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Dynamic Contact Problems in Bone Neoplasm Analyses and the Primal-Dual Active Set (PDAS) Method
Nedoma, Jiří
In the contribution growths of the neoplasms (benign and malignant tumors and cysts), located in a system of loaded bones, will be simulated. The main goal of the contribution is to present the useful methods and efficient algorithms for their solutions. Because the geometry of the system of loaded and possible fractured bones with enlarged neoplasms changes in time, the corresponding mathematical models of tumor’s and cyst’s evolutions lead to the coupled free boundary problems and the dynamic contact problems with or without friction. The discussed parts of these models will be based on the theory of dynamic contact problems without or with Tresca or Coulomb frictions in the visco-elastic rheology. The numerical solution of the problem with Coulomb friction is based on the semi-implicit scheme in time and the finite element method in space, where the Coulomb law of friction at every time level will be approximated by its value from the previous time level. The algorithm for the corresponding model of friction will be based on the discrete mortar formulation of the saddle point problem and the primal-dual active set algorithm. The algorithm for the Coulomb friction model will be based on the fixpoint algorithm, that will be an extension of the PDAS algorithm for the Tresca friction. In this algorithm the friction bound is iteratively modified using the normal component of the Lagrange multiplier. Thus the friction bound and the active and inactive sets are updated in every step of the iterative algorithm and at every time step corresponding to the semi-implicit scheme.
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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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On the quality of local flux reconstructions for guaranteed error bounds
Vejchodský, Tomáš
In this contribution we consider elliptic problems of a reaction-diffucion type discretized by the finite element method and study the quality of guaranteed upper bounds of the error. In particular, we concentrate on complementary error bounds whose values are determined by suitable flux reconstructions. We present numerical experiments comparing the performance of the local flux reconstruction of Ainsworth and Vejchodský [2] and the reconstruction of Braess and Schröberl [5]. We evaluate the efficiency of these flux reconstructions by their comparison with the optimal flux reconstruction computed as a global minimization problem.
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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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On the number of stationary patterns in reaction-diffusion systems
Rybář, Vojtěch ; Vejchodský, Tomáš
We study systems of two nonlinear reaction-diffusion partial differential equations undergoing diffusion driven instability. Such systems may have spatially inhomogeneous stationary solutions called Turing patterns. These solutions are typically non-unique and it is not clear how many of them exists. Since there are no analytical results available, we look for the number of distinct stationary solutions numerically. As a typical example, we investigate the reaction-diffusion systém designed to model coat patterns in leopard and jaguar.
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Why quintic polynomial equations are not solvable in radicals
Křížek, Michal ; Somer, L.
We illustrate the main idea of Galois theory, by which roots of a polynomial equation of at least fifth degree with rational coefficients cannot general be expressed bz radicals, i.e., by the operations +, -, ., :, and .... Therefore, higher order polynomial equations are usually solved by approximate methods. They can also be solved algebraically by means of ultraradicals.
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