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Grid Generation and Algebraic Solvers

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Springer Handbook of Semiconductor Devices

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Abstract

This work focuses on some of the most relevant numerical issues in the solution of the drift-diffusion model for semiconductor devices. The drift-diffusion model consists of an elliptic and two parabolic partial differential equations which are nonlinearly coupled. A reliable numerical approximation of this model unavoidably leads to choose a suitable tessellation of the computational domain as well as specific solvers for linear and nonlinear systems of equations. These are the two main issues tackled in this work, after introducing a classical discretization of the drift-diffusion model based on finite elements. Numerical experiments are also provided to investigate the performances both of up-to-date and of advanced numerical procedures.

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Notes

  1. 1.

    We can relax the regularity assumption on σ, by choosing σ ∈ L2( Ω). By exploiting inequality \(\| uv\|{ }_{L^2(\Omega )}\le \| u\|{ }_{L^4(\Omega )}\| v\|{ }_{L^4(\Omega )}\) and the Sobolev immersion theorem, we have \(\big | a_{\text{NLP}}(u, v) \big |\le \big [ \| \varepsilon \|{ }_{L^\infty (\Omega )} + \| \sigma \|{ }_{L^2(\Omega )}\big ] \| u\|{ }_V \| v\|{ }_V\), i.e., the continuity property with \(M=\| \varepsilon \|{ }_{L^\infty (\Omega )} + \| \sigma \|{ }_{L^2(\Omega )}\).

  2. 2.

    The basic geometric concept behind this approach has been discussed already by Dirichlet in 1850 [42].

  3. 3.

    faculty.cse.tamu.edu/ davis/suitesparse.html.

  4. 4.

    http://mumps.enseeiht.fr/.

  5. 5.

    https://www.pardiso-project.org/.

  6. 6.

    The computations have been run on a portable PC equipped with INTEL i5 processor 2.4 GHz and 8 GB of RAM.

References

  1. Quarteroni, A., Valli, A.: Numerical approximation of partial differential equations. In: Springer Series in Computational Mathematics, vol. 23. Springer, Berlin (1994)

    Google Scholar 

  2. Renardy, M., Rogers, R.C.: An Introduction to Partial Differential Equations, 2nd edn. Springer, New York (2004)

    MATH  Google Scholar 

  3. Salsa, S.: Partial Differential Equations in Action—From Modelling to Theory. Springer, Milan (2008)

    MATH  Google Scholar 

  4. Van Roosbroeck, W.: Theory of flow of electrons and holes in germanium and other semiconductors. Bell System Tech. J. 29, 560–607 (1950)

    Article  MATH  Google Scholar 

  5. Jerome, J.W.: Analysis of Charge Transport, 1st edn. Springer, Berlin (1996)

    Book  Google Scholar 

  6. Jüngel, A.: Quasi-Hydrodynamic Semiconductor Equations, 1st edn. Birkhäuser, Basel (2001)

    Book  MATH  Google Scholar 

  7. Markowich, P.A.: The Stationary Semiconductor Device Equations, 1st edn. Springer, Wien (1986)

    Book  Google Scholar 

  8. Smith, G.D.: Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd edn. Oxford University, Oxford (1985)

    Google Scholar 

  9. Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z.: The Finite Element Method: Its Basis and Fundamentals, 7th edn. Elsevier/Butterworth Heinemann, Amsterdam (2013)

    MATH  Google Scholar 

  10. Leveque, R.J.: Finite volume methods for hyperbolic problems. In: Cambridge Text in Applied Mathematics. Cambridge University, Cambridge (2002)

    Book  MATH  Google Scholar 

  11. Frey, P.J., George, P.-L.: Mesh Generation: Application to Finite Elements, 2nd edn. Wiley, Hoboken (2008)

    Book  MATH  Google Scholar 

  12. George, P.L., Borouchaki, H.: Delaunay Triangulation and Meshing-Application to Finite Element. Editions Hermes, Paris (1998)

