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  • 1.
    Almqvist, Andreas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Fundamentals of Physics-Informed Neural Networks Applied to Solve the Reynolds Boundary Value Problem2021In: Lubricants, E-ISSN 2075-4442, Vol. 9, no 8, article id 82Article in journal (Refereed)
    Abstract [en]

    This paper presents a complete derivation and design of a physics-informed neural network (PINN) applicable to solve initial and boundary value problems described by linear ordinary differential equations. The objective with this technical note is not to develop a numerical solution procedure which is more accurate and efficient than standard finite element- or finite difference-based methods, but to give a fully explicit mathematical description of a PINN and to present an application example in the context of hydrodynamic lubrication. It is, however, worth noticing that the PINN developed herein, contrary to FEM and FDM, is a meshless method and that training does not require big data which is typical in machine learning.

  • 2.
    Almqvist, Andreas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Homogenization of the Reynolds equation governing hydrodynamic flow in a rotating device2011In: Journal of tribology, ISSN 0742-4787, E-ISSN 1528-8897, Vol. 133, no 2, p. 021705-1Article in journal (Refereed)
    Abstract [en]

    In this paper, a method facilitating the analysis of the effects of surface roughness on the lubrication of a rotating device is presented. The analysis utilizes homogenization—a suitable technique for averaging the effects of roughness as modeled by the Reynolds equation. The originality of this work lies in a novel way of deriving the so called local problems, also known as microbearing problems. It is clearly shown how this increases the computational efficiency by eliminating the dependence of the global coordinates on the formulation of these local problems. This does not only speed up the computation, it also means that the derived flow factors or flow tensors require less storage space. To provide for good usability, alongside the flow factors for the averaged Reynolds equation, the correction factors for the averaged friction torque (and force) and the expression for averaged load carrying capacity are presented here.

  • 3.
    Almqvist, Andreas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Modelling and simulation elastohydrodynamic lubrication of tilted-pad bearings2018Conference paper (Refereed)
  • 4.
    Almqvist, Andreas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    On the effects of surface roughness in lubrication2006Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Tribology is a multidisciplinary field defined as the science and technology of interacting surfaces in relative motion, and embraces the study of friction, wear and lubrication. A typical tribological application is the rolling element bearing. Tribological contacts may also be found in other types of bearings, cam-mechanisms, gearboxes and hydraulic systems. Examples of bearings inside the human body are the operation of the human hip joint and the contact between teeth during chewing. To fully understand the operation of this type of application one has to understand the couplings between the lubricant fluid dynamics, the structural dynamics of the bearing material, the thermodynamical aspects and the resulting chemical reactions. This makes modeling tribological applications an extremely delicate task. Because of the multidisciplinary nature, such theoretical models lead to mathematical descriptions generally in the form of non-linear integro-differential systems of equations. Some of these systems of equations are sufficiently well posed to allow numerical solutions to be carried out, resulting in accurate predictions on performance. In this work, the influence on performance of a surface microscopical nature, the surface roughness, in contact interfaces between different types of machine element components is the subject of study. An example is the non-conformal lubricated contact between one of the rollers and the inner ring in a rolling element bearing. The tribological contact controlling the operation of the human hip joint is also very similar to this. Another example of a non-conformal contact occurs when driving on rainy roads, where the hydrodynamic action of the water separates the tire. To enable investigations of these types of problems, different theoretical models were studied; for the selected model, a numerical solution technique was developed within this project. This model is based on the Reynolds equation coupled with the film thickness equation. The numerical solution technique involves a multilevel technique to facilitate the solution process. Results presented in this thesis, utilizing this approach, study elementary surface features such as ridges and indentations passing each other inside the lubricated conjunction. The Reynolds equation is derived under the assumptions of thin fluid film and creeping flow, and considers in its most general form shear thinning of the lubricant. This type of equation describes the hydrodynamic action of the lubricant flow and may be used when the interfaces consist of either conformal or non-conformal conjunctions. Examples of applications having conformal interfaces are thrust- and journal- bearings or the contact between the eye and a (optical) contact lens. In such types of applications the load carried by the interface is distributed over a fairly large area that under certain circumstances helps to prevent mechanical deformation of the contacting surfaces. Such applications are said to operate in the hydrodynamic lubrication (HL) regime. Lubricant compressibility and cavitation are important aspects and have received some attention. However, the main objective when modeling HL has been to investigate and develop methods that enable the influence of surface roughness to be to be studied efficiently. Homogenization is a rigorous mathematical concept that when applied to a certain problem may be regarded as an averaging technique as well as it provides information about the induced effects of local surface roughness. Homogenization inflicts no restrictions on the surface roughness representation other than the representative part of the chosen surface roughness being assumed periodically distributed and of course the assumptions of thin film flow made through the Reynolds equation. The homogenization process leads to a two sets of equations one for the local scale describing surface roughness, scale and one for the global scale describing application geometry. The unequivocally determined coefficients of the global problem, which may be regarded as flow factors, are obtained through the solution of local problems. This makes homogenization an eminent approach to be used investigating the influence of surface roughness on hydrodynamic performance. In the present work, homogenization has been used to derive computationally feasible forms of problems originating from incompressible and compressible Reynolds type equations that describe stationary and unstationary flows in both cartezian and cylindrical co-ordinates. This technique enables simulations of surface roughness induced effects when considering surface roughness descriptions originating from measurements. Moreover, the application of homogenization facilitates the interpretation of results. Numerical investigations following the homogenization process have been carried out to verify the applicability of homogenization in hydrodynamic lubrication. Homogenization has also been shown here to enable efficient analysis of rough hydrodynamically lubricated problems. Also of note, in connection to the scientific contribution within tribology, collaboration with a group in applied mathematics has lead to the development of novel techniques in that area. These ideas have also been successfully applied, with some results presented in this thesis. At start-ups, the contact in a rolling element bearing could be both starved and drained from lubricant. In this case the hydrodynamic action becomes negligible in terms of load carrying capacity. The load is carried exclusively by surface asperities, the tribo film, or both. This is hereby modeled as the unlubricated frictionless contact between rough surfaces, i.e. a contact mechanical approach. A variational principle was used in which the real area of contact and the contact pressure distribution minimize the total complementary potential energy. The material model is linear elastic-perfectly plastic and the energy dissipation due to plastic deformation is accounted for. The numerics of this contact mechanical approach involve the fast Fourier transformation (FFT) technique in order to facilitate the solution process. Investigation results of the contact mechanics of realistic surfaces are presented in this thesis. In this investigation the variation in the real area of contact, the plasticity index and some surface roughness parameters due to applied load were studied.

