The visualization of power density and temperature are critical in electronic-chip industries. Taking the advantages of mathematical algorithm that is based on elliptic equation of partial differential equation (PDE), the two-dimensional simulation has successfully been generated. The numerical iterative methods that are being used results in efficient elliptic PDE solvers. Numerical analysis is done in terms of execution time, number of iterations and computational complexity of the algorithm being used. The comparison is done between two iterative methods which are Gauss-Seidel and Red-Black Gauss Seidel. The elliptic solver is developed using C language and executed in Linux operating system's environment

A. Ghaffar, Zarith Safiza

Alias, Norma

Satam, Noriza

English

Universiti Teknologi Malaysia Institutional Repository

140THE VISUALIZATION OF POWER DENSITY ANDTEMPERATURE DISTRIBUTION OF ELECTRONIC SYSTEMUSING ELLIPTIC PDE SOLVERNorma Alias', Noriza Satam2Zarith Safiza Abd. Ghaffar2,'Institute ofibnu SinaUniversity Teknologi Malaysia81300 Skudai, Johor2Department of Mathematics,Faculty of Science,University Teknologi Malaysia81300 Skudai, JohorEmail: norma@ibnusina.utm.myAbstract: The visualization of power density and temperature are critical in electronic-chipindustries. Taking the advantages of mathematical algorithm that is based on elliptic equationof partial differential equation (PDE), the two-dimensional simulation has successfully beengenerated. The numerical iterative methods that are being used results in efficient ellipticPDE solvers. Numerical analysis is done in terms of execution time, number of iterations andcomputational complexity of the algorithm being used. The comparison is done between twoiterative methods which are Gauss-Seidel and Red-Black Gauss Seidel. The elliptic solver isdeveloped using C language and executed in Linux operating system's environment.Keywords: full-chip thermal simulation, elliptic equation, power density distribution,temperature visualization, multilayered architecture.1. INTRODUCTIONThe increasing needs in managing energy efficiency in electronic-chip industries drives theroute of research to produce efficient tools for predicting on-chip temperature behavior.The power and temperature visualization involves large scale of computational complexityand consuming lots of execution time. In this paper, the visualization of power densitybehavior and temperature distribution for multi-layer full-chip structure is being focused on.Figure 1 shows the packaging of multilayer full-chip structure including the heat spreader andheat sink are explicitly shown in the model below.Jilid 20, Bil. 3 (Disember 2008) Jurnal Teknologi Maklumat141Full-chippackagingFigure 1: Simplified model of the multilayered full-chip structureThe solution that is based on numerical computational mathematics proposed thedevelopment of partial differential equation based on Poisson's equation of tWO dimensionalelliptic equations. The equation is discretized using Finite Difference Method (FDM) in orderto transfonn it into matrix fonn.2. MATHEMATICAL MODELINGTemperature distribution inside the multilayered chip is governed by the following Poisson'sequation.Where !(x,y) is the function introduced in [2]. The function introduced is as follows:(I)!(x,y) = g(x,y)k,(x,y)(2)With g(x, y) represent power density distribution on the multilayer full-chip and k,(x,y) isthennal conductivity of the layer at pointx,y. Whereas, the discretization of (I) results inh 2Where p = --,-k-U i+1 J -2uij· +U,_lj U'j+1 -2ui J' +U, J-I ( ), ; , +' ; , = f x,yh kUi+l. J +ur-I,J +PUr,J+1 -(2 + 2p)u"j + PU',J-l =!