Dynamics of Very High Dimensional Systems

The text is written in September 2008 for the http://groups.google.com/group/mor4ansys group.

Dynamics of Very High Dimensional Systems by Earl H. Dowell, Deman Tang, 2003

The book is pretty systematic and it uses modal analysis as the basic tool. This is what I am actually missing, as I am not a mechanical engineer. I am considering reading it more carefully, as it may help me to understand engineers better. It should be especially mentioned that the authors consider reduced order modeling and reduced models in the book.

My current problem is that I do not understand why mechanical engineers are so fixed on mode superposition. From the model reduction viewpoint this is not the best method and this one can easily observe in the work of mathematicians. They just do not consider it seriously. The reason is simple – modal superposition does not work for most dynamic systems, to name a few: an electrical circuit, a thermal system, an advective-convective problem and so on.

Actually I know only one case where mode superposition works – structural mechanics. Yet, this is the background for many engineers and they seem do not want to believe that there could be something better as mode superposition. I have observed it many times. It is enough to mention that I know a model reduction method that is better than mode superposition. After that a person looses the interest to the talk completely. Presumably he thinks – one more nuts.

I should confess that mode superposition has one strong advantage. The method does not depend on the load vector and the reduced model is based only on system matrices. On the other hand, model reduction based on Krylov subspaces is faster and more accurate. However here it is necessary to limit the excitation to a few load vectors.

Back to the book. It looks really good. What I could additionally recommend after that is the Antoulas book Approximation of Large-Scale Dynamical Systems

One can see clearly a gap. In the Antoulas book there are just a couple of pages about mode superposition and in the book in the subject mode superposition plays the major role. I guess that what is missing is the understanding why actually mode superposition works for mechanical models. I mean that the system matrices describing a mechanical model seem to posses some special mathematical properties and it would be good to define them explicitly. After all structural models are met most often in engineering and even this is a special case, it is very important one.

Well, if we make one step from pure structural mechanics, then mode superposition does not work, as I have already said. A good example is modeling of fluid-structural interaction considered in the book in subject. In this case, the model reduction based on Krylov subspaces is just better – see Srini’s thesis

http://modelreduction.com/Applications/Acoustics.html

at the bottom of the page.