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  • 5.
    Almqvist, Andreas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    On the effects of surface roughness in lubrication2009 (ed. 2)Book (Other academic)
  • 6.
    Almqvist, Andreas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Preface2021In: Proceedings of the Institution of mechanical engineers. Part J, journal of engineering tribology, ISSN 1350-6501, E-ISSN 2041-305X, Vol. 235, no 12, p. 2504-2505Article in journal (Other academic)
  • 7.
    Almqvist, Andreas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Rough surface elastohydrodynamic lubrication and contact mechanics2004Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    In the field of tribology, there are numerous theoretical models that may be described mathematically in the form of integro-differential systems of equations. Some of these systems of equations are sufficiently well posed to allow for numerical solutions to be carried out resulting in accurate predictions. This work has focused on the contact between rough surfaces with or without a separating lubricant film. The objective was to investigate how surface topography influences contact conditions. For this purpose two different numerical methods were developed and used. For the lubricated contact between rough surfaces the Reynolds equation were used as a basis. This equation is derived under the assumptions of thin fluid film and creeping flow. In highly loaded, lubricated, non- conformal contacts of surfaces after running-in, the load concentration no longer results in plastic deformations, however large elastic deformations will be apparent. It is the interaction between the hydrodynamic action of the lubricant and the elastic deformations of the surfaces that, in certain applications, enable the lubricant film to fully separate the surfaces. This is commonly referred to as full film elastohydrodynamic (EHD) lubrication. Typical machine elements that operates in the full film EHD lubrication (FL) regime include rolling element bearings, cams and gears. Unfortunately, a cost effective way of machining engineering surfaces seldom results in a surface topography that influence contact conditions in the same way as a surface after running-in. Such topographies may prevent the lubricant from fully separating the surfaces because of deteriorated hydrodynamic action. In this case the applied load is carried in part by the lubricant and in part by surface asperities and/or surface active lubricant additives. This could also be the case in lubricant starved contacts, which is a common situation in not only grease lubricated contacts but also in many liquid lubricated contacts, such as high speed operating rolling element bearings. The load sharing between the highly compressed lubricant and the surface and/or surface active lubricant additives is the reason why this lubrication regime is most commonly referred to as mixed EHD lubrication (ML). Machine elements that while running operate in the FL regime may experience a transition into the ML regime at stops or due to altered operating conditions. It is not possible to simulate direct contact between the surfaces using a numerical method based on Reynolds equation. A parameter study, of elementary surface features passing each other inside the EHD lubricated conjunction, was performed. The results obtained, even though no direct contact could be simulated, does indicate that a transition from the FL to the ML regime would occur for certain combinations of the varied parameters. At start-ups, the contact in a rolling element bearing could be both starved and drained from lubricant. In this case the hydrodynamic action becomes negligible in terms of load carrying capacity. The load is carried exclusively by surface asperities and/or surface active lubricant additives. This regime is referred to as boundary lubrication (BL). Operation conditions could also make both FL and ML impossible to achieve, for example, in the case in a low rpm operating rolling element bearing. The BL regime is in this work modeled as the unlubricated frictionless contact between rough surfaces, i.e., a dry contact approach. A variational principle was used in which the real area of contact and contact pressure distribution are those which minimize the total complementary energy. A linear elastic-perfectly plastic deformation model in which energy dissipation due to plastic deformation is accounted for was used. The dry contact method was applied to the contact between four different profiles and a plane. The variation in the real area of contact, the plasticity index and some surface roughness parameters due to applied load were investigated. The surface roughness parameters of the profiles differed significantly.

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  • 8.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Burtseva, Evgeniya
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Rajagopal, K.
    Department of Mechanical Engineering, Texas AM University, Texas, United States.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On lower-dimensional models in lubrication, Part B: Derivation of a Reynolds type of equation for incompressible piezo-viscous fluids2021In: Proceedings of the Institution of mechanical engineers. Part J, journal of engineering tribology, ISSN 1350-6501, E-ISSN 2041-305X, Vol. 235, no 8, p. 1703-1718Article in journal (Refereed)
    Abstract [en]

    The Reynolds equation is a lower-dimensional model for the pressure in a fluid confined between two adjacent surfaces that move relative to each other. It was originally derived under the assumption that the fluid is incompressible and has constant viscosity. In the existing literature, the lower-dimensional Reynolds equation is often employed as a model for the thin films, which lubricates interfaces in various machine components. For example, in the modelling of elastohydrodynamic lubrication (EHL) in gears and bearings, the pressure dependence of the viscosity is often considered by just replacing the constant viscosity in the Reynolds equation with a given viscosity-pressure relation. The arguments to justify this are heuristic, and in many cases, it is taken for granted that you can do so. This motivated us to make an attempt to formulate and present a rigorous derivation of a lower-dimensional model for the pressure when the fluid has pressure-dependent viscosity. The results of our study are presented in two parts. In Part A, we showed that for incompressible and piezo-viscous fluids it is not possible to obtain a lower-dimensional model for the pressure by just assuming that the film thickness is thin, as it is for incompressible fluids with constant viscosity. Here, in Part B, we present a method for deriving lower-dimensional models of thin-film flow, where the fluid has a pressure-dependent viscosity. The main idea is to rescale the generalised Navier-Stokes equation, which we obtained in Part A based on theory for implicit constitutive relations, so that we can pass to the limit as the film thickness goes to zero. If the scaling is correct, then the limit problem can be used as the dimensionally reduced model for the flow and it is possible to derive a type of Reynolds equation for the pressure.