    MATH  Google Scholar 

  13. Liseikin, V.D.: Grid generation methods. In: Scientific Computation. Springer, Switzerland (2017)

    Google Scholar 

  14. Thompson, J.F., Soni, B.K., Weatherill, N.P.: Handbook of Grid Generation, 1st edn. CRC Press, New York (1998)

    Book  MATH  Google Scholar 

  15. Hawken, D.F., Gottlieb, J.J., Hansen, J.S.: Review of some adaptive node-movement techniques in finite-element and finite-difference solutions of partial differential equations. J. Comput. Phys. 95(2), 254–302 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. Löhner, R.: Adaptive remeshing for transient problems. Comput. Methods Appl. Mech. Eng. 75(1–3), 195–214 (1989)

    Article  MATH  Google Scholar 

  17. Heiser, G., Baltes, H.: Design and Implementation of a Three-Dimensional General Purpose Semiconductor Device Simulator. Ph.D. Thesis, ETH, Zurich (1991)

    Google Scholar 

  18. Mohankumar, N.: Device simulation using ISE-TCAD. In: Technology Computer Aided Design. Simulation for VLSI MOSFET, CK Sarkar Edition (2017)

    Google Scholar 

  19. Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics, 2nd edn. Springer, Berlin (2007)

    Google Scholar 

  20. Bank, R.E., Coughran, W.M., Fichtner, W., Grosse, E.H., Rose, D.J., Smith, R.K.: Transient simulation of silicon devices and circuits. IEEE T. Comput. Aid D. 4(4), 436–451 (1985)

    Article  Google Scholar 

  21. Saad, Y.: Iterative Method for Sparse Linear System, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2003)

    Book  MATH  Google Scholar 

  22. Muller, R.S., Kamins, T.I.: Device Electronics for Integrated Circuits, 3rd edn. Wiley, New York (2002)

    Google Scholar 

  23. De Mari, A.: An accurate numerical steady state one-dimensional solution of the p-n junction. Solid-State Electron. 11(1), 33–58 (1968)

    Article  Google Scholar 

  24. Selberherr, S.: Analysis and Simulation of Semiconductor Devices, 1st edn. Springer, Wien (1984)

    Book  Google Scholar 

  25. Brezzi, F., Marini, L.D., Micheletti, S., Pietra, P., Sacco, R., Wang, S.: Discretization of semiconductor device problems (I). In: Handbook of Numerical Analysis, vol. XIII, pp. 317–441. North-Holland, Amsterdam (2005)

    Google Scholar 

  26. Synopsys, Inc, Mountain View, California. https://www.synopsys.com/silicon/tcad.html

  27. Gummel, H.K.: A self-consistent iterative scheme for one-dimensional steady state transistor calculations. IEEE T. Electron Dev. 11(10), 455–465 (1964)

    Article  Google Scholar 

  28. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems, 1st edn. North-Holland Publishing Co., Amsterdam (1978)

    MATH  Google Scholar 

  29. Slotboom, J.W.: Computer-aided two-dimensional analysis of bipolar transistors. IEEE T. Electron Dev. 20(8), 669–679 (1973)

    Article  Google Scholar 

  30. Mauri, A.G., Bortolossi, A., Novielli, G., Sacco, R.: 3D finite element modeling and simulation of industrial semiconductor devices including impact ionization. J. Math. Ind. 5(1), 1–18 (2015)

    Article  MathSciNet  Google Scholar 

  31. Silvaco TCAD Modeling. https://www.silvaco.com

  32. Ross, H.G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems, 1st edn. Springer, Berlin (1996)

    Google Scholar 

  33. Gummel, H.K., Scharfetter, D.: Large-signal analysis of a silicon read diode oscillator. IEEE T. Electron Dev. 16(1), 64–77 (1969)

    Article  Google Scholar 

  34. Zikatanov, L.T., Lazarov, R.D.: An exponential fitting scheme for general convection-diffusion equations on tetrahedral meshes. Comput. Appl. Math. 1(92), 60–69 (2012)

    Google Scholar 

  35. Pinto, M.R., Rafferty, C.S., Dutton, R.W.: Iterative methods in semiconductor device simulation. IEEE T. Electron Dev. 32(10), 2018–2027 (1985)

