Read Back Signals in Magnetic Recording - Research Group Fidler

Numerical Methods
Aij = ∫ σ() r ∇ϕ j () r ∇ϕi()
r dV , (4.13)
V
si =−∫s() r ϕi()
r dV , (4.14)
V
∂u()
r
bi = ∫ σ() r ∇u() r ϕ i() r dA= ∫ σ() r ϕi()
r dA
∂n
� � (4.15)
∂V ∂V
(4.12) can be written as
N
∑ Au ij j
j=
1
= bi + si = ri
. (4.16)
Here the system matrix A = ( Aij
) is symmetric and positive definite. s i is the source term,
and b i is due to boundary conditions and only contributes, if the i-th node is situated at the
surface ∂ V ( r i ∈∂V
). In this case we have to determine the integral in (4.15) by taking
boundary conditions into account. Here we have to distinguish between Dirichlet and
Neumann boundary conditions. r i denotes the right hand side and is the sum of the boundary
and the source term.
4.1.1 Dirichlet Boundary Conditions
In case of Dirichlet boundary conditions the value of the i-th node u( r ) = b is known. So the
whole i-th equation can be replaced by
i i
ui = bi.
(4.17)
This would destroy the symmetry of the matrix A ij . To maintain its symmetry, which is
convenient for solving a system of linear equations, the matrix entry A ii is just multiplied by a
6
large number ( ~10 ). Then the right hand side is set to b i multiplied with the new coefficient
A ii .
4.1.2 Neumann Boundary Conditions
In case of Neumann boundary conditions the normal component of σ() r ∇u()
r , that is
n⋅σ() r ∇ u() r =− j()
r , is given at the boundary of the mesh. Here j() r can be interpreted as a
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