1. 
 Almqvist, Andreas, et al.
(författare)

New concepts of homogenization applied in rough surface hydrodynamic lubrication
 2007

Ingår i: International Journal of Engineering Science.  Elsevier Inc..  00207225. ; 45:1

Tidskriftsartikel (refereegranskat)abstract
 This work introduces a new concept of homogenization that enables efficient analysis of the effects of surface roughness representations obtained by measurements in applications modeled by the Reynolds equation. Examples of such applications are trust and journalbearings. The numerical analysis of these types of applications requires an extremely dense computational mesh in order to resolve the surface roughness, suggesting some type of averaging. One such method is homogenization, which has been applied to Reynolds type equations with success recently. This approach is similar to the technique proposed by Patir and Cheng, who introduced flow factors determining the hydrodynamic action due to surface roughness. The difference is, however, that the present technique has a rigorous mathematical support. Moreover, the recipe to compute the averaged coefficients is simple without any ambiguities. Using either the technique proposed by Patir and Cheng or homogenization, the coefficients determining the averaged Reynolds equation are obtained by solving differential equations on a local scale. Unfortunately, this is detrimental when investigating the effects induced by real, measured, surface roughness, even though these local problems may be solved in parallel. The present work presents a solution by applying the technique based on bounds. This technique transforms the stationary Reynolds equation into two computationally feasible forms, one for the upper bound and one for the lower bound, where the flow factors are obtained by straightforward integration. Together with the preciseness of these bounds, the bounds approach becomes an eminent tool suitable for investigating the effect of real, measured, surface roughness on hydrodynamic performance.


2. 


3. 
 Lukkassen, Dag, et al.
(författare)

A new approach for estimating the friction in thin film lubrication
 2010

Ingår i: Mathematical Problems in Engineering.  Hindawi Publishing Corp..  1024123X . 1026707. ; 2010

Tidskriftsartikel (refereegranskat)abstract
 An important problem in the theory of lubrication is to model and analyze the effect of surface roughness on, for example, the friction and load carrying capacity. A direct numerical computation is often impossible since an extremely fine mesh is required to resolve the surface roughness. This suggests that one applies some averaging technique. The branch in mathematics which deals with this type of questions is known as homogenization. In this paper we present a completely new method for computing the friction. The main idea is that we study the variational problem corresponding to the Reynolds equation. We prove that the homogenized variational problem is closely related to the homogenized friction. Finally we use bounds on the homogenized Lagrangian to derive bounds for the friction. That these bounds can be used to efficiently compute the friction is demonstrated in a typical example


4. 
 Lukkassen, Dag, et al.
(författare)

Analysis of the effects of rough surfaces in compressible thin film flow by homogenization
 2011

Ingår i: International Journal of Engineering Science.  Elsevier Inc..  00207225. ; 49:5

Tidskriftsartikel (refereegranskat)abstract
 A classical problem in lubrication theory is to predict the pressure distribution in a thin fluid film between two surfaces which are in relative motion. If one of the surfaces is rough, then the distance between the surfaces is rapidly oscillating. This leads to that the governing Reynolds partial differential equation involves rapidly oscillating coefficients. The branch in mathematics which considers such types of equations is known as homogenization. In this paper we study the effects of surface roughness for a special type of compressible fluid. In particular, we derive homogenization results connected to the friction force and the load carrying capacity


5. 


6. 
 Lukkassen, Dag, et al.
(författare)

Homogenization of a class of nonlinear variational inequalities with applications in fluid film flow
 2011

Ingår i: Chinese Annals of Mathematics. Series B.  Springer.  02529599 . 1000831. ; 32:3

Tidskriftsartikel (refereegranskat)abstract
 The authors consider the homogenization of a class of nonlinear variational inequalities, which include rapid oscillations with respect to a parameter. The homogenization of the corresponding class of differential equations is also studied. The results are applied to some models for the pressure in a thin fluid film fluid between two surfaces which are in relative motion. This is an important problem in the lubrication theory. In particular, the analysis includes the effects of surface roughness on both faces and the phenomenon of cavitation. Moreover, the fluid can be modeled as Newtonian or nonNewtonian by using a Rabinowitsch fluid model


7. 
 Lukkassen, Dag, et al.
(författare)

Homogenization of some variational problems connected to the theory of lubrication
 2009

Ingår i: International Journal of Engineering Science.  Elsevier Inc..  00207225. ; 47:1

Tidskriftsartikel (refereegranskat)abstract
 An important problem in the theory of lubrication is to model and analyze the effects of surface roughness on the hydrodynamic performance. An efficient method to do this is homogenization. In this paper we prove a general homogenization result which allows us to consider unstationary variational problems, related to Reynolds type equations, where the lubricant may be Newtonian or nonNewtonian. Recently, the idea of finding upper and lower bounds on the effective behavior, obtained by homogenization, was applied for the first time in tribology. The homogenization result in this work may therefore also serve as a rigorous starting point for developing these successful results to unstationary problems.


8. 
 Lukkassen, Dag, et al.
(författare)

Mathematical analysis and homogenization of the torsion problem
 2008

Ingår i: Journal of Function Spaces and Applications.  Scientific Horizon.  09726802. ; 6:2

Tidskriftsartikel (refereegranskat)abstract
 In this paper we address the limitations of the classical formulation of the torsion problem and give a selfcontained survey of the function spaces and formulations that are more suitable for analysing the torsional behavior of composite materials. We also prove some homogenization results for the torsion problem in both the periodic and stochastic setting. Our theoretical results are illustrated by numerical examples.


9. 
 Lukkassen, Dag, et al.
(författare)

Multiscale homogenization of monotone operators
 2008

Ingår i: Discrete and Continuous Dynamical Systems. Series A.  American Institute of Mathematical Sciences.  10780947 . 15535231. ; 22:3

Tidskriftsartikel (refereegranskat)abstract
 In this paper we prove a generalization of the iterated homogenization theorem for monotone operators, proved by Lions et al. in [ ] and [ ]. Our results enable us to homogenize more realistic models of multiscale structures.


10. 
 Lukkassen, Dag, et al.
(författare)

On the role of homogenization in lubrication theory
 2014

Ingår i: 10th International conference on mathematical problems in engineering, aerosace and sciences : ICNPAA 2014 : Narvik, Norway, 15–18 July 2014.  Melville, NY : American Institute of Physics.  9780735412767 ; 11701176

Konferensbidrag (refereegranskat)abstract
 Tribology is an interdisciplinary subject dealing with science and technology of interacting surfaces in relative motion. We discuss some new mathematical tools which may be useful in the further development of this field.

