×

Coupled simulation of transient heat flow and electric currents in thin wires: application to bond wires in microelectronic chip packaging. (English) Zbl 1443.80021

Summary: 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 J. Numer. Anal. 50, No. 1, 194–215 (2012; Zbl 1246.65215)]. 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.

MSC:

80A21 Radiative heat transfer
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
78A55 Technical applications of optics and electromagnetic theory

Citations:

Zbl 1246.65215

Software:

GitHub
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Umashankar, K. R.; Taflove, A.; Beker, B., Calculation and experimental validation of induced currents on coupled wires in an arbitrary shaped cavity, IEEE Trans. Antennas and Propagation, 35, 11, 1248-1257 (1987)
[2] Noda, T.; Yokoyama, S., Thin wire representation in finite difference time domain surge simulation, IEEE Trans. Power Deliv., 17, 3, 840-847 (2002)
[3] Wu, H.; Cangellaris, A. C., Efficient finite element electromagnetic modeling of thin wires, Microw. Opt. Tech. Lett., 50, 2, 350-354 (2008)
[4] De Rham, G., Differentiable Manifolds (1984), Springer: Springer Berlin, Germany · Zbl 0534.58003
[5] Auchmann, B.; Kurz, S., De Rham currents in discrete electromagnetism, COMPEL, 26, 3, 743-757 (2007) · Zbl 1142.78005
[6] D’Angelo, C., Multiscale Modelling of Metabolism and Transport Phenomena in Living Tissues (2007), Ph.D. thesis
[7] Cattaneo, L.; Zunino, P., Numerical investigation of convergence rates for the FEM approximation of 3D-1D coupled problems, (Abdulle, A.; etal., ENUMATH 2013 (2014), Springer International Publishing: Springer International Publishing Switzerland), 727-734 · Zbl 1334.92103
[8] D’Angelo, C.; Quarteroni, A., On the coupling of 1D and 3D diffusion-reaction equations. Application to tissue perfusion problems, M3AS, 18, 8, 1481-1504 (2008) · Zbl 1359.35200
[9] L. Heltai, A. Caiazzo, Multiscale modeling of vascularized tissues via non-matching immersed methods, 2018. http://dx.doi.org/10.20347/wias.preprint.2555.
[10] ter Maten, E. J.W., Nanoelectronic COupled problems solutions - nanoCOPS: Modelling, multirate, model order reduction, uncertainty quantification, fast fault simulation, J. Math. Ind., 7, 2 (2016)
[11] Casper, T., Electrothermal simulation of bonding wire degradation under uncertain geometries, (Proceedings of the 2016 Design, Automation & Test in Europe Conference & Exhibition (DATE) (2016), IEEE), 1297-1302, arXiv:1610.04303
[12] Casper, T.; Römer, U.; Schöps, S., Determination of bond wire failure probabilities in microelectronic packages, (Bailey, C.; etal., 22nd International Workshop on Thermal Investigations of ICs and Systems (THERMINIC 2016) (2016), IEEE: IEEE Budapest, Hungary), 39-44, arXiv:1609.06187
[13] Duque, D. J.; Schöps, S.; Wieers, A., Fast and reliable simulations of the heating of bond wires, (Russo, G.; etal., Progress in Industrial Mathematics At ECMI 2014. Progress in Industrial Mathematics At ECMI 2014, The European Consortium for Mathematics in Industry, vol. 22 (2014), Springer: Springer Berlin)
[14] Weiland, T., A discretization method for the solution of Maxwell’s equations for six-component fields, AEÜ, 31, 116-120 (1977)
[15] Weiland, T., Time domain electromagnetic field computation with finite difference methods, Int. J. Numer. Model. Electron. Network. Dev. Field, 9, 4, 295-319 (1996)
[16] Clemens, M., Self-consistent simulations of transient heating effects in electrical devices using the finite integration technique, IEEE Trans. Magn., 37, 5, 3375-3379 (2001)
[17] Noda, T., Error in propagation velocity due to staircase approximation of an inclined thin wire in FDTD surge simulation, IEEE Trans. Power Deliv., 19, 4, 1913-1918 (2004)
