# 470-6502/02 – Mathematical Modelling and FEM (MMMKP)

 Gurantor department Department of Applied Mathematics Credits 10 Subject guarantor doc. Ing. Dalibor Lukáš, Ph.D. Subject version guarantor doc. Ing. Dalibor Lukáš, Ph.D. Study level postgraduate Requirement Choice-compulsory type B Year Semester winter + summer Study language English Year of introduction 2015/2016 Year of cancellation Intended for the faculties FEI, FS, HGF, FMT Intended for study types Doctoral
Instruction secured by
BLA19 prof. RNDr. Radim Blaheta, CSc.
LUK76 doc. Ing. Dalibor Lukáš, Ph.D.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Examination 28+0
Part-time Examination 28+0

### Subject aims expressed by acquired skills and competences

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.

### Teaching methods

Lectures
Individual consultations
Project work

### Summary

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.

### Compulsory literature:

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.

### Recommended literature:

C. Johnson: Numerical solution of partial differential equations by the finite element method, Cambridge Univ. Press, 1995

### Way of continuous check of knowledge in the course of semester

Continuous monitoring of study activities and assigned tasks during regular consultations. If some publication activity will be a part of the student's tasks, the relevant article would be presented in the course.

### Other requirements

There are not defined other requirements for student.

### Prerequisities

Subject has no prerequisities.

### Co-requisities

Subject has no co-requisities.

### Subject syllabus:

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.

### Conditions for subject completion

Part-time form (validity from: 2015/2016 Winter semester)
Min. number of pointsMax. počet pokusů
Examination Examination   3
Mandatory attendence participation:

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Conditions for subject completion and attendance at the exercises within ISP:

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2015/2016 (P1807) Computer Science, Communication Technology and Applied Mathematics (1801V001) Informatics P English Ostrava Choice-compulsory study plan
2015/2016 (P1807) Computer Science, Communication Technology and Applied Mathematics (1801V001) Informatics K English Ostrava Choice-compulsory study plan
2015/2016 (P3924) Materials Science and Engineering P English Ostrava Choice-compulsory study plan

### Occurrence in special blocks

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