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  • 1.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Jonasson, Pernilla
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Olsson Lindberg, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Lobkova, Tatiana
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Homogenization of a Hyperbolic-Parabolic Problem with Three Spatial Scales2017In: Progress in Industrial Mathematics at ECMI 2016 / [ed] Quintela, P., Barral, P., Gómez, D., Pena, F.J., Rodríguez, J., Salgado, P., Vázquez-Mendéz, M.E., Springer, 2017, p. 617-623Conference paper (Refereed)
    Abstract [en]

    We study the homogenization of a certain linear hyperbolic-parabolic problem exhibiting two rapid spatial scales {ε; ε2}. The homogenization is performed by means of evolution multiscale convergence, a generalization of the concept of two-scale convergence to include any number of scales in both space and time. In particular we apply a compactness result for gradients. The outcome of the homogenization procedure is that we obtain a homogenized problem of hyperbolic-parabolic type together with two elliptic local problems, one for each rapid scale, for the correctors.

  • 2.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Olsson Lindberg, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    A strange term in the homogenization of parabolic equations with two spatial and two temporal scales2012In: Journal of Function Spaces and Applications, ISSN 0972-6802, E-ISSN 1758-4965, p. Art. no. 643458-Article in journal (Refereed)
    Abstract [en]

    We study the homogenization of a parabolic equation with oscillations in both space and time in the coefficient a((x/()),(t/²)) in the elliptic part and spatial oscillations in the coefficient ((x/())) that is multiplied with the time derivative ∂_{t}u^{}. We obtain a strange term in the local problem. This phenomenon appears as a consequence of the combination of the spatial oscillation in ((x/())) and the temporal oscillation in a((x/()),(t/²)) and disappears if either of these oscillations is removed.

  • 3.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Olsson Lindberg, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Homogenization of parabolic equations with an arbitrary number of scales in both space and time2014In: Journal of Applied Mathematics, ISSN 1110-757X, E-ISSN 1687-0042, p. Art. no. 101685-Article in journal (Refereed)
    Abstract [en]

    The main contribution of this paper is the homogenization of the linearparabolic equationtu (x, t) − ·axq1, ...,xqn,tr1, ...,trmu (x, t)= f(x, t)exhibiting an arbitrary finite number of both spatial and temporal scales.We briefly recall some fundamentals of multiscale convergence and providea characterization of multiscale limits for gradients in an evolution settingadapted to a quite general class of well-separated scales, which we nameby jointly well-separated scales (see Appendix for the proof). We proceedwith a weaker version of this concept called very weak multiscale convergence.We prove a compactness result with respect to this latter typefor jointly well-separated scales. This is a key result for performing thehomogenization of parabolic problems combining rapid spatial and temporaloscillations such as the problem above. Applying this compactnessresult together with a characterization of multiscale limits of sequences ofgradients we carry out the homogenization procedure, where we togetherwith the homogenized problem obtain n local problems, i.e. one for eachspatial microscale. To illustrate the use of the obtained result we apply itto a case with three spatial and three temporal scales with q1 = 1, q2 = 2and 0 < r1 < r2.MSC: 35B27; 35K10

  • 4.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Olsson Lindberg, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Two-scale convergence: Some remarks and extensions2013In: Pure and Applied Mathematics Quarterly, ISSN 1558-8599, E-ISSN 1558-8602, Vol. 9, no 3, p. 461-486Article in journal (Refereed)
    Abstract [en]

    We first study the fundamental ideas behind two-scale conver-

    gence to enhance an intuitive understanding of this notion. The classical

    definitions and ideas are motivated with geometrical arguments illustrated

    by illuminating figures. Then a version of this concept, very weak two-scale

    convergence, is discussed both independently and brie°y in the context of

    homogenization. The main features of this variant are that it works also

    for certain sequences of functions which are not bounded in

    L2  and at

    the same time is suited to detect rapid oscillations in some sequences which

    are strongly convergent in

    L2 . In particular, we show how very weak

    two-scale convergence explains in a more transparent way how the oscilla-

    tions of the governing coe±cient of the PDE to be homogenized causes the

    deviation of the

    G-limit from the weak L2 NxN-limit for the sequence of

    coe±cients. Finally, we investigate very weak multiscale convergence and

    prove a compactness result for separated scales which extends a previous

    result which required well-separated scales.

