Page 55 - Computational Modeling in Biomedical Engineering and Medical Physics

Physical, mathematical, and numerical modeling 41

resides on the interface then the finite size “jump” of the components thus selected
equals that source—for example, sheet current for the tangent and surface charge
for the normal.
These forms are related either to continuity conditions at the interfaces separating
regions with different properties or surfaces, curves, points with field sources—charges
and currents (Mocanu, 1981).
The vector fields are presented through force or field lines (called streamlines
in fluid mechanics), which are the geometric loci of the lines to which the vector
field is tangent. From this perspective, a vector field with a divergence-type of
source has open lines that start from positive sources and end on negative (sink)
sources. In turn, a vector filed produced by a curl-type source shows off closed
lines (eddies). Of course, should the sources be situated outside the ROI (diver-
gence-free or curl-free vector fields) and if the ROI is under the action of that
vector field then the field lines inside the ROI are open—start and land on the
boundary.
The physical model of a vector field problem has to consider the laws that
provide for the two possible types of the vector field sources—a vector source,
R(r), and a scalar source, ρ(r). For instance, for the EMF problems,
Maxwell Hertz laws apply. Consider the particular case of the electrostatic field
problem (in immobile media). The electric field quantities are the electric field
strength, E, and the electric flux density, D. Maxwell laws that present the diver-
gence and the curl of the electric field are the electromagnetic induction
(Faraday) law
r 3 E 5 0: ðA1:9Þ

and the electric flux (Gauss) law, which in the absence of volume electrical charge
density is as follows:
rUD 5 0: ðA:1:10Þ

Apparently Eqs. (A1.7) and (A1.8) imply two different vector fields (E and D);
therefore a law that provides for a supplementary relation between them is needed.
The substance inside the system enters scene through a material (continuity) law.
Assuming a homogeneous, linear, isotropic medium without polarization (Mocanu,
1981), the constitutive relation is as follows:

D 5 εE; ðA:1:11Þ

where ε [F/m] is the electric permittivity, a material property. At this point, it is worth
to note that there are as many constitutive laws as many different media may exist,
and from this perspective Maxwell’s system of laws (equations) is open.