  • 9.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Burtseva, Evgeniya
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Rajagopal, Kumbakonam
    J. Mike Walker’66 Department of Mechanical Engineering, Texas A&M University, 100 Mechanical Engineering, Office Building, 3123 TAMU, College Station, TX 77843-3123, TX, USA.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On flow of power-law fluids between adjacent surfaces: Why is it possible to derive a Reynolds-type equation for pressure-driven flow, but not for shear-driven flow?2023In: Applications in Engineering Science, ISSN 2666-4968, Vol. 15, article id 100145Article in journal (Refereed)
    Abstract [en]

    Flows of incompressible Navier–Stokes (Newtonian) fluids between adjacent surfaces are encountered in numerous practical applications, such as seal leakage and bearing lubrication. In seals, the flow is primarily pressure-driven, whereas, in bearings, the dominating driving force is due to shear. The governing Navier–Stokes system of equations can be significantly simplified due to the small distance between the surfaces compared to their size. From the simplified system, it is possible to derive a single lower-dimensional equation, known as the Reynolds equation, which describes the pressure field. Once the pressure field is computed, it can be used to determine the velocity field. This computational algorithm is much simpler to implement than a direct numerical solution of the Navier–Stokes equations and is therefore widely employed by engineers. The primary objective of this article is to investigate the possibility of deriving a type of Reynolds equation also for non-Newtonian fluids, using the balance of linear momentum. By considering power-law fluids we demonstrate that it is not possible for shear-driven flows, whereas it is feasible for pressure-driven flows. Additionally, we demonstrate that in the full 3D model, a normal stress boundary condition at the inlet/outlet implies a Dirichlet condition for the pressure in the Reynolds equation associated with pressure-driven flow. Furthermore, we establish that a Dirichlet condition for the velocity at the inlet/outlet in the 3D model results in a Neumann condition for the pressure in the Reynolds equation.

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  • 10.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Burtseva, Evgeniya
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Rajagopal, Kumbakonam
    Department of Mechanical Engineering, Texas A&M University, Texas, USA.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On lower-dimensional models in lubrication, Part A: Common misinterpretations and incorrect usage of the Reynolds equation2021In: Proceedings of the Institution of mechanical engineers. Part J, journal of engineering tribology, ISSN 1350-6501, E-ISSN 2041-305X, Vol. 235, no 8, p. 1692-1702Article in journal (Refereed)
    Abstract [en]

    Most of the problems in lubrication are studied within the context of Reynolds’ equation, which can be derived by writing the incompressible Navier-Stokes equation in a dimensionless form and neglecting terms which are small under the assumption that the lubricant film is very thin. Unfortunately, the Reynolds equation is often used even though the basic assumptions under which it is derived are not satisfied. One example is in the mathematical modelling of elastohydrodynamic lubrication (EHL). In the EHL regime, the pressure is so high that the viscosity changes by several orders of magnitude. This is taken into account by just replacing the constant viscosity in either the incompressible Navier-Stokes equation or the Reynolds equation by a viscosity-pressure relation. However, there are no available rigorous arguments which justify such an assumption. The main purpose of this two-part work is to investigate if such arguments exist or not. In Part A, we formulate a generalised form of the Navier-Stokes equation for piezo-viscous incompressible fluids. By dimensional analysis of this equation we, thereafter, show that it is not possible to obtain the Reynolds equation, where the constant viscosity is replaced with a viscosity-pressure relation, by just neglecting terms which are small under the assumption that the lubricant film is very thin. The reason is that the lone assumption that the fluid film is very thin is not enough to neglect the terms, in the generalised Navier-Stokes equation, which are related to the body forces and the inertia. However, we analysed the coefficients in front of these (remaining) terms and provided arguments for when they may be neglected. In Part B, we present an alternative method to derive a lower-dimensional model, which is based on asymptotic analysis of the generalised Navier-Stokes equation as the film thickness goes to zero.

  • 11.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Burtseva, Evgeniya
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Rajagopal, Kumbakonam
    Department of Mechanical Engineering, Texas A&M University, College Station, TX, USA.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On lower-dimensional models of thin film flow, Part C: Derivation of a Reynolds type of equation for fluids with temperature and pressure dependent viscosity2023In: Proceedings of the Institution of mechanical engineers. Part J, journal of engineering tribology, ISSN 1350-6501, E-ISSN 2041-305X, Vol. 237, no 3, p. 514-526Article in journal (Refereed)
    Abstract [en]

    This paper constitutes the third part of a series of works on lower-dimensional models in lubrication. In Part A, it was shown that implicit constitutive theory must be used in the modelling of incompressible fluids with pressure-dependent viscosity and that it is not possible to obtain a lower-dimensional model for the pressure just by letting the film thickness go to zero, as in the proof of the classical Reynolds equation. In Part B, a new method for deriving lower-dimensional models of thin-film flow of fluids with pressure-dependent viscosity was presented. Here, in Part C, we also incorporate the energy equation so as to include fluids with both temperature and pressure dependent viscosity. By asymptotic analysis of this system, as the film thickness goes to zero, we derive a simplified model of the flow. We also carry out an asymptotic analysis of the boundary condition, in the case where the normal stress is specified on one part of the boundary and the velocity on the remaining part.

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  • 12.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Burtseva, Evgeniya
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Ràfols, Francesc Pérez
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    New insights on lubrication theory for compressible fluids2019In: International Journal of Engineering Science, ISSN 0020-7225, E-ISSN 1879-2197, Vol. 145, article id 103170Article in journal (Refereed)
    Abstract [en]

    The fact that the film is thin is in lubrication theory utilised to simplify the full Navier–Stokes system of equations. For incompressible and iso-viscous fluids, it turns out that the inertial terms are small enough to be neglected. However, for a compressible fluid, we show that the influence of inertia depends on the (constitutive) density-pressure relationship and may not always be neglected. We consider a class of iso-viscous fluids obeying a power-law type of compressibility, which in particular includes both incompressible fluids and ideal gases. We show by scaling and asymptotic analysis, that the degree of compressibility determines whether the terms governing inertia may or may not be neglected. For instance, for an ideal gas, the inertial terms remain regardless of the film height-to-length ratio. However, by means of a specific modified Reynolds number that we define we show that the magnitudes of the inertial terms rarely are large enough to be influential. In addition, we consider fluids obeying the well-known Dowson and Higginson density-pressure relationship and show that the inertial terms can be neglected, which allows for obtaining a Reynolds type of equation. Finally, some numerical examples are presented in order to illustrate our theoretical results.

  • 13.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Campañá, C
    University of Ottawa, Department of Chemistry, Ottawa, Canada K1N6N5.
    Prodanov, N
    Sumy State University, 2 Rimskii-KorsakovStr., 40007 Sumy, Ukraine.
    Persson, B N J
    IFF, Jülich.
    Interfacial separation between elastic solids with randomly rough surfaces: Comparison between theory and numerical techniques2011In: Journal of the mechanics and physics of solids, ISSN 0022-5096, E-ISSN 1873-4782, Vol. 59, no 11, p. 2355-2369Article in journal (Refereed)
    Abstract [en]

    We study the distribution of interfacial separations at the contact region between two elastic solids with randomly rough surfaces. An analytical expression is derived for the distribution of interfacial separations using Persson's theory of contact mechanics, and is compared to numerical solutions obtained using (a) a half-space method based on the Boussinesq equation, (b) a Green's function molecular dynamics technique and (c) smart-block classical molecular dynamics. Overall, we find good agreement between all the different approaches.