  • 5.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    A myriad shades of green2009In: Proceedings of Bridges 2009, Banff, Alberta, Canada, 2009Conference paper (Refereed)
    Abstract [en]

    We discuss the possible application of techniques inspired by the theories of G-convergence and homogenization to understand mixtures of colors and how they appear as observed by the human eye.  The ideas are illustrated by pictures describing the equivalent of a convergence process     for different kinds of mixtures of colors.

  • 6.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Detection of scales of heterogeneity and parabolic homogenization applying very weak multiscale convergence2011In: Annals of Functional Analysis, ISSN 2008-8752, E-ISSN 2008-8752, Vol. 2, no 1, p. 84-99Article in journal (Refereed)
    Abstract [en]

    We apply a new version of multiscale convergence named very weak multiscale convergence to find possible frequencies of oscillation in an unknown coefficient of a diffeential equation from its solution. We also use thís notion to study homogenization of a certain linear parabolic problem with multiple spatial and temporal scales

  • 7.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    On the determination of effective properties of certain structures with non-periodic temporal oscillations2009In: MATHMOD 2009 - 6th Vienna International Conference on Mathematical Modelling, Wien: Vienna University Press (WUV), 2009, p. 2627-2630Conference paper (Refereed)
    Abstract [en]

    We investigate the homogenization of an evolution problem modelled by a parabolic equation, where the coefficient describing the structure is periodic in space but may vary in time in a non-periodic way. This is performed applying a generalization of two-scale convergence called λ-scale convergence. We give a result on the characterization of the λ-scale limit of gradients under certain boundedness assumptions. This is then applied to perform the homogenization procedure. It turns out that, under a certain condition on the rate of change of the temporal variations, the effective property of the given structure can be determined the same way as in periodic cases.

  • 8.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Very weak multiscale convergence2010In: Applied Mathematics Letters, ISSN 0893-9659, E-ISSN 1873-5452, Vol. 23, no 10, p. 1170-1173Article in journal (Refereed)
    Abstract [en]

    We briefly recall the concept of multiscale convergence, which is a generalization of two-scale convergence. Then we investigate a related concept, called very weak multiscale convergence, and prove a compactness result with respect to this type of convergence. Finally we illustrate how this result can be used to study homogenization problems with several scales of oscillations.

  • 9.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Silfver, Jeanette
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    A model for the heat transfer between steel edge and running base in alpine racing skis2006In: WSEAS transactions on heat and mass transfer.,ISSN 1790-5044, 2006, Vol. 1, no 3, p. 256-261p. 256-261Conference paper (Refereed)
  • 10.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Silfver, Jeanette
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    On the convergence and determination of limits for some sequences of differential operators: Presented at The Ninth International Conference on Integral Methods in Science and Engineering, Niagara Falls, Ontario, Canada, July 23-27 20062006Conference paper (Other academic)
  • 11.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Silfver, Jeanette
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Some remarks on homogenization in perforated domains2008In: Integral Methods in Science and Engineering: Techniques and Applications, Boston: Birkhäuser Verlag, 2008, p. 89-97Chapter in book (Other academic)
  • 12.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Silfver, Jeanette
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Catrakis, R. J.
    On a heat transfer problem related to alpine ski racing2006In: Proceedings of the WSEAS HEAT '06, FLUID '06 : Elounda, Agios Nikolaos, Crete Island, Greece, August 21 - 23, 2006 ; 4th IASME/WSEAS International Conference on Heat Transfer, Thermal Engineering and Environment, 4th IASME/WSEAS International Conference on Fluid Mechanics and Aerodynamics: CD-rom, Elounda, Greece: WSEAS press , 2006, , p. 301-304Conference paper (Refereed)
  • 13.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Silfver, Jeanette
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Troch, Inge
    On some convergemce results and their relation to the impact of impurities on effective heat conduction properties2006In: Proceedings / 5th MATHMOD Vienna : February 8 - 10, 2006, Vienna University of Technology, Austria, Wien, 2006, , p. 16-1-16-7Conference paper (Refereed)
  • 14.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Svanstedt, Nils
    Reiterated homogeneization of monotone parabolic problems2007In: Annali dell'Universita di Ferrara, ISSN 0430-3202, Vol. 53, no 2, p. 217-232Article in journal (Refereed)
    Abstract [en]

