1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2023/2024
4. Mathematics - drugiego stopnia z tyt. magistra
5. Mathematical Modelling 2

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Mathematical Modelling 2 - opis przedmiotu

Cel przedmiotu

The aim of the course is to familiarize students (at a basic level) with the nature, scope and stages of mathematical modeling. The lecture will include a broad overview of the mathematical models and methods used in technical problems, such as heat transfer processes or deformations of viscoelastic bodies. The goal of the laboratory classes is to simulate the presented models using a selected mathematical package (Matlab, Octave or Scilab). After completing this course, the student should be prepared to create simple mathematical models using computers and their mathematical knowledge.

Wymagania wstępne

Zapisz zmiany

Students should pass: Introduction to Numerical Methods, Differential Equations.

Zakres tematyczny

Lecture

Methods of asymptotic analysis in mathematical modeling

1. Introduction to mathematical modeling of technical problems. (2 hrs.)
2. Dimensional analysis - units, scaling, dimensionlessness. (2 hrs.)
3. Regular perturbation in differential equations. (2 hrs.)
4. Singular perturbation in differential equations. (2 hrs.)
5. Application of perturbation methods in modeling of technical problems. (2 hrs.)

Mathematical models of thermal processes

1. One-dimensional models of heat conduction. Construction of stationary and non-stationary models. Methods of determining solutions. Examples of mathematical models. (4 hrs.)
2. Two-dimensional models of heat conduction. Construction of stationary and non-stationary models. Methods of determining solutions. Examples of mathematical models. (4 hrs)

Modeling of viscoelastic materials

1. Fundamentals of vector and tensor calculus. (2 hrs.)
2. Kinematics of solids. (4 hrs.)
3. Law of conservation of mass, momentum and energy. Equation of motion. (2 hrs.)
4. Constitutive equations of viscoelastic materials. (4 hrs.)

Laboratory

Methods of asymptotic analysis in mathematical modeling

1. Introduction to a mathematical package (Matlab, Octave or Scilab). (2 hrs)
2. Dimensional analysis - units, scaling, dimensionlessness. (2 hrs.)
3. Regular perturbation in differential equations. (3 hrs.)
4. Singular perturbation in differential equations. (2 hrs.)
5. Application of perturbation methods in modeling of technical problems - construction of models, determination of solutions, interpretation and visualization of results, use of a mathematical package in the modeling process. (4 hrs.)
6. Colloquium (1 hr)

Mathematical models of thermal processes

1. One-dimensional heat conduction models - construction and analysis of models, determination of solutions, interpretation and visualization of results, use of mathematical package in the modeling process. (6 hrs)

Modeling of deformation of viscoelastic materials

1. Fundamentals of vector and tensor calculus. (4 hrs.)
2. Kinematics of solids. (2 hrs.)
3. Models of viscoelastic materials - construction of models, determination of solutions, interpretation and visualization of results, use of a mathematical package in the modeling process. (3 hrs.)
4. Colloquium (1 hr).

Metody kształcenia

Lectures using multimedia devices. Laboratory exercises in which students solve computational problems analytically and using a given mathematical package (Matlab, Octave or Scilab).

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

Laboratory grade is based on tests (80%) and class activity (20%).

The course grade consists of the laboratory grade (50%) and the exam grade (50%).

The condition for passing the course is positive grades from the laboratory and the exam.

Literatura podstawowa

1. B. Burnes, G. R. Fulford, Mathematical modeling with case studies, Taylor and Francis, 2002.
2. G. R. Fulford, P. Forrester, A. Jones, Modelling with Differential and Difference Equations, Cambridge University Press, 1997.
3. D. Kincaid, W. Cheney, Numerical Analysis. Mathematics of Scientific Computing, The University of Texas at Austin, 2002.
4. A. S. Wineman, K. R. Rajagopal, Mechanical response of polymers. An introduction., Cambridge University Press, 2000.
5. G. A. Holtzapfel, Nonlinear Solid Mechanics – A Continuum Approach for Engineering., Wiley, New York, 2000.

Literatura uzupełniająca

1. J. D. Logan,  Applied mathematics, a contemporary approach, John Wiley and Sons, New York, 2001.
2. J. D. Logan, An Introduction to Nonlinear PDE, John Wiley and Sons, New York, 1994.
3. A. Björck, G. Dahlquist, Numerical Methods in Scientific Computing, SIAM, 2008.
4. G. R. Fulford, P. Broadbridge, Industrial Mathematics, Cambridge University Press, 2002.

Uwagi

Zmodyfikowane przez dr Ewa Sylwestrzak-Maślanka (ostatnia modyfikacja: 10-04-2024 16:18)