Mathematical modeling for engineers

Innovative teaching and learning strategies in open modelling and simulation environment for

studentcentered engineering education Projekt-Nr.: 573751EPP-1-2016-1-DE-EPPKA2-CBHE-JP

*The European Commission's support for the production of this publication does not constitute an endorsement of the contents, which reflect the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

Pag

e1

Mathematical modeling for engineers

Autumn semester, 2018-2019

Study Program 010300 – Fundamental informatics and informational technologies 02.03.02- Informatics and Computer Science

Study Program Cooordinator

Yuri Senichenkov (Peter the Great St. Petersburg Polytechnic University, Russia)

Credits 4 ECTS (required course), 64 in-class hours

Lector Yuri Senichenkov (Peter the Great St. Petersburg Polytechnic University, Russia)

Level BSc

Host institution Peter the Great St. Petersburg Polytechnic University, Russia

Course duration September, 4 –December, 25, 2018

Summary Mathematical modeling and mathematical models are the basis of computer models,

which can be built with the help of modern tools for modeling and simulation of complex real world and technical systems. In this course students study special class of mathematical models named complex dynamical systems. Complex dynamical system is the generalization of classical dynamical systems. Classical dynamical system is widely used for modelling real world systems and designing new technical systems. Dynamical systems may be discrete, continuous, and event-driven. An engineer should understand that any real technical system is subjected to perturbation, so it is very important to know what it means stability, bifurcation, numerical experiment and how to use computer experiment in day-to-day engineering activity. Discipline «Mathematical modeling for engineers» is the final in the series of disciplines of the bachelor curriculum. The discipline teach students to use modern tool for modeling and simulation complex dynamical systems.

Target student audiences

This course is recommended to all future engineers irrelatively their concrete specialty Students study basic discipline «Mathematical modeling for engineers» during 7-th semester. Discipline «Mathematical modeling for engineers» is basic discipline of Block 2 (part 2) of bachelor curriculum «Informatics and Computer Science». Prerequisites Required courses (or equivalents):

- Linear algebra




Pag

e2

- Differential equations, - Numerical Methods, - Theory of algorithms and automata

Competences of graduating student

code Competences of graduating student

GPC-1 Ability to use basic knowledges of natural sciences, mathematic and informatics, basic facts, conceptions, principals of theories, connected with fundamental informatics and informational technologies.

GPC-3 Ability to find algorithmic and program solutions for problems in the field of system and applied programming, mathematical, informational and simulation models, to design informational resources for wide-area networks, education content, applied databases, tests and tools for testing and verification compliance with standard and requirements

GPC-4 Ability to solve standard professional problems using informational and bibliographical data with the help of Information, Communications Technologies and taking in account basic requirements of information security

PC-2 Ability to understand, modify, and use modern mathematical apparatus, fundamental conceptions and systems methodologies, international and professional standards in the field of computer science and information technologies.

PC-3 Ability to use modern workbench and computing facilities.

PC-7 Ability to design and realize life-cycle processes of informational systems and their services, software, and methods and mechanisms of estimation and analyzing functioning tools and systems of informational technologies.

Desired course outcomes (Estimated results of discipline learning):

Knowledge: Basis of mathematical modeling of complex dynamical systems. Abilities and experiences: Ability to design, debug and test models of complex dynamical systems, to carry out computer experiments with them using modern tools for modeling and simulation (Matlab, Mathematica, Maple, Rand Model Designer). Activities: Designing computer models of complex dynamical systems, using tools for visual modeling technical systems, building new instruments for analyzing models, using and modifying numerical software.

Overview of sessions and teaching methods




Pag

e3

The course based on hybrid learning methodology. The course include face-to-face (classroom) and distance training using e-learning technologies. The classroom and distance training activities include individual and group methods of education. Elements of the project-oriented approach used in carrying out individual practical assignments.

Course workload The table below summarizes course workload distribution:

Activities

Learning outcomes Assessment Estimated workload (hours)

In-class activities

Lectures Understanding theories, concepts, methodology and tools

Class participation

32

In-class assignments / Virtual laboratories

Ability to build a basic mathematical models of complex dynamical systems, perform a computational experiment, analyze the obtained results

Class participation and preparedness for assignments; Virtual laboratories report

32

Individual work

Learning how to do computer experiments in Mathematica (Maple). Topics Linear Algebra, Algebraic equations, Ordinarily differential equations

Ability to use Mathematica (Maple). Help. User Guide.

