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→Problem Formulation

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~~\Sum_~~{{m}=1}^{~~\nel~~} \left[+~~\end{aligned}~~+

<math>

<math>

F_{mn} =

F_{mn} =

−A_{mn} \dot~~{\mathbf~~{r~~}~~}_{m} + J_{mn} \left( ~~\mathbf~~{e}, ~~\mathbf~~{r}_{m} \right)

+ A_{mn} \dot{r}_{m} + J_{mn} \left({e}, {r}_{m} \right)

\qquad

\qquad

−\begin{~~aligned~~}

+\begin{array}{l}

−n ~~&~~=~~& ~~1, \, \ldots \, ,N \\

+n = 1, \, \ldots \, ,N \\

−m ~~&~~=~~& ~~1, \, \ldots \, ,M

+m = 1, \, \ldots \, ,M

−\end{~~aligned~~}

+\end{array}

</math>

</math>

−Notice that the variables <math>{~~\mathbf~~{r}}_{m}</math> only appear in the equations defining the fluxes relative to the m-th element, for this reason they are sometimes referred to as internal variables of the m-th element.

+Notice that the variables <math>{{r}}_{m}</math> only appear in the equations defining the fluxes relative to the m-th element, for this reason they are sometimes referred to as internal variables of the m-th element.

−The full MNA model is finally obtained by substituting the current definitions in the KCL and complementing it with a suitable number <~~matj~~>I_{m}</~~matj~~> of constitutive relations for the internal variables of each element

+The full MNA model is finally obtained by substituting the current definitions in the KCL and complementing it with a suitable number <math>I_{m}</math> of constitutive relations for the internal variables of each element

<math>

<math>

−\~~begin{gather}~~

+\sum_{{m}=1}^{M} \left[

−\ A_{mn} \dot{{r}}_{m} +

−\ A_{mn} \dot{~~\mathbf~~{r}}_{m} +

+J_{mn} \left( {e}, {r}_{m} \right)

−J_{mn} \left( ~~\mathbf~~{e}, ~~\boldsymbol{\theta}, \mathbf~~{r}_{m} \right)

\right] = 0

\right] = 0

−\qquad {n} = 1, \, \ldots \, ,~~\nnodes ~~ \\ ~~\label{eq:abstractmnab}~~

+\qquad {n} = 1, \, \ldots \, ,N \\

−B_{mi} \dot{~~\mathbf~~{r}}_{m} +

+B_{mi} \dot{{r}}_{m} +

−Q_{mi}\ \left( ~~\mathbf~~{e}, \~~boldsymbol{\theta}, \mathbf~~{r}_{m} \right) = 0.

+Q_{mi}\ \left( {e}, \{r}_{m} \right) = 0.

\qquad

\qquad

−\begin{~~aligned~~}

+\begin{array}{l}

{i} = 1, \, \ldots \, ,{I}_m \\

{i} = 1, \, \ldots \, ,{I}_m \\

−{m} = 1, \, \ldots \, ,~~\nel~~

+{m} = 1, \, \ldots \, ,M

−\end{array}

−\end{~~gather~~}

</math>

</math>

Notice that the assumption that only time derivatives of internal variables appear above and that terms involving such derivatives are linear does not impose restrictions on the applicability of the model.

Notice that the assumption that only time derivatives of internal variables appear above and that terms involving such derivatives are linear does not impose restrictions on the applicability of the model.

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