  • 14.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Dasht, Johan
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    The homogenization process of the Reynolds equation describing compressible liquid flow2006In: Tribology International, ISSN 0301-679X, E-ISSN 1879-2464, Vol. 39, no 9, p. 994-1002Article in journal (Refereed)
    Abstract [en]

    This paper summarizes the homogenization process of rough, hydrodynamic lubrication problems governed by the Reynolds equation used to describe compressible liquid flow. Here, the homogenized equation describes the limiting result when the wavelength of a modeled surface roughness goes to zero. The lubricant film thickness is modeled by one part describing the geometry/shape of the bearing and a periodic part describing the surface topography/roughness. By varying the periodic part as well as its wavelength, we can try to systematically investigate the applicability of homogenization on this type of problem. The load carrying capacity is the target parameter; deterministic solutions are compared to homogenized by this measure. We show that the load carrying capacity rapidly converges to the homogenized results as the wavelength decreases, proving that the homogenized solution gives a very accurate representation of the problem when real surface topographies are considered

  • 15.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Dasht, Johan
    Glavatskih, Sergei
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Larsson, Roland
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Marklund, Pär
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Persson, Lars-Erik
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Sahlin, Fredrik
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Homogenization of the Reynolds equation2005Report (Other academic)
  • 16.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Essel, Emmanuel Kwame
    Department of Mathematics and Statistics, University of Cape Coast.
    Fabricius, John
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Multiscale homogenization of a class of nonlinear equations with applications in lubrication theory and applications2011In: Journal of Function Spaces and Applications, ISSN 0972-6802, E-ISSN 1758-4965, Vol. 9, no 1, p. 17-40Article in journal (Refereed)
    Abstract [en]

    We prove a homogenization result for monotone operators by using the method of multiscale convergence. More precisely, we study the asymptotic behavior as epsilon -> 0 of the solutions u(epsilon) of the nonlinear equation div a(epsilon)(x, del u(epsilon)) = div b(epsilon), where both a(epsilon) and b(epsilon) oscillate rapidly on several microscopic scales and a(epsilon) satisfies certain continuity, monotonicity and boundedness conditions. This kind of problem has applications in hydrodynamic thin film lubrication where the bounding surfaces have roughness on several length scales. The homogenization result is obtained by extending the multiscale convergence method to the setting of Sobolev spaces W-0(1,p)(Omega), where 1 < p < infinity. In particular we give new proofs of some fundamental theorems concerning this convergence that were first obtained by Allaire and Briane for the case p = 2.

  • 17.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Essel, Emmanuel Kwame
    Fabricius, John
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Reiterated homogenization applied in hydrodynamic lubrication2008In: Proceedings of the Institution of mechanical engineers. Part J, journal of engineering tribology, ISSN 1350-6501, E-ISSN 2041-305X, Vol. 222, no 7, p. 827-841Article in journal (Refereed)
    Abstract [en]

    This work is devoted to studying the combined effect that arises due to surface texture and surface roughness in hydrodynamic lubrication. An effective approach in tackling this problem is by using the theory of reiterated homogenization with three scales. In the numerical analysis of such problems, a very fine mesh is needed, suggesting some type of averaging. To this end, a general class of problems is studied that, e.g. includes the incompressible Reynolds problem in both artesian and cylindrical coordinate forms. To demonstrate the effectiveness of the method several numerical results are presented that clearly show the convergence of the deterministic solutions towards the homogenized solution.Moreover, the convergence of the friction force and the load carrying capacity of the lubricant film is also addressed in this paper. In conclusion, reiterated homogenization is a feasible mathematical tool that facilitates the analysis of this type of problem.

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  • 18.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Essel, Emmanuel Kwame
    Fabricius, John
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Reiterated homogenization of a nonlinear Reynolds-type equation2008Report (Other academic)
  • 19.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Essel, Emmanuel Kwame
    Department of Mathematics and Statistics, University of Cape Coast.
    Fabricius, John
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Variational bounds applied to unstationary hydrodynamic lubrication2008In: International Journal of Engineering Science, ISSN 0020-7225, E-ISSN 1879-2197, Vol. 46, no 9, p. 891-906Article in journal (Refereed)
    Abstract [en]

    This paper is devoted to the effects of surface roughness in hydrodynamic lubrication. The numerical analysis of such problems requires a very fine mesh to resolve the surface roughness, hence it is often necessary to do some type of averaging. Previously, homogenization (a rigorous form of averaging) has been successfully applied to Reynolds type differential equations. More recently, the idea of finding upper and lower bounds on the effective behavior, obtained by homogenization, was applied for the first time in tribology. In these pioneering works, it has been assumed that only one surface is rough. In this paper we develop these results to include the unstationary case where both surfaces may be rough. More precisely, we first use multiple-scale expansion to obtain a homogenization result for a class of variational problems including the variational formulation associated with the unstationary Reynolds equation. Thereafter, we derive lower and upper bounds corresponding to the homogenized (averaged) variational problem. The bounds reduce the numerical analysis, in that one only needs to solve two smooth problems, i.e. no local scale has to be considered. Finally, we present several examples, where it is shown that the bounds can be used to estimate the effects of surface roughness with very high accuracy.

  • 20.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Essel, Emmanuel Kwame
    Persson, Lars-Erik
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Homogenization of the unstationary incompressible Reynolds equation2007In: Tribology International, ISSN 0301-679X, E-ISSN 1879-2464, Vol. 40, no 9, p. 1344-1350Article in journal (Refereed)
    Abstract [en]

    This paper is devoted to the effects of surface roughness during hydrodynamic lubrication. In the numerical analysis a very fine mesh is needed to resolve the surface roughness, suggesting some type of averaging. A rigorous way to do this is to use the general theory of homogenization. In most works about the influence of surface roughness, it is assumed that only the stationary surface is rough. This means that the governing Reynolds type equation does not involve time. However, recently, homogenization was successfully applied to analyze a situation where both surfaces are rough and the lubricant is assumed to have constant bulk modulus. In this paper we will consider a case where both surfaces are assumed to be rough, but the lubricant is incompressible. It is also clearly demonstrated, in this case that homogenization is an efficient approach. Moreover, several numerical results are presented and compared with those corresponding to where a constant bulk modulus is assumed to govern the lubricant compressibility. In particular, the result shows a significant difference in the asymptotic behavior between the incompressible case and that with constant bulk modulus.

