190 research outputs found
Inhomogeneous poly-scale refinement type equations and Markov operators with perturbations
Given measure spaces (Ω1,A1,μ1), . . . , (ΩN,AN,μN), functions φ1 : Rm×Ω1 Rm, . . . ,φN : Rm×ΩN Rm and g : Rm R, we present results on the existence of solutions f : Rm R of the inhomogeneous poly-scale refinement type equation of the form f(x) = ΣN n=1 ∫ Ωn det(φn)′x(x, ωn) f (φn(x, ωn)) dμn(ωn) + g(x)
in some special classes of functions. The results are obtained by Banach fixed point theorem applied to a perturbed Markov operator connected with the considered inhomogeneous poly-scale refinement type equation.
Mathematics Subject Classification. Primary 37H99, 37N99; Secondary 39B12
Coupled Simulation of Transient Heat Flow and Electric Currents in Thin Wires: Application to Bond Wires in Microelectronic Chip Packaging
This work addresses the simulation of heat flow and electric currents in thin
wires. An important application is the use of bond wires in microelectronic
chip packaging. The heat distribution is modeled by an electrothermal coupled
problem, which poses numerical challenges due to the presence of different
geometric scales. The necessity of very fine grids is relaxed by solving and
embedding a 1D sub-problem along the wire into the surrounding 3D geometry. The
arising singularities are described using de Rham currents. It is shown that
the problem is related to fluid flow in porous 3D media with 1D fractures [C.
D'Angelo, SIAM Journal on Numerical Analysis 50.1, pp. 194-215, 2012]. A
careful formulation of the 1D-3D coupling condition is essential to obtain a
stable scheme that yields a physical solution. Elliptic model problems are used
to investigate the numerical errors and the corresponding convergence rates.
Additionally, the transient electrothermal simulation of a simplified
microelectronic chip package as used in industrial applications is presented.Comment: all numerical results can be reproduced by the Matlab code openly
available at https://github.com/tc88/ETwireSi
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