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4.2.1 Discretization of Laplace's Equation

The electric potential is calculated from equation (3.71). Multiplying it by a test function $ {\symShapeFun}_{p}={\symShapeFun}_{p}(\vec r)$, and integrating over the domain $ \symDomain$ one obtains

$\displaystyle \ensuremath{\int_{\symDomain}{\left[\ensuremath{\nabla\cdot{(\sym...
...symElecPot}})}}\right]{\symShapeFun}_{p}}\ d{\symDomain}} = 0,\qquad p=1,...,N.$ (4.25)

Using Green's formula [152]

$\displaystyle \ensuremath{\int_{\symDomain}{v\ensuremath{\nabla^2 w}}\ d{\symDo...
...mBoundary}{v\ensuremath{\frac{\partial w}{\partial \vec n}}}\ d{\symBoundary}},$ (4.26)

where $ \partial{w}/\partial{\vec n}$ represents the normal derivative in the outward normal direction to the boundary $ \symBoundary$, and assuming a Neumann boundary condition, i.e. vanishing normal derivatives, on $ \symBoundary$, equation (4.25) can be written as

$\displaystyle -\ensuremath{\int_{\symDomain}{\symElecCond\ensuremath{\nabla{\sy...
...}\ensuremath{\nabla{{\symShapeFun}_{p}}}}\ d{\symDomain}} = 0,\qquad p=1,...,N.$ (4.27)

Since the electrical conductivity $ \symElecCond$ depends on the temperature according to (3.73) and varies along the simulation domain, it is part of the integrand in (4.27). However, in a single element the conductivity is assumed to be constant, being determined by the average of the temperature on the element nodes, i.e. $ \symElecCond = \symElecCond(\bar\T)$ with

$\displaystyle \bar\T = \frac{{\T}_{1} + {\T}_{2} + {\T}_{3} + {\T}_{4}}{4},$ (4.28)

for a tetrahedral element. Applying the discretization for the electric potential as in (4.6),

$\displaystyle \symElecPot(\vec r, t=t_n) = \symElecPot^n = \ensuremath{\sum_{i=1}^{4}{\symElecPot_i^n}}{\symShapeFun}_{i}(\vec r),$ (4.29)

where $ \symElecPot_i^n$ is the electric potential of the node $ i$ at a time $ t_n$, equation (4.27) for a single element becomes

$\displaystyle -\symElecCond\ensuremath{\sum_{i=1}^{4}{\symElecPot_i^n}} \ensure...
...}\ensuremath{\nabla{{\symShapeFun}_{p}}}}\ d{\symDomain}} = 0,\qquad p=1,...,4.$ (4.30)

Using the shorthand notation

$\displaystyle K_{ip}$ $\displaystyle = \ensuremath{\int_{T}{\ensuremath{\nabla{{\symShapeFun}_{i}}}\ensuremath{\cdot}\ensuremath{\nabla{{\symShapeFun}_{p}}}}\ d{\symDomain}}$    
  $\displaystyle = \det(\mathbf{J})\int_{0}^{1}\int_{0}^{1-\xi}\int_{0}^{1-\xi-\et...
...\ensuremath{\cdot}\mathbf{\Lambda}\nabla^t \symShapeFun_p^t\ d\zeta d\eta d\xi,$ (4.31)

where the last term is the calculation in the transformed coordinate system, (4.30) is rewritten as

$\displaystyle -\symElecCond\ensuremath{\sum_{i=1}^{4}{\symElecPot_i^n}}K_{ip} = 0, \qquad p=1,...,4,$ (4.32)

which corresponds to a discrete system of 4 equations with 4 unknowns, i.e. the electric potential at each node of the tetrahedral element.


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R. L. de Orio: Electromigration Modeling and Simulation