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Mathematical analysis and numerical simulation of bone remodeling
Čížek, Pavel ; Hron, Jaroslav (referee) ; Maršík, František (advisor)
In the presented work, we study a model of bone deformation caused by action of external forces. After the biological introduction, in which we describe a bone remodelling process and its connection with bone deformation, we deal with conservation laws and material characteristics of a bone. We define weak formulation. Further, we linearize the equations and prove existence and uniqueness of the solution. Consecutively, we describe the finite element method and the programming of this method for the computing of the approximate solution of the problem using the logic of object-oriented programming. We also adduce a description of the used data structures. In the conclusion, we present numerical results reached by the described program that represent deformation of femur and deformation of vertebras and intervertebral foramina.
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Application of spectral element method in fluid dynamics
Pech, Jan ; Hron, Jaroslav (referee) ; Maršík, František (advisor)
This work presents application of spectral element method (SEM) for solving partial differential equations. This method can be seen as combination of spectral method (SM) and finite element method (FEM). Computational domain is decomposed to smaller elements, what enable description of more general geometries. On each element is then used the spectral method, which brings high accuracy. In this work the basic theory is presented for SM and SEM with emphasize to facilitate application of this method in developing computer programs. Numerical scheme is presented on several appropriate examples. The result of this work is in comparison of solution of chosen problems obtained by SM and SEM from written program. Comparison with professional software is also done and shows power of spectral methods.
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Metoda vrstevnic a modelování proudění nemísitelných nestlačitelných tekutin
Čapek, Marek ; Maršík, František (referee) ; Hron, Jaroslav (advisor)
The focus of this thesis are three-dimensional simulations of flow of two immiscible incompressible viscous fluids and distribution of concentration transported by the flow where the free interface between the fluids is realized by level-set method. It is applied to blood flow and coagulation. The blood flow is described using Navier-Stokes equations for incompressible fluids, numerical method is done by finite element method with Q2 - Q1 elements and the problem is solved as coupled. The distribution of concentration is described using transport equation, discretization is done using Q1 elements.
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Modelování nestlačitelných ionizovaných směsí
Havrda, Jaroslav ; Roubíček, Tomáš (advisor) ; Bulíček, Miroslav (referee) ; Hron, Jaroslav (referee)
In the present work we study the model of incompressible ionized mixtures from point of view of mathematical analysis and practical implementations. The model consists of the continuity equation, Navier-Stokes equations, Nernst-Planck equations and Poisson equation for the electric potential. We proove the existence of a weak solution for the model in three space dimensions. We also present a simple program for computing the solutions of Nernst-Planck equations and Poisson equation and use it to examine the problem that arrise during the computation. The results in thermodynamics of mixtures are refered as the physical part of the thesis.
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