  • 21.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Fabricius, John
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Larsson, Roland
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    A new approach for studying cavitation in lubrication2014In: Journal of tribology, ISSN 0742-4787, E-ISSN 1528-8897, Vol. 136, no 1, article id 11706Article in journal (Refereed)
    Abstract [en]

    The underlying theory, in this paper, is based on clear physical arguments related to conservation of mass flow and considers both incompressible and compressible fluids. The result of the mathematical modeling is a system of equations with two unknowns, which are related to the hydrodynamic pressure and the degree of saturation of the fluid. Discretization of the system leads to a linear complementarity problem (LCP), which easily can be solved numerically with readily available standard methods and an implementation of a model problem in matlab code is made available for the reader of the paper. The model and the associated numerical solution method have significant advantages over today's most frequently used cavitation algorithms, which are based on Elrod-Adams pioneering work

  • 22.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Fabricius, John
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Larsson, Roland
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Reynolds equation flow factor estimates by means of homogenization2010In: ASIATRIB 2010: Frontiers in tribology - knowledge & friendship . proceedings of the fourth Asia International Conference on Tribology, 5-9 December 2010, Perth, Western Australia, 2010, p. 185-Conference paper (Refereed)
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  • 23.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Fabricius, John
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Lundström, Staffan
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Flow in thin domains with a microstructure: Lubrication and thin porous media2017In: AIP Conference Proceedings, ISSN 0094-243X, E-ISSN 1551-7616, Vol. 1798, article id 020172Article in journal (Refereed)
    Abstract [en]

    This paper is devoted to homogenization of different models of flow in thin domains with a microstructure. The focus is on applications connected to the effect of surface roughness in full film lubrication, but a parallel to flow in thin porous media is also discussed. Mathematical models of such flows naturally include two small parameters. One is connected to the fluid film thickness and the other to the microstructure. The corresponding asymptotic analysis is a delicate problem, since the result depends on how fast the two small parameters tend to zero relative to each other. We give a review of the current status in this area and point out some future challenges.

  • 24.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Fabricius, John
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Spencer, Andrew
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Similarities and differences between the flow factor method by Patir and Cheng and homogenization2011In: Journal of tribology, ISSN 0742-4787, E-ISSN 1528-8897, Vol. 133, no 3, p. 031702-1Article in journal (Refereed)
    Abstract [en]

    Different averaging techniques have proved to be useful for analyzing the effects of surface roughness in hydrodynamic lubrication. This paper compares two of these averaging techniques, namely the flow factor method by Patir and Cheng (P&C) and homogenization. It has been rigorously proved by many authors that the homogenization method provides a correct solution for arbitrary roughness. In this work it is shown that the two methods coincide if and only if the roughness exhibits certain symmetries. Hence, homogenization is always the preferred method.

  • 25.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Fabricius, John
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Homogenization of a Reynolds equation describing compressible flow2011Report (Other academic)
  • 26.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Fabricius, John
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Homogenization of a Reynolds equation describing compressible flow2012In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 390, no 2, p. 456-471Article in journal (Refereed)
    Abstract [en]

    We homogenize a Reynolds equation with rapidly oscillating film thickness function hε, assuming a constant compressiblity factor in the pressure-density relation. The oscillations are due to roughness on the bounding surfaces of the fluid film. As shown by previous studies, homogenization is an effective approach for analyzing the effects of surface roughness in hydrodynamic lubrication. By two-scale convergence theory we obtain the limit problem (homogenized equation) and strong convergence in L2 for the unknown density ρε. By adding a small corrector term we also obtain strong convergence in the Sobolev norm.

  • 27.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Glavatskih, Sergei
    Larsson, Roland
    Marklund, Pär
    Sahlin, Fredrik
    Dasht, Johan
    Persson, Lars-Erik
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Homogenization of Reynolds equation2005Report (Other academic)
  • 28.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Hardell, Jens
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Prakash, Braham
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Selected papers from those presented at the 3rd International Tribology Symposium of IFToMM2014In: Lubrication Science, ISSN 0954-0075, E-ISSN 1557-6833, Vol. 26, no 5, p. 273-275Article in journal (Other academic)
    Abstract [en]

    Selected papers presented at the 3rd International Tribology Symposium of the International Federation for the Promotion of Mechanism and Machine Science (IFToMM) are discussed. The 3rd International Tribology Symposium of IFToMM was organized by the Division of Machine Elements of Luleå University of Technology in Sweden from March 19 21, 2013. The symposium was held in the university campus, providing a perfect forum for the tribologists for presenting their latest research and exchange of ideas. The participants were also given the opportunity to experience the snowy landscape during a memorable outdoor dinner. The symposium attracted 146 participants from 26 countries along with 92 papers spread over 32 sessions. These papers covered the fundamental and applied aspects of wide ranging topics such as friction, wear, lubrication, lubricants, tribology in hostile environment, tribomaterials, solid lubricants, surface engineering, and tribotesting.

  • 29. Almqvist, Andreas
    et al.
    Larsson, Roland
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    The effect of two-sided roughness in rolling/sliding EHL line contacts2004In: Transient processes in tribology: proceedings of the 30th Leeds-Lyon Symposium on Tribology / [ed] Gerard Dalmaz, Amstersam: Elsevier, 2004Conference paper (Refereed)
    Abstract [en]

    In most theoretical studies carried out to date on the effect of surface roughness in elastohydrodynamic lubrication (EHL) one surface is considered smooth and one as being rough. In real tribological contacts however, both surfaces normally have similar roughness heights. When modelling a rolling contact it is possible to simply sum the roughness of the two contact surfaces but in a sliding EHL contact, a continuously changing effective surface roughness occurs. The aim of this work was to investigate the influence of elementary surface features such as dents and ridges on the film thickness and pressure. This was done numerically using transient non-Newtonian simulations of an EHL line contact using a coupled smoother combined with a multilevel technique. Four different "overtaking" phenomena were investigated; ridge-ridge, dent-ridge, ridge-dent, and dent-dent. It was shown that the minimum film-thickness produced by a ridge is further reduced in a dent-ridge overtaking event. The squeeze effect seen in the ridge-ridge case resulted in large deformations and film-thickness heights comparable to the corresponding smooth case just before the overtaking event occurred. These local effects arising from simulating two-sided roughness were compared to simulations using a traditional "one-sided rough surface contacting a perfectly smooth surface.".

