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Read Back Signals in Magnetic Recording - Research Group Fidler

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Numerical Methods<br />

ϕ ( x ) =δ . (4.8)<br />

i j ij<br />

Each basis function ϕ i refers to the i-th node of the tetrahedral mesh. Figure 4.1 shows such<br />

functions for a 2-dimensional mesh, which are also called “hat functions” due to their<br />

appearance.<br />

Figure 4.1: Typical basis functions of two neighbor<strong>in</strong>g nodes of a 2-dimensional mesh.<br />

For our simulations we need to solve Laplace and Poisson equations of the form<br />

( )<br />

∇ σ() r ∇ u() r = s()<br />

r . (4.9)<br />

Here s( r ) is the source term, which vanishes <strong>in</strong> case of a Laplace equation. The additional<br />

factor σ( r ) also <strong>in</strong>cludes the generalized Laplace equation for <strong>in</strong>homogeneous conductors<br />

(see (2.31)). Now we apply the Galerk<strong>in</strong> Method, which leads to<br />

( ( ) )<br />

∫ ∇ σ() r ∇u() r −s() r ϕ i () r dV = 0,<br />

(4.10)<br />

V<br />

Partial <strong>in</strong>tegration yields<br />

∫ ∫ ∫<br />

σ() r ∇u() r ∇ϕ i() r dV = � σ() r ∇u() r ϕi() r dA− s() r ϕi()<br />

r dV . (4.11)<br />

V ∂V<br />

V<br />

We take <strong>in</strong>to account that the solution u is a l<strong>in</strong>ear comb<strong>in</strong>ation of all basis functions ϕ i :<br />

N<br />

∑ ∫ ∫ ∫<br />

u σ() r ∇ϕ () r ∇ϕ () r dV = � σ() r ∇u() r ϕ () r dA− s() r ϕ () r dV . (4.12)<br />

j j i i i<br />

j= 1 V ∂V<br />

V<br />

With the def<strong>in</strong>itions<br />

43

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