Read Back Signals in Magnetic Recording - Research Group Fidler

Numerical Methods
ϕ ( x ) =δ . (4.8)
i j ij
Each basis function ϕ i refers to the i-th node of the tetrahedral mesh. Figure 4.1 shows such
functions for a 2-dimensional mesh, which are also called “hat functions” due to their
appearance.
Figure 4.1: Typical basis functions of two neighboring nodes of a 2-dimensional mesh.
For our simulations we need to solve Laplace and Poisson equations of the form
( )
∇ σ() r ∇ u() r = s()
r . (4.9)
Here s( r ) is the source term, which vanishes in case of a Laplace equation. The additional
factor σ( r ) also includes the generalized Laplace equation for inhomogeneous conductors
(see (2.31)). Now we apply the Galerkin Method, which leads to
( ( ) )
∫ ∇ σ() r ∇u() r −s() r ϕ i () r dV = 0,
(4.10)
V
Partial integration yields
∫ ∫ ∫
σ() r ∇u() r ∇ϕ i() r dV = � σ() r ∇u() r ϕi() r dA− s() r ϕi()
r dV . (4.11)
V ∂V
V
We take into account that the solution u is a linear combination of all basis functions ϕ i :
N
∑ ∫ ∫ ∫
u σ() r ∇ϕ () r ∇ϕ () r dV = � σ() r ∇u() r ϕ () r dA− s() r ϕ () r dV . (4.12)
j j i i i
j= 1 V ∂V
V
With the definitions
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