J OU R NAL O F XI D IAN U N IV E R S I T Y

Transcription

1 ( ) J OU R NAL O F XI D IAN U N IV E R S I T Y J un Vol. 35 No. 3 / B i C GS t a b,,, (, ) : /.,, ; BiCGStab,,., HSPICE ; HSPICE 95 %, 75 %.,,, /. : / ; ;BiCGStab : TP :A : (2008) Fast method for the large2scale power and ground net work analysis based on the compressed BiCGStab algorithm S U H ao2han g, Z H A N G Yi2men, Z H A N G Yu2mi ng, M A N J i n2cai ( Ministry of Education Key Lab. of Wide Band2Gap Semiconductor Materials and Devices, School of Microelectronic, Xidian Univ., Xi an , China) Abstract : An effective method is proposed to perform static and transient simulations for the large2scale power and ground network circuit s with a good result obtained. This method compresses the large coefficient matrix by only storing the non2zero element s with the column coordinate index which avoids the row coordinate index and eases the burden of memory usage. Then it uses the BiCGStab algorithm to analyze the large network which avoids the inverse matrix computing. Extensive experimental result s on the large2scale power and ground network show that the presented method is over two orders of magnitude faster than HSPICE in transient simulations. Furthermore, our method reduces over 95 % memory usage than HSPICE and 75 % memory usage than Incomplete Cholesky Conjugate Gradient while the accuracy is not compromised. The presented method has more powerf ul capability to deal with the increasing size of power grids in modern microprocessors than general2purpose circuit simulators with significant memory and run2time advantages. Key Words : power and ground networks ;circuit simulations ;BiCGStab algorithm, /,,,, / VL SI. / CPU 3, /. : [ 1 ] [2 ] [3 ] [4 6 ]. (ICCG) Krylov2subspace, /. /, ICCG : : (2007F20) : (19792),,,E2mail : com.

2 3 : / BiCGStab 509,., /, n ( ),,., BiCGStab (BiConjugate Gradient Stabilized),,,, /. 1 / MNA,, /.. /,,. /, /. /, IR, hot point,,. / : ( ), ( / ) ; /,, ; ;., / RL C, 1. MNA (Modified Nodal Analysis), : gg = 1 / gg x + gc x = gb, (1) G A l - A T l 0, gc = C 0 0 L, x = v( t) i l ( t) G, C L ; Al ; v( t), i l ( t) ; b( t)., RL, / RL, RL, 2. RL C, RL,, gb = b( t) 0, 2 RL, 2 (a) ( ). (b), RL = 2L/ h, I ( t + h) = I ( t) + V ( t + h) / RL, h., (c) (d). RL C RC, RL,,., (1) : G x( t) + C x ( t) = b( t), (2)

3 ( ) G, C, x( t), b( t). : ( G + C/ h) x( t) = b( t) + ( C/ h) x( t - h). (3) (3). G + C/ h.,,, (3) /,., /. 2 BiCGStab BiCGStab (3). BiCGStab : BiCGStab. 211 / G + C/ h, 7,,. /,,.,,. G + C/ h,,., 7, 1,, ;,, G + C/ h,,.,,. 212 BiCGStab /,, /,. : [1 ],,, ;,,,. ICCG [4-6 ] PGMRES(Precondition Generalized Minimum RESidual) [4 ] Jacobi [2 ] Gauss2Seidel [2 ]. ICCG CG,, /,, :..,,, ;,,.,,ICCG /., ( GMRES),,,., GMRES P GMRES, P GMRES,. [4 ],,P GMRES ICCG. J acobi Gauss2Seidel,. /. BiCGStab /. BiCGStab, /. BiCGStab CGS(Conjugate Gradient Squared) BCG(Bi2Conjugate Gradient) GMRES,,,

