it is more important that readers have a complete understanding of special cases that convey essential ideas. A companion to Kelley's book, Iterative Methods for Linear and Nonlinear Equations (SIAM, 1995), this book contains many exercises and examples and can be used as a text, a tutorial for self-study, or a reference. Iterative Methods for Optimization does more than cover traditional gradient-based optimization: it is the first book to treat sampling methods, including the Hooke& Jeeves, implicit filtering, MDS, and Nelder& Mead schemes in a unified way.

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TL;DR: In this article, the authors present a text for a first-year course in analysis, which covers the same topics as elementary calculus -linear algebra, differentiation, integration -but treated in a manner suitable for people who will be using it in further mathematical investigations.
Abstract: This book is meant as a text for a first-year graduate course in analysis. In a sense, the subject matter covers the same topics as elementary calculus - linear algebra, differentiation, integration - but treated in a manner suitable for people who will be using it in further mathematical investigations. The book begins with point-set topology, essential for all analysis. The second part deals with the two basic spaces of analysis, Banach and Hilbert spaces. The book then turns to the subject of integration and measure. After a general introduction, it covers duality and representation theorems, some applications (such as Dirac sequences and Fourier transforms), integration and measures on locally compact spaces, the Riemann-Stjeltes integral, distributions, and integration on locally compact groups. Part four deals with differential calculus (with values in a Banach space). The next part deals with functional analysis. It includes several major spectral theorems of analysis, showing how one can extend to infinite dimensions certain results from finite-dimensional linear algebra; a discussion of compact and Fredholm operators; and spectral theorems for Hermitian operators. The final part, on global analysis, provides an introduction to differentiable manifolds. The text includes worked examples and numerous exercises, which should be viewed as an integral part of the book. The organization of the book avoids long chains of logical interdependence, so that chapters are as independent as possible. This allows a course using the book to omit material from some chapters without compromising the exposition of material from later chapters.

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Q1. What are the contributions in "About an inverse problem for a free boundary compressible problem in hydrodynamic lubrication" ?

In this paper an inverse problem is considered for a non coercive partial differential equation, issued from a mass conservation cavitation model for a slightly compressible fluid.