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Ting-Yuan Wang Charlie Chung-Ping Chen Electrical and Computer Engineering

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## Ting-Yuan Wang Charlie Chung-Ping Chen Electrical and Computer Engineering

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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**Motivation**• 1999 International Technology Roadmap for Semiconductor (ITRS) • Maximum power • Number of metal layers • Wire current density**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**Finite-Difference Formulation of the Heat Conduction on a**Chip • Space Domain • Time Domain**Heat Conduction Equation**where : Temperature : Material density : Specific heat : Heat generation rate : Time : Thermal conductivity**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**Space Domain Discretization**• Heat Conduction Equation • Central-Finite-Difference Approximation**Time domain discretization**• Heat Conduction Equation • Simple Explicit Method • Simple Implicit Method • Crank-Nicolson Method**Simple Explicit Method**• Accuracy: • Stability Constraint: • No matrix inversion but time steps are limited by space discretization**Simple Implicit Method**Accuracy: Unconditionally Stable No limits on time step but involves with large scale matrix inversion**Crank-Nicolson Method**Accuracy: Unconditionally stable No limits on time step but involves with large scale matrix inversion**Analysis of Crank-Nicolson Method**e.x. m=4,n=4 Total node number N = mn n m Matrix size = NxN**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**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**Peaceman-Rachford Algorithm**• Step I • Step II**Step I**Step II Douglas-Gunn Algorithm**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**Analysis of ADI Method**X-direction implicit Tridiagonal Matrix n … 2xnxm = 2nm =2N 2 2 steps n matrices tridaigonal matrix j = 1 i = 1 2 … m Time complexity: O(N)**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) (i,j-1) (i,j-1) (i,j-1)**Results – Stability Constraint**Gamma is the stability limit for simple explicit method