Looking at it from some distance now, my scientific life consisted of three different periods: an introductory and exploratory phase, my successful flirt with finite differences and finally my long struggle with the unruly subject of finite elements. On this page I hope to describe each of these parts in a way that can be enjoyed by both the electromagnetic lay(wo)man and my former colleagues. I also have a link to some new ideas about finite elements for electromagnetics. Finally I will tell a few things about my life after science.
After graduating in Electrical Engineering in 1970, I got a
job at the
Delft University of Technology in
For proving one is able to do scientific work, a PhD thesis had to be written. Before writing my thesis I first built a general picture of the field of interest and did some work on various subjects that were hot topics at the time, like holography and fibre optics. I also studied various finite-difference techniques, with an application to hyperthermia in mind. These tools were still too immature at the time and we did not have enough computing power and memory available yet. My explorations of the subject area finally led to compiling the thesis:
1. G. Mur, Computation
of electromagnetic fields in
inhomogeneous media: Scattering and Guiding properties, PhD
This thesis, that consists of a number of papers I published in the 1970's, is still available. You can contact me about it by E-mail . The title of my thesis has accurately predicted the central theme of my scientific career, a subject that is usually referred to as computational electromagnetics, the research concerned with the question of how to efficiently, accurately and reliably compute electromagnetic fields in complex geometries. Because of the very fast development of computer power and memory capacity over the past decades this was a rapidly developing science over that period.
At the time I did my PhD it was customary in our research
subsequently widen your view by spending a year doing research abroad.
a place at the Department of Electrical and Electronic
2. G. Mur, Absorbing Boundary Conditions for the finite-difference approximation of the time-domain electromagnetic-field equations, IEEE Transactions on Electromagnetic Compatibility, vol. 23, no. 4, pp. 377-382, 1981.
3. G. Mur, The modeling of singularities in the finite-difference approximation of the time-domain electromagnetic-field equations, IEEE Transactions on Microwave Theory and Techniques, vol. 29, no. 10, pp. 1073-1077, 1981.
The paper describing Absorbing Boundary Conditions (ABCs)
proved to be
a lasting success. The terms first order Murand second
order Mur were coined by their users to indicate which one of
methods described in that paper they applied for solving their
those terms as search strings in Google proves that the methods I
almost 30 years ago now are still very popular, you may even find exams
subject. At a much later date I published an inhomogeneous version of
the ABC, thus
work on this subject.
In 2007 the IEEE Transactions on Electromagnetic Compatibility (EMC) celebrated its 50th anniversary. On that occasion they republished the paper on the ABCs because of it being the Second Most-ReferencedIEEE Transactions on EMC paper in the 50 years of their existence.Click this line for the PDF of the relevant EMC Society Newsletter, together with the PDF of the paper.
Recently I discovered that a lecture on a one-dimensional version of my ABCs, given by professor Cynthia Furse of the Department of Electrical and Computer Engineering,
Meanwhile, having returned to
4. G. Mur and J. Baan, Computation of the input impedances of a catheter for cardiac volumetry, IEEE Transactions on Biomedical Engineering, vol. 31, pp. 448-453, 1984.
Another interesting activity during the period mentioned above was
contract-research for firms interested in our kind of expertise. This
research made us aware of the state of the art in the Finite-Element
(FEM) for electromagnetics which, next to FDTD, is a very
computational tool. This brought me back to the subject of computational
electromagnetics. It struck us (=de Hoop and me) that
the Finite-Element world was talking about the so-called edge
elementsthat seemed to solve all their problems. Edge elements are a
of finite element that is defined on simplices, i.e. on triangles in 2D
tetrahedra in 3D (see the picture at the top or bottom of this page).
edge elements then had, and possibly for many former colleagues still
very much of a panacea, but we did not see that then, we joined the
Professor Aad (A.T.) de Hoop and I discussed edge elements at length, aiming at a more accurate and efficient, a so-called fully consistent, version of them. After a few weeks we both went home some day and independently designed new edge elements. Comparing our results the next day our approaches proved to be fundamentally different, this difference very much reflecting our different approaches to mathematical science, but after analyzing the final results we could show that we had designed exactly the same fully consistent elements! Using them, a simple finite-element programme was build by me to demonstrate our ideas and they were published in the paper below.
5. G. Mur and A. T. de Hoop, A finite-element method for computing three-dimensional electromagnetic fields in inhomogeneous media, IEEE Transactions on Magnetics, vol. 21, no. 6, pp. 2188-2191, 1985.
When using our new elements for practical problems it became clear to us that, although being more efficient than the existing tools, they still were computationally rather expensive and that edge elements proved not to be entirely without problems. Successive improvements regarding both the choice of elements and the formulation of the finite-element method were reported in the following series of papers.
6. G. Mur, Optimum choice of finite elements for computing three-dimensional electromagnetic fields in inhomogeneous media, IEEE Transactions on Magnetics, vol. 24, no. 1, pp. 330-333, 1988.
7. G. Mur, Finite-element modeling of three-dimensional electromagnetic fields in inhomogeneous media, Radio Science, vol. 26, no. 1, pp. 275-280, 1991.
