Read Back Signals in Magnetic Recording - Research Group Fidler

Numerical Methods
within the volume V. Outside the magnetic field H 1 is set to zero
out
H 1 = 0 . (4.54)
Then the boundary condition writes as
∂H
∂n
in
1
=− n× j
∂V
. (4.55)
Now H 1 can be calculated within the volume V as described in Section 4.1. The source term
is determined by
∫
s = curl jϕ ( r ) dV . (4.56)
i i
V
Here the source term is a vector. Each component refers to one of the three Poisson equations
in (4.53). Partial integration gives
si = � ∫ n× jϕi() r dA − ∫j×
grad ϕi()
r dV . (4.57)
∂V
V
The boundary integral becomes
in
∂H1
bi = ∫ ϕ i() r dA =− × ϕi()
dA
∂ ∫ n j r
n
� � . (4.58)
∂V ∂V
So the first integral in the source term and the boundary term cancel each other. The right
hand side is therefore
∫
r =− j× grad ϕ(
r ) dV . (4.59)
i i
V
The gradient of the test function and the current density are constant for each tetrahedron, so
the integral (4.59) can be calculated as sum over all tetrahedrons, which are enclosed to the
i-th node.
The second part H 2 has to fulfill
and further
Δ H = 0, Δ H = 0
(4.60)
in out
2 2
53