  • 30. Almqvist, Andreas
    et al.
    Larsson, Roland
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Sahlin, Fredrik
    Development of a lubrication simulation model2009In: Svenska mekanikdagarna: Södertälje 2009, Stockholm: Svenska nationalkommittén för mekanik , 2009, p. 74-Conference paper (Other academic)
  • 31. Almqvist, Andreas
    et al.
    Larsson, Roland
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    The homogenization process of the time dependent Reynolds equation describing compressible liquid flow2006Report (Other academic)
  • 32.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Larsson, Roland
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    The homogenization process of the time dependent Reynolds equation describing compressible liquid flow2007In: Tribologia : Finnish Journal of Tribology, ISSN 0780-2285, Vol. 26, no 4, p. 30-44Article in journal (Refereed)
    Abstract [en]

    To increase the hydrodynamic performance in different machine elements during lubrication, e.g. journal bearings and thrust bearings, it is important to understand the influence of surface roughness. In this connection one encounters different approaches commonly based on some form of the Reynolds equation. They may generally be divided into deterministic- and averaging- techniques. The former regards all surface roughness information and provides a detailed understanding of the local effects that arise. The latter method is suitable when investigating how the surface roughness affects performance of the machine element as a whole. Homogenization is a rigorous mathematical concept that when applied to a certain problem may be thought of as an averaging technique also providing information about local effects. In this work the compressible time dependent Reynolds equation is homogenized. Related problems have recently been analyzed by homogenization techniques under various assumptions. In the present paper the compressibility is modeled assuming a constant lubricant bulk modulus. The formal method of multiple scale expansion is used to derive a so-called homogenized equation and a numerical solution method to solve both the deterministic problem and the homogenized problem is implemented. The numerical results clearly show that the solution of the homogenized equation is a suitable approximation to the solution of the deterministic problem. It is also demonstrated that for small values of the roughness wavelength, the homogenization technique is superior, since the solution of the deterministic problem requires an extremely fine discretization mesh. More over, the solution of the time dependent homogenized problem may in some cases be reduced to solve a stationary problem that facilitates the solution process and interpretation of results.

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  • 33.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Lukkassen, Dag
    Meidell, Annette
    Narvik University College, 8505 Narvik, Norway.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    New concepts of homogenization applied in rough surface hydrodynamic lubrication2007In: International Journal of Engineering Science, ISSN 0020-7225, E-ISSN 1879-2197, Vol. 45, no 1, p. 139-154Article in journal (Refereed)
    Abstract [en]

    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 journal-bearings. 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.

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  • 34.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Pellegrini, Barbara
    Department of Neurosciences, Biomedicine and Movement Sciences, University of Verona, Verona, Italy; CeRiSM, Sport Mountain and Health Research Centre, University of Verona, Verona, Italy.
    Lintzén, Nina
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics.
    Emami, Nazanin
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Holmberg, H-C
    Luleå University of Technology, Department of Health, Learning and Technology, Health, Medicine and Rehabilitation. School of Kinesiology, University of British Columbia, Vancouver, BC, Canada.
    Larsson, Roland
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    A Scientific Perspective on Reducing Ski-Snow Friction to Improve Performance in Olympic Cross-Country Skiing, the Biathlon and Nordic Combined2022In: Frontiers in Sports and Active Living, E-ISSN 2624-9367, Vol. 4, article id 844883Article in journal (Refereed)
    Abstract [en]

    Of the medals awarded at the 2022 Winter Olympics in Beijing, 24% were for events involving cross-country skiing, the biathlon and Nordic combined. Although much research has focused on physiological and biomechanical characteristics that determine success in these sports, considerably less is yet known about the resistive forces. Here, we specifically describe what is presently known about ski-snow friction, one of the major resistive forces. Today, elite ski races take place on natural and/or machine-made snow. Prior to each race, several pairs of skis with different grinding and waxing of the base are tested against one another with respect to key parameters, such as how rapidly and for how long the ski glides, which is dependent on ski-snow friction. This friction arises from a combination of factors, including compaction, plowing, adhesion, viscous drag, and water bridging, as well as contaminants and dirt on the surface of and within the snow. In this context the stiffness of the ski, shape of its camber, and material composition and topography of the base exert a major influence. An understanding of the interactions between these factors, in combination with information concerning the temperature and humidity of both the air and snow, as well as the nature of the snow, provides a basis for designing specific strategies to minimize ski-snow friction. In conclusion, although performance on “narrow skis” has improved considerably in recent decades, future insights into how best to reduce ski-snow friction offer great promise for even further advances.

  • 35.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Ràfols, Francesc Pérez
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Modelling Flows in Lubrication2020In: Modeling and Simulation of Tribological Problems in Technology / [ed] Marco Paggi & David Hills, Springer, 2020, 1, p. 229-278Chapter in book (Other academic)
    Abstract [en]

    This chapter introduces the reader to lubrication theory and describes the governing equations, models and methods that can be used to simulate various types of lubricated systems. It starts with an introduction to the tribological contact and to the different lubrication regimes. The basis for the classical lubrication theory is then given and thereafter follows a presentation of how to obtain the Reynolds equation by means of scaling and asymptotic analysis of the Navier–Stokes equations. After having obtained the Reynolds equation, a quite elaborate presentation of cavitation algorithms is given. It includes discretisation and presents the analytical solution for a pocket bearing as a benchmark model problem. Then, the concept of homogenisation of surface roughness is introduced. This starts from the simplest iso-viscous and incompressible case, expands to include compressibility with a constant bulk modulus constitutive relation and then also addresses the case of ideal gases. Thereafter, the relation between homogenised coefficients and the Patir and Cheng flow factors is described and finally it is shown how to incorporate the effect of mixed lubrication into the model.

  • 36.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Ràfols, Francesc Pérez
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Scientific Computing with Applications in Tribology: A course compendium2022Other (Other academic)
    Abstract [en]

    This compendium comprises models and numerical solution procedure for tribological interfaces. It describes the tribological contact and the classical lubrication regimes. A thorough derivation of the Reynolds equation, governing the fluid pressure, from the Navier-Stokes momentum equations and the continuity equation for conservation of mass, is presented along with its analytical solution for the infinitely wide linear slider bearing.

