A Novel Technique for Incremental Analysis of Onchip Power Distribution Networks

1
A Novel Technique for Incremental Analysis of
On-chip Power Distribution Networks

• Yuhong Fu, Rajendran Panda,
• Ben Reschke, Savithri Sundareswaran,
• and Min Zhao
• Freescale Semiconductor Inc.
• Austin, TX, USA
• ( Currently with Magma Design Automation)

2
Outline

• Need for incremental analysis
• Prior work
• Fictitious domain method (FDM)
• Power grid what-if with FDM
• Results and conclusions

3
Need for incremental analysis

• Power network design is an iterative task
• Build ? analyze ? modify ? analyze ?
• Will have to be modified and re-analyzed
  repeatedly to fix problems or to accommodate
  design changes
• Power/ground networks are huge ( 106 - 108
  parasitic elements), and hence take very long run
  time and large memory to extract, build model and
  simulate.
• Example A 100-million elements network takes
  several hours and GBs memory on a 64 bit Opteron.
• Incremental analysis is extremely valuable for
  this iterative task

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Example what-if modifications
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Nature of modifications

• Some modifications do not change original network
  topology
• Example Change of wire width and deletion of
  wires (in some cases)
• Do not change the nodes and connectivity
• Preserve the size of circuit matrix
• Many modifications change the topology
• Example addition/deletion of vias and wires
• Topology changes pose problem for incremental
  matrix solution since the matrix size has changed

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Prior Work

• Through computation of Large Change Sensitivity
  (LCS)
• Vlach, J., Singhal., K., Computer Methods for
  Circuit Analysis and Design. Van Nostrand
  Reinhold, 1983.
• Pillage, L., Electronic Circuit System
  Simulation Methods. McGraw-Hill Professional,
  1999.
• Through computation of macro-model
• Zhao M., Panda, R.V., Sapatnekar, S.S., and
  Blaauw, D.T., Hierarchical Analysis of Power
  Distribution Networks, IEEE Transactions on
  Computer-Aided Design, vol. 21, 2002.

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Large Change Sensitivy Method

• Reuses factors of the original matrix, G.
• n resistance elements modified between existing
  nodes of the network
• is the connection vector that designates the
  node pair where a resistance is added or removed
• Required computation when n elements are
  modified
• n backward/forward substitutions, plus
• Solution of an n x n system.
• Not suitable when topology changes
• Very expensive for large n

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Large Change Sensitivity Method
Example modification

• Applicable when
• modifications are few
• modifications do not change topology

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Macro-modeling Method

• Reduced models for macros A, B, C, etc. are
  computed and solved together with the remainder
  (global) network model
• If global part is modified, all macro-models
  can be re-used
• If a macro is modified, only its model needs
  re-computation
• Final reduced matrix is still quite large and
  dense
• Very expensive if too many macros are modified

Original (Flat) Matrix
Reduced Matrix (with macro-models)
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Macro-modeling Method

• Applicable when
• modifications are confined to hierarchical blocks
  that are macro-modeled
• the model need to be regenerated even if only a
  few modifications inside are made
• Allows topology changes within the macros

Macro-models for A and B need to be recomputed
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Proposed Solution

• Based on a Fictitious Domain Method
• Used successfully to solve PDEs from
  linear/non-linear, inhomogeneous, problems
• Podnos, E.G., Applications of Fictitious Domain
  Method to Analysis of Composite Materials, Ph.D.
  Dissertation, the University of Texas at Austin,
  1999.
• Babuska, I., Podnos, E.G. and Rodin, G.J., New
  Fictitious Domain Methods Formulation and
  Analysis, Mathematical Models and Methods in
  Applied Sciences, vol. 15, no. 10, 2005
• Hybrid of direct and iterative solution
  techniques
• Iterated over the sub-problems
• Direct solver for efficient solving of
  sub-problems
• To our knowledge, this is the first practical
  method to incrementally re-analyze after very
  large number of power grid modifications
• Handles topology changes naturally

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Fictitious Domain Method (Proposed)

• Large number of changes (typically involving
  change of thousands or millions of circuit
  elements)
• Changes can span the global network as well as
  the networks in several macros
• Allows topology changes
• No Approximation

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Outline

• Need for incremental analysis
• Prior work
• Fictitious domain method (FDM)
• Power grid what-if with FDM
• Results and conclusions

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Fictitious Domain Method

• Domain Decomposition Methods
• Overlapping domains Non-overlapping domains
• Fictitious Domain Methods
• (completely overlapping domains)

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Simple Example
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FDM to Power Grid Analysis
I1
I2
R
R
R cluster of wires need to be modified R
modified wire clusters I1 interface current in
original design I2 interface current in modified
design
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View of Unmodified Design Part
I1
I2
R
?II2-I1

• Two solutions for the unmodified part are
  equivalent if additional ?I current is applied in
  original design
• The solution inside R can be obtained after the
  port voltage is calculated

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Iterative Procedure
P1
?Ik

• T is the relaxation parameter
• Complexity O(mN)O(Mß) (1ltßlt1.5)
• N original size M number of
  modifications
• m number of iterations

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Relaxation to Speed up Convergence

• Lin,C.C., Lawton, E.C., Caliendo,J.A., and
  Anderson, L.R., An Iterative Finite
  Element-Boundary Element Algorithm, Computers
  Structures, Vol. 59, No.5, 1996.

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Experimental Results

• 6 benchmark designs of size from 1.5M to 140M
  elements
• 1 to 30 of the circuit elements were modified
  in these designs

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Runtime Comparison

• Compared with
• Complete re-analysis Direct solver (Cholesky
  factors)
• Complete re-analysis Iterative solver
  (Pre-conditioned Conjuage Gradient method)

2X to 18X Speed-up
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Comparison with LCS

• How many modifications can be handled by the LCS
  method (for the same CPU time as FDM)?

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Convergence Behavior w/ Relaxation
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Impact of Coating and Relaxation

• Coating helps to converge, cuts down iterations
  and permits topology changes
• Relaxation cuts down iterations further

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Superior Convergence

• No. of FDM iterations compared to PCG iterations

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Conclusions

• To our knowledge, this is the first practical
  approach to incremental power network analysis
• Efficient for analyzing the impact of very large
  number of power grid modifications
• Earlier methods can handle only a handful of
  changes incrementally
• Proposed solution can deal with topology changes
  occuring when power grid is modified
• Well-suited for parallel implementation and
  hierarchical analysis

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• Thank you!