Abstract
Numerical approximation has played a valuable supporting role in VLSI device simulation. Examples include (1) tensor product variation diminishing splines for models of transistor charges and currents and (2) continuation to find a safe operating region avoiding avalanche breakdown. More recently, quadratic box splines in three variables have been studied for use in Monte Carlo solution of the Boltzmann transport equation. The bivariate Zwart-Powell element does not directly generalize, but another particular box spline is constructed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. de Boor and K. Höllig, B-splines from parallelepipeds, J. d'Anal. Math. 42 (1982) 99–115.
C. de Boor and K. Höllig, Bivariate box splines and smooth pp functions on a three-direction mesh, J. Comp. Appl. Math. 9 (1983) 13–28.
D. Brust, Electronic spectra of crystalline germanium and silicon, Phys. Rev. 134 (1964) A1337–1347.
J. Bude, E. Grosse and R.K. Smith, Phase-space simplex Monte Carlo for semiconductor transport, in preparation (1993).
C.K. Chui and H. Diamond, A natural formulation of quasi-interpolation by multivariate splines, Proc. Amer. Math. Soc. 99 (1987) 643–646.
C.K. Chui and R.-H. Wang, On a bivariate B-spline basis, Tech. Rep. 7, Texas A&M University, Center for Approximation Theory (1981).
W.M. Coughran, Jr., W. Fichtner and E. Grosse, Extracting transistor charges from device simulations by gradient fitting, IEEE Trans. Comp. Aided Des. CAD-8 (1989) 380–394.
W.M. Coughran, Jr., E. Grosse and D.J. Rose, Variation diminishing splines in simulation, SIAM J. Sci. Statist. Comp. 7 (1986) 696–705.
.M. Coughran, Jr., E.H. Grosse and M.R. Pinto, Computing folds and bifurcations in current-voltage characteristics of semiconductor devices, in:Workshop on Numerical Modeling of Processes and Devices for Integrated Circuits: NUPAD IV, Technical Digest, IEEE (1992) pp. 149–153.
W. Dahmen and C.A. Micchelli, Some results on box splines, Bull. (New Series) Amer. Math. Soc. 11 (1984) 147–150.
T. Duff, Interval arithmetic and recursive subdivision for implicit functions and constructive solid geometry, Comp. Graph. (SIGGRAPH '92 Proc.) 26 (1992) 131–138.
M.V. Fischetti and S.E. Laux, Monte Carlo analysis of electron transport in small semiconductor devices including band-structure and space-charge effects, Phys. Rev. B 38 (1988) 9721–9745.
A.C. Hearn,REDUCE User's Manual (The Rand Corporation, Los Angeles, 1987).
K. Hess (ed.),Monte Carlo Device Simulation: Full Band and Beyond (Kluwer Academic, 1991).
D. Lee, A note on bivariate box splines on a k-direction mesh, J. Comp. Appl. Math. 15 (1986) 117–122.
M. Lundstrom,Fundamentals of Carrier Transport (Addison-Wesley, 1990).
M.J.D. Powell, Piecewise quadratic surface fitting for contour plotting, in:Software for Numerical Analysis, ed. D.J. Evans (Academic Press, 1974) pp. 253–271.
F.P. Preparata and M.I. Shamos,Computational Geometry: An Introduction (Springer, 1985).
W.E. Rheinboldt,Numerical Analysis of Parametrized Nonlinear Equations (Wiley-Interscience, 1986).
L.L. Schumaker, On spaces of piecewise polynomials in two variables, in:Approximation Theory and Spline Functions, eds. S.P. Singh et al. (Reidel, 1984) pp. 151–197.
A.J. Worsey and B. Piper, A trivariate Powell-Sabin interpolant, Comp. Aided Geom. Des. 5 (1988) 177–186.
P.B. Zwart, Multivariate splines with nondegenerate partitions, SIAM J. Numer. Anal. 10 (1973) 665–673.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Grosse, E. Approximation in VLSI simulation. Numer Algor 5, 591–601 (1993). https://doi.org/10.1007/BF02221586
Issue Date:
DOI: https://doi.org/10.1007/BF02221586