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
Thesis,

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
group to
subsequently widen your view by spending a year doing research abroad.
I found
a place at the Department of Electrical and Electronic
Engineering,

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 Mur*and *second
order Mur* were coined by their users to indicate which one of
the two
methods described in that paper they applied for solving their
problems. Using
those terms as search strings in Google proves that the methods I
developed
almost 30 years ago now are still very popular, you may even find exams
on the
subject. At a much later date I published an inhomogeneous version of
the ABC, thus
completing my
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-Referenced*IEEE 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
performing
contract-research for firms interested in our kind of expertise. This
contract-
research made us aware of the state of the art in the Finite-Element
Method
(FEM) for electromagnetics which, next to FDTD, is a very
popular
computational tool. This brought me back to the subject of *computational
electromagnetics*. It struck us (=de Hoop and me) that
everybody in
the Finite-Element world was talking about the so-called *edge
elements*that seemed to solve all their problems. Edge elements are a
special kind
of finite element that is defined on simplices, i.e. on triangles in 2D
and
tetrahedra in 3D (see the picture at the top or bottom of this page).
Those
edge elements then had, and possibly for many former colleagues still
have,
very much of a panacea, but we did not see that then, we joined the
herd.

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
1997
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
I
presented, but I was complimented by many, envied by some and I was
also warned
by a concerned colleague saying: "*Dear Gerrit, Riomay be a dangerous city but from now on this conference room is even
more
dangerous for you.*" After a nice 4 weeks holiday in various
corners
of Brazil, during which we even met the legendary (in Holland), or
infamous (in
Germany), Harm Dost in his very nice Pousada "Nova Holanda", about
13km. from Canguaretama, Rio Grande do Norte.

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
disappointing
conclusion that the type of finite-element method we had in mind, with
or
without edge elements, could not be made fully reliable. Because of
this the
possibility of obtaining, possibly undetected, *erroneous
results*, in
this context they are often kindly referred to as "*spurious
solutions*"
or "*spurious modes*", remains. This absence
of
reliability has far-reaching consequences that many still prefer to,
mostly
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
independent
conditions on a vector having only 3 components, or alternatively to
have more
unknowns than independent equations. But didn't we all learn that long
ago?

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
my work
because of a
heavy fatigue that forced me to stay at home almost permanently. It
turned out
to be a new symptom of the Primary Progressive version
of Multiple
Sclerosis that I was diagnosed with the year before.
For overcoming
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
and
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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