| 1. |
- Flodén, Lotta, et al.
(författare)
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A discussion of a homogenization procedure for a degenerate linear hyperbolic-parabolic problem
- 2017
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Ingår i: ICNPAA 2016 World Congress. - Melville, USA : American Institute of Physics (AIP). - 9780735414648
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Konferensbidrag (refereegranskat)abstract
- 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.
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| 2. |
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| 3. |
- Holmbom, Anders
(författare)
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Some modes of convergence and their application to homogenization and optimal composites design
- 1996
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 4. |
- Flodén, Liselott, 1967-
(författare)
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G-Convergence and Homogenization of some Sequences of Monotone Differential Operators
- 2009
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis mainly deals with questions concerning the convergence of some sequences of elliptic and parabolic linear and non-linear operators by means of G-convergence and homogenization. In particular, we study operators with oscillations in several spatial and temporal scales. Our main tools are multiscale techniques, developed from the method of two-scale convergence and adapted to the problems studied. For certain classes of parabolic equations we distinguish different cases of homogenization for different relations between the frequencies of oscillations in space and time by means of different sets of local problems. The features and fundamental character of two-scale convergence are discussed and some of its key properties are investigated. Moreover, results are presented concerning cases when the G-limit can be identified for some linear elliptic and parabolic problems where no periodicity assumptions are made.
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| 5. |
- Flodén, Liselott, 1967-, et al.
(författare)
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On the determination of effective properties of certain structures with non-periodic temporal oscillations
- 2009
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Ingår i: MATHMOD 2009 - 6th Vienna International Conference on Mathematical Modelling. - Wien : Vienna University Press (WUV). - 9783901608353 ; , s. 2627-2630
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Konferensbidrag (refereegranskat)abstract
- 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.
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| 6. |
- Persson, Jens, 1978-
(författare)
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Homogenization of Some Selected Elliptic and Parabolic Problems Employing Suitable Generalized Modes of Two-Scale Convergence
- 2010
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The present thesis is devoted to the homogenization of certain elliptic and parabolic partial differential equations by means of appropriate generalizations of the notion of two-scale convergence. Since homogenization is defined in terms of H-convergence, we desire to find the H-limits of sequences of periodic monotone parabolic operators with two spatial scales and an arbitrary number of temporal scales and the H-limits of sequences of two-dimensional possibly non-periodic linear elliptic operators by utilizing the theories for evolution-multiscale convergence and λ-scale convergence, respectively, which are generalizations of the classical two-scale convergence mode and custom-made to treat homogenization problems of the prescribed kinds. Concerning the multiscaled parabolic problems, we find that the result of the homogenization depends on the behavior of the temporal scale functions. The temporal scale functions considered in the thesis may, in the sense explained in the text, be slow or rapid and in resonance or not in resonance with respect to the spatial scale function. The homogenization for the possibly non-periodic elliptic problems gives the same result as for the corresponding periodic problems but with the exception that the local gradient operator is everywhere substituted by a differential operator consisting of a product of the local gradient operator and matrix describing the geometry and which depends, effectively, parametrically on the global variable.
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| 7. |
- Persson, Jens, 1978-
(författare)
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λ-scale convergence applied to the stationary heat conduction equation with non-periodic thermal conductivity matrix
- 2009
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Ingår i: MATHMOD 2009 - 6th Vienna International Conference on Mathematical Modelling. - Wien : Vienna University Press (WUV). - 9783901608353 ; , s. 2720-2723
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Konferensbidrag (refereegranskat)abstract
- In this contribution we study the homogenization of non-periodic stationary heat conduction problems with homogeneous Dirichlet boundary data by applying the recently developed λ-scale convergence technique developed by Holmbom and Silfver. λ-scale convergence can be seen as either being a special case of scale convergence (developed by Mascarenhas and Toader) or of “generalized” two-scale convergence (developed by Holmbom, Silfver, Svanstedt and Wellander). From either viewpoint, it is a possibly powerful generalization of Nguetseng’s classical, periodic two-scale convergence method. We give a definition of the concept of λ-scale convergence, which is then used to claim a main theorem on homogenization of certain non-periodic stationary heat conduction problems. The original part of the contribution starts by defining a two-dimensional “toy model”. We show that the “toy model” satisfies the right conditions such that the aforementioned main theorem on the homogenization can be employed. In this way we derive the homogenized problem, i.e. the homogenized thermal conductivity matrix, and the local problem. The contribution is concluded by giving a numerical example where we explicitly compute the homogenized thermal conductivity matrix.
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| 8. |
- Andrist B., Rafael, et al.
(författare)
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Holomorphic Automorphisms of Danielewski Surfaces II: Structure of the Overshear Group
- 2015
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Ingår i: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 25:3, s. 1859-1889
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Tidskriftsartikel (refereegranskat)abstract
- We apply Nevanlinna theory for algebraic varieties to Danielewski surfaces and investigate their group of holomorphic automorphisms. Our main result states that the overshear group, which is known to be dense in the identity component of the holomorphic automorphism group, is a free product.
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| 9. |
- Wikström, Frank
(författare)
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The Dirichlet problem for maximal plurisubharmonic functions on analytic varieties in C^n
- 2009
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Ingår i: International Journal of Mathematics. - 0129-167X. ; 20:4, s. 521-528
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Tidskriftsartikel (refereegranskat)abstract
- Let Ω be a B-regular domain in ℂn and let V be a locally irreducible analytic variety in Ω. Given a continuous function , we prove that there is a unique maximal plurisubharmonic function u on V with boundary values given by ϕ and furthermore that u is continuous on .
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| 10. |
- Mamontov, Eugen, 1955, et al.
(författare)
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Nonsearch paradigm for large-scale parameter-identification problems in dynamical systems related to oncogenic hyperplasia
- 2006
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Ingår i: Systems, Control, Modeling and Optimization. IFIP International Federation for Information Processing. - : Springer US. - 1571-5736. - 9780387338811 - 0387338810 ; 202, s. 269-278
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Konferensbidrag (refereegranskat)abstract
- In many engineering and biomedical problems there is a need to identify parameters of the systems from experimental data. A typical example is the biochemical-kinetics systems describing oncogenic hyperplasia where the dynamical model is nonlinear and the number of the parameters to be identified can reach a few hundreds. Solving these large-scale identification problems by the local- or global-search methods can not be practical because of the complexity and prohibitive computing time. These difficulties can be overcome by application of the non-search techniques which are much less computation- demanding. The present work proposes key components of the corresponding mathematical formulation of the nonsearch paradigm. This new framework for the nonlinear large-scale parameter identification specifies and further develops the ideas of the well-known approach of A. Krasovskii. The issues are illustrated with a concise analytical example. The new results and a few directions for future research are summarized in a dedicated section.
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