8. I. E. Lager and G. Mur, Compatibility relations for time-domain and static electromagnetic field problems, Applied Computational Electromagnetics Society Journal, vol. 9, no. 2, pp. 25-29, 1994. PDF.
9. G. Mur, Edge elements, their advantages and their disadvantages, IEEE Transactions on Magnetics, vol. 30, no. 5, pp. 3552-3557, 1994.PDF
In the above paper I summarized our findings until then. I continued my quest, from now on being skilfully supported by Hansi (I.E.) Lager who wrote his PhD thesis on the static and stationary version of the subject.
10. I. E. Lager and G. Mur, The finite element modeling of static and stationary electric and magnetic fields, IEEE Transactions on Magnetics, vol. 32, no. 3, pp. 631-634, 1996. PDF
11. I. E. Lager and G. Mur, Finite element method for stationary and static electromagnetic fields, Journal of Applied Physics, vol. 81, no. 8, pp. 4079-4081, 1997.
Our research led to mounting evidence against edge elements and at the
COMPUMAG Conference on the Computation of Electromagnetic Fields in
The analysis I presented at the Panel Session was later published in print as:
12. G. Mur, The fallacy of edge elements, IEEE Transactions on Magnetics, vol. 34, no. 5, pp. 3244-3247, 1998. PDF
Obviously not everybody was very pleased with the critical conclusions
presented, but I was complimented by many, envied by some and I was
by a concerned colleague saying: "Dear Gerrit,
I arrived home unharmed to continue my work on finite elements. The very explicit conclusions of this paper still remain unrefuted. We continued our work with the intention to find solutions to the fundamental reliability problems we had encountered.
13. I. E. Lager and G. Mur, Generalized Cartesian finite elements, IEEE Transactions on Magnetics, vol. 34, nr. 4, p. 2220-2227, 1998. PDF
14. G. Mur and I. E. Lager, A finite-element method for the modeling of electromagnetic fields using generalized Cartesian elements, IEEE Transactions on Magnetics, vol. 35, no. 3, pp. 1376-1379, 1999. PDF
15. G. Mur and I.E. Lager, A finite element package for electromagnetics using generalized Cartesian elements, Software for electrical Engineering. Analysis and Design IV, A. Konrad and C.A. Brebbia (editors), WITPRESS, Southampton, pp. 203-212, 1999.
Unfortunately our scientific quest has ended with the
conclusion that the type of finite-element method we had in mind, with
without edge elements, could not be made fully reliable. Because of
possibility of obtaining, possibly undetected, erroneous
this context they are often kindly referred to as "spurious
or "spurious modes", remains. This absence
reliability has far-reaching consequences that many still prefer to,
silently but sometimes even explicitly, ignore.
Some additional explicit observations on the causes of spurious solutions were presented in:
16. G. Mur and I. E. Lager, On the causes of spurious solutions in electromagnetics, Electrotechnics, vol. 22, no. 4, pp. 357-367, 2002. PDF
To summarize my sad story about the finite edge elements it
can be said
that we rediscovered that it is very unwise to impose 4
conditions on a vector having only 3 components, or alternatively to
unknowns than independent equations. But didn't we all learn that long
New formulations of a finite-element method for computational electromagnetics that aim at avoiding the problems mentioned above were proposed in:
17. I. E. Lager and G. Mur, Application of the Domain-Integrated Field Relations Method to the Solution of Large Scale Static and Stationary Magnetic Field Problems, IEEE Transactions on Magnetics, vol. 38, no. 2, pp. 465-468, 2002. PDF
18. I. E. Lager, E. Tonti, A. T. de Hoop, G. Mur, and M. Marrone, Finite Formulation and Domain-Integrated Field Relations in Electromagnetics—A Synthesis , IEEE Transactions on Magnetics, vol. 39, no. 3, pp. 1199-1202, 2003. PDF
I did not actively contribute to these last two papers, but the authors who did, expressed their appreciation for my work on the subject by making me their co-author. I wish to use this opportunity to thank them for this honor.
NEW: Some of my recent thoughts about the finite-element modeling of electromagnetic fields can be found here. Do have a look there if you are familiar with edge elements.
From early 2001 I was unable to continue
because of a
heavy fatigue that forced me to stay at home almost permanently. It
to be a new symptom of the Primary Progressive version
Sclerosis that I was diagnosed with the year before.
this fatigue a companion in distress advised my to start biking, which
fortunately is, and always was, my type of sport. I followed his advise
started biking, twice a week, in summer on the roads around
After my retirement in 2002 I first started modeling in clay and wax and subsequently I started sculpting in stone, first in relatively soft materials like alabaster and serpentine and nowadays mainly in harder materials like the beautiful black Noir de Mazy (Belgian marble) and often in marble. Sculpting is a very physical outdoors (because of the dust) activity that keeps me physically fit, without needing to bike all day. It requires about the same set of skills as my scientific work did, i.e. a good spatial insight, a lot of stamina and finally also some creativity. Now that you have come this far, do also have a look at these more recent results of my efforts, I hope you like some of them, and please don't hesitate to comment using E-mail. Below you can choose between a Dutch and an English version of the presentation of my sculptures.
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