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Finite element solution of axially loaded bars using linear element
Plucnar, Tomáš ; Návrat, Tomáš (referee) ; Vaverka, Jiří (advisor)
This bachelor thesis deals with the finite element method for axially loaded bars using linear basis functions. The theoretical part briefly describes the theory of axially loaded bars and states the individual steps that lead from the initial differential equation to a system of linear algebraic equations. A Weak formulation of the differential equation is used to derive the system. Using the theory described in the first part, an algorithm is created in Matlab, which is used to solve four problems. The results are then compared with the analytical solution and with the model in Ansys.
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Finite element solution of axially loaded bars using quadratic element
Janáčik, Lukáš ; Halabuk, Dávid (referee) ; Vaverka, Jiří (advisor)
This bachelor thesis describes an algorithm for programming finite-element model with quadratic elements for axial loaded bar. In the introduction, we define the basic concepts of mechanics of materials, which are used in this thesis and are necessary to understand problems, for which finite element method was formulated. The thesis clarifies a transition from basic differential equation to weak formulation, which is the base of finite element method. We define element matrices and describe transition to global matrices, relating to the whole body. Then describe implementation of boundary conditions and postprocessing of the results, necessary for calculation and displaying of other unknowns. In the practical part, 3 illustrative problems are presented and calculated numerically in FEM solver using Matlab, analytically and in software ANSYS Workbench. Results are then compared and evaluated. Problems have different boundary conditions (linear axial load, tempered cross section, statically indeterminate fixation). Results of displacement and normal stress for programmed solver are identical to those from Ansys (using the same settings) and analytical solution (after more elements are added, if necessary). Problem with tempered cross section was simulated in Ansys using plain stress, because the program can’t define bar with tempered cross section. This revealed sheer stress contained in parts of cross section further from centreline, which are not calculated in our FEM solver and in some cases might be significant.
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Finite element solution of axially loaded bars using quadratic element
Janáčik, Lukáš ; Halabuk, Dávid (referee) ; Vaverka, Jiří (advisor)
This bachelor thesis describes an algorithm for programming finite-element model with quadratic elements for axial loaded bar. In the introduction, we define the basic concepts of mechanics of materials, which are used in this thesis and are necessary to understand problems, for which finite element method was formulated. The thesis clarifies a transition from basic differential equation to weak formulation, which is the base of finite element method. We define element matrices and describe transition to global matrices, relating to the whole body. Then describe implementation of boundary conditions and postprocessing of the results, necessary for calculation and displaying of other unknowns. In the practical part, 3 illustrative problems are presented and calculated numerically in FEM solver using Matlab, analytically and in software ANSYS Workbench. Results are then compared and evaluated. Problems have different boundary conditions (linear axial load, tempered cross section, statically indeterminate fixation). Results of displacement and normal stress for programmed solver are identical to those from Ansys (using the same settings) and analytical solution (after more elements are added, if necessary). Problem with tempered cross section was simulated in Ansys using plain stress, because the program can’t define bar with tempered cross section. This revealed sheer stress contained in parts of cross section further from centreline, which are not calculated in our FEM solver and in some cases might be significant.
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Finite element solution of axially loaded bars using linear element
Plucnar, Tomáš ; Návrat, Tomáš (referee) ; Vaverka, Jiří (advisor)
This bachelor thesis deals with the finite element method for axially loaded bars using linear basis functions. The theoretical part briefly describes the theory of axially loaded bars and states the individual steps that lead from the initial differential equation to a system of linear algebraic equations. A Weak formulation of the differential equation is used to derive the system. Using the theory described in the first part, an algorithm is created in Matlab, which is used to solve four problems. The results are then compared with the analytical solution and with the model in Ansys.
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Weak formulation of equations describing fluid flows
Dostalík, Mark ; Pokorný, Milan (advisor) ; Kaplický, Petr (referee)
The standard way of deriving the weak formulation of balance equations of continuum mechanics is derived from their localized form, and thus requires differentiability of functions involved in the corresponding balance law. However, the existence of classical solutions of these equations is often not known. It would be suitable to find a transition to the weak formulation of balance laws without the need of their differential form. The aim of this work is to show that the initial integral form of balance equations of continuum mechanics, provided relatively weak assumptions, directly implies their weak formulation, and thus that the weak solution is for these equations a more natural notion than the classical solution is.
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Weak formulation of equations describing fluid flows
Dostalík, Mark ; Pokorný, Milan (advisor) ; Kaplický, Petr (referee)
The standard way of deriving the weak formulation of balance equations of continuum mechanics is derived from their localized form, and thus requires differentiability of functions involved in the corresponding balance law. However, the existence of classical solutions of these equations is often not known. It would be suitable to find a transition to the weak formulation of balance laws without the need of their differential form. The aim of this work is to show that the initial integral form of balance equations of continuum mechanics, provided relatively weak assumptions, directly implies their weak formulation, and thus that the weak solution is for these equations a more natural notion than the classical solution is.
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