| Course Unit Code | 470-6502/01 |
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| Number of ECTS Credits Allocated | 10 ECTS credits |
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| Type of Course Unit * | Choice-compulsory |
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| Level of Course Unit * | Third Cycle |
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| Year of Study * | |
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| Semester when the Course Unit is delivered | Winter, 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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| KUC14 | prof. RNDr. Radek Kučera, Ph.D. |
| LUK76 | doc. Ing. Dalibor Lukáš, Ph.D. |
| Summary |
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The lectures concern the boundary value problems arising in the mathematical modelling of
heat transfer, elasticity and other phenomena (diffusion, electric and magnetic fields, etc.).
The differential and variational formulations of these problems are derived and the numerical solution by the finite element method is considered in detail, including analysis of the discretization error. The lectures also touch the principles of proper use of mathematical models for solving engineering problems.
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| Learning Outcomes of the Course Unit |
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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. |
| Course Contents |
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Lectures:
Mathematical modeling. Purpose and general principles of modeling. Benefits mathematical modeling. Proper use of mathematical models.
Differential formulation of mathematical models. One-dimensional task heat conduction and its mathematical formulation. Generalization of the model. Input data linearity, existence and uniqueness of solutions. Discrete input data. One-dimensional problems of elasticity and other models. Multivariate models.
Variational formulation of boundary problems. Weak formulation of boundary value problems and its relationship to classical solutions. Energy functional and energy formulation. Coercivity and boundedness. Uniqueness, continuous dependence solution to the input data. Existence and smoothness of the solution.
Ritz - Galerkin (RG) method. RG method. Method konenčných elements (FEM) as a special case of the RG method. History FEM.
Algorithm finite element method. Assembling the stiffness matrix and load vector. Taking into account the boundary conditions. Numerical solution of the system linear algebraic equations. Different types of finite elements.
Accuracy of finite element solutions. Priori estimate of the discretization errors. Convergence, h-and p-version FEM. Posteriori estimates. Design network FEM, adaptive technology and optimal network.
Software for FEM and its use for MM. Preprocessing and postprocessing. Commercial software systems. Solutions particularly challenging and special tasks. Principles of mathematical modeling using FEM. |
| Recommended or Required Reading |
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| Required Reading: |
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K. Rektorys: Variational Methods in Mathematics, Science and Engineering. (Translation from Czech). Dortrecht—London—Boston, Reidel Publ. Co, 1st edition 1977,
2nd edition 1979. 566 pages
J. Nečas, I. Hlaváček: Mathematical Theory of Elastic and Elasto-Plastic Bodies: An. Introduction. Elsevier, Amsterdam-Oxford-New York, 1981; (Translation from Czech).
R. D. Cook: Finite element modelling for stress analysis, J. Wiley, New York, 1995.
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K. Rektorys: Variační metody v inženýrských problémech a v problémech
matematické fyziky, Academia 1999.
J. Nečas, I. Hlaváček: Úvod do matematické teorie pružných a pružně
plastických těles, SNTL Praha 1983.
R. D. Cook: Finite element modelling for stress analysis, J. Wiley, New
York, 1995.
C. Johnson: Numerical solution of partial differential equations by the
finite element method, Cambridge Univ. Press, 1995
D. Braess.: Finite elements. Cambridge University Press, 2001
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| Recommended Reading: |
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| C. Johnson: Numerical solution of partial differential equations by the finite element method, Cambridge Univ. Press, 1995 |
| C. Johnson: Numerical solution of partial differential equations by the finite element method, Cambridge Univ. Press, 1995 |
| Planned learning activities and teaching methods |
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| Lectures, Individual consultations, Project work |
| 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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| Examination | Examination | | |