    Article  Google Scholar 

  36. Fortin, M.: Personal communication, EPFL, Losanna (1999)

    Google Scholar 

  37. Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element methods on polytopal meshes. Int. J. Numer. Anal. Model. 12(1), 31–53 (2015)

    MathSciNet  MATH  Google Scholar 

  38. Liseikin, V.D.: Techniques for generating three-dimensional grids in aerodynamics (review). Probl. At. Sci. Tech. Ser. Math. Model. Phys. Process 3, 31–45 (1991)

    Google Scholar 

  39. Thompson, J.F., Weatherill, N.P.: Aspects of numerical grid generation: current science and art. AIAA Paper 93, 3539 (1993)

    Google Scholar 

  40. Eriksson, L.E.: Practical three-dimensional mesh generation using transfinite interpolation. SIAM J. Sci. Statist. Comput. 6(3), 712–741 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  41. Yerry, M.A., Shephard, M.S.: A modified-quadtree approach to finite element mesh generation. IEEE Comput. Graph. Appl. 3(1), 39–46 (1983)

    Article  Google Scholar 

  42. Dirichlet, G.L.: Uber die reduction der positiven quadratischen formen mit drei underestimmten ganzen zahlen. Z. Reine Angew. Mathematics 40(3), 209–227 (1850)

    MATH  Google Scholar 

  43. Delaunay, B.: Sur la sphère vide, Bulletin de l’Acadèmie des Sciences de l’URSS. Classe des Sci. Mathèmatiques et Naturelles 6, 793–800 (1934)

    MATH  Google Scholar 

  44. Baker, T.J.: Automatic mesh generation for complex three-dimensional regions using a constrained Delaunay triangulation. Eng. Comput. 5(3–4), 161–175 (1989)

    Article  Google Scholar 

  45. Murphy, M., Mount, D.M., Gable, C.W.: A point-placement strategy for conforming Delaunay tetrahedralization. Internat. J. Comput. Geom. Appl. 11(6) 669–682 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  46. Lo, S.H.: Volume discretization into tetrahedra—II. 3D triangulation by advancing front approach. Comp. Struct. 39(5), 501–511 (1991)

    Google Scholar 

  47. Löhner, R., Parikh, P.: Generation of three-dimensional unstructured grids by the advancing-front method. Int. J. Numer. Methods Fluids 8(10), 1135–1149 (1988). AIAA Paper AIAA-88-0515

    Google Scholar 

  48. Nakahashi, K., Obayashi, S.: Viscous flow computations using a composite grid. In: Proceedings of the 8th Computational Fluid Dynamics Conference, p. 1128 (1987). AIAA Paper, AIAA-87-1128

    Google Scholar 

  49. Weatherill, N.P.: On the combination of structured-unstructured meshes. In: Sengupta, S., Hauser, J., Eiseman, P.R., Thompson, J.F. (eds.) Numerical Grid Generation in CFD ’88, pp. 729–739. Pineridge Press, Swansea (1988)

    Google Scholar 

  50. Lilja, K., Moroz, V., Wake, D.: A 3D mesh generation method for the simulation of semiconductor processes and devices. In: MSM’98, the International Conference on Modeling and Simulation of Microsystems Semiconductors, Sensors and Actuators, Santa Clara, CA, USA, pp. 334–338 (1998)

    Google Scholar 

  51. Tanaka, K., Notsu, A., Matsumoto, H.: A new approach to mesh generation for complex 3D semiconductor device structures. In: SISPAD’96, the International Conference on Simulation of Semiconductor Processes and Devices, pp. 167–168. Business Center for Academic Societies Japan, Tokyo (1996)

    Google Scholar 

  52. Garretòn, G., Villablanca, L., Strecker, N., Fichtner, W.: A hybrid approach for building 2D and 3D conforming Delaunay meshes suitable for process and device simulation. In: De Meyer, K., Biesemans, S. (eds.) Simulation of Semiconductor Processes and Devices, pp. 185–188. Springer, Wien, New York (1998)