    Reiterated homogenization is studied for divergence structure parabolic problems of the form ∂uε ∂t − div a x ε , x ε2 , t, Duε = f . It is shown that under standard assumptions on the function a (y1, y2, t, ξ) the sequence {uε} of solutions converges weakly in L p(0, T ;W1,p 0 (Ω)) to the solution u of the homogenized problem ∂u ∂t − div (b (t, Du)) = f

  • 15.
    Flodén, Lotta
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Jonasson, Pernilla
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Lobkova, Tatiana
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Olsson Lindberg, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Zhang, Ye
    Örebro universitet.
    A discussion of a homogenization procedure for a degenerate linear hyperbolic-parabolic problem2017In: AIP Conference Proceedings / [ed] Sivasundaram, S, American Institute of Physics (AIP), 2017, Vol. 1798, article id UNSP 020177Conference paper (Refereed)
    Abstract [en]

    We study the homogenization of a hyperbolic-parabolic PDE with oscillations in one fast spatial scale. Moreover, the first order time derivative has a degenerate coefficient passing to infinity when ϵ→0. We obtain a local problem which is of elliptic type, while the homogenized problem is also in some sense an elliptic problem but with the limit for ϵ-1∂tuϵ as an undetermined extra source term in the right-hand side. The results are somewhat surprising and work remains to obtain a fully rigorous treatment. Hence the last section is devoted to a discussion of the reasonability of our conjecture including numerical experiments.

  • 16.
    Gulliksson, Marten
    et al.
    Univ Orebro, Sch Sci & Technol, Dept Math, SE-70182 Örebro, Sweden.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Zhang, Ye
    Univ Orebro, Sch Sci & Technol, Dept Math, SE-70182 Orebro, Sweden.
    A separating oscillation method of recovering the G-limit in standard and non-standard homogenization problems2016In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 32, no 2, article id 025005Article in journal (Refereed)
    Abstract [en]

    Reconstructing the homogenized coefficient, which is also called the G-limit, in elliptic equations involving heterogeneous media is a typical nonlinear ill-posed inverse problem. In this work, we develop a numerical technique to determine G-limit that does not rely on any periodicity assumption. The approach is a technique that separates the computation of the deviation of the G-limit from the weak L-2-limit of the sequence of coefficients from the latter. Moreover, to tackle the ill-posedness, based on the classical Tikhonov regularization scheme we develop several strategies to regularize the introduced method. Various numerical tests for both standard and non-standard homogenization problems are given to show the efficiency and feasibility of the proposed method.

  • 17.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    A Darcy law for quasi-periodic porous media and optimal design of composite materials1996In: Proceedings of international Conference on Composites Engineering ICCE/3: 21-25/7 -96, New Orleans, LA, USA, 1996, p. 363-364Conference paper (Other academic)
  • 18.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    A model for non-linear nonstationary heat conduction in composites with many small nonperiodic perforations1997In: Proceedings of International Conference on Composites Engineering ICCE/4: 6-12/7, 1997, Hawaii, USA, 1997, p. 435-344Conference paper (Other academic)
  • 19.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Homogenization applied to some problems in mechanics1993Licentiate thesis, monograph (Other scientific)
  • 20.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Homogenization in various kinds of quasi-periodic domains1995Report (Other academic)
  • 21.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Homogenization of parabolic equations: an alternative approach and some corrector type results.1997In: Applications of Mathematics, ISSN 0862-7940, Vol. 42, no 5, p. 321-343Article in journal (Refereed)
    Abstract [en]

    We extend and complete some quite recent results by Nguetseng and Allaire concerning two-scale convergence. In particular, a compactness result for a certain class of parameterdependent functions is proved and applied to perform an alternative homogenization procedure for linear parabolic equations with coefficients oscillating in both their space and time variables. For different speeds of oscillation in the time variable, this results in three cases. Further, we prove some corrector-type results and benefit from some interpolation properties of Sobolev spaces to identify regularity assumptions strong enough for suck results to hold.

