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Cosmology on Small Scales 2022: Dark Energy and the Local Hubble Expansion Problem
Křížek, Michal ; Dumin, Y. V.
One hundred years ago, Russian mathematician and physicist Alexander A. Friedmann applied the system of Einstein equations to the three-dimensional sphere with a time varying radius. In this way, he obtained a nonlinear ordinary differential equation which is called the Friedmann equation after him and serves now as a cornerstone of the standard cosmological model. Unfortunately, it is well known that this model exhibits a number of paradoxes. Thus, the main goal of the CSS 2022 Conference Proceedings is to discuss whether and how the Friedmann equation can be applied at the various spatial scales, from our local cosmic neighborhood up to the whole Universe, and if the existence of dark matter and dark energy are merely artifacts of the excessive extrapolations.
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100 years of the Friedmann equation
Křížek, Michal
In 1922, Alexander Friedmann applied Einstein’s equations to a three-dimensional sphere to describe the evolution of our universe. In this way he obtained a nonlinear ordinary differential equation (called after him) for the expansion function representing the radius of that sphere. At present, the standard cosmological ΛCDM model of the universe is based just on the Friedmann equation. It needs a significant amount of dark matter, about six times that of the usual baryonic matter, besides an even larger amount of dark energy to be consistent with the real universe. But to date, both dark matter and dark energy have remained without concrete evidence based on direct physical measurements. We present several arguments showing that such a claimed amount of dark matter and dark energy can only be the result of vast overestimation, incorrect extrapolations, and that it does not correspond to the real universe. The spatial part of our universe seems to be locally flat and thus it can be locally modeled by the Euclidean space. However, Friedmann did not consider the flat space with zero curvature. Therefore, in the second part of this paper we will derive a general form of the corresponding metric tensor satisfying Einstein’s equations with zero right-hand side.
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Mathematical Analysis and Numerical Computation of Volume-Constrained Evolutionary Problems Involving Free Boundaries
Švadlenka, Karel ; Feistauer, Miloslav (advisor) ; Křížek, Michal (referee) ; Knobloch, Petr (referee)
The object of study of the present thesis are evolutionary problems satisfying volume preservation condition, i.e., problems whose solution have a constant value of the integral of their graph. In particular, the following types of problems with volume constraint are dealt with: parabolic problem (heat-type), hyperbolic problem (wave-type), parabolic free-boundary problem (heat-type with obstacle) and hyperbolic free-boundary problem (degenerate wave-type with obstacle). The key points are design of equations, proof of existence of weak solutions to them and development of numerical methods and algorithms for such problems. The main tool in both the theoretical analysis and the numerical computation is the discrete Morse flow, a variational method consisting in discretizing time and stating a minimization problem on each time-level. The volume constraint appears in the equation as a nonlocal nonlinear Lagrange multiplier but it can be handled elegantly in discrete Morse flow method by restraining the set of admissible functions for minimization. The theory is illustrated with results of numerical experiments.
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Fluid-structure interaction of compressible flow
Hasnedlová, Jaroslava ; Feistauer, Miloslav (advisor) ; Křížek, Michal (referee) ; Kozel, Karel (referee) ; Rannacher, Rolf (referee)
Title: Fluid-structure interaction of compressible flow Author: RNDr. Jaroslava Hasnedlová Department: Department of Numerical Mathematics, Institute of Applied Mathematics Supervisors: Prof. RNDr. Miloslav Feistauer, DrSc., Dr. h. c., Prof. Dr. Dr. h. c. Rolf Rannacher Supervisors' e-mail addresses: feist@karlin.mff.cuni.cz, rannacher@iwr.uni-heidelberg.de Abstract: The presented work is split into two parts. The first part is devoted to the theory of the discontinuous Galerkin finite element (DGFE) method for the space-time discretization of a nonstationary convection-diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The DGFE method is applied sep- arately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time discretization. The main result is the proof of error estimates in L2 (L2 )-norm and in DG-norm formed by the L2 (H1 )-seminorm and penalty terms. The second part of the thesis deals with the realization of fluid-structure interaction problem of the compressible viscous flow with the elastic structure. The time-dependence of the domain occupied by the fluid is treated by the ALE (Arbitrary Lagrangian-Eulerian) method, when the compress- ible Navier-Stokes equations are formulated in...
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Computation of an anisotropic and nonlinear magnetic field by the finite element method
Kunický, Zdeněk ; Vejchodský, Tomáš (advisor) ; Křížek, Michal (referee)
In the present work we study the modelling of stationary magnetic fields in nonlinear anisotropic media by FEM. The magnetic characteristics of such materials are thoroughly examined and eventually applied to the construction of a full 2D model of an anisotropic steel sheet. Some improvements in the construction in comparison with the ones previously published are achieved. We also present an extension of a 3D model of steel and dielectric laminations for anisotropic sheets. We point out that the standard formulations and the subsequent theorems for the boundary value problems in fact do not correspond with the physical situation. Instead, we propose new formulations that reflect the real physical properties of matter. General existence and uniqueness theorems for the obtained boundary value problems are proved as well as the convergence theorems for the discrete solutions. Finally, the conventional and full 2D model of an anisotropic steel sheet are compared in two transformer core models using the adaptive Newton-Raphson iterative scheme and the obtained results are presented and analysed.
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