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Partition-Driven Standard Cell Thermal Placement Guoqiang Chen Synopsys Inc. Sachin Sapatnekar Univ of Minnesota For ISPD 2003 Outline Introduction Thermal Placement Simplified Thermal Model for Partitioning Partition-Driven Thermal Placement Experimental Results Conclusion

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partition driven standard cell thermal placement

Partition-Driven Standard Cell Thermal Placement

Guoqiang Chen

Synopsys Inc.

Sachin Sapatnekar

Univ of Minnesota

For ISPD 2003

outline
Outline
  • Introduction
  • Thermal Placement
    • Simplified Thermal Model for Partitioning
    • Partition-Driven Thermal Placement
  • Experimental Results
  • Conclusion
why thermal placement
Why Thermal Placement?
  • Thermal problem is projected to be a major bottleneck for the next-generation circuits
  • Placement is the natural starting point in the design process where we take thermal problem into consideration

Courtesy Intel

typical heat conduction environment of the wafer
Typical Heat Conduction Environment of the Wafer

+

~

...

...

heat sources

wafer

y

z

x

...

...

ambient temperature

+

~

thermal equation
Thermal Equation
  • Partial differential equation
    • We will consider the steady state version: Poison Equation
  • Applying finite difference method and eliminating internal mesh nodes yields

G T = P

    • G is the thermal conductance matrix
    • T and P are the temperature and power density vector over mesh nodes on the top surface of the wafer
thermal placement
Thermal Placement
  • Minimize max. temperature variation
  • Problem formulation:

Find permutation  of Pi: {1, …,n}  {1,…n}

such that δT = max|Ti-Ti,neighbor| is minimized

(No wire-length/timing considerations)

  • This is a NP-hard problem already
  • Previous work
    • Chu and Wong, TCAD98: matrix synthesis
    • Tsai and Kang, TCAD00: simulated annealing based
partition driven thermal placement
Partition-Driven Thermal Placement
  • Partition based placement methods are powerful methods to solve the placement problems
  • Could we easily extend Tsai and Kang’s work for partition driven placement?
two obvious approache s
Two Obvious Approaches
  • Use equation G T=P directly at each partitioning step
    • During the early partitioning stages, we do not know where the cells will be eventually located
    • Too expensive to compute
  • Compute the desired power distribution, and try to match the power distribution during partition stages
    • Difficult to get an exact budget for the power distribution
    • We are not optimizing the temperature directly
outline9
Outline
  • Introduction
  • Thermal Placement
    • Simplified Thermal Model for Partitioning
    • Partition-Driven Thermal Placement
  • Experimental Results
  • Conclusion
simplified thermal model for partitioning
Simplified Thermal Model for Partitioning
  • It is known that Poisson equation can be solved with Multigrid method effectively
  • Our model is motivated by one interpretation of the mutligrid methods
multigrid solver for poisson equation
Multigrid Solver for Poisson Equation
  • The multi-grid method solves different spatial frequency components at different levels of mesh.
    • Low-frequency components: coarse mesh
    • High-frequency components: fine mesh
  • The temperature distribution across the chip can be considered as a superposition of low spatial frequency components and high spatial frequency components
top down partition process
Top-down Partition Process

Thermal grids

Standard cell

This process can be considered as series of

operations on a gradually refined meshes

basic ideas
Basic Ideas
  • At each partition level, we are only concerned about the spatial distribution of the temperature corresponding to the current coarse grids
  • In the early stage of partitioning, we are mainly concerned about the low frequency components
  • As the mesh is refined, higher frequency terms, corresponding to local variation of temperature, will be considered
simplified thermal model
Simplified Thermal Model
  • At each partition level, we are only interested in controlling the temperature differences between the current coarse grids.
  • We will assume that the temperature within each grid is same.
the first step in top down partitioning
The First Step in Top-down Partitioning

N

TL

TR

  • Original Equation:GT=P

T, P are N2 x 1 vector, and G is a N2 x N2 matrix

  • Now the equation is simplified to:

1

1

N

extension to general case
Extension to General Case
  • This process can be extended to general case where we partition the chip into k regions
  • Resulting G matrix is a k x k matrix and it is positive definite
slide17

Outline

  • Introduction
  • Thermal Placement
    • Simplified Thermal Model for Partitioning
    • Partition-Driven Thermal Placement
  • Experimental Results
  • Conclusion
before partitioning a new level
Before Partitioning a New Level
  • Compute the simplified thermal conductivity matrix G
  • Prepare the matrix for incremental update
before partitioning of a block
Before Partitioning of a Block
  • Generate multiple initial solutions and compute δT
  • Set the thermal budget for the current partition to be (1-α) δTmin +

α δTmax

  • Pick the initial partition with lowest δTas the initial solution for partitioner
when we move one cell
When We Move One Cell
  • Compute power changes induced by the cell movement
  • Compute temperature changes for blocks that are affected by the move.
  • Compute δTfor the current move, and check against the budget to see if we will accept the move or not
slide21

Outline

  • Introduction
  • Thermal Placement
    • Simplified Thermal Model for Partitioning
    • Partition-Driven Thermal Placement
  • Experimental Results
  • Conclusion
conclusion
Conclusion
  • We presented an simplified thermal model to take temperature directly as partition constraints.
  • The basic idea is we want to control different spatial frequency of the temperature variation at different partition level
  • We proposed a top-down partition-driven placement scheme to use the simplified model
slide26

End

Thank You