# 470-8743/03 – Mathematical Modelling and FEM (MMMKP)

 Gurantor department Department of Applied Mathematics Credits 4 Subject guarantor doc. Ing. Dalibor Lukáš, Ph.D. Subject version guarantor doc. Ing. Dalibor Lukáš, Ph.D. Study level undergraduate or graduate Requirement Choice-compulsory Year 1 Semester summer Study language Czech Year of introduction 2016/2017 Year of cancellation Intended for the faculties FMT, FEI, USP Intended for study types Follow-up Master
Instruction secured by
LUK76 doc. Ing. Dalibor Lukáš, Ph.D.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 3+1
Part-time Credit and Examination 15+5

### Subject aims expressed by acquired skills and competences

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.

Lectures
Tutorials

### Summary

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.

### Compulsory literature:

- 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 literature:

- 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.

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

Written and oral exam.

### E-learning

Lecture notes as a pdf-file are available at am.vsb.cz.

### Other requirements

Additional requirements for students are not.

### Prerequisities

Subject has no prerequisities.

### Co-requisities

Subject has no co-requisities.

### Subject syllabus:

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.

### Conditions for subject completion

Full-time form (validity from: 2016/2017 Winter semester)
Min. number of pointsMax. počet pokusů
Credit and Examination Credit and Examination 100 (100) 51
Credit Credit 30  10
Examination Examination 70  21 3
Mandatory attendence participation: Attendance at discussions is obligatory (70 %). Attendance at lectures is expected.

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Conditions for subject completion and attendance at the exercises within ISP: Completion of all mandatory tasks within individually agreed deadlines.

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### Occurrence in study plans

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2023/2024 (N0719A270002) Nanotechnology MM2 P Czech Ostrava 1 Choice-compulsory type A study plan
2022/2023 (N0719A270002) Nanotechnology MM2 P Czech Ostrava 1 Choice-compulsory type A study plan
2021/2022 (N0719A270002) Nanotechnology MM2 P Czech Ostrava 1 Choice-compulsory type A study plan
2020/2021 (N0719A270002) Nanotechnology MM2 P Czech Ostrava 1 Choice-compulsory type A study plan
2019/2020 (N0719A270002) Nanotechnology MM2 P Czech Ostrava 1 Choice-compulsory type A study plan
2018/2019 (N2658) Computational Sciences (2612T078) Computational Sciences P Czech Ostrava 1 Choice-compulsory study plan
2017/2018 (N2658) Computational Sciences (2612T078) Computational Sciences P Czech Ostrava 1 Choice-compulsory study plan
2016/2017 (N2658) Computational Sciences (2612T078) Computational Sciences P Czech Ostrava 1 Choice-compulsory study plan

### Occurrence in special blocks

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner

### Assessment of instruction

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