  • 22.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    On homogenization and correctors for elliptic equations with mixed boundary conditions in periodic domains1995Report (Other academic)
  • 23.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    On homogenization applied to Stokes flow in porous media. A brief discussion of geometrical modelling and possible applications1992In: Proceedings of the Industrial Mathematics Week: Trondheim August 1992, Department of Mathematical Sciences, NTH, 1992, p. 152-159Conference paper (Other academic)
  • 24.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    On homogenization theory and its application to optimal micro-design of composite materials1994Report (Other academic)
  • 25.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    On homogenization theory and its application to optimal micro-design of composite materials1994In: Proceedings of International Conference on Composites Engineering ICCE/1: New Orleans, LA, USA, Aug 28-31, 1994, 1994, p. 749-750Conference paper (Other academic)
  • 26.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Some modes of convergence and their application to homogenization and optimal composites design1996Doctoral thesis, monograph (Other scientific)
    Abstract [en]

    In this thesis, we develop, extend, apply, and discuss a number of methods for the study of limits of sequences of functions and operators. The connection between the notion of two-scale convergence and more general concepts of convergence is investigated and some alternative classes of admissible test functions are characterized. These techniques are extended into compactness results suitable to prove homogenization and corrector results for linear parabolic equations. A further refinement of these methods, together with a characterization of the limits of certain sequences of parameter-dependent functions which has been subject to extension from a quite general class of periodic domains, is introduced. This provides an efficient tool for the homogenization of e.g. nonlinear evolution heat conduction in heterogeneous materials which vibrate with high frequencies or are perforated by periodically arranged nonconducting holes. Moreover, we prove compactness and homogenization for sequences of solutions of linear elliptic and monotone parabolic equations defined in some classes of nonperiodic domains and derive a Darcy's law for a type of nonperiodic porous media. In the linear elliptic case the convergence is strengthened by means of correctors. Finally, we present some numerical results for homogenized stiffness of fibre composites and demonstrate how homogenization techniques for elasticity in composite materials and for liquid flow in porous media can be combined with recent optimization techniques to obtain optimal layout of composite materials.

  • 27.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    The concept of parabolic two-scale convergence, a new compactness result and its application to homogenization of evolution partial differential equations1994Report (Other academic)
  • 28.
    Holmbom, Anders
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Lobkova, Tatiana
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Homogenization of a hyperbolic-parabolic problem in a perforated domain2016Conference paper (Other academic)
  • 29.
    Holmbom, Anders
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Nilsson, L
    Reinisch, G
    Theorie des skis paraboliques1998In: L'Entraineur de ski alpin, ISSN 0983-9305, Vol. 27, no septArticle in journal (Other academic)
  • 30.
    Holmbom, Anders
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Olsson Lindberg, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Flodén, Liselott
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    A note on parabolic homogenization with a mismatch between the spatial scales2013In: Abstract and Applied Analysis, ISSN 1085-3375, E-ISSN 1687-0409, p. Art. no. 329704-Article in journal (Refereed)
    Abstract [en]

    We consider the homogenization of the linear parabolic problem rho(x/epsilon(2))partial derivative(t)u(epsilon)(x,t) - del . (a(x/epsilon(1), t/epsilon(2)(1))del u(epsilon) (x,t)) = f(x,t) which exhibits a mismatch between the spatial scales in the sense that the coefficient a(x/epsilon(1), t/epsilon(2)(1)) of the elliptic part has one frequency of fast spatial oscillations, whereas the coefficient rho(x/epsilon(2)) of the time derivative contains a faster spatial scale. It is shown that the faster spatialmicroscale does not give rise to any corrector termand that there is only one local problemneeded to characterize the homogenized problem. Hence, the problem is not of a reiterated type even though two rapid scales of spatial oscillation appear.

