Page 54 - Computational Modeling in Biomedical Engineering and Medical Physics

40 Computational Modeling in Biomedical Engineering and Medical Physics

and if on the surface Σ either its normal component or its tangential component are
specified
nUFnðÞ 5 F n ; n 3 FnðÞ 5 F t : ðA1:4Þ

The curl and divergence are local physical quantities and the mathematical models,
to which they lead, may be higher order partial differential equations (PDEs) of differ-
ent types—Laplace, Helmholtz, and so on.
The integral form of the theorem (A.1.3) requires that the circulation, R(r), and
the flux, Q(r), of the vector field F(r) are specified
I ð I ð
FUds 5 ð r 3 FÞUdA 5 R; FdA 5 ð rUFÞdv 5 Q: ðA1:5Þ
Γ A Γ Σ v Σ

The two integral relations, Stokes and Gauss, that relate the circulation and
flux integrals to surface and volume integrals, respectively, are used as discussed
earlier.
The PDEs are solved numerically by using domain (interior) methods, such as
finite differences and finite element (Bathe and Wilson, 1976; Peyret and Taylor,
1983; Morega, 1998). The mathematical models derived using the integral forms
may be solved numerically using boundary numerical methods (Brebia et al.,
1984).
Remarkably, the integral forms (3) are used to define the divergence and the
curl, which are invariant with respect to the system of coordinates. The curl is
given by
H H
Γ Fds Σ ð n 3 FÞdA
r 3 F 5 n lim ; or r 3 F 5 n lim ; ðA:1:6Þ
ΔA Γ -0 ΔA Γ Δv Σ -0 Δv Σ
j
where n is the unit vector of the direction along the which r 3 Fj is maximum, A Γ
is the area of an open surface A that is bounded by the closed curve Γ, and v A is the
volume bounded by A. The divergence is defined by

H
F ndA
Σ
r F 5 lim ðA1:7Þ
Δv Σ -0 Δv Σ
The special forms of the curl and divergence for interfaces between media with
different material properties are as follows:
r S 3 F 5 n 12 3 F 2 2 F 1 Þ; r S F 5 n 12 F 2 2 F 1 Þ; ðA1:8Þ
ð
ð
where ( ) 1,2 denote the two different media separated by the interface, and n 12 is
the normal to the interface, oriented from ( ) 1 to ( ) 2 . If a superficial source