Read Back Signals in Magnetic Recording - Research Group Fidler

Numerical Methods
current flowing out of the volume. Normally the Neumann boundary conditions are given
either for each boundary triangle
j() r = j , ∀r ∈Δ
(4.18)
ijk ijk
or for each boundary node of the mesh
j( r) = j , ∀r ∈∂V
. (4.19)
i i i
For the first case we get for the boundary integral term in (4.15)
1
σ() r ∇u() r ϕ () r dA=− j() r ϕ () r dA=−∑ j ϕ () r dA=− ∑A
j
3
�∫ � ∫ ∫
. (4.20)
i i ijk i ijk ijk
∂V ∂V Δijk Δijk
j< k
Δijk
j< k
For the second case we get for the current j by linear interpolation between the sampling
points (points at the boundary) using the test functions i ϕ
∑
j() r = j ⋅ϕ () r . (4.21)
i
i i
Then the result of the boundary integral is
�∫ �∫
σ() r ∇u() r ϕ () r dA=− j() r ϕ () r dA=
i i
∂V ∂V
1
−∑ ∫ ( jiϕ i() r + jjϕ j() r + jkϕk() r ) ϕ i() r dA=− ∑Aijk
( 2 ji + jj + jk)
.
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Δijk Δ
Δ
ijk
ijk
j< k j< k
4.1.3 Linear equation solver
The FEM method leads to a system of linear equations (4.16)
(4.22)
Au ij j = ri
⇔ Au ⋅ = r (4.23)
with a sparse symmetric and positive definite matrix A. To solve this system of linear
equations a Cholesky factorization method for sparse matrices seemed to be practicable [23].
The advantage of this method over iterative methods is the insensibility of “bad” meshes. The
convergence fails with iterative methods if the tetrahedrons show much difference from
regular tetrahedrons, so that they have small angles. This is problematic for meshes of thin
layers, as used in GMR elements.
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