  • 31.
    Holmbom, Anders
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Silfver, Jeanette
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    On the determination of limits for some sequences of functions and differential operators2006In: Canadian Applied Mathematics Quarterly, ISSN 1073-1849, Vol. 14, no 3, p. 299-329Article in journal (Refereed)
    Abstract [en]

    This article deals with homogenization methods, namely two-scale convergence and Hconvergence, with the aim of comparing their efficiency in studying composite materials. Analytic examples are given to illustrate both methods. While compensated compactness allows one to determine the limit of a product of two weakly converging sequences under additional assumptions on the derivatives, two-scale convergence takes advantage of underlying oscillations of solution sequences and H-convergence is concerned with the operator behaviour. Special attention is paid to periodic homogenization as this case yields the most involved results. Numerical experiments are used to investigate open questions in H-convergence dealing with the possible relaxation of some convergence hypotheses. Partial results are established in this respect.

  • 32.
    Holmbom, Anders
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Persson, L.E
    Svanstedt, N
    A homogenization procedure for computing effective moduli and micro stresses in elastic composite materials1992In: Composites Part B: Engineering, ISSN 1359-8368, Vol. 2, no 4, p. 249-259Article in journal (Refereed)
  • 33.
    Holmbom, Anders
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Silfver, Jeanette
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    On a convergence result for sequences of functions with multiple scales2006In: Proceedings of the 9th WSEAS international conference on applied mathematics, Istanbul. Turkey., Aten: WSEAS press , 2006Conference paper (Refereed)
  • 34.
    Holmbom, Anders
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Silfver, Jeanette
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    On the characterization of effective properties in highly heterogeneous media2005In: IASME transactions / International ASsociation of Mechanical Engineers, ISSN 1790-031X, Vol. 2, p. 177-179Article in journal (Refereed)
    Abstract [en]

    We discuss methods related to homogenization theory and G-convergence for the computation of effective properties in heterogeneous materials. Especially we investigate a method to detect deviations from the arithmetic mean in heterogeneous materials that are not necessarily periodic.

  • 35.
    Holmbom, Anders
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Silfver, Jeanette
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    On the characterization of G-limits by means of some generalized two-scale techniques: Paper presented at the 26th Midwest-Pacific Differential Equations Conference, October 15-17, 2005, University of Alberta Edmonton, Alberta, Canada2005Conference paper (Refereed)
    Abstract [en]

    G-convergence usually deals with the convergence of sequences of elliptic or parabolic operators. When the convergence of the sequence of operators is strong enough, it is trivial to determine the $G$-limit. Other cases need sophisticated techniques for this aim, where the most well investigated case is periodic homogenization. The main tool of today for this purpose has become the so called two-scale convergence method by Nguetseng. This approach relies on a fundamental compactness result which says that for any bounded sequence $\left\{ u_{h}\right\} $ in $L^{2}\left( \Omega \right) $ there is $u_{0}\in L^{2}\left( \Omega \times Y\right) $ such that \begin{equation*} \dint\nolimits_{\Omega }u_{h}(x)\tau _{h}v(x)dx\rightarrow \dint\nolimits_{\Omega} \dint\nolimits_{Y}u_{0}(x,y)v(x,y)dxdy\end{equation*}% for any $v\in X=L^{2}(\Omega ;C_{\sharp }(Y))$ up to a subsequence, where% \begin{equation*} \tau _{h}v(x)=v(x,\frac{x}{\varepsilon _{h}}),\varepsilon _{h}\rightarrow 0% \text{.}\end{equation*} For gradients of sequences $\left\{ u_{h}\right\} $ bounded in $H^{1}\left( \Omega \right) $ the deviation from the week limit can be made explicit in terms of a local gradient $\nabla _{y}u_{1}$, $u_{1}\in L^{2}(\Omega;H_{\sharp }^{1}(Y))$, and this is the key to the characterization of the $G$%-limit. Similar techniques can be developed for other choices of the maps $\tau _{h}$ and admissible spaces $X$ which do not necessarily depend on any periodicity assumptions. We study such examples with respect to the possible appearance of residual terms corresponding to $\nabla _{y}u_{1}$ in periodic homogenization. In particular $G$-limits for problems, where the matrices defining the operators are generated by a kind of modified Hilbert-Schmidt operators, are investigated with respect to such deviations.

