# Applied and industrial mathematics in Italy II : selected by Vincenzo Cutello; Giorgio Fotia; Luigia Puccio; Società By Vincenzo Cutello; Giorgio Fotia; Luigia Puccio; Società Italiana di Matematica Applicata e Industriale (eds.)

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Extra resources for Applied and industrial mathematics in Italy II : selected contributions from the 8th SIMAI Conference : Baia Samuele (Regusa), Italy, 22-26 May 2006

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Am,). 4 . 2 . Thermal-electric coupling First, let us consider the thermal-to-electric coupling. This coupling occurs because some components of the electric network may depend on temperature. '^^'^ We will discuss later the case of semiconductor devices. Here, we concentrate briefly on resistors. A resistance may have a quadratic dependence on t e m p e r a t ~ r e . 'We ~ can state this dependence by writing: (25) j R =jR(AAu, ar We recall that j R defines the matrix G = -(AAu), which appears in awR the network equations.

Klv-1) E NN-' be given and set Rk := ( 0 , k l ) x * " x (0,kN-1). For any E Z N , let fc : Z N x Rd 3 [O,+oo) be such that f t ( ( . , a N ) , z ) is %&-periodic for any CYN E Z and z E Rd. Then we consider of the following form fi f \$ ( a , z ) := f t &). (1) In this case, the growth conditions (2) and (3) and hypotheses ( H l ) and (H2) can be rewritten as follows: f"l(CY,Z) 2 Cl(lZ1" - l), vi E { l , .. (a) = S(a,) if (a, E > 0 and S E M d x N - l , + [-E,E]~-') n A" # 8 ) . 31 The following homogenization theorem holds true.

Bartel, M. Giinther, M. Schulz, Modeling and Discretization of a ThermalElectric Test Circuit, in Modeling, Simulation and Optimization of Integrated Circuits, eds. K. Antreich, R. Bulirsch, A. Gilg, P. Rentrop, Int. Series Num. Math. Vol. 146 (Birkhauser, Basel 2003), pp. 187-201. 20. A. Chryssafis, W. Love, Solid-State Electron 22, 249 (1979). it A. it M. CICALESE Dipartimento d i Matematica e Applicazioni ‘R. it We review some results concerning the variational analysis of the dimension reduction problem for discrete thin films drawing a parallel with the analogous problem in the continuum and highlighting some new features.