# Numerical Methods for Analysis and Optimization of Mechanical Systems

Summerterm

## Content of the lecture

Introduction to numerical methods used for investigating dynamic systems. General principles of numerical calculations, machine numbers, error estimation, numerical stability, linear algebra: Cholesky-decomposition, Gaussian elimination, LU-decomposition, QR-decomposition, iterative methods, least square problem, eigenvalue problem: general basics, normal forms, power methods, QR-algorithm, computaion of eigenvectors, initial value problem: ordinary differential equations, Runge-Kutta methods with step size control, extrapolation methods, linear multistep methods, -applications, programme libraries, comparison of methods for analytical investigations with computational methods.

## Course information

###### Lecture and Exercises

For students of Mechanical engineering, Mechatronics, Mathematics and the interdisciplinary Graduate Study Programme COMMAS.

Monday, 11.30 a.m. - 1.00 p.m., V55.01.
Thursday, 11.30 a.m. - 1.00 p.m., V47.04.

First lecture in the summer term 2024 is on Monday, 8. April 2024.

###### Language

The course is tought in German.

###### Institute

The Institute of Engineering and Computational Mechanics is located in Pfaffenwaldring 9, 3rd and 4th floor.

###### Contact
• 1. Introduction
• 2. General principles of numerical calculations
• 2.1 Definitions
• 2.2 Numerical principals
• 2.3 Machine numbers
• 2.4 Error estimation
• 3. Systems of linear algebraic equations
• 3.1 Motivation
• 3.2 Direct methods for square coefficient matrix
• 3.3 Cholesky-decomposition
• 3.4 Gauss-elimination
• 3.5 LR-decomposition
• 3.6 QR-decomposition
• 3.7 Iterative methodes for square coefficient matrix
• 3.8 Determinant of a matrix
• 3.9 Matrix inversion
• 3.10 Least square problem
• 3.11 Tools and numerical libraries for linear algebraic equations
• 4. Eigenvalue problems
• 4.1 General basics
• 4.2 Normal forms
• 4.3 Power methods
• 4.4 QR-algorithm
• 4.5 Calculation of eigenvectors
• 4.6 Practical solution of eigenvalue problems
• 4.7 Tools and numerical libraries for eigenvalue problems
• 5. Initial value problem for ordinary differantial equations
• 5.1 Motivation
• 5.2 Basic remarks
• 5.3 Single-step methods
• 5.4 Extrapolation methods
• 5.5 Multistep methods
• 5.6 Comparison of the different methods
• 5.7 Tools and numerical libraries for initial value problems

The lecture is supplemented by exercises. The exercises are discussed in class by Dr.-Ing. Pascal Ziegler. The documents are avaliable on the german webpage of the course.

### Current information

Current information can be found on the German Webpage.