  • 36.
    Holmbom, Anders
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Silfver, Jeanette
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    On the convergence of some pairs of weakly convergent sequences: Presented at the Applmath05 Fourth conference on applied mathematics and scientific computing June 19-24 Brijuni Island, Croatia2005Conference paper (Other academic)
  • 37.
    Holmbom, Anders
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Silfver, Jeanette
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    On the convergence of some sequences of oscillating functionals2006In: WSEAS transactions on mathematics, ISSN 1109-2769, Vol. 5, no 8, p. 951-956Article in journal (Refereed)
    Abstract [en]

    We study an intermediate case between the two-scale convergence of Nguetseng and the more general concept of scale convergence of Mascarenhas and Toader. Suitable assumptions to provide scale convergence with some of the most essential properties of two-scale convergence are identified. Some aspects of the characterization of limits for sequences of gradients are discussed briefly.

  • 38.
    Holmbom, Anders
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Silfver, Jeanette
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Lazakidou, Athina
    On the characterization of effective properties in highly heterogeneous media (Proc.)2005In: Proceedings of the WSEAS HEAT '06, FLUID '06 : Elounda, Agios Nikolaos, Crete Island, Greece, August 21 - 23, 2006 ; 4th IASME/WSEAS International Conference on Heat Transfer, Thermal Engineering and Environment, 4th IASME/WSEAS International Conference on Fluid Mechanics and Aerodynamics Elounda, Greece, 2006, Elounda, Greece,, 2005, , p. -Conference paper (Refereed)
    Abstract [en]

    We discuss methods related to homogenization theory and G-convergence for the computation of effective properties in heterogeneous materials. Especially we investigate a method to detect deviations from the arithmetic mean in heterogeneous materials that are not necessarily periodic.

  • 39.
    Holmbom, Anders
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Silfver, Jeanette
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Svanstedt, N
    Wellander, N
    On the relationship between some weak compactnesses with different numbers of scales2003Report (Other academic)
    Abstract [en]

    A general concept of two-scale convergence is introduced and two-scale compactness theorems are stated and proved for some classes of non-periodic bounded functions in L²(). Further the relation to the classical notion of compensated compactness and the recent concept of two-scale compensated compactness is discussed and a defect measure for two-scale convergence is introduced.

  • 40.
    Holmbom, Anders
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Silfver, Jeanette
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Svanstedt, Nils
    Wellander, Niklas
    On two-scale convergence and related sequential compactness topics2006In: Applications of Mathematics, ISSN 0862-7940, Vol. 51, no 3, p. 247-262Article in journal (Refereed)
    Abstract [en]

    A general concept of two-scale convergence is introduced and two-scale compactness theorems are stated and proved for some classes of sequences of bounded functions in L 2(Ω) involving no periodicity assumptions. Further, the relation to the classical notion of compensated compactness and the recent concepts of two-scale compensated compactness and unfolding is discussed and a defect measure for two-scale convergence is introduced

  • 41.
    Holmbom, Anders
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Svanstedt, Nils
    Wellander, N
    Asymptotic analysis of parabolic problems with multiple time and spatial scales2005In: Proceedings of the Conference on Applied Mathematics and Scientific Computing, 4th, June 19-24 Brijuni Island, Croatia, 2005Conference paper (Other academic)
    Abstract [en]

    We study the homogenization of linear parabolic equations with multiple scales for different speeds of oscillation in the time variable and identify local and homogenized problems

  • 42.
    Holmbom, Anders
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Svanstedt, Nils
    Wellander, N
    Multi-scale convergence and reiterated homogenization of parabolic problems2005In: Applications of Mathematics, ISSN 0862-7940, Vol. 50, no 2, p. 131-151Article in journal (Refereed)
    Abstract [en]

    Reiterated homogenization is studied for divergence structure parabolic problems. It is shown that under standard assumptions the sequence of solutions converges weakly in a suitable evoloution sense to the solution u of the corresponding homogenized problem

1 - 42 of 42
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