Page 461 - Maxwell House

APPROACH TO NUMERICAL SOLUTION OF EM PROBLEMS 441

Introduction
In the chapter that follows, we will attempt to breach the numerical fence hiding the solutions
of Maxwell’s equations. Depending on the problem’s nature this fence may be a see-through
2
1
veil like shown in Figure 9.1.1a or a protective suit depicted in Figure 9.1.1c .

a) b) c) d)

Figure 9.1.1 a) See-through veil, b) Venus de Milo, c) Protective suit, d) Man Ray, Fair
Weather (Le Beau Temps), 1940

The design process of the analytical, numerical, or semi-analytical model based on the see-
through veil image is evident. For example, merely transfer the measurement data from 3D
recognition optical camera into your computer, slightly correct the object shape and sizes using
3
Photoshop and the 3D underlying model of the beautiful statue, shown in Figure 9.1.1b , is
created. Translated this scenario to the computer model, our design becomes “well-posed or
correctly-set” meaning that “small” variations in input parameters lead to “small” changes in
following computer simulation results. Meanwhile, if your concept was covered by a protective
4
suit, as shown in Figure 9.1.1c, the human models may look something like in Figure 9.1.1d .
This is certainly a piece of art but too abstract to recognize a real lady or man. Furthermore, the
fundamental assumption that the “creature” in Figure 9.1.1c is a woman might be wrong. It
means the model is ill-posed and changes drastically under the influence of just one of the input
parameters, e.g. gender.

Now let us return to the more prosaic theme of electrodynamics. As an example illustrating the
design process, we chose a simple parabolic antenna or dish. Keep in mind that the described
path to a numerical solution is far from being unique and depends on your expertise, prior
experience, creativity, and intuition.

Unfortunately, the computer simulation quite seldom increases our understanding the problem
and physics behind it. According to [7], “… it often works the other way. People with the most
prodigious codes and supercomputers may not have the cutest physical insight. This is because
they must spend so much time buying cutting-edge hardware, improving graphical user-

1 Public Domain Image, source: http://www.metmuseum.org/art/collection/search/247004
2 Public Domain Image, source: https://www.remm.nlm.gov/percutaneous_ppe.htm
3 Public Domain Image, source: https://www.inbodyusa.com/blogs/inbodyblog/78720577-is-there-an-
ideal-body-composition
4 Public Domain Image, source: https://www.pinterest.com/pin/422986589972115635/