Read Back Signals in Magnetic Recording - Research Group Fidler

Numerical Methods
∫ ∫ ∫
si =− div Mϕ i( r) dV = M⋅grad ϕi( r) dV −� Mϕi( r) dA,
(4.31)
∫
V V ∂V
bi = � Mϕi() r dA.
(4.32)
∂V
Both terms contribute to the right hand side, resulting in
N
r = M⋅grad ϕ ( r) dV = M ϕ ( r)grad ϕ ( r ) dV . (4.33)
∫ ∑ ∫
i i j j i
V j=
1 V
Here the integral can be calculated analytically for each tetrahedron of the mesh.
Since the total potential S S,1 S,2
satisfy the Laplace equation
in out
S,2 S,2
Ψ =Ψ +Ψ must fulfill equations (4.25) and (4.26), Ψ ,2 must
ΔΨ = 0, ΔΨ = 0
(4.34)
with boundary conditions
and
out in
∂Ψ S,2 ∂Ψ S,2
− = 0
∂n ∂n
∂V
out in in
S,2 S,2 S,1
∂V
S
(4.35)
Ψ −Ψ =Ψ . (4.36)
Fortunately the last equations also describe the magnetic scalar potential of a dipole layer with
moment Ψ S ,1 at the surface ∂ V . The scalar potential of such a dipole layer is known:
1
r−r′ Ψ () r = � Ψ ( r′ ) dA′
. (4.37)
∫
S,2 4π
∂V
in
S,13
r−r′ i
In practice the magnetic potential Ψ S ,2 can be evaluated at each node S,2 S,2 i
i
with FEM. Generally the potential vector ( S ,2 )
j
with the vector ( Ψ S ,1 ) ,
i j
S,2 ij S,1
Ψ =Ψ ( r ) , if used
Ψ can be calculated by a matrix multiplication
Ψ = M Ψ . (4.38)
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