## Numerical Analysis of Physical Processes

Home Institution: Lublin University of Technology, Poland

Course pre-requisites: basic knowledge of high school math and physics, ability to use the computer for studying, intermediate (or higher) level of technical english.

Course Overview
The course is designed for students on bachelor and master academic degrees which represents any engineering field of studies. It is well known that the students learn best when they are motivated by problems. The main idea of course is to familiarize students with problem-solving approaches based on numerical methods with using of mathematics, logics and computation, which will improve their skills, knowledge and experience in engineering.

The course begins from introduction to numerical analysis and methods – current state of knowledge about the approach and discussion about its using in different aspects of engineering to describe physical processes. Then the description of basic principles and definitions of numerical methods with giving suitable examples as well as available non-computing and software tools needed for its implementation will be presented. The mathematical background of the course will contain: functions, arrays, differentiation and integration, systems of linear equations, nonlinear algebra, interpolation and approximation. Students will be able to practice in solving of such issues by using well known computer tools for numerical analysis as MatLab or SciLab. They will have an opportunity to provide graphical interpretations of problem-solving as flowcharts, 2D and 3D plotting. Understanding of mathematical basics is necessary part of proper development and using of numerical methods in engineering.

Second part of course is related to numerical investigations of physical problems such as: magnetostatic, electrostatic, heat flow and current flow. It will be also possible to examine electromagnetic field distribution and its individual components in systems designed by students. Participants will learn basics of lightweight script programming, so-called Lua, which will be used for dynamical analysis of magnetic systems. All physical investigations will be provided in FEMM computer environment.

Last part of course will be summarizing and drawing conclusions. The importance of numerical analysis in physical processes will be defined. In addition, main advantages and disadvantages of numerical methods as problem-solving approach will be determined. At the very end of course, potential applications of computer aided numerical analysis in different aspects of science and engineering will be proposed.

Learning Outcomes
By the end of this course the following effects are expected:

• students should have gained basic understanding of the importance of problem solving approach based on computer aided numerical analysis,
• participants should understand the concepts of numerical analysis and the opportunities of its implementation in physical investigations,
• students should be able to identify the problem, analyze it and propose the numerical approach for its solving,
• students should understand the importance of graphical interpretation of physical processes,
• participants will be capable of providing numerical investigation of physical phenomena by using computer simulation environments,
• students should be able to develop structured computer programs on the basis of pseudocode, flowcharts, or other forms of algorithms,
• participants should become familiar with the computer tools being used during the course; students will be comfortable in using them to solve numerical problems in further studying.

Course Content
The main topics to be addressed in course:

• An introduction to the application of numerical analysis in engineering: the importance of numerical research, the numerical experiment design, characterization of numerical methods.
• Basic principles and definitions of numerical methods: functions, arrays, differentiation and integration, systems of linear equations, nonlinear algebra, interpolation and approximation.
• Graphical interpretations of engineering problems: flowcharts, 2D and 3D plotting.
• The main concepts, classification and mathematical background of physical problems: magnetostatic problem, electrostatic problem, heat flow and current flow problems. Physical meaning of equations describing problems.
• The modeling in continuous media: definition of continuous media, basic algebraic equations describing material science problems.
• The mechanics of continuous field problem, the conjugate field problem.
• Introduction to the Boundary Element Method: methodology, implementation, element approximation, integration, area discretization, discrete equations, boundary conditions, algebraic equations.
• Introduction to Finite Element Method: basic concepts, geometry, planar and axisymmetric problems, visualization, discretization, multi-dimensional problems.
• Description and characterization of computer environments for numerical problem-solving in engineering: Matlab, Scilab, Mathcad, FEMM, Lua programming.

Instructional Method
The course includes the next types of lessons:

• Lectures – 8×2h = 16h,
• Computer laboratory exercises – 8×3h = 24h,
• Examination (final test) – 2h.

Total course duration is 42h.

Required Course Materials
Literature recommended for use during the course:

• C. Chapra, R. P. Canale, “Numerical Methods for Engineers. 7th edition”, McGraw-Hill Education, 2015.
• W. Hamming, “Numerical Methods for Scientists and Engineers. 2nd edition”, Dover Publications, 2012.
• S. Sastry, “Introductory methods of numerical analysis”, Prentice-Hall of India, 2006.

Necessary software: FEMM v.4.2 (Finite Element Method Magnetics), Matlab v. 2012 (or higher) or Sci-Lab v. 6.02 (free equivalent of Matlab).

Additional equipment: slide projector, laser pointer.

Assessment
On the very beginning (during first lecture) short quiz with multiple-choice questions related to basics of high school mathematics will be completed. It’ll allow to gauge the students prior knowledge of the subject and build further strategy of teaching. At the end of each lecture some short quizzes or minute papers will be given for participants to complete – it will help me to evaluate how well the students are grasping the material. The laboratory classes will be divided onto 2 sets of computer exercises linked thematically. During the labs I’ll provide a discussion related to the topic in way of ask students a questions and giving them enough time to respond. Important part of our interaction will be asking students for questions to be sure that the participants understand material properly. Students will be required to compile reports on each set of exercises. The lectures will end on final test related to main aspects of the course.

To assign the final grade I’ll take into account: student’s learning achievement (quizzes and final test results, reports evaluation), general performance (quality of the reports, proper way of thinking), and effort (commitment to the course, interacting with lecturer, asking questions, demonstration of initiative).

The European ECTS grading system will be used: A = excellent (> 90%), B = very good (80-90%), C = good (70-80%), D = satisfied (60-70%), E = pass (50-60%), F = fail (<50%).

ECTS credits are divided in the next way: lectures = 3 credits, labs = 3 credits, which gives 6 credits in total.