[18] Bossavit, A.; Kettunen, L., Yee-like schemes on staggered cellular grids: a synthesis between FIT and FEM approaches, IEEE Trans. Magn., 36, 4, 861-867 (2000)
[19] Bossavit, A.; Kettunen, L., Yee-like schemes on a tetrahedral mesh, with diagonal lumping, Int. J. Numer. Model. Electron. Network. Dev. Field, 12, 1-2, 129-142 (1999) · Zbl 0936.78011
[20] Bondeson, A.; Rylander, T.; Ingelström, P., (Computational Electromagnetics. Computational Electromagnetics, Texts in Applied Mathematics (2005), Springer) · Zbl 1111.78001
[21] Brezzi, F.; Lipnikov, K.; Shashkov, M., Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes with curved faces, M3AS, 16, 02, 275-297 (2006) · Zbl 1094.65111
[22] Vandekerckhove, S., Mimetic discretisation and higher order time integration for acoustic, electromagnetic and elastodynamic wave propagation, J. Comput. Appl. Math., 259, 65-76 (2014) · Zbl 1291.65274
[23] D’Angelo, C., Finite element approximation of elliptic problems with Dirac measure terms in weighted spaces: Applications to one- and three-dimensional coupled problems, SIAM J. Numer. Anal., 50, 1, 194-215 (2012) · Zbl 1246.65215
[24] Cessenat, M., Mathematical Methods in Electromagnetism. Linear Theory and Applications, Vol. 41 (1996), World Scientific Publishing Co. Inc: World Scientific Publishing Co. Inc River Edge, NJ · Zbl 0917.65099
[25] Tucker, R., Differential form valued forms and distributional electromagnetic sources, J. Math. Phys., 50, 3, Article 033506 pp. (2009) · Zbl 1202.35065
[26] Bossavit, A., Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements (1998), Academic Press: Academic Press San Diego · Zbl 0945.78001
[27] Clemens, M.; Weiland, T., Discrete electromagnetism with the finite integration technique, PIER, 32, 65-87 (2001)
[28] Tarhasaari, T.; Kettunen, L.; Bossavit, A., Some realizations of a discrete Hodge operator: a reinterpretation of finite element techniques, IEEE Trans. Magn., 35, 3, 1494-1497 (1999)
[29] Vijalapura, P. K.; Strain, J.; Govindjee, S., Fractional step methods for index-1 differential-algebraic equations, J. Comput. Phys., 203, 1, 305-320 (2005) · Zbl 1063.65074
[30] Hiptmair, R., Discrete Hodge operators, Numer. Math., 90, 2, 265-289 (2001) · Zbl 0993.65130
[31] Apel, T., A priori mesh grading for an elliptic problem with Dirac right-hand side, SIAM J. Numer. Anal., 49, 3, 992-1005 (2011) · Zbl 1229.65203
[32] Babus̆ka, I., Error-bounds for finite element method, J. Numer. Math., 16, 4, 322-333 (1969) · Zbl 0214.42001
[33] Chung, E. T.; Engquist, B., Convergence analysis of fully discrete finite volume methods for Maxwell’s equations in nonhomogeneous media, SIAM J. Numer. Anal., 43, 1, 303-317 (2005) · Zbl 1128.78013
[34] Quarteroni, A.; Valli, A., (Numerical Approximation of Partial Differential Equations. Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, vol. 23 (2008), Springer: Springer Berlin, Germany) · Zbl 1151.65339
[35] Thomée, V., (Galerkin Finite Element Methods for Parabolic Problems. Galerkin Finite Element Methods for Parabolic Problems, Series in Computational Mathematics, vol. 25 (2006), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin) · Zbl 1105.65102
[36] T. Casper, et al. Efficient thin wire simulator for 3D electrothermal problems, 2018. https://github.com/tc88/ETwireSim.
[37] Apel, T.; Sändig, A.-M.; Whiteman, J. R., Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains, Math. Methods Appl. Sci., 19, 1, 63-85 (1996) · Zbl 0838.65109
[38] Serret, J. A., Sur quelques formules relatives à la théorie des courbes à double courbure, J. Math. Pures Appl., 16, 193-207 (1851)
[39] Frenet, J. F., Sur les courbes à double courbure, J. Math. Pures Appl., 17, 437-447 (1852)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.