    Google Scholar 

  53. Albone, C.M., Swift, V.J.: Resolution of high Reynolds number flow features using dynamically-overlying meshes. In: Soni, B.K., Thompson, J.F., Hauser, J., Eiseman, P.R. (eds.) Numerical Grid Generation in Computational Field Simulations, pp. 855–864. Mississippi State University, Mississippi (1996)

    Google Scholar 

  54. Patis, C.C., Bull, P.W.: Generation of grids for viscous flows around hydrodynamic vehicles, Numerical Grid Generation in Computational Field Simulations. In: Soni, B.K., Thompson, J.F., Hauser, J., Eiseman, P.R. (eds.) Numerical Grid Generation in Computational Field Simulations, pp. 825–834. Mississippi State University, Mississippi (1996)

    Google Scholar 

  55. Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York (2000)

    Book  MATH  Google Scholar 

  56. Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations, 1st edn. Birkhäuser, Basel (2003)

    Book  MATH  Google Scholar 

  57. Verfürth, R.: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, New York (1996)

    MATH  Google Scholar 

  58. Zienkiewicz, O.C., Zhu, J.Z.: A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Methods Engrg. 24(2), 337–357 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  59. Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates. II: Error estimates and adaptivity. Int. J. Numer. Methods Engrg. 33, 1365–1382 (1992)

    MathSciNet  MATH  Google Scholar 

  60. Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates. I: The recovery technique. Int. J. Numer. Methods Engrg. 33, 1331–1364 (1992)

    MathSciNet  MATH  Google Scholar 

  61. Micheletti, S., Perotto, S., Farrell, P.E.: A recovery-based error estimator for anisotropic mesh adaptation in CFD. SeMA J. 50(1), 115–137 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  62. Freitag, L.A., Ollivier-Gooch, C.: Tetrahedral mesh improvement using swapping and smoothing. Int. J. Numer. Methods Engrg. 40(21), 3979–4002 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  63. Dompierre, J., Vallet, M.G.,Bourgault, Y., Fortin, M., Habashi, W.G.: Anisotropic mesh adaptation: towards user-independent, mesh-independent and solver-independent CFD.Part III. Unstructured meshes. Int. J. Numer. Methods Fluids 39(8), 675–702 (2002)

    Article  MATH  Google Scholar 

  64. Peraire, J., Peirò, J, Morgan, K.: Adaptive remeshing for three-dimensional compressible flow computations. J. Comput. Phys. 103(2), 269–285 (1992)

    Article  MATH  Google Scholar 

  65. Zienkiewicz, O.C., Craig, A.W.: Adaptive mesh refinement and a posteriori error estimation for the p-version of the finite element method. In: Adaptive Computational Methods for Partial Differential Equations, pp. 33–56. SIAM, Philadelphia (1983)

    Google Scholar 

  66. Berzins, M.: A solution-based triangular and tetrahedral mesh quality indicator. SIAM J. Sci. Comput. 19(6), 2051–2060 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  67. Knupp, P.M.: Algebraic mesh quality metrics. SIAM J. Sci. Comput. 23(1), 193–218 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  68. Shewchuk, J.R.: What is a good linear element? Interpolation, conditioning, and quality measures. In: Proceedings of the 11th International Meshing Roundtable, pp. 115–126. Sandia National Laboratories, Albuquerque (2002)

    Google Scholar 

  69. Apel, T.: Interpolation of non-smooth functions on anisotropic finite element meshes. M2AN Math. Model. Numer. Anal. 33(6), 1149–1185 (1999)

    Google Scholar 

  70. Formaggia, L., Perotto, S.: New anisotropic a priori error estimates. Numer. Math. 89(4), 641–667 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  71. Belhamadia, Y., Fortin, A., Bourgault, Y.: On the performance of anisotropic mesh adaptation for scroll wave turbulence dynamics in reaction-diffusion systems. J. Comput. Appl. Math. 271, 233–246 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  72. Carpio, J., Prieto, J.L., Vera, M.: A local anisotropic adaptive algorithm for the solution of low-Mach transient combustion problems. J. Comput. Phys. 306, 19–42 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  73. El khaoulani, R., Bouchard, P.O.: An anisotropic mesh adaptation strategy for damage and failure in ductile materials. Finite Elem. Anal. Des. 59, 1–10 (2012)

