next up previous contents
Next: 4.3.4 Calculation of the Up: 4.3 Simulation in FEDOS Previous: 4.3.2 Assembly of the


4.3.3 Assembly of the Vacancy Dynamics Problem

The vacancy dynamics problem requires the solution of

  $\displaystyle f_p = \sum_{k=1}^{4} C_{v,k}^n M_{kp} - \sum_{k=1}^{4}C_{v,k}^{n-...
...B\bar\T} \sum_{i=1}^4\sum_{k=1}^4 \symElecPot_i^n C_{v,k}^n \Theta_{ipk}\right.$    
  $\displaystyle \left. - \frac{\Q}{\kB\bar\T^2} \sum_{j=1}^4\sum_{k=1}^4 \T_j^n C...
...ar\T} \sum_{l=1}^4\sum_{k=1}^4 \symHydStress_l^n C_{v,k}^n \Theta_{lpk} \right]$ (4.87)
  $\displaystyle + \frac{\Delta t_n}{\symVacRelTime}\left( q\sum_{m=1}^4 C_{im,m}^n M_{mp} - \Ceq V_p\right) = 0,$    

and

$\displaystyle g_{p+4} = \sum_{m=1}^{4} C_{im,m}^n M_{mp} - \sum_{m=1}^{4}C_{im,...
...\sum_{m=1}^4 C_{im,m}^n M_{mp} - \frac{\Delta t_n}{\symVacRelTime}\Ceq V_p = 0,$ (4.88)

for $ p=1,...,4$.

Applying Newton's method the Jacobian matrix has the form

$\displaystyle \mathbf{J_N} = \begin{bmatrix}\displaystyle\ensuremath{\frac{\par...
...isplaystyle\ensuremath{\frac{\partial g_8}{\partial C_{im,4}^n}} \end{bmatrix},$ (4.89)

where the entries are computed by

  $\displaystyle \ensuremath{\frac{\partial f_p}{\partial C_{v,i}^n}} = M_{ip} + \...
...c{\vert\Z\vert\ee}{\kB\bar\T} \sum_{j=1}^4 \symElecPot_j^n \Theta_{jpi} \right.$    
  $\displaystyle \qquad\qquad\qquad\qquad\qquad\left. - \frac{\Q}{\kB\bar\T^2} \su...
...tor\symAtomVol}{\kB\bar\T} \sum_{j=1}^4 \symHydStress_j^n \Theta_{jpi} \right],$ (4.90)
  $\displaystyle \ensuremath{\frac{\partial f_p}{\partial C_{im,i}^n}} = q\frac{\Delta t_n}{\symVacRelTime} M_{ip},$ (4.91)

and

  $\displaystyle \ensuremath{\frac{\partial g_{p+4}}{\partial C_{v,i}^n}} = 0$ (4.92)
  $\displaystyle \ensuremath{\frac{\partial g_{p+4}}{\partial C_{im,i}^n}} = \left(1 + q\frac{\Delta t_n}{\symVacRelTime}\right) M_{ip}, \qquad i,p = 1,\dots,4.$ (4.93)

The corresponding residuals are given by

  $\displaystyle R_p = - \sum_{k=1}^{4} C_{v,k}^{n-1} M_{kp} + \sum_{k=1}^{4}C_{v,k}^{n-2} M_{kp} - \Delta t_n \DV \left[ \sum_{k=1}^{4}C_{v,k}^{n-1} K_{kp} \right.$    
  $\displaystyle + \frac{\vert\Z\vert\ee}{\kB\bar\T} \sum_{i=1}^4\sum_{k=1}^4 \sym...
...rac{\Q}{\kB\bar\T^2} \sum_{j=1}^4\sum_{k=1}^4 \T_j^n C_{v,k}^{n-1} \Theta_{jpk}$ (4.94)
  $\displaystyle + \left. \frac{\symVacRelFactor\symAtomVol}{\kB\bar\T} \sum_{l=1}...
...a t_n}{\symVacRelTime}\left( q\sum_{m=1}^4 C_{im,m}^n M_{mp} - \Ceq V_p\right),$    
  $\displaystyle R_{p+4} = -\sum_{m=1}^{4} C_{im,m}^{n-1} M_{mp} + \sum_{m=1}^{4}C...
...\sum_{m=1}^4 C_{im,m}^{n-1} M_{mp} + \frac{\Delta t_n}{\symVacRelTime}\Ceq V_p,$ (4.95)

$ p=1,\dots,4$.


next up previous contents
Next: 4.3.4 Calculation of the Up: 4.3 Simulation in FEDOS Previous: 4.3.2 Assembly of the

R. L. de Orio: Electromigration Modeling and Simulation