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HP-FEM for Coupled Problems in Fluid Dynamics
Dubcová, Lenka ; Feistauer, Miloslav (advisor) ; Segeth, Karel (referee) ; Dolejší, Vít (referee)
The thesis is concerned with the solution of multiphysics problems described by partial differential equations using higher-order finite element method (hp-FEM). Basics of hp-FEM are described, together with some practical details and challenges. The hp-adaptive strategy, based on the reference solution and meshes with arbitrary level hanging nodes, is discussed. The thesis is mainly concerned with the extension of this strategy to monolithical solution of coupled multiphysics problems, where each physical field exhibits different qualitative behavior. In such problems, each physical field is discretized on an individual mesh automatically obtained by the adaptive algorithm to suit the best the corresponding solution component. Moreover, the meshes can change in time, following the needs of the solution components. All described methods and technologies are demonstrated on several examples throughout the thesis, where comparisons with traditionally used approaches are shown.
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A particular smooth interpolation that generates splines
Segeth, Karel
There are two grounds the spline theory stems from -- the algebraic one (where splines are understood as piecewise smooth functions satisfying some continuity conditions) and the variational one (where splines are obtained via minimization of some quadratic functionals with constraints). We use the general variational approach called $it smooth interpolation$ introduced by Talmi and Gilat and show that it covers not only the cubic spline and its 2D and 3D analogues but also the well known tension spline (called also spline with tension). We present the results of a 1D numerical example that characterize some properties of the tension spline.
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Spatial interpolation and soil erosion modeling
Bek, Stanislav ; Ježek, Josef (advisor) ; Segeth, Karel (referee) ; Dostál, Tomáš (referee)
The doctoral thesis deals with selected methods of spatial interpolation and their applications to numerical modeling of the earth's surface, in particular soil erosion. The first part contains the description of the studied methods. Firstly and foremost, the method called regularized spline with tension (RST) is introduced. It has proven to be useful in interpolating elevation data. In the thesis, RST is presented in depth with the derivation of its radial basis function and its links to kriging. Further on the mathematics of digital terrain models and the tools for terrain geometric analysis are covered. The following chapter deals with the description of the soil erosion process and of the selected erosion models. The second part of the thesis summarizes five of the author's research articles which include applications of the described mathematical methods. The first two articles are devoted to the problem of elevation data interpolation and the building of digital elevation models. They deal with the optimization of the RST method for particular input data and target erosion models. The third article analysis the spatial structure of the soil data and the pedogenesis of the Žofínský prales natural forest. The last two articles deal with the spatial properties of heavy rainfalls and the mapping of...
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Approximation, numerical realization and qualitative analysis of contact problems with friction
Ligurský, Tomáš ; Haslinger, Jaroslav (advisor) ; Segeth, Karel (referee) ; Rohan, Eduard (referee)
Title: Approximation, numerical realization and qualitative analysis of contact problems with friction Author: Tomáš Ligurský Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Jaroslav Haslinger, DrSc., Department of Numerical Mathe- matics Abstract: This thesis deals with theoretical analysis and numerical realization of dis- cretized contact problems with Coulomb friction. First, discretized 3D static contact prob- lems with isotropic and orthotropic Coulomb friction and solution-dependent coefficients of friction are analyzed by means of the fixed-point approach. Existence of at least one solution is established for coefficients of friction represented by positive, bounded and con- tinuous functions. If these functions are in addition Lipschitz continuous and upper bounds of their values together with their Lipschitz moduli are sufficiently small, uniqueness of the solution is guaranteed. Second, properties of solutions parametrized by the coefficient of friction or the load vector are studied in the case of discrete 2D static contact problems with isotropic Coulomb friction and coefficient independent of the solution. Conditions under which there exists a local Lipschitz continuous branch of solutions around a given reference point are established due to two variants of the...
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Automatic hp-adaptivity on Meshes with Arbitrary-Level Hanging Nodes in 3D
Kůs, Pavel ; Vejchodský, Tomáš (advisor) ; Segeth, Karel (referee) ; Dolejší, Vít (referee)
The thesis is concerned with theoretical and practical aspects of the hp- adaptive finite element method for solving elliptic and electromagnetic prob- lems described by partial differential equations in three spatial dimensions. Besides the standard element refinements, the hp-adaptivity allows indepen- dent adaptation of degrees of the polynomial approximation as well. This leads to exponentially fast convergence even for problems with singularities. The efficiency of the hp-adaptivity is enhanced even more by the ability of the algorithm to work with meshes with arbitrary-level hanging nodes. This generality, however, leads to great complexity of the implementation. There- fore, the thesis concentrates on the mathematical analysis of algorithms that have led to successful implementation of the method. In addition, the the- sis discusses the numerical integration in 3D and the implementation of the method itself. Finally, numerical results obtained by this new implemen- tation are presented. They confirm advantages of hp-adaptivity on meshes with arbitrary-level hanging nodes. 1
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Proceedings of the International Conference Applications of Mathematics 2015 : Prague, November 18-21, 2015
Brandts, J. ; Korotov, S. ; Křížek, Michal ; Segeth, Karel ; Šístek, Jakub ; Vejchodský, Tomáš
Professors Ivo Babuška, Milan Práger, and Emil Vitásek are renowned experts in numerical analysis and computational methods. Their fruitful scientific careers started in Prague, at the Institute of Mathematics of the Czechoslovak Academy of Sciences (now Czech Academy of Sciences). They collaborated there on various projects including the computational analysis of the construction technology for Orlík Dam. In 1966 they published their joint book entitled Numerical Processes in Differential Equations. It is an honor for the Institute of Mathematics to host a conference on the occasion of their birthdays.
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Programs and Algorithms of Numerical Mathematics 17 : Dolní Maxov, June 8-13, 2014 : Proceedings of Seminar
Chleboun, J. ; Přikryl, Petr ; Segeth, Karel ; Šístek, Jakub ; Vejchodský, Tomáš
This volume comprises peer-reviewed papers that are based on invited lectures, survey lectures, short communications, and posters presented at the 17th seminar Programs and Algorithms of Numerical Mathematics (PANM) held in Dolní Maxov, Czech Republic, June 8–13, 2014. The seminar was organized by the Institute of Mathematics of the Academy of Sciences of the Czech Republic. It continued the previous seminars on mathematical software and numerical methods held (with only one exception) biannually in\nAlšovice, Bratříkov, Janov nad Nisou, Kořenov, Lázně Libverda, Dolní Maxov, and Prague in the period 1983–2012. The objective of this series of seminars is to provide a forum for presenting and discussing advanced topics in numerical analysis, singleor multi-processor applications of computational methods, and new approaches to mathematical modeling.
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A note on tension spline
Segeth, Karel
Spline theory is mainly grounded on two approaches: the algebraic one (where splines are understood as piecewise smooth functions) and the variational one (where splines are obtained via minimization of quadratic functionals with constraints). We show that the general variational approach called smooth interpolation introduced by Talmi and Gilat covers not only the cubic spline but also the well known tension spline (called also spline in tension or spline with tension). We present the results of a 1D numerical example that show the advantages and drawbacks of the tension spline.
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Smooth approximation spaces based on a periodic system
Segeth, Karel
A way of data approximation called smooth was introduced by Talmi and Gilat in 1977. Such an approach employs a (possibly infinite) linear combination of smooth basis functions with coefficients obtained as the unique solution of a minimization problem. While the minimization guarantees the smoothness of the approximant and its derivatives, the constraints represent the interpolating or smoothing conditions at nodes. In the contribution, a special attention is paid to the periodic basis system $exp(-ii kx)$. A 1D numerical example is presented.
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