1. Thermal-ADI: a Linear-Time Chip-Level Dynamic Thermal Simulation Algorithm Based on Alternating-Direction-Implicit(ADI) Method
Good afternoon! The topic that I am going to talk today is “Thermal-ADI: a Linear-Time Chip-Level Dynamic Thermal Simulation Algorithm Based on Alternating-Direction-Implicit(ADI) Method.” My name is Ting-Yuan Wang. This work was supervised by my advisor, Professor Charlie Chen. We are from Electrical and Computer Engineering at University of Wisconsin-Madison.
Ting-Yuan Wang
Charlie Chung-Ping Chen
Electrical and Computer Engineering
University of Wisconsin-Madison
April

2. Motivation
1999 International Technology Roadmap for Semiconductor (ITRS)
Maximum power
Number of metal layers
Wire current density
Due to the relentless push for high speed, high performance, and high component density, the power density and the on-chip temperature in the high-end VLSI circuits rise significantly. The 1999 International Technology Roadmap for Semiconductors (ITRS) shows that the maximum power, number of metal layers, and the wire current density will significantly increase for the future high-performance Microprocessor Unit (MPU). This trend shows the importance of thermal issues on VLSI design.
April

3. Maximum number of metal level
1999 ITRS
Year
1999
2000
2001
2002
2003
2004
2005
2008
2011
2014
Technology Node (nm)
180 -
130
100 70 50 35
Maximum Power (W) 90
115
140
150
160
170
174
183
On-chip,
across-chip clock
(MHz)
1200
1321
1454
1600
1724
1857
2500
3004
3600
Maximum number of metal level 7 8 9 10
5.8e5
7.1e5
8.0e5
9.5e5
1.1e6
1.3e6
1.4e6
2.1e6
3.7e6
4.6e6
This table shows the trend. What’s the problem for high temperature? High temperature not only causes timing failures for both transistors and interconnects but also degrades chip reliability. For example, electromigration (EM) effect for the interconnects is exponentially proportional to the temperature. So, how do we effectively analyze the thermal distributions and locate the hot spots are very important.
April

4. 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
April

5. Finite-Difference Formulation of the Heat Conduction on a Chip
Space Domain
Time Domain
For a given chip, there are two steps to establish the finite-difference method. First step is to discretize the continuous space domain. The second step is to discretize the time domain. During the following discussions of discretization, we will concern the accuracy and stability issues.
April

6. Heat Conduction Equation
where
: Temperature
: Material density
: Specific heat
: Heat generation rate
: Time
: Thermal conductivity
This is the heat conduction equation. It is a second-order parabolic partial differential equation. Where ……..
April

7. Increasing rate of stored
Energy Conservation
Before talking the time domain discretization, let’s look at the energy conservation of the heat conduction equation. It can be explained physically as the increasing rate of the stored energy in a control unit volume being equal to the net rate of energy transferring into the volume and heat generation rate from the volume.
Increasing rate of stored
energy which causes
temperature increase
Net rate of energy
transferring into
the volume
Heat generation rate
in the volume
April

8. Space Domain Discretization
Heat Conduction Equation
Central-Finite-Difference Approximation
Let’s see this heat conduction equation. It is a second-order parabolic partial differential equation. For space domain discretization, we use central-finite-deference approximation to represent the second order partial derivative term in order to have a second order accuracy. Next, we replace partial derivative terms by the difference term.
April

9. Time domain discretization
Heat Conduction Equation
Simple Explicit Method
Simple Implicit Method
Crank-Nicolson Method
So, the concerns are coming from what time step will be used to update this term? N or n+1? There are three choices of time updating: simple explicit method, simple implicit method, and Crank-Nicolson method.
April

10. Simple Explicit Method
Accuracy:
Stability Constraint:
No matrix inversion but time steps are limited by space discretization
For the simple explicit method, we apply the explicit update on the right-hand side of the equation. This method has second order accuracy on space and first order accuracy on time. However, this method need to satisfy the stability constraint in order to avoid fluctuation.
April

11. Simple Implicit Method
Accuracy:
Unconditionally Stable
No limits on time step but involves with large scale matrix inversion
For the simple implicit method, we apply the implicit update on the right-hand side of the equation. This method has second second-order accuracy on space and first order accuracy on time. But this method is unconditionally stable.
April

12. Crank-Nicolson Method
For the Crank-Nicolson method, we take the average of simple explicit and implicit on the right hand side of the equation. This method not only has second order accuracy on space but also on time. Fortunately, this method also sustains unconditional stability.
Accuracy:
Unconditionally stable
No limits on time step but involves with large scale matrix inversion
April

13. Analysis of Crank-Nicolson Method
e.x. m=4,n=4
Total node number N = mn n
For a two-dimensional mesh with total number of nodes N = mn, it requires a matrix with size NxN. To solve the equations Ax = b by LU decomposition or Cholesky decomposition, the runtime and memory requirement are superlinear with sparse matrix techniques. Therefore, we will face the difficulty of computational intense. m
Matrix size = NxN
April

14. 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
The ADI (Alternating Direction Implicit) method is a process to reduce the two-dimensional or three-dimensional problems to a succession of two or three one-dimensional problems. This algorithm separates the time step from n to n+1 into two sub time steps: from n to n+1/2 and from n+1/2 to n+1.During Step I, it applies the implicit update in the x-direction and the explicit update in the y-direction. During Step II, it applies the explicit update in the x-direction and the implicit update in the y-direction. We have two different approaches to ADI method: Peaceman-Rachford and Douglas-Gunn Algorithm.
April

