# Numerical Modelling of Physical Phenomena in Engineering, Biology and Medicine

## Course description

Lectures:

A brief overview of the basic procedures of modelling in engineering and biology; determination of the system and its surroundings, the selection and mathematical description of variables that describe the system and the observation time.

Numerical methods for solving systems of linear algebraic equations and nonlinear algebraic equations.

The optimization procedures.

Numerical methods for solving ordinary differential equations.

Formulation of partial differential equations with appropriate initial and boundary conditions.

Numerical solution of partial differential equations; basics of finite difference method and finite element method.

The basics of cellular automata and Monte Carlo methods.

Laboratory work:

Solving of various problems in biology and medicine using Matlab, its toolboxes (Partial Differential Equation Toolbox) and Comsol Multiphysics program.

## Objectives and competences

During this course students will gain knowledge about the modeling and the use of numerical methods for solving problems in engineering, biology and medicine. They will learn the basic procedures of mathematical model construction on the basis of typical examples from technique, medicine and biology. They will learn the basics of cellular automata modeling,  Monte Carlo methods and optimization procedures.

Mostly, the course deals with numerical methods for solving partial differential equations. Students learn the theoretical basis of the finite difference and finite element method. On the basis of solving simple problems at the beginning and more complex problems during the course the students familiarize with the advantages and limitations of numerical methods. An important part of the learning process is an analysis of the calculated values and their comparison with the experimentally obtained values in cases where the results of the corresponding measurements are available.

The diversity of cases offers students an useful knowledge in the wider field of engineering and science.

## Learning and teaching methods

Lectures; solving typical problems in the context of laboratory work; solving complex tasks in the context of laboratory work and independent work at home.

## Intended learning outcomes

After successful completion of the course, students should be able to:

• describe the concepts of modelling in technology, biology in medicine
• choose the appropriate method for numerical solving of mathematical formulations
• state the theory of finite element method as the main method for numerical solving of partial differential equations
• use numerical methods for solving simple and more complex cases

## Reference nosilca

1. Maček Lebar A, Damjanić F, Antolič V, Iglič A, Srakar F, Brajnik D. Nepravilnosti v cementnem plašču : analiza z metodo končnih elementov = Cement filling defects : a finite element analysis. Farmacevtski vestnik 47(3): 311-314, 1996.
2. Iglič A, Kralj-Iglič V, Daniel M…, Maček Lebar A. Computer determination of contact stress distribution and size of weight bearing area in the human hip joint. Computer methods in biomechanics and biomedical engineering 5(2): 185-192, 200
3. Šel D, Maček Lebar A, Miklavčič D. Feasibility of employing model-based optimization of pulse amplitude and electrode distance for effective tumor electropermeabilization. IEEE T. Biomed. Eng. 54: 773-781, 2007.
4. Jelenc J, Jelenc J, Miklavčič D, Maček Lebar A. Low-frequency sonoporation in vitro: experimental system evaluation. Strojn. Vestn. 58: 319-326, 2012.
5. Maček Lebar A, Velikonja A, Kramar P, Iglič A: Internal configuration and electric potential in planar negatively charged lipid head group region in contact with ionic solution, Bioelectrochemistry, 111: 49–56, 2016.

## Study materials

1. Dunn SM, Constantinides A, Moghe PV. Numerical methods in biomedical engineering, Elsevier 2006
2. Reddy J.N. Introduction to the Finite Element Method, McGraw-Hill 1993
3. Fagan MJ. Finite Element Analysis – Theory and Practice, Longman 1992
4. Kwon YW, Bang H. The finite element method using Matlab, CRC Press 2000
5. Comsol Multiphysics – User's Guidebook, Comsol AB., 2004
6. Schiff JL. Cellular Automata: A Discrete View of the World, Wiley-Interscience 2008

## Bodi na tekočem

Univerza v Ljubljani, Fakulteta za elektrotehniko, Tržaška cesta 25, 1000 Ljubljana