    Google Scholar 

  74. Ferro, N., Micheletti, S., Perotto, S.: Anisotropic mesh adaptation for crack propagation induced by a thermal shock in 2D. Comput. Methods Appl. Mech. Eng. 331, 138–158 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  75. Formaggia, L., Micheletti, S., Perotto, S.: Anisotropic mesh adaption with application to CFD problems. In: Proceedings of the WCCM V (Fifth World Congress on Computational Mechanics), pp. 1481–1493 (2002)

    Google Scholar 

  76. Lèye, B., Koko, J., Kane, S., Sy, M.: Numerical simulation of saltwater intrusion in coastal aquifers with anisotropic mesh adaptation. Math. Comput. Simulation 154, 1–18 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  77. Loseille, A., Dervieux, A., Alauzet, F.: Fully anisotropic goal-oriented mesh adaptation for 3D steady Euler equations. J. Comput. Phys. 229(8), 2866–2897 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  78. Micheletti, S., Perotto, S., Soli, L.: Topology optimization driven by anisotropic mesh adaptation: towards a free-form design. Comput. Struct. 214(1), 60–72 (2019)

    Article  Google Scholar 

  79. Porta, G.M., Perotto, S., Ballio, F.: Anisotropic mesh adaptation driven by a recovery-based error estimator for shallow water flow modeling. Internat. J. Numer. Methods Fluids 70(3), 269–299 (2012)

    Google Scholar 

  80. Stein, E., Ohnimus, S.: Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems. Comput. Methods Appl. Mech. Eng. 176(1–4), 363–385 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  81. Fleischmann, P.: Mesh Generation for Technology CAD in Three Dimensions, Ph.D. Thesis. Technischen Universität, Wien (1999)

    Google Scholar 

  82. Garretòn, G., Villablanca, L., Strecker, N., Fichtner, W.: Unified grid generation and adaptation fo device simulation. In: Ryssel, H., Pichler, P. (eds.) Simulation of Semiconductor Devices and Processes, vol. 6. Springer, Berlin (1995), 468–471

    Chapter  Google Scholar 

  83. Kumashiro, S., Yokota, I.: A triangular mesh generation method suitable for the analysis of complex MOS device structures. In: Proceedings of International Workshop on Numerical Modeling of Processes and Devices for Integrated Circuits NUPAD V, Honolulu, pp. 167–170 (1994)

    Google Scholar 

  84. Axelrad, V.: Grid quality and its influence on accuracy and convergence in device simulation. IEEE T. Comput. Aid. D. 17(2), 149–157 (1998)

    Article  Google Scholar 

  85. Moroz, V., Motzny, S., Lilja, K.: A boundary conforming mesh generation algorithm for simulation of devices with complex geometry. In: SISPAD’97, the International Conference on Simulation of Semiconductor Processes and Devices, pp. 293–295. Cambridge, Massachusetts (1997)

    Google Scholar 

  86. Conti, P., Tomizawa, M., Yoshii, A.: Generation of oriented three-dimensional Delaunay grids suitable for the control volume integration method. Int. J. Numer. Methods Engrg. 37(19), 3211–3227 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  87. Sever, M.: Delaunay partitioning in three dimensions and semiconductor models. COMPEL 5(2), 75–93 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  88. Hitschfeld, N., Villablanca, L., Krause, J., Rivara, M.C.: Improving the quality of meshes for the simulation of semiconductor devices using Lepp-based algorithms. Int. J. Numer. Meth. Engng 58, 333–347 (2003)

    Article  MATH  Google Scholar 

  89. Hitschfeld, N., Conti, P., Fichtner, W.: Mixed element trees: a generalization of modified octrees for the generation of meshes for the simulation of complex 3-D semiconductor device structures. IEEE T. Comput. Aid. D. 12(11), 1714–1725 (1993)

    Article  Google Scholar 

  90. El Boukili, A., Madrane, A., Vaillancourt, R.: Unstructured grid adaptation for convection-dominated semiconductor equations. Can. Appl. Math. Q. 10(4), 447–472 (2002)