15. Alternating Direction Implicit Method
Step I:
x-direction implicit
y-direction explicit
Step II:
x-direction explicit
y-direction implicit n
The ADI (Alternating Direction Implicit) method is a process to reduce the two-dimensional or three-dimensional problems to a succession of two or three one-dimensional problems. This algorithm separates the time step from n to n+1 into two sub time steps: from n to n+1/2 and from n+1/2 to n+1.During Step I, it applies the implicit update in the x-direction and the explicit update in the y-direction. During Step II, it applies the explicit update in the x-direction and the implicit update in the y-direction. We have two different approaches to ADI method: Peaceman-Rachford and Douglas-Gunn Algorithm.
Peaceman-Rachford Algorithm
Douglas-Gunn Algorithm
April

16. Peaceman-Rachford Algorithm
Step I
Step II
Peaceman-Rachford Algorithm. After rearranging the heat conduction equation, we have the form like this. For step I, we take the implicit term of x and explicit term of y, as shown in the figure with green color. For step II, we take the implicit term of y and explicit term of x, as shown in the figure with yellow color. This algorithm has second-order accuracy both in space and time domain.
April

17. Douglas-Gunn Algorithm
Step I
Step II
For Douglas-Gunn algorithm, the other scheme was used. The heat conduction can be rearranged like this. For step I, we apply the implicit update with x, and keep y direction with explicit update. During step II, we apply the explicit update in x direction, but this term is from step I. Also we apply the implicit update with y. This algorithm has a second accuracy in both space and time domains.
April

18. Illustration for ADI Step I Step II X-direction implicit
Y-direction implicit n n … …
This is the illustration of ADI method. In step I,for every j, there are m equations for the corresponding (i,j) points. Since each point (i,j) is related to two points (i-1,j) and (i+1,j), the coefficient matrix for each row is in tridiagonal form which can be solved with time complex O(m). There is a similar procedure for step II. 2 2
j = 1
j = 1
i = … m
… m
April

19. Analysis of ADI Method X-direction implicit Tridiagonal Matrix n …
2xnxm = 2nm =2N
This is the analysis of ADI method. For every j, we has a tridiagonal matrix like this. So totally we need 2 steps, each step need to solve n matrices, and each matrix needs time m. So the total time complex is O(N). 2
2 steps
n matrices
tridaigonal matrix
j = 1
i = … m
Time complexity: O(N)
April

20. Boundary Condition April 4 2001
Boundary conditions. So far we have not concern about the boundary conditions yet. The difference equations we discussed so far are only for the nodes inside this region. How do we deal with the nodes on the boundary and still have second order accuracy? Let’s look at the points on the boundary, region I and II.
April

21. Boundary Conditions (cont’d)
We introduce virtual points here outside the boundary to help deriving the equations.
April

22. Convection Boundary Condition
Central-finite-difference
approximation
Virtual points
For the convection boundary condition, we have the equations like this. By the central-finite-difference approximation, we can get the difference equation like this. This will keep the second order accuracy.
This is the heat difference equation with point on the boundary. By solving these simultaneous equations, we can eliminate the virtual points and get the difference equations for the nodes on the boundary.
April

23. Flowchart
Flowchart.
April

24. Three Different Locations of Node
(case I)
(case II)
(case III)
(i,j+1)
(i,j+1)
(i,j+1) Si Si
Heat Source
Heat Source
Heat Source
(i-1,j)
(i,j)
(i+1,j)
(i-1,j)
(i,j)
(i+1,j)
(i-1,j)
(i,j)
(i+1,j)
For the implementation, we need to consider three different situations of layout extraction. For case I, the node is on the corner. For case II, the node is at the boundary between two materials. For case III, the node is inside the heat source. For case I and II, we need to modify the difference equations we had discussed. But we don’t talk about the details here.
(i,j-1)
(i,j-1)
(i,j-1)
April

25. Results comparison April 4 2001
The temperature of transient thermal simulation at a random chosen point is shown here. The error of the Douglas-Gunn algorithm compared to the Crank-Nicolson method is less than 0.1.However, the error of the Peaceman-Rachford algorithm compared to the Crank-Nicolson method is 2.25 %. Of course, the error also depends on the delta t we choose.
April

26. Result – Run Time Comparison I
The runtime comparison of the Crank-Nicolson method, Peaceman-Rachford algorithm, and Douglas-Gunn algorithm is shown here. The runtime of the Douglas-Gunn and Peaceman-Rachford algorithm is linearly proportional to the number of nodes as shown with size up to $10^{8}$. However, the runtime of Crank-Nicolson method increases dramatically.
5000X
April

27. Result – Run Time Comparison II
April

28. Results – Memory Usages I
the memory usages of the Douglas-Gunn and Peaceman-Rachford Algorithms are linearly proportional to the number of nodes up to $10^{8}$. However, the memory usage of Crank-Nicolson Method increases dramatically.
April

29. Results – Memory Usages II
April

30. Results – Stability Constraint
The stability constraint gamma was varied from 0.4 to 20 as shown here, where 1/2 is the stability limit. We can see it is stable.
Gamma is the stability limit for simple explicit method
April

31. Results – Error v.s. Time Interval
As illustrated here, the larger $\Delta t$ of the Douglas-Gunn algorithm turns out to have the bigger deviation away from the curve of the Crank-Nicolson method.
April

32. Thank you for your attention!
April