6

Learning how to do computer experiments in Rand Model Designer (Continuous and hybrid models)

Ability to use Rand Model Designer Help. User Guide.

12

Learning how to do computer experiments

Ability to use Rand Model Designer Help. User Guide.

12




Pag

e4

in Rand Model Designer (modeling language, Animation, standard computer experiment)

Individual work on a course materials in e-learning system

The ability to work effectively with e-learning resources, the ability of group communication in the study of educational materials

Quantitative and qualitative statistics of student work in the e-learning system Sakai

20

Individual study of the main and additional literature on the course and preparation for lectures and practical classes

The ability of self-organization and self-education; Familiarity with and ability to critically and creatively discuss key concepts, tools and methods as presented in the literature

Quality of completed assignments for the course

12

Preparation for intermediate control (exam)

The ability to correctly present the material studied in the course of the course to the teacher, to show possession of theoretical and practical skills

High knowledge of the course materials

18

Total 144

Grading The students’ performance based on the following: - Lectures at the training course (10%); - Performance of virtual laboratory work (20%); - Individual work, Learning RMD, Mathematica (Maple) (20%); - Presentation of completed virtual laboratory works (10%); - Work with training resources in the Sakai system (20%); - Control testing (10%); - Level of readiness to participate in discussions in the classroom and Sakai system (10%)

Course schedule

Topic title Lesson form

Lesson content Hours Day/ Time

Lector

Module 1. Mathematical

modeling

Lecture History of modeling. Material

and abstracts models. 2 4.09.18

Thursday Yu.Senichenkov




Pag

e5

Classification of

models

Numerical

experiments.

Software for

mathematical

modeling.

Mathematical models.

Computer models. Models

based on differential and

difference equations. Natural

and numerical experiments.

Stages of numerical

experiments. Virtual reality.

Models based differential

equations with partial

derivatives. Probabilistic

models. Tools for mathematical

modeling: Matlab, Maple,

Mathematica.

12:00-13.40

Module 2. Dynamical

systems Continues

dynamical

systems

Discrete

dynamical

systems

Properties of

dynamical

systems

Lecture Knowledge:

Continuous time. Linear and

non-linear differential

equations with initial

conditions. Singular points and

their classification. Solution of

linear systems. Phase portrait.

Discrete time. Linear and non-

linear difference equations.

Fixed points. Solution of linear

systems.

Two-dimensional dynamical

systems and their behavior.

6 11.09.18 Thursday

12:00-13.40

Yu.Senichenkov

18.09.18 Thursday

12:00-13.40

25.09.18 Thursday

12:00-13.40

Module 3 Stability od

dynamical

systems

Lyapunov’v

stability

Lyapunov's first

and second

methods for

stability

Lecture Knowledge:

Definition of Lyapunov’s

stability. Theorems about

stability. Stability of linear and

non-linear systems.

Linearization and connection

between stability of non-liner

and linearized systems.

Lyapunov functions. Lyapunov

stability criterion.

6 2.10.19 Thursday

10:10-11.40

Yu.Senichenkov

9.10.19 Thursday

10:10-11.40

23.10.19 Thursday

10:10-11.40

Module 4




Pag

e6

Theory of

bifurcation

Bifurcation of

continues

systems

Bifurcation of

discrete systems

Strange attractors

Lecture Knowledge:

Bifurcation in dynamical

systems. Periodic orbits or

other invariant sets. Types and

examples of bifurcation.

Bifurcation diagrams. Lameray

diagram. Chaos. Lorenz

attractor.

4 30.10.19 Thursday

10:10-11.40

Yu.Senichenkov

6.11.19 Thursday

10:10-11.40

Module 5 Examples of

dynamical

systems

Theory of

oscillations

Markovian

processes

Lecture Knowledge:

Oscillation of two-dimensional

nonlinear autonomous systems.

Autooscillations. Effect of

harmonic force.

Markov chains.

4 13.11.19 Thursday

10:10-11.40

Yu.Senichenkov

20.11.19 Thursday

10:10-11.40

Module 6 Numerical

experiments. Visualization of

behavior

Statistical

experiments

Lecture Knowledge:

Numerical methods for ODE.

