| Course Unit Code | 470-8743/03 |
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| Number of ECTS Credits Allocated | 4 ECTS credits |
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| Type of Course Unit * | Choice-compulsory type A |
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| Level of Course Unit * | Second Cycle |
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| Year of Study * | First Year |
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| Semester when the Course Unit is delivered | Summer Semester |
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| Mode of Delivery | Face-to-face |
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| Language of Instruction | Czech |
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| Prerequisites and Co-Requisites | Course succeeds to compulsory courses of previous semester |
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| Name of Lecturer(s) | Personal ID | Name |
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| LUK76 | doc. Ing. Dalibor Lukáš, Ph.D. |
| Summary |
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| The course should prepare the students to be able to formulate the boundary value problems arising in mathematical modelling of heat conduction, elasticity and other physical processes. The students should be also able to derive differential and variational formulation of these problems and understand the mathematical principles of their numerical solution, especially by the finite element method. The course will also touch the principles of proper use of mathematical modelling methods for solving engineering problems. |
| Learning Outcomes of the Course Unit |
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| Students will be able to formulate the boundary value problems arising in mathematical modeling of heat conduction, elasticity, and other phenomena (diffusion, electro and magnetostatics, etc.). It will also be able to derive the differential and variational formulation of the task and numerical solution of the finite element method. They will know the principles of proper use of mathematical models for solving engineering problems. |
| Course Contents |
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Mathematical modeling. Purpose and general principles of modeling. Benefits
mathematical modeling. Proper use of mathematical models.
Differential formulation of mathematical models. One-dimensional heat conduction problem and its mathematical formulation. Generalizing the model. The input linearity,
existence and uniqueness of solutions. Discrete input data. One-dimensional task
flexibility and other models. Multivariate models.
Variational formulation of boundary problems. Weak formulation of boundary problems and its relationship to the classical solutions. Energy and energy functional formulation.
Coercivity and boundedness. Uniqueness, continuous dependence of solutions
input data. Existence and smoothness of the solution.
Ritz - Galerkin (RG) method. RG method. Konenčných element method (FEM)
as a special case of the RG method. History MLP.
Algorithm finite element method. Assembling the stiffness matrix and vector
load. Taking into account the boundary conditions. Numerical solution of linear systems algebraic equations. Different types of finite elements.
The accuracy of finite element solutions. Priori estimate of the discretization error.
Convergence, h-and p-version FEM. Posteriori estimates. Network design for MLP
adaptive technology and optimal network.
FEM software and its use for MM. Preprocessing and postprocessing. Commercial
software systems. Solutions particularly difficult and special problems. Principles
Mathematical modeling using FEM. |
| Recommended or Required Reading |
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| Required Reading: |
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- GROSSMANN, Christian a ROOS, Hans-Görg. Numerical treatment of partial differential equations. Přeložil Martin STYNES. Universitext. Berlin: Springer, c2007. ISBN 978-3-540-71582-5.
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- BLAHETA, Radim. Matematické modelování a MKP. Ostrava: VŠB-TU, 2012. http://mi21.vsb.cz
- FEISTAUER, Miloslav a KUČERA, Václav. Základy numerické matematiky. Praha: Matfyzpress, 2014. ISBN 978-80-7378-264-1.
- GROSSMANN, Christian a ROOS, Hans-Görg. Numerical treatment of partial differential equations. Přeložil Martin STYNES. Universitext. Berlin: Springer, c2007. ISBN 978-3-540-71582-5.
|
| Recommended Reading: |
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| - QUARTERONI, Alfio a VALLI, Alberto. Numerical approximation of partial differential equations. Springer series in computational mathematics, 23. Berlin: Springer, c2008. ISBN 978-3-540-85267-4. |
- LUKÁŠ, Dalibor. Matematické modelování elektromagnetických polí. Ostrava: VŠB-TU, 2011. http://mi21.vsb.cz
- DRÁBEK, Pavel a HOLUBOVÁ, Gabriela. Parciální diferenciální rovnice: úvod do klasické teorie. Plzeň: Západočeská univerzita, 2001. ISBN 80-7082-766-1.
- QUARTERONI, Alfio a VALLI, Alberto. Numerical approximation of partial differential equations. Springer series in computational mathematics, 23. Berlin: Springer, c2008. ISBN 978-3-540-85267-4. |
| Planned learning activities and teaching methods |
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| Lectures, Tutorials |
| Assesment methods and criteria |
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| Task Title | Task Type | Maximum Number of Points
(Act. for Subtasks) | Minimum Number of Points for Task Passing |
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| Credit and Examination | Credit and Examination | 100 (100) | 51 |
| Credit | Credit | 30 | 10 |
| Examination | Examination | 70 | 21 |