4.3.3 Assembly of the Vacancy Dynamics Problem

The vacancy dynamics problem requires the solution of

$$f_p = \sum_{k=1}^{4} C_{v,k}^n M_{kp} - \sum_{k=1}^{4}C_{v,k}^{n-... ...B\bar\T} \sum_{i=1}^4\sum_{k=1}^4 \symElecPot_i^n C_{v,k}^n \Theta_{ipk}\right.$$

$$\left. - \frac{\Q}{\kB\bar\T^2} \sum_{j=1}^4\sum_{k=1}^4 \T_j^n C... ...ar\T} \sum_{l=1}^4\sum_{k=1}^4 \symHydStress_l^n C_{v,k}^n \Theta_{lpk} \right]$$ (4.87)

$$+ \frac{\Delta t_n}{\symVacRelTime}\left( q\sum_{m=1}^4 C_{im,m}^n M_{mp} - \Ceq V_p\right) = 0,$$

and

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

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

Applying Newton's method the Jacobian matrix has the form

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

where the entries are computed by

$$\ensuremath{\frac{\partial f_p}{\partial C_{v,i}^n}} = M_{ip} + \... ...c{\vert\Z\vert\ee}{\kB\bar\T} \sum_{j=1}^4 \symElecPot_j^n \Theta_{jpi} \right.$$

$$\qquad\qquad\qquad\qquad\qquad\left. - \frac{\Q}{\kB\bar\T^2} \su... ...tor\symAtomVol}{\kB\bar\T} \sum_{j=1}^4 \symHydStress_j^n \Theta_{jpi} \right],$$ (4.90)

$$\ensuremath{\frac{\partial f_p}{\partial C_{im,i}^n}} = q\frac{\Delta t_n}{\symVacRelTime} M_{ip},$$ (4.91)

and

$$\ensuremath{\frac{\partial g_{p+4}}{\partial C_{v,i}^n}} = 0$$ (4.92)

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

The corresponding residuals are given by

$$R_p = - \sum_{k=1}^{4} C_{v,k}^{n-1} M_{kp} + \sum_{k=1}^{4}C_{v,k}^{n-2} M_{kp} - \Delta t_n \DV \left[ \sum_{k=1}^{4}C_{v,k}^{n-1} K_{kp} \right.$$

$$+ \frac{\vert\Z\vert\ee}{\kB\bar\T} \sum_{i=1}^4\sum_{k=1}^4 \sym... ...rac{\Q}{\kB\bar\T^2} \sum_{j=1}^4\sum_{k=1}^4 \T_j^n C_{v,k}^{n-1} \Theta_{jpk}$$ (4.94)

$$+ \left. \frac{\symVacRelFactor\symAtomVol}{\kB\bar\T} \sum_{l=1}... ...a t_n}{\symVacRelTime}\left( q\sum_{m=1}^4 C_{im,m}^n M_{mp} - \Ceq V_p\right),$$

$$R_{p+4} = -\sum_{m=1}^{4} C_{im,m}^{n-1} M_{mp} + \sum_{m=1}^{4}C... ...\sum_{m=1}^4 C_{im,m}^{n-1} M_{mp} + \frac{\Delta t_n}{\symVacRelTime}\Ceq V_p,$$ (4.95)

$p=1,\dots,4$.