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Mathematical Aspects of Object-Oriented Modeling and Simulation

FDA167, 2004VT

Status Archive
School Computer and Information Science (CIS)
Division SAS
Owner Peter Fritzson

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Course plan

No of lectures

24 hours (12 lectures)

Recommended for

Ph.D. students or practitioners in computer science or systems engineering.

The course was last given

New course


The investigation of dynamical systems in mechanical, electrical or chemical engineering usually requires a mathematical modeling of the system behavior. This graduate course provides an understanding on the mathematical aspects of object-oriented modeling and simulation based on Modelica. However it differs from traditional numerical analysis courses in that it focuses on the understanding of symbolic and numerical techniques necessary to build simulation language compilers rather than on isolated detailed analyses on them. Building simulation environments requires sometimes a different approach compared to approaches used for traditional programming language compilers. At the end of the course the participants should acquire the necessary knowledge of the symbolic transformation and numerical algorithms for building a simple prototype Modelica Compiler.


Some elementary knowledge in numerical analyses and compiler construction.




1) OO Modeling and Simulation Environments.
a. The equation-based modeling and simulation paradigm
b. Principles of OO Modeling & Simulation with Modelica
c. Design of symbolic and numerical simulation engines.
2) Numerical methods for ordinary differential equations with initial conditions:
a. single-step and multi-step algorithms
• convergence, discretization error, step size control
b. special methods for stiff systems
• backward difference schemes
c. event handling
• root -finding methods
3) Structural Analyses and symbolic manipulation
a. higher-index problems
• index reduction, assignment algorithms, dummy derivative methods
b. simple symbolic transformation
• eliminating trivial equations
c. Error handling and recovery
• Structural and numerical inconsistencies.
4)Solution of linear and nonlinear system of equations
a. direct methods for small systems
• LU (QR) - decomposition
b. iterative methods for large systems
• Jacobi- bzw. Gauß-Seidel methods
c. Newton-methods, quasi-Newton-methods
5) Line methods for partial differential equations (PDEs)
a. Difference methods
• Convergence and Stability


Relevant papers will be distributed during the course as needed.
[1] Brenan Kathryn Eleda, S. L. Campbell, and L.R. Petzold. (1996). Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations., SIAM's Classics in Applied Mathematics. Society for Industrial & Applied Mathematics, Philadelphia, 1996.
[2] Cellier François E. (1991). Continuous System Modeling. Springer-Verlag, New York, 1991.
[3] Cheney Ward and Devid Kincaid. (2004). Numerical Mathematics and Computing - Fifth Edition. Thomson Learning Ltd, 2004.
[4] Fritzson Peter. (2003). Principles of Object-Oriented Modeling and Simulation - with Modelica 2.1. IEEE Press and John Willey, 2003.
[5] Health Michael T. (2002). Scientific Computing. An Introductory Survey - Second Edition. McGraw-Hill Higher Education, 2002.
[6] Mattsson Sven Erik and Gustav Söderlind. (1993) "Index Reduction in Differential-Algebraic Equations Using Dummy Derivatives." SIAM Journal on Scientific Computing, vol. 14, pp. 677-692, 1993.
[7] Pantelides Costas. (1988) "The consistent initialization of differential algebraic systems." SIAM Journal on Scientific and Statistical Computing, vol. 9: 2, pp. 213--231, 1988.
[8] Tarjan R.E. (1972) "Depth First Search and Linear Graph Algorithms." SIAM Journal of Computting, vol. 1, pp. 146-160, 1972.


Bernhard Bachmann
Peter Bunus


Peter Fritzson
Bernhard Bachmann


Homework assignments


3 credits


Prof. Dr. Bernhard Bachmann received his Ph.D. in Mathematics from the University of Zürich, Switzerland in 1994. After working several years in industry, he has been appointed in 1999 Professor of mathematics at the University of Applied Sciences in Bielefeld, Germany. His research areas are numerical analysis, scientific computation, object-oriented modeling and power systems simulation. Dr. Bachmann is a member of the Modelica Association.

Preliminary Schedule

1. Monday, April 26 10.00h - 12.00h
2. Tuesday, April 27 13.00h - 15.00h
3. Tuesday, May 4 13.00h - 15.00h
4. Monday, May 10 10.00h - 12.00h
5. Tuesday, May 11 13.00h - 15.00h
6. Monday, May 17 10.00h - 12.00h
7. Tuesday, May 18 13.00h - 15.00h
8. Tuesday, June 1 13.00h - 15.00h
9. Monday, June 7 10.00h - 12.00h
10. Tuesday, June 8 13.00h - 15.00h
11. Monday, June 14 10.00h - 12.00h
12. Tuesday, June 15 13.00h - 15.00h

Page responsible: Director of Graduate Studies
Last updated: 2012-05-03