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## Very Fast Chip-level Thermal Analysis

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**Very Fast Chip-level Thermal Analysis**Budapest, Hungary, 17-19 September 2007 Keiji Nakabayashi†, Tamiyo Nakabayashi‡, and Kazuo Nakajima* †Graduate School of Information Science, Nara Institute of Science and Technology Keihanna Science City, Nara, Japan, keiji-n@is.naist.jp ‡Graduate School of Humanities and Sciences, Nara Women’s University Kitauoyahigashi-machi, Nara, Japan, nakaba@ics.nara-wu.ac.jp *Dept. of Electrical and Computer Engineering, University of Maryland College Park, MD 20742, USA, nakajima@umd.edu**Abstarct**We present a new technique of VLSI chip-level thermal analysis. We extend a newly developed method of solving two dimensional Laplace equations to thermal analysis of four adjacent materials on a mother board. We implement our technique in C and compare its performance to that of a commercial CAD tool. Our experimental results show that our program runs 5.8 and 8.9 times faster while keeping smaller residuals by 5 and 1 order of magnitude, respectively.**1. Introduction**• Thermal phenomena is very important factor in VLSI and board design in the post-90 nm era. • Thermal analysis is modeling of thermal conduction by a Laplace equation and its solution by finite difference method (FDM). • We developed a new, very fast chip-level thermal analysis technique.**2. Problem**• Consider a multi-layer structure, where four layers of materials p,q,r,and s of thermal conductivities kp, kq, kr, and ks, respectively, are stacked together. • Heat travels through a heat transfer pass consisting of the chip die, the adhesive, the heat spreader, and the heat sink, and goes out to the ambient air. • Our problem is to find temperature distribution through two dimensional steady-state thermal conduction analysis.**3. Method**• Recently, a new efficient direct method, called Symbolic Partial Solution Method (S-PSM) was developed in the area of computational fluid dynamics. • S-PSM-based solution process goes through many levels of repeated operations of decomposition and merging. • We extend this S-PSM-based Laplace equation solver to a multi-layer structure.**h**X Tleft=20(℃) (Reference) Tright=20(℃) (Reference) 2048um Tamb=55(℃) (Ambient temperature) np+2 128um Heak sink kp= 260 [W/(m・K)] nq+2 128um Heak spreader kq= 230 [W/(m・K)] nr+2 g 64um Adhesive kr= 16 [W/(m・K)] 256um ns+2 Chip die ks= 100 [W/(m・K)] Y (0,0) Tj=120(℃) （Heat source: Si bulk temperature) Cell h : number of Y-axis grid points g : number of X-axis grid points where g = (np+2)+(nq+2)+(nr+2)+(ns+2) Cell size : = = 2048(um) / h Boundary Value Problem (BVP)**The arrangement of interior grid and boundary points**for the four material domains. kp kq kr ks**Laplace Equation and Finite Difference Method for each**material u (=p, q, r, s): Finite Difference Method Final Solution for each material :**thermal conductivity of the boundary between adjacent**materials From the viewpoint of material q, the following difference equation holds at its boundary with material p (first order approximation) : Similarly, at its boundary with material r,**matrix-vectors form of equations for four materials (three**boundaries)**For materials p, q, r, swithkqr**Final solutions: Eq. (3-35) ↑ Final equations: Eq. (3-33) For materials p, qwithkpq For materials r, s withkrs {q5, r0} Solutions: Eq. (3-31) ↑ Equations: Eq. (3-28) Solutions: Eq. (3-32) ↑ Equations: Eq. (3-29) Back substitution {p0, p5, q0} {r5, s0, s5} Solutions: Eq. (3-24) ↑ Equations: Eq. (3-20) Solutions: Eq. (3-25) ↑ Equations: Eq. (3-21) Solutions: Eq. (3-26) ↑ Equations: Eq. (3-22) Solutions: Eq. (3-27) ↑ Equations: Eq. (3-23) For material p withkp For material qwithkq For material r withkr For material s withks {p1, p4} {q1, q4} {r1, r4} {s1, s4} Decompose Equations Eq. (3-19) into 4 subsystems { } : Variables with known values. : Merging operation for adjacent partial solutions.**System Decomposition and Partial Solutions**• for Each Subsystem/Material (3-20) (3-21) Decomposition (3-22) (3-23)**(3-24)**(3-25) Partial solution (3-26) (3-27)**(3-28)**Merge (3-29) where**(3-31)**Partial solution (3-32) Merge (3-33) Final solution (3-35)**Temperature distributions of steady-state heat conduction**for four layers of materials : our program vs. commercial tool Raphael Our Program Solver: S-PSM Raphael [7] Solver : Iteration method**4. Results and Discussion**• Table I shows the CPU times required and the residuals produced by our program and Raphael. • The results demonstrate that for the largest grid, our program ran 5.8 and 8.9 times faster while keeping smaller residuals by 5 and 1 order of magnitudes, than Raphael [7]**5. Conclusions**• We have proposed a new technique of solving two dimensional Laplace equations to thermal analysis for multi-layer VLSI chips. • Our program is superior to a commercial CAD tool, Raphael (iteration method) [7]. • Further Work : extension to Poisson equation (heat generation), three dimensional, transient heat conduction analysis, and the case of complex shapes and boundary conditions of materials.