4 3 : / BiCGStab 511,,, ICCG [7 ],,.,,BiCGStab., ( ),. n v n, t n BiCGStab : (1) x 0,, n = 0 ; (2) r 0 = b - Ax 0, 0 = 0 = 0 = 0, v 0 = p 0 = 0 ; (3) r 0, r 0,., (4) ; (4) n = r T 0 r n- 1, = n n, p n- 1 n = r n- 1 + n ( p n- 1 n- 1 = As n, n = tt n s n t T, x n = x n- 1 + n p n n t n (5) r n,., (6) ; (6) n = n + 1, (4). ICCG : (1) x 0,, k = 0 ; (2) r 0 = b - Ax 0, p 0 = (LL T ) - 1 r 0 ; (3) r 0, r 0,., (4) ; + n s n, r n = s n - n t n ; (4) k = ( r k, (LL T ) - 1 r k ), x ( p k, Ap k ) k+1 = x k + k p k, r k+1 = r k - k A p k ; (5) r k+1, r k+1,., (6) ; (6) k = ( r k+1, (LL T ) - 1 r k+1 ) ( r k, (LL T ) - 1, p k+1 = (LL T ) - 1 r k+1 + k p k ; r k ) (7) k = k + 1, (4). - n- 1 v n- 1 ), v n = Ap n, n = n r T, s n = r n- 1-0 v n BiCGStab, /, BiCGStab. 3 C + +, Pentium,CPU 2166 GHZ, 1 GB., 118 V., HSPICE. 6 ns, 120, ,, ICCG [4 ] HSPICE, 3 (a)

5 ( ) HSPICE 95 %, ICCG 75 % , HSPICE 8211 MB, ICCG 1316 MB, BiCGStab CG 316 MB 314 MB ; , HSPICE, ICCG MB, BiCGStab CG 2515 MB 2314 MB. BiCGStab CG,, 3 (b), BiCGStab CG.,, 1. HSPICE, BiCGStab, HSPICE. [5 ],, BiCGStab ICCG,, ICCG. /. [ 5,6 ],, /. 540, HSPICE,, HSPICE, 6 mv, 4 mv. 1 HSPICE 2CG 2BICGStab # # # # # # s 4 BiCGStab /., BiCGStab,,, HSPICE, HSPICE 95 %, ICCG 75 %. BiCGStab 10 min. /. : [ 1 ] Zhao M, Panda R V, Sapatnekar S S. Hierarchical Analysis of Power Distribution Networks [ J ]. IEEE Trans on Computer2Aided Design of Integrated Circuit s and Systems, 2002, 21 (2) : [ 2 ] Kozhaya J N, Nassif S R, Najm F N. A Multigrid2Like Technique for Power Grid Analysis[J ]. IEEE Trans on Computer2 Aided Design of Integrated Circuits and Systems, 2002, 21 (10) : [ 3 ] Qian H, Nassif S R, Sapatnekar S S. Hierarchical Random2walk Algorithm, ms for Power Grid Analysis[J ]. IEEE Trans on Computer2Aided Design of Integrated Circuits and Systems, 2005, 24 (8) : [ 4 ] Chen Tsung2Hao, Chen Charlie Chung2Ping. Efficient Large2scale Power Grid Analysis Based on Preconditioned Krylov2 Subspace Iterative Methods[ C]/ / Proceeding of Design Automation Conference. Las Vegas : ACM, 2001 : [5 ],,. / [J ]., 2005, 17 (4) : Cai Yici, Pan Zhu, L uo Zuying. Fast Analysis of Power/ Ground Net Works Via Circuit Reduction[J ]. Chinese Journal of Semi Conductors, 2005, 17 (4) : [ 6 ] Cai Yici. Fast Analysis of Power/ Ground Networks via Circuit Reduction[J ]. Chinese Journal of Semiconductors, 2005, 26 (7) : [ 7 ] Top sakal E, Kindt R, Sertel K, et al. Evaluation of the BICGSTAB ( l) Algorithm for the Finite2element/ boundary2 integral Method[J ]. IEEE Antennas and Propagation Magazine, 2001, 43 (6) : ( : )