    The compilation of the compendium was conducted by the first author during his tenure as Professor at the Division of Machine Elements, Department of Engineering Sciences and Mathematics, Luleå University of Technology and by the second author during his tenure as a postdoctoral researcher at the same division.

    Although the compilation of this text is the work solely of the authors, the models and solution procedure presented herein is joint development of many good colleagues and co-authors. Our sincere gratitude is extended towards them all.

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    Compendium v9
  • 37.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Sahlin, Fredrik
    Larsson, Roland
    An Abbott curve based rough surface contact mechanics approach2005In: Proceedings of the World Tribology Congress III - 2005: presented at ..., September 12 - 16, 2005, Washington, DC, New York: American Society of Mechanical Engineers , 2005, Vol. Paper no 64038, p. 397-398Conference paper (Refereed)
    Abstract [en]

    In this way all the height information of the surface profile is preserved and not only a few parameters, like Ra, Rq, Rz, Rsk, etc. The aim of this work is to investigate how classes of surfaces based on a single Abbott curve perform in terms of contact mechanical parameters like the real area of contact. The result shows that surfaces taken from a class of random surfaces generated from a specific Abbott curve behaves similar in a contact mechanics simulation. That is, the distribution of for example the real area of contact within such a class is compact, having a small deviation from its mean.This implies that it is possible to simulate classes of surfaces based on Abbott curves and to use the results to predict contact mechanical properties of real surface topographies.

  • 38.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Sahlin, Fredrik
    Larsson, Roland
    Glavatskih, Sergei
    On the dry elasto-plastic contact of nominally flat surfaces2004In: Proceedings of the 11th Nordic symposium on tribology: NORDTRIB 2004 : Tromsø, Harstad, Hurtigruten, [Bodø], Norway, June [1 - 5], 2004, 2004, p. 753-762Conference paper (Refereed)
  • 39.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Sahlin, Fredrik
    Larsson, Roland
    Glavatskih, Sergei
    On the dry elasto-plastic contact of nominally flat surfaces2007In: Tribology International, ISSN 0301-679X, E-ISSN 1879-2464, Vol. 40, no 4, p. 574-579Article in journal (Refereed)
    Abstract [en]

    A model to be used for numerical simulation of the contact of linear elastic perfectly plastic rough surfaces was developed. Energy dissipation due to plastic deformation is taken into account. Spectral theory and an FFT-techique are used to facilitate the numerical solution process. Results of simulations using four two-dimensional profiles with different topographies in contact with a rigid plane for a number loads are reported. From the results it is clear that the real area of contact (Ar) changes almost linearly with load and is only slightly affected by the difference in topography. A plasticity index is defined as the ratio of plastically deformed area (Ap) and Ar. Plastic deformation occurs even at low loads and there is a significant difference in plasticity index between the surface profiles considered. An investigation on how the spectral content of the surface profile influences the results presented is also performed. This is to ensure that the metrological limitations of the optical profiler used to measure the surfaces do not have a significant influence. It is concluded that the highest frequencies of the measured profile have a negligible influence on the real area of contact.

  • 40. Almqvist, Andreas
    et al.
    Taylor, R.I.
    Shell Global Solutions, UK.
    Larsson, Roland
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Simulation of piston ring: cylinder liner lubrication considering layered fluid films2009In: Tribologia : Finnish Journal of Tribology, ISSN 0780-2285, Vol. 28, no 3-4, p. 44-58Article in journal (Refereed)
    Abstract [en]

    During the operation of hydrodynamically lubricated devices a fully formulated lubricant has the ability to form layers at the surfaces. A friction modifier's task is to adjust the interaction between lubricant and the surface so that friction is lowered. An antiwear additive creates a protective layer on the surface and this definitely influence the performance of the lubricated device. To gain fundamental understanding, models that address the modified liquid - solid interaction due to the formation of layers, but also models that may be used to study the effects of layers already formed on the contacting surfaces are required. In this paper, two non-Newtonian lubricant rheology models that may be used to simulate reacted layers resembling those created by lubricant additives are adopted for the simulation of the piston ring - cylinder liner lubrication problem. The possibility of layer to layer interaction, which is likely to occur in the convex conjunction between the ring and the liner, is considered and this extends the models found in the literature. The effects induced by this type of layering are studied by using a modified Reynolds' equation where the coefficients have been corrected with factors that accounts for the layer properties. This enables, effectively, studies of layers resembling those created by lubricant additives during the operation of the lubricated conjunction between a piston ring and a cylinder liner.

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  • 41.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Taylor, Robert Ian
    Shell Global Solutions, UK.
    Larsson, Roland
    Simulation of piston ring: cylinder liner lubrication considering layered fluid films2008In: Proceedings of the 35th Leeds-Lyon Symposium on Tribology, 2008, 2008Conference paper (Other academic)
    Abstract [en]

    During the operation of hydrodynamically lubricated devices a fully formulated lubricant has the ability to form layers at the surfaces. Such layers alter the interaction between the lubricant and the surface that definitely will influence the performance of the lubricated device.To gain fundamental understanding, models that address the formation of layers and the altered liquid – solid interaction, but also models that may be used to study the effects of existing layers are required. In this paper, non-Newtonian lubricant rheology models that may be used to resemble layers of variable shear strength – wall-slip specifically – are considered for the simulation of the piston ring - cylinder liner lubrication problem.The effects induced by this type of layering are studied by using a modified Reynold’s equation where the coefficients have been corrected with factors that accounts for layer properties. This enables, effectively, studies of immobile layers as well as wall-slip in the lubricated conjunction between a piston ring and a cylinder liner.