    MathSciNet  MATH  Google Scholar 

  91. Liu, W., Yuan, Y.: A finite difference scheme for a semiconductor device problem on grids with local refinement in time and space. Numer. Math. J. Chin. Univ. (Engl. Ser.) 15(3), 278–288 (2006)

    Google Scholar 

  92. Liu, W., Yuan, Y.: A finite difference scheme for two-dimensional semiconductor device of heat conduction on composite triangular grids. Appl. Math. Comput. 218(11), 6458–6468 (2012)

    MathSciNet  MATH  Google Scholar 

  93. Kuprat, A., George, D., Linnebur, E., Trease, H., Smith, R.K.: Moving adaptive unstructured 3-D meshes in semiconductor process modeling applications. VLSI Design 6(1–4), 373–378 (1998)

    Article  Google Scholar 

  94. Micheletti, S., Perotto, S.: Anisotropic mesh adaptivity via a dual-based a posteriori error estimation for semiconductors. In: Scientific Computing in Electrical Engineering, vol. 9. Mathematics in Industry. Springer, Berlin (2006)

    Google Scholar 

  95. Geuzaine, C., Remacle, J.-F.: Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Meth. Eng. 79(11), 1309–1331 (2009)

    Article  MATH  Google Scholar 

  96. Rudolf, F., Weinbub, J., Rupp, K., Selberherr, S.: The meshing framework ViennaMesh for finite element applications. J. Comput. Appl. Math. 270, 166–177 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  97. https://www.silvaco.com/products/tcad/victory_mesh/Victory_ Mesh.html

    Google Scholar 

  98. Nocedal, J., Wright, S.J.: Numerical optimization. In: Springer Series in Operations Research. Springer, New York (1999)

    Google Scholar 

  99. Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice Hall, Englewood Cliffs (1983)

    MATH  Google Scholar 

  100. Kelley, C.T.: Iterative methods for linear and nonlinear equations. In: Frontiers in Applied Mathematics. SIAM, New York (1995)

    Google Scholar 

  101. Bank, R.E., Rose, D.J.: Global approximate Newton methods. Numer. Math. 37, 279–295 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  102. Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal. 19, 400–408 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  103. Eisenstat, S.C., Walker, H.F.: Choosing the forcing terms in an inexact Newton method. SIAM J. Sci. Comput. 17(1), 16–32 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  104. Pernice, M., Walker, H.F.: NITSOL: A Newton iterative solver for nonlinear systems. SIAM J. Sci. Comput. 19(1), 302–318 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  105. Bellavia, S., Morini, B.: A globally convergent Newton-GMRES subspace method for systems of nonlinear equations. SIAM J. Sci. Comput. 23(3), 940–960 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  106. Eisenstat, S.C., Walker, H.F.: Globally convergent inexact Newton method. SIAM J. Optim. 4(2), 393–422 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  107. Davis, T.: Direct methods for sparse linear systems. In: Fundamentals of Algorithms. SIAM, New York (2006)

    Google Scholar 

  108. Davis, T., Gilbert, J.R., Larimore, S.I., Ng, E.G.: A column approximate minimum degree ordering algorithm. ACM T. Math. Software 30(3), 353–376 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  109. Amestoy, P.R., Duff, I.S., L’Excellent, J.Y., Koster, J.: A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23(1), 15–41 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  110. Amestoy, P.R., Buttari, A., L’Excellent, J.Y., Mary, T.: Performance and scalability of the block low-rank multifrontal factorization on multicore architectures. ACM T. Math. Software 45(1), 1–26 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  111. Petra, C.G., Schenk, O., Anitescu, M.: Real-time stochastic optimization of complex energy systems on high-performance computers. Comput. Sci. Eng. 16, 32–42 (2014)

    Article  Google Scholar 

  112. Petra, C.G., Schenk, O., Lubin, M., Gartner, K.: An augmented incomplete factorization approach for computing the Schur complement in stochastic optimization. SIAM J. Sci. Comput. 36(2), C139 (2014)

