| 1. |
- Berntsson, Fredrik, 1971-, et al.
(författare)
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A Modification to the Kirchhoff Conditions at a Bifurcation and Loss Coefficients
- 2018
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- One dimensional models for fluid flow in tubes are frequently used tomodel complex systems, such as the arterial tree where a large numberof vessels are linked together at bifurcations. At the junctions transmission conditions are needed. One popular option is the classic Kirchhoffconditions which means conservation of mass at the bifurcation andprescribes a continuous pressure at the joint.In reality the boundary layer phenomena predicts fast local changesto both velocity and pressure inside the bifurcation. Thus it is not appropriate for a one dimensional model to assume a continuous pressure. In this work we present a modification to the classic Kirchhoff condi-tions, with a symmetric pressure drop matrix, that is more suitable forone dimensional flow models. An asymptotic analysis, that has beencarried out previously shows that the new transmission conditions hasen exponentially small error.The modified transmission conditions take the geometry of the bifurcation into account and can treat two outlets differently. The conditions can also be written in a form that is suitable for implementationin a finite difference solver. Also, by appropriate choice of the pressuredrop matrix we show that the new transmission conditions can producehead loss coefficients similar to experimentally obtained ones.
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| 2. |
- Berntsson, Fredrik, et al.
(författare)
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Numerical Solution of the Cauchy Problem for the Helmholtz Equation
- 2014
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- The Cauchy problem for the Helmholtz equation appears in applications related to acoustic or electromagnetic wave phenomena. The problem is ill–posed in the sense that the solution does not depend on the data in a stable way. In this paper we give a detailed study of the problem. Specifically we investigate how the ill–posedness depends on the shape of the computational domain and also on the wave number. Furthermore, we give an overview over standard techniques for dealing with ill–posed problems and apply them to the problem.
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| 3. |
- Ghosh, Arpan, et al.
(författare)
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A Two Dimensional Model of the Thin Laminar Wall of a Curvilinear Flexible Pipe
- 2018
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We present a two dimensional model describing the elastic behaviour of the wall of a curved pipe to model blood vessels in particular. The wall has a laminate structure consisting of several anisotropic layers of varying thickness and is assumed to be much smaller in thickness than the radius of the vessel which itself is allowed to vary. Our two-dimensional model takes the interaction of the wall with the surrounding material and the fluid flowing inside into account and is obtained via a dimension reduction procedure. The curvature and twist of the vessel axis as well as the anisotropy of the laminate wallpresent the main challenges in applying the dimension reduction procedure so plenty of examples of canonical shapes of vessels and their walls are supplied with explicit systems of dierential equations at the end.
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| 4. |
- Kozlov, Vladimir, et al.
(författare)
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Asymptotic Models of Anisotropic Heterogeneous Elastic Walls of Blood Vessels
- 2015
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Using the dimension reduction procedure in the three-dimensional elasticity system, we derive a two-dimensional model for elastic laminate walls of a blood vessel. The wall of arbitrary cross-section consists of several (actually three) elastic, anisotropic layers. Assuming that the wall’s thickness is small compared with the vessel’s diameter and length, we derive a system of the limit equations. In these equations, the wall’s displacements are unknown given on the two-dimensional boundary of a cylinder, whereas the equations themselves constitute a second order hyperbolic system. This system is coupled with the Navier–Stokes equations through the stress and velocity, i.e. dynamic and kinematic conditions at the interior surface of the wall. Explicit formulas are deduced for the effective rigidity tensor of the wall in two natural cases. The first of them concerns the homogeneous anisotropic laminate layer of constant thickness like that in the wall of a peripheral vein, whereas the second case is related to enforcing of the media and adventitia layers of the artery wall by bundles of collagen fibers. It is also shown that if the blood flow stays laminar, then the describing cross-section of the orthotropic homogeneous blood vessel becomes circular.
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| 5. |
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| 6. |
- Kozlov, Vladimir, 1954-, et al.
(författare)
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Pressure drop matrix for a bifuration of an artery with defects
- 2018
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We consider a bifurcation of an artery. The influence of defects of the vessel's wall near the bifurcation point on the pressure drop matrix is analyzed. The elements of this matrix are included in the modified Kirchhoff transmission conditions, which were introduced earlier in [1], [2], and which describe adequately the total pressure loss at the bifurcation point of the flow passed through it.
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| 7. |
- Kozlov, Vladimir, 1954-, et al.
