A Parallel and Incremental Extraction of Variational Capacitance With Stochastic Geometric Moments

This paper presents a parallel and incremental solver for stochastic capacitance extraction. The random geometrical variation is described by stochastic geometrical moments, which lead to a densely augmented system equation. To efficiently extract the capacitance and solve the system equation, a parallel fast-multipole-method (FMM) is developed in the framework of stochastic geometrical moments. This can efficiently estimate the stochastic potential interaction and its matrix-vector product (MVP) with charge. Moreover, a generalized minimal residual (GMRES) method with incremental update is developed to calculate both the nominal value and the variance. Our overall extraction show is called piCAP. A number of experiments show that piCAP efficiently handles a large-scale on-chip capacitance extraction with variations. Specifically, a parallel MVP in piCAP is up 3 × to faster than a serial MVP, and an incremental GMRES in piCAP is up to 15× faster than non-incremental GMRES methods.

[1]  Hao Yu,et al.  A provably passive and cost-efficient model for inductive interconnects , 2005, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[2]  Marcel J. M. Pelgrom,et al.  Matching properties of MOS transistors , 1989 .

[3]  D. A. Dunnett Classical Electrodynamics , 2020, Nature.

[4]  Raminderpal Singh FastCap: A Multipole Accelerated 3D Capacitance Extraction Program , 2002 .

[5]  Kapur,et al.  IES/sup 3/: a fast integral equation solver for efficient 3-dimensional extraction , 1997, 1997 Proceedings of IEEE International Conference on Computer Aided Design (ICCAD).

[6]  Wei Cai,et al.  A Sparse Grid based Spectral Stochastic Collocation Method for Variations-Aware Capacitance Extraction of Interconnects under Nanometer Process Technology , 2007, 2007 Design, Automation & Test in Europe Conference & Exhibition.

[7]  John A. Board,et al.  Efficient parallel implementations of multipole based n-body algorithms , 1999 .

[8]  Charles A. Brau,et al.  Modern Problems in Classical Electrodynamics , 2003 .

[10]  Valeria Simoncini,et al.  Recent computational developments in Krylov subspace methods for linear systems , 2007, Numer. Linear Algebra Appl..

[11]  Lexing Ying,et al.  A New Parallel Kernel-Independent Fast Multipole Method , 2003, ACM/IEEE SC 2003 Conference (SC'03).

[12]  Yiyu Shi,et al.  Fast Analysis of a Large-Scale Inductive Interconnect by Block-Structure-Preserved Macromodeling , 2010, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

[13]  Michael S. Warren,et al.  A parallel hashed oct-tree N-body algorithm , 1993, Supercomputing '93. Proceedings.

[14]  G. Stewart Matrix Algorithms, Volume II: Eigensystems , 2001 .

[15]  Jacob K. White,et al.  FastSies: a fast stochastic integral equation solver for modeling the rough surface effect , 2005, ICCAD-2005. IEEE/ACM International Conference on Computer-Aided Design, 2005..

[16]  Sheldon X.-D. Tan,et al.  Variational capacitance modeling using orthogonal polynomial method , 2008, GLSVLSI '08.

[17]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[18]  Weiping Shi,et al.  A fast hierarchical algorithm for 3-D capacitance extraction , 1998, DAC.

[19]  Tiejun Yu,et al.  A fast hierarchical algorithm for 3-D capacitance extraction , 1998, Proceedings 1998 Design and Automation Conference. 35th DAC. (Cat. No.98CH36175).

[20]  Sarma B. K. Vrudhula,et al.  Hermite Polynomial Based Interconnect Analysis in the Presence of Process Variations , 2006, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[21]  Jacob K. White,et al.  FastCap: a multipole accelerated 3-D capacitance extraction program , 1991, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[22]  Ying Liu,et al.  Impact of interconnect variations on the clock skew of a gigahertz microprocessor , 2000, DAC.

[23]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[24]  Yang Yi,et al.  Impedance extraction for 3-D structures with multiple dielectrics using preconditioned boundary element method , 2007, 2007 IEEE/ACM International Conference on Computer-Aided Design.

[25]  David E. Long,et al.  IES3: a fast integral equation solver for efficient 3-dimensional extraction , 1997, ICCAD.

[26]  Yu Cao,et al.  Modeling the electrical effects of metal dishing due to CMP for on-chip interconnect optimization , 2004, IEEE Transactions on Electron Devices.

[27]  S. Gratton,et al.  Incremental spectral preconditioners for sequences of linear systems , 2007 .

[28]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[29]  G. W. Stewart,et al.  Matrix algorithms , 1998 .