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4.2.3 Discretization of the Vacancy Balance Equation
The combination of (3.74) with (3.75) and (3.76) yields the vacancy balance equation





(4.41) 
for which the weak formulation of the form
is obtained under the assumption of a Neumann boundary condition.
Applying the electric potential and the temperature discretization, (4.29) and (4.36), respectively,
together with

(4.43) 

(4.44) 

(4.45) 
and the backward Euler time discretization, the vacancy balance equation discretized in a single element is given by
under the assumption that
, , and are constant inside an element.
Using the shorthand notation (4.31), (4.38), (4.39), and

(4.47) 
(4.46) is written as
In the above derivation the vacancy diffusivity is treated as a scalar diffusion coefficient. In order to take into account the anisotropy of diffusivity due to the mechanical stress, as presented in Section 3.2.2, a diffusivity tensor must be applied. This requires a slight modification of (4.48) to
where the diffusivity tensor
is now incorporated into and
, given by

(4.50) 
and

(4.51) 
At material interfaces and grain boundaries the trapped vacancy concentration is governed by (3.76), rewritten here as

(4.52) 
The finite element formulation of this equation follows the same procedure described above, which yields the discretization





(4.53) 
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R. L. de Orio: Electromigration Modeling and Simulation