(författare)
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Trapped modes in armchair graphene nanoribbons
- 2018
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We study scattering on an ultra-low potential in armchair graphene nanorib bon. Using the continuous Dirac model and including a couple of articial waves in the scattering process, described by an augumented scattering matrix, we derive a condition for the existence of a trapped mode. We consider the threshold energies, where the the multiplicity of the continuous spectrum changes and show that a trapped mode may appear for energies slightly less than a thresold and its multiplicity does not exceed one. For energies which are higher than a threshold, there are no trapped modes, provided that the potential is suciently small.
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| 8. |
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Proceedings of Workshops on Inverse Problems, Data, Mathematical Statistics and Ecology
- 2011
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Proceedings (redaktörskap) (övrigt vetenskapligt/konstnärligt)abstract
- Processes in Nature may be considered as deterministic or/and random. We are observing global problems such as climate changes (e.g. warming and extreme weather conditions), pollutions (e.g. acidification, fertilization, the spread of many types of pollutants through air and water) and whole ecosystems that are under pressure (e.g. the Baltic sea and the Arctic region). To understand the processes in Nature and (predict) understand what might occur it is not enough with empirical studies. One needs theoretical fundaments including models and theories to perform correct actions against different threats or at least to carry out appropriate simulation studies. For example, extreme value theory can explain some of the observed phenomena, classical risk analysis may be of help, different types of multivariate and high-dimensional analysis can explain data, time series analysis is essential, for forthcoming studies the theory of experimental designs is of interest, data assimilation together with inverse problem technique is useful for adjustment of data into mathematical models and the list can be made much longer. Behind all these approaches mathematics is hidden, sometimes at a very advanced level. Chemical and physical processes influence all observations but the challenge is to do appropriate approximations so that mathematical/statistical models can be applied. The main aim of this project is to present state of the art knowledge concerning the modelling of Nature with focus on mathematical modelling, in particular "inverse and ill-posed problems", as well as spatiotemporal models. Inverse and ill-posed problems are characterized by the property that the solutions are extremely sensitive to measurement and modelling errors. There are established connections between inverse problems and Bayesian inference but very little has been carried out with focus on parametric inference such as the likelihood approach. Concerning spatio-temporal models these are usually extensions of classical time series models or/and classical multivariate analysis models. From the Nordic Council of Ministers, within the program Nordic - Russian Cooperation in Education and Research we asked for funding of 3 preparatory meetings where the plan was to create a series of events taking place during 2011-2013. Partner organizations were Institute of Problems of Mechanical Engineering, St. Petersburg St. Petersburg State University Helsinki University Swedish Agricultural University Stockholm University Linköping University However, there were also some other participants from other universities. The planned events should be connected to the following fields: applied mathematics, biophysics and mathematical statistics. Within applied mathematics: mathematical modelling and partial differential equations, inverse and ill-pose problems, data assimilation, dynamical systems, linear algebra, matrix theory; within biophysics; neural networks and inverse modelling of objects; within mathematical statistical; analyses of stochastic processes, spatio-temporal modelling, experimental design, where considered. There exists a wide overlap between these areas and it is challenging to systemize this overlap and transmit this knowledge to students and stakeholders. However, due to unsure funding it was decided to discuss what can be presented during a one-year program. Moreover, due to practical reasons only 2 meetings/workshops were held: Workshop on Inverse Problems, Data, Mathematical Statistics and Ecology: May 20-21, 2010 at Department of Mathematics, Linköping University. Workshop on Inverse Problems, Data, Mathematical Statistics and Ecology, Part II: August 25-27, 2010 at Department of Mathematics, Helsinki University. The output from the above events can be summarized as follows: We have identified a number of different areas which can be taught on from different perspectives depending on students background of mathematics. We have learned to know many interesting researchers who are willing to share there experiences when for example creating a summer school. There is no doubt that we can organize cross-disciplinary summer/winter schools with focus on either the Baltic or Archtic regions. This booklet is also part of the deliverables. It comprizes extended abstracts of the majority of the talks of the participants showing their great interest. It is in some way a unique cross-disciplinary document which has joined researchers from different areas from Russia, Finland and Sweden. We are extremely grateful for the support given by the Nordic Council of Ministers (NCM-RU-PA-2009/10382) and all the enthusiastic contributions by the participants, including our host in Helsinki, professor Lassi Päivärinta. Vladimir Kozlo, Linköping University Martin Ohlso, Linköping University Dietrich von Rosen, Linköping University/SLU
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