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Řešení 3D elektrostatických problémů se singulaturou s použitím adaptivní hp-FEM
Kůs, Pavel ; Šolín, Pavel ; Doležel, Ivo
For most numerical methods, accurate resolution of singularities occurring at sharp re-entrant corners or edges of electrically conductive objects is highly problematic. Finite differences are known for their inability to treat complex geometries, and traditional low-order (piecewise-linear or quadratic) finite element methods (FEM) exhibit extremely poor convergence. Nowadays, the best numerical method for the solution of most singular problems is the adaptive hp-version of the FEM (hp-FEM). This method is based on spatial refinements toward the singularities combined with optimal variation of polynomial degrees on the elements. The hp-FEM has mathematically proven exponential convergence, and also in practical computations typically it is by several orders of magnitudes faster than standard FEM.
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Zlepšování podmíněnosti v hp verzi metody konečných prvků
Vejchodský, Tomáš ; Šolín, Pavel
We present the problem of choice of higher-order basis functions for hp-version of the finite element method (hp-FEM) with respect to the condition number of the resulting stiffness matrix. We compare numerically the condition numbers for several popular sets of basis functions. We show that the best conditioning have the basis functions that are orthogonal in the energetic sense on the reference element. Moreover, any two sets of basis functions with this property have identical condition number of both stiffnes and mass matrices.
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Diskrétní Greenova funkce a princip maxima
Vejchodský, Tomáš ; Šolín, Pavel
In this paper the discrete Green´s function (DGF) is introduced and its fundamental properties are proven. Further it is indicated how to use these results to prove the discrete maximum principle for 1D Poisson equation discretized by the hp-FEM with pure Dirichlet or with mixed Dirichlet-Neumann boundary conditions and with piecewise constant coefficient
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Aproximace hraničních vrstev konečnými prvky
Segeth, Karel ; Šolín, Pavel
The numerical solution of convection-diffusion problems with a dominant role of convection plays important role in many scientific and engineering applications, such as viscous flow, fluid-structure interaction transport models, and others. These problems typically exhibit steep gradients, e.g., in the vicinity of solid walls, which are called boundary layers. In this study we investigate the potential of the hp-FEM to facilitate the numerical treatment of this class of problems.
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O některých výsledcích aposteriorních odhadů chyby v metodě přímek
Segeth, Karel ; Šolín, Pavel
We present an (incomplete) historical survey of some basic results of residual type estimation procedures. Recently we witness a rapidly increasing use of the hp-FEM which is due to the well-established theory. However, the conventional a posteriori error estimates (in the form of a single number per element) are not enough here, more complex estimates are needed.
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Tři způsoby interpolace na konečných prvcích
Šolín, Pavel ; Segeth, Karel
Interpolation on finite elements usually occurs in a Hilbert space setting, which means that interpolation techniques involving orthogonal projection are an alternative for the traditional Lagrange nodal interpolation schemes. In addition to the Lagrange interpolation, this paper discusses the global orthogonal projection and the projection-based interpolation. These techniques are compared from the point of view of quality, efficiency, sensitivity to input parameters and other aspects.
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Techniky zpracování tuhých a tekutých kovů založené na elektromagnetické indukci
Doležel, Ivo ; Šolín, Pavel ; Musil, Ladislav ; Ulrych, B. ; Karban, P. ; Barglik, J.
A lot of up-to-date industrial technologies associated with treatment of solid and liquid metals are based on thermal and force effects of electromagnetic field. The fundamental process is induction heating of metals that precedes a number of consequent operations such as tempering, drying, melting, stirring, hardening, hot pressing etc. The paper summarizes physical essence of the above processes (that usually represent complex coupled problems) and presents their mathematical and computer models as well as possible methods of their solutions.
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