  • 42.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    A new concept in cavitation modelling2013In: Tribo Lyon 2013: book of abstracts : a joint event of WTC 2013, Lyon, 2013, p. 170-Conference paper (Refereed)
  • 43.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Homogenization applied in rough surface hydrodynamic lubrication2007In: Svenska Mekanikdagar 2007: Program och abstracts / [ed] Niklas Davidsson; Elianne Wassvik, Luleå: Luleå tekniska universitet, 2007, p. 31-Conference paper (Other academic)
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    FULLTEXT01
  • 44.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Homogenization of the Reynolds equation2013In: Encyclopedia of Tribology, Berlin: Springer-Verlag New York Inc. , 2013, p. 1685-1690Chapter in book (Refereed)
  • 45.
    Almqvist, Andreas
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Wall, Peter
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Modelling cavitation in (elasto)hydrodynamic lubrication2016In: Advances in tribology / [ed] Pranav H. Darji, Croatia: INTECH, 2016, p. 198-213Chapter in book (Refereed)
    Abstract [en]

    In this chapter we will present a derivation of a mathematical model describing how cavitation influences the pressure distribution in a thin lubricant film between two moving surfaces. The main idea in the derivation is to first describe the influence of cavitation on the mass flow and thereafter using a conservation law for the mass. This leads to a nonlinear system with two complementary variables: one is the pressure distribution and the other is related to the density, i.e. a nonlinear complementarity problem (NLCP). The proposed approach is used to derive a mass conserving cavitation model considering that density, viscosity and film thickness of the lubricant depend on the pressure. To demonstrate the applicability and evaluate the proposed model and the suggested numerical implementation, a few model problems are analysed and presented.

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    Modelling Cavitation in (Elasto)Hydrodynamic Lubrication
  • 46. Almqvist, Torbjörn
    et al.
    Almqvist, Andreas
    Larsson, Roland
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    A comparison between computational fluid dynamic and Reynolds approaches for simulating transient EHL line contacts2004In: Tribology International, ISSN 0301-679X, E-ISSN 1879-2464, Vol. 37, no 1, p. 61-69Article in journal (Refereed)
    Abstract [en]

    When simulating elastohydrodynamic lubrication (EHL), the Reynolds equation is the predominating partial differential equation for prediction of the fluid flow. Also very few attempts have been carried out using the full momentum and continuity equations separately. The aim of this investigation is to compare two different approaches for simulation of EHL line contacts where a single ridge travels through an EHL conjunction. One of the approaches is based on the Reynolds equation, addressing the coupling between the pressure and the film thickness. The solver uses the advantages of multilevel techniques to speed up the convergence rate. The other approach is based on commercial CFD software. The software uses the momentum and continuity equations in their basic form, enabling numerical simulations outside the contact regions, as well as in the thin film region to be carried out. The numerical experiments show that, under the running conditions chosen, only small deviations between the two approaches can be observed. The results are encouraging from several viewpoints: validation of the codes, the possibilities of further developments of the CFD approach and the justification of using a Reynolds approach under the running conditions chosen

  • 47.
    Andersson, Joel
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Almqvist, Andreas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Larsson, Roland
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Numerical simulation of a wear experiment2011In: Wear, ISSN 0043-1648, E-ISSN 1873-2577, Vol. 271, no 11-12, p. 2947-2952Article in journal (Refereed)
    Abstract [en]

    A wear model including a deterministic FFT-accelerated contact mechanical tool to calculate pressure and elastic-plastic deformation, is employed to simulate the time dependent wear in a sphere on flat contact. The results of the wear simulations compared to experimental results from a reciprocating test in a ball on disk tribometer. The conditions of the simulations and the experiments are independently adjusted to match up. Similarities and differences shows upon the usefulness and limitation of wearmodelling of this type.

  • 48.
    Andersson, Joel
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Almqvist, Andreas
    Larsson, Roland
    Simulation of a wear experiment2010In: 14th Nordic Symposium on Tribology: NORDTRIB 2010 : Storforsen, Sweden, June 8-11, 2010, Luleå: Luleå tekniska universitet, 2010Conference paper (Refereed)
    Abstract [en]

    By using a deterministic FFT-accelerated contact mechanical tool to calculate pressure and elastic-plastic deformation, a wear model is utilized to simulate the time dependent wear from a sphere on at contact. The results of the simulated wear are compared to experimental results form a SRV ball on disk tribometer, from which worn surfaces are optically measured. The conditions of the simulation and the experiments are independently adjusted to match. Agreement and diversity shows upon the usefulness and limitation of wear modeling of this type.

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  • 49.
    Andersson, Joel
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Larsson, Roland
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Almqvist, Andreas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Grahn, Mattias
    Luleå University of Technology, Department of Civil, Environmental and Natural Resources Engineering, Sustainable Process Engineering.
    Minami, Ichiro
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Semi-deterministic chemo-mechanical model of boundary lubrication2012In: Faraday discussions, ISSN 1359-6640, E-ISSN 1364-5498, Vol. 156, p. 343-360Article in journal (Refereed)
    Abstract [en]

    A model for tribofilm growth is developed. The model is used in combination with numerical contact mechanics tools to enable evaluation of the combined effects of chemistry and contact mechanics. The model is tuned with experimental data and is thereafter applied to rough surfaces. The growth of the tribofilm is evaluated for 3 different contact cases and short-term tribofilm growth behaviour is analyzed. The results show how tribofilms grow in patches. The model is expected to be used as a tool for analysis of the interaction between rough surfaces.

  • 50.
    Avan, Evan Y.
    et al.
    Leonardo Centre for Tribology, Department of Mechanical Engineering, University of Sheffield.
    Spencer, Andrew
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Dwyer-Joyce, Rob S.
    Leonardo Centre for Tribology, Department of Mechanical Engineering, University of Sheffield.
    Almqvist, Andreas
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Larsson, Roland
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
    Experimental and numerical investigations of oil film formation and friction in a piston ring–liner contact2013In: Proceedings of the Institution of mechanical engineers. Part J, journal of engineering tribology, ISSN 1350-6501, E-ISSN 2041-305X, Vol. 227, no 2, p. 126-140Article in journal (Refereed)
    Abstract [en]

    The piston ring–cylinder liner contact is a major source of the total parasitic losses in an internal combustion engine. The lubrication process of this contact highly influences the amount of friction, oil consumption and wear that occurs. In this work, a reciprocating test rig combined with an ultrasonic film thickness measurement system was developed and then used for tribological investigation of the piston ring–cylinder liner contact under idealised cold conditions. A special piston ring and cylinder liner holder were designed and five sensors were glued on to the back side of the liner specimen. Ultrasonic reflections captured by the sensors, used to obtain the film thickness, and friction were continuously recorded as the piston ring section reciprocated over the liner. Several experiments were performed at different speed and load conditions. Furthermore, a numerical model has been developed to predict film thickness and friction in all lubrication regimes. The experimentally measured film thickness and friction were compared with the output from the numerical model and good correlation was found. The parameters affecting the accuracy of the ultrasound measurements and numerical simulations of film thickness and friction are then discussed.

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