next up previous contents
Next: 4.2.3 Discretization of the Up: 4.2 Discretization of the Previous: 4.2.1 Discretization of Laplace's


4.2.2 Discretization of the Thermal Equation

For convenience, the thermal equation (3.72) is rewritten here as

$\displaystyle \symMatDens\symSpecHeat\dot\T - \ensuremath{\nabla\cdot{(\symTher...
...abla{\T}})}} - \symElecCond\left(\ensuremath{\nabla{\symElecPot}}\right)^2 = 0,$ (4.33)

where the notation $ \dot\T = \partial{\T}/\partial{t}$ is used. Following the same procedure as for Laplace's equation, one multiplies (4.33) by $ {\symShapeFun}_{p}(\vec r)$ and integrates over the domain $ \symDomain$, which together with Green's formula (4.26) and a Neumann boundary condition yields the weak formulation

$\displaystyle \ensuremath{\int_{\symDomain}{\symMatDens\symSpecHeat\dot\T{\symS...
...bla{\symElecPot}}\right)^2 \symShapeFun_p}\ d{\symDomain}} = 0,\quad p=1,...,N.$ (4.34)

The thermal equation (4.33) corresponds to a parabolic problem. Besides the spatial discretization, a discretization in time has also to be performed. A simple choice is the backward Euler method

$\displaystyle \dot\T = \frac{\T^n - \T^{n-1}}{\Delta t_n},$ (4.35)

where $ \Delta t_n = t^n - t^{n-1}$ is the time step. Applying now the spatial discretization for the temperature variable

$\displaystyle \T^n = \ensuremath{\sum_{j=1}^{4}{\T_j^n}}{\symShapeFun}_{j}(\vec r),$ (4.36)

and the electric potential discretization (4.29), equation (4.34) is written for a single element as

  $\displaystyle \symMatDens\symSpecHeat \sum_{j=1}^{4}\T_j^n \int_{T} \symShapeFu...
...mShapeFun_j}}\ensuremath{\cdot}\ensuremath{\nabla{\symShapeFun_p}}\ d\symDomain$    
  $\displaystyle - \Delta t_n \symElecCond \sum_{i=1}^4\sum_{q=1}^4 \symElecPot_i^...
...nabla{\symShapeFun_q}}\right) \symShapeFun_p\ d\symDomain = 0,\qquad p=1,...,4.$ (4.37)

Using the integral notations (4.31),

$\displaystyle M_{jp} = \int_{T}\symShapeFun_j \symShapeFun_p\ d\symDomain = \de...
...\xi}\int_{0}^{1-\xi-\eta} \symShapeFun_j^t \symShapeFun_p^t\ d\zeta d\eta d\xi,$ (4.38)

and

$\displaystyle \Theta_{iqp}$ $\displaystyle = \int_{T} \left(\ensuremath{\nabla{\symShapeFun_i}}\ensuremath{\cdot}\ensuremath{\nabla{\symShapeFun_q}}\right) \symShapeFun_p\ d\symDomain$    
  $\displaystyle =\det(\mathbf{J})\int_{0}^{1}\int_{0}^{1-\xi}\int_{0}^{1-\xi-\eta...
...f{\Lambda}\nabla^t \symShapeFun_q^t\right)\symShapeFun_p^t \ d\zeta d\eta d\xi,$ (4.39)

(4.37) can be expressed as

$\displaystyle \symMatDens\symSpecHeat \sum_{j=1}^{4}\T_j^n M_{jp} - \symMatDens...
...=1}^{4}\T_j^{n-1} M_{jp} + \Delta t_n \symThermCond \sum_{j=1}^{4}\T_j^n K_{jp}$    
$\displaystyle - \Delta t_n \symElecCond \sum_{i=1}^4\sum_{q=1}^4 \symElecPot_i^n\symElecPot_q^n \Theta_{iqp} = 0,\qquad p=1,...,4.$ (4.40)

This is the discrete system of equations, which has to be solved each time step in order to obtain the temperature at each node of the element.


next up previous contents
Next: 4.2.3 Discretization of the Up: 4.2 Discretization of the Previous: 4.2.1 Discretization of Laplace's

R. L. de Orio: Electromigration Modeling and Simulation