    Article  MathSciNet  Google Scholar 

  113. Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7(3), 856–869 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  114. van der Vorst, H.A.: BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13(2), 631–644 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  115. Freund, R.W.: A transpose-free quasi-minimum residual algorithm for nonHermitian linear systems. SIAM J. Sci. Comput. 14(2), 470–482 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  116. Nachtingal, N.M., Reddy, S.C., Thefethen, L.N.: How fast are nonsymmetric matrix iterations. SIAM J. Matrix Anal. Appl. 13(3), 778–795 (1992)

    Article  MathSciNet  Google Scholar 

  117. Saad, Y., Wu, K.: DQGMRES: a direct quasi-minimal residual algorithm based on incomplete orthogonalization. Numer. Linear Algebra Appl. 3(4), 329–343 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  118. Knoll, D.A., Keyes, D.E.: Jacobian-free Newton-Krylov methods: a survey of approaches and applications. J. Comput. Phys. 193, 357–397 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  119. Brown, P.N., Saad, Y.: Convergence theory of nonlinear Newton–Krylov algorithms. SIAM J. Optim. 4(2), 297–330 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  120. Mauri, A.G., Sacco, R., Verri, M.: Electro-thermo-chemical computational models for 3D heterogeneous semiconductor device simulation. Appl. Math. Model. 39(14), 4057–4074 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  121. Masetti, G., Severi, S.: Modeling of carrier mobility against carrier concentration in arsenic-, phosphorus-, and boron-doped silicon. IEEE T. Electron. Dev. 7, 764–769 (1983)

    Article  Google Scholar 

  122. Canali, C.: Electron and hole drift velocity measurements in silicon and their empirical relation to electric field and temperature. IEEE T. Electron. Dev. 22, 1045–1047 (1975)

    Article  Google Scholar 

  123. Deuflhard, P.: A modified Newton method for the solution of ill-conditioned system of nonlinear equations with application to multiple shooting. Numer. Math 22(4), 289–315 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  124. Amestoy, P.R., Davis, T.A., Duff, I.S.: An approximate minimum degree ordering algorithm. SIAM J. Matrix Anal. Appl. 17(4), 886–905 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  125. Demmel, J., Eisenstat, S., Gilbert, J., Li, X., Liu, J.: A supernodal approach to sparse partial pivoting. SIAM J. Matrix Anal. and Appl. 20(3), 720–755, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  126. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes The Art of Scientific Computing, 3rd edn. Cambridge University, New York (2007)

    MATH  Google Scholar 

  127. Eigen v3 (2010). http://eigen.tuxfamily.org

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Acknowledgements

The financial support of INdAM-GNCS Projects 2020 is gratefully acknowledged by the second and by the third authors.

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Authors and Affiliations

  1. Polytechnic University of Milan, Department of Mathematics, Milan, Italy

    Aurelio Giancarlo Mauri & Simona Perotto

  2. Department of Industrial Engineering, University of Florence, Florence, Italy

    Benedetta Morini

  3. Department of Mathematics, University of Bologna, Bologna, Italy

    Fiorella Sgallari

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  1. Aurelio Giancarlo Mauri
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  2. Benedetta Morini
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  3. Simona Perotto
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  4. Fiorella Sgallari
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Correspondence to Fiorella Sgallari .

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  1. Department of Electrical, Electronic and Information Engineering and Advanced Research Center on Electronic Systems, University of Bologna, Bologna, Italy

    Massimo Rudan

  2. Department of Physics, Informatics and Mathematics - FIM, University of Modena and Reggio Emilia, Modena, Italy

    Rossella Brunetti

  3. Department of Electrical, Electronic and Information Engineering and Advanced Research Center on Electronic Systems, University of Bologna, Bologna, Italy

    Susanna Reggiani

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Mauri, A.G., Morini, B., Perotto, S., Sgallari, F. (2023). Grid Generation and Algebraic Solvers. In: Rudan, M., Brunetti, R., Reggiani, S. (eds) Springer Handbook of Semiconductor Devices . Springer Handbooks. Springer, Cham. https://doi.org/10.1007/978-3-030-79827-7_38

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  • DOI: https://doi.org/10.1007/978-3-030-79827-7_38

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