2D- and 3D animation.

Standard experiments

4 27.11.19 Thursday

10:10-11.40

Yu.Senichenkov

4.12.19 Thursday

10:10-11.40

Module 7 Competitive

analysis of tools

for

mathematical

modeling

Lecture Knowledge:

Comparative analysis of Maple,

Mathematica. Building isolated

systems in Rand Model

Designer and Open Modelica.

4 11.12.19 Thursday

10:10-11.40

Yu.Senichenkov

18.12.19 Thursday

10:10-11.40

Lab1 One dimensional

dynamical

system

(difference and

differential

equation)

Lab Tools for mathematical


Mathematica. Plots

RMD: continuous systems

Skills: To be able to use mathematical

tools for analyzing dynamical

systems.

4 7.09.18 Friday

14.00-15.40

Yu.Senichenkov

13.09.18 Friday

14.00-15.40

Lab2




Pag

e7

Two-dimensional

dynamical

system

(difference and

differential

equation)

Lab Tools for mathematical


Mathematica. ODE, Phase

Portraits RMD: hybrid systems

Skills: To be able to use mathematical

tools for analyzing hybrid

systems (state-machines)

6 19.09.18 Friday

14.00-15.40

Yu.Senichenkov

26.09.18 Friday

14.00-15.40

5.10.18 Friday

14.00-15.40

Lab3 Lyapunov’v

stability Lab Skills:

To be able to analyze stability

of non-linear systems with the

help of mathematical tools.

6 12.10.18 Friday

14.00-15.40

Yu.Senichenkov

19.10.18 Friday

14.00-15.40

26.10.18 Friday

14.00-15.40

Lab4 Bifurcation. Lab Tools for mathematical


Mathematica. ODE, Phase

Portraits Bifurcation of Continuous and

Discrete systems

6 2.11.18 Friday

14.00-15.40

Yu.Senichenkov

9.11.18 Friday

14.00-15.40

16.11.18 Friday

14.00-15.40

Lab5 Queuing systems. Lab Markov equations.

Continuous systems and

Discrete systems

6 24.11.18 Friday

14.00-15.40

Yu.Senichenkov

30.11.18 Friday

14.00-15.40

7.12.18 Friday

14.00-15.40

Lab6




Pag

e8

Computer

experiment.

Debugging and

testing.

Lab Skills:

To use mathematical tools for

carrying out computer

experiments.

4 14.12.18 Friday

14.00-15.40

Yu.Senichenkov

21.12.18 Friday

14.00-15.40

Total 62

Course assignments Course assignments are include: - Performance of virtual laboratory work; - Working with Helps and User’ guides - Using different tool for computer experiments Tasks for virtual labs: 1. Senichenkov Yu.B., Ampilova N.B., Timofeev E. Collection of tasks for the course "Mathematical modeling of complex dynamic systems". - St. Petersburg: SPbPU, 2017. 2. https://sakai.dcn.icc.spbstu.ru/portal/ List of virtual Lab in Sakai

https://sakai.dcn.icc.spbstu.ru/portal/




Pag

e9

Literature

Basic literature

1. Kolesov Yu.B., Senichenkov Yu.B. Mathematical modeling of complex dynamic systems. - St. Petersburg: SPbPU, 2018.

2. Senichenkov Yu.B., Ampilova N.B., Timofeev E. Collection of tasks for the course

"Mathematical modeling of complex dynamic systems". - St. Petersburg: SPbPU, 2017. Additional literature

3. Shornokov Yu. V., Dostovalov D. N. Fundamentals of event- continuous systems simulation theory/ Novosibirsk, 2018




Pag

e10

4. Kolesov Yu. B., Senichenkov Yu. B. Mathematical modeling of hybrid dynamic systems. - St. Petersburg: SPbPU, 2014.

5. Kolesov Yu. B., Senichenkov Yu. B. Object-Oriented Modeling in Rand Model Designer 7.

St. Petersburg: Prospekt, 2016

Internet resources required for studying the course 1. https://sakai.dcn.icc.spbstu.ru/portal/ 2. https://www.mvstudium.com/

https://sakai.dcn.icc.spbstu.ru/portal/

https://www.mvstudium.com/

Documents

Mathematical modeling for engineers