ThermalADI: a LinearTime ChipLevel Dynamic Thermal Simulation Algorithm Based on AlternatingDirectio

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Thermal-ADI a Linear-Time Chip-Level Dynamic
Thermal Simulation Algorithm Based on
Alternating-Direction-Implicit(ADI) Method

• Ting-Yuan Wang
• Charlie Chung-Ping Chen
• Electrical and Computer Engineering
• University of Wisconsin-Madison

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Motivation

• 1999 International Technology Roadmap for
  Semiconductor (ITRS)
• Maximum power
• Number of metal layers
• Wire current density

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1999 ITRS
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Existing Thermal Simulation Methods

• Finite Difference Method
• Easy, good for regular geometry, fast
• Finite Element Method
• More complicated, good for irregular geometry
• Equivalent RC Model (S.M. Kang)
• Compatible with SPICE model, need to solve large
  scale matrix

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Finite-Difference Formulation of the Heat
Conduction on a Chip

• Space Domain
• Time Domain

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Heat Conduction Equation

• where
• Temperature
• Material density
• Specific heat
• Heat generation rate
• Time
• Thermal conductivity

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Energy Conservation
Increasing rate of stored energy which
causes temperature increase
Net rate of energy transferring into the volume
Heat generation rate in the volume
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Space Domain Discretization

• Heat Conduction Equation
• Central-Finite-Difference Approximation

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Time domain discretization

• Heat Conduction Equation
• Simple Explicit Method
• Simple Implicit Method
• Crank-Nicolson Method

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Simple Explicit Method

• Accuracy
• Stability Constraint
• No matrix inversion but time steps are limited by
  space discretization

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Simple Implicit Method

• Accuracy
• Unconditionally Stable
• No limits on time step but involves with large
  scale matrix inversion

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Crank-Nicolson Method

• Accuracy
• Unconditionally stable
• No limits on time step but involves with large
  scale matrix inversion

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Analysis of Crank-Nicolson Method
e.x. m4,n4
Total node number N mn
n
m
Matrix size NxN
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Alternating Direction Implicit Method

• Solves higher dimension problem by successive
  Lower dimension methods
• Accuracy
• Unconditionally stable
• No limits on time step and no large scale matrix
  inversion

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Alternating Direction Implicit Method
Step I x-direction implicit
y-direction explicit Step II x-direction
explicit y-direction implicit
n

• Peaceman-Rachford Algorithm
• Douglas-Gunn Algorithm

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Peaceman-Rachford Algorithm

• Step I
• Step II

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Douglas-Gunn Algorithm

• Step I
• Step II

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Illustration for ADI
Step I
Step II
X-direction implicit
Y-direction implicit
n
n


2
2
j 1
j 1
i 1 2 m
1 2 m
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Analysis of ADI Method
Tridiagonal Matrix
X-direction implicit
n

2xnxm 2nm 2N
2
2 steps
n matrices
tridaigonal matrix
j 1
Time complexity O(N)
i 1 2 m
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Boundary Condition
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Boundary Conditions (contd)
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Convection Boundary Condition
Central-finite-difference approximation
Virtual points
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Flowchart
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Three Different Locations of Node
(case I)
(case II)
(case III)
(i,j1)
(i,j1)
(i,j1)
Si
Si
Heat Source
Heat Source
Heat Source
(i,j)
(i1,j)
(i,j)
(i-1,j)
(i1,j)
(i,j)
(i-1,j)
(i-1,j)
(i1,j)
(i,j-1)
(i,j-1)
(i,j-1)
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Results comparison
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Result Run Time Comparison I
5000X
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Result Run Time Comparison II
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Results Memory Usages I
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Results Memory Usages II
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Results Stability Constraint
Gamma is the stability limit for simple explicit
method
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Results Error v.s. Time Interval
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Thank you for your attention!