(x,y)(3)(4)Assume that the chip is M x N of grid sIze and from [2] the on-chip power densitydistribution can be referred aslilid 20, Bil. 3 (Disember 2008) lurnal Teknologi Maklumat142M-IN-I (i1l(2m+l») (i 1l(2n+l»)Pij = L Leos cosm=On=O 2M 2N3. NUMERICAL METHODS(5)To numerically solve equation (4), we use the iterative methods which are Gauss-Seidel andRed-Black Gauss-Seidel. A brief description for both methods is given below.3.1 Gauss Seidel Method (GS)Gauss Seidel method represents an improved version of 1acobi method in solving the systemof linear equations, AU;f. This method coined by the German mathematicians Carl FriedrichGauss and Philipp Ludwig von Seidel. By using previously computed results as soon as theybecome available, this implementation is explicitly shown in the following pseudocode.Set number of iterationsCalculations of equation (5)For round: = 1 increase 1 until numbers of iterations doInitial valuesFor timestep :=1 until convergence criterion met doFor i=lincrease 1 until matrix size doFor j=l increase until matrix size doU I . + U I . + U ·+1 - 4u . + U . I =f (x, Y)1+ • .1 1- •.1 1./ 1•.1 '.1-End loop jEnd loop iCheck convergenceEnd loop timestepEnd loop roundFigure 2: Pseudocode of Gauss-Seidel Iteration Method3.2 Red-Black Gauss Seidel MethodRed-Black Gauss-Seidel method represents the same computational way as Gauss Seidelmethod. The only difference is that this method divides the computation into two sub-regionsJilid 20, Bi\. 3 (Disember 2008) lurnal Teknologi Maklumatwhich are odd and even. The divisions of the regions are as shown in the followingpseudocode:Set number of iterationsCalculations of powerFor round: ~ 1 increase 1 until maximum number of iterations (k)doInitial valuesFor timestep :=1 until convergence criterion met doFor i= 1 increase until matrix size doFor j=l increase until matrix size doIf odd, doU 1 . + u 1 + U 1- 4u . + U . I =!(x,y)1+ .j 1- .j I.j+ I.j I,j-End loop jEnd loop iFor i=lincrease 1 until matrix size doFor j=l increase until matrix size doIf even, doU 1· +U 1· +U ·1 -4u . +u . I = !(x,y)1+ ,j ,- .j I.j+ I.j I.}-End loop jEnd loop iCheck convergenceEnd loop timestepEnd loop roundFigure 3: Pseudocode of Red-Black Gauss-Seidel Iteration MethodJilid 20, Bi!. 3 (Disember 2008) Jumal Teknologi MakJumat1444. TEMPERATURE ALGORITHMBoundary condition and Power Density CalculationNumerical Methodunder considerationFigure 4: Sequential algorithm of parabolic equationFigure 4 shows the computational flow in calculating the temperature of multilayer full-chiparchitecture. The item count is the number of iterations taken in order to fulfill the stoppingcriterion. Variable round is being referred as the number of time step in calculating U'J value.5. NUMERICAL RESULTS AND ANALYSISThe two-dimensional visualization of power and temperature distribution for multilayer full-chip architecture is shown in figure 5 and 6. Visualization of graph (figure 5) is based onequation (5) and the graph of temperature (figure 6) is the result gained from equation (1).While figure 7 shows the three-dimensional visualization of the temperature profile usingCOMSOL Multiphysics software.Jilid 20, Bil. 3 (fJisember 2008) JurnalTeknologi Maklumat145Table 1: Power density distribution based on grid valuexiy 0 I 2 3 4 5 6 7 8 90 201.95 203.71 205.11 206.00 206.31 206.00 205.11 203.71 201.95 200.00I 201.76 203.35 204.61 205.42 205.70 205.42 204.61 203.35 201.76 200.002 201.40 202.66 203.66 204.30 204.52 204.30 203.66 202.66 201.40 200.003 200.90 201.71 202.35 202.76 202.90 202.76 202.35 201.71 200.90 200.004 200.31 200.59 200.81 200.95 201.00 200.95 200.81 200.59 200.31 200.005 199.69 199.41 199.19 199.05 199.00 199.05 199.19 199.41 199.69 200.006 199.10 198.29 197.65 197.24 197.10 197.24 197.65 198.29 199.10 200.007 198.60 197.34 196.34 195.70 195.48 195.70 196.34 197.34 198.60 200.008 198.24 196.65 195.39 194.58 194.30 194.58 195.39 196.65 198.24 200.009 198.05 196.29 194.89 194.00 193.69 194.00 194.89 196.29 198.05 200.00Table 2: Temperature distribution based on grid valuexiv 0 I 2 3 4 5 6 7 8 90 28.00 28.00 28.00 28.00 28.00 28.00 28.00 28.00 28.00 28.00I 28.00 133.48 181.42 196.61 204.82 203.60 202.44 195.48 179.93 142.962 28.00 180.60 248.17 281.48 286.85 290.82 285.59 276.13 250.61 194.513 28.00 194.60 279.80 309.06 324.17 322.73 320.49 307.88 279.03 214.804 28.00 201.61 283.24 321.92 330.37 334.29 329.42 318.00 287.64 221.295 28.00 199.41 285.48 318.24 331.79 332.58 330.04 318.28 288.54 222.046 28.00 197.66 279.03 314.17 324.64 327.52 324.51 313.88 284.92 219.707 28.00 190.65 269.08 300.50 311.59 313.68 311.56 301.69 274.67 212.198 28.00 175.63 244.03 271.69 280.69 282.84 281.02 272.82 249.07 193.429 28.00 139.87 189.55 208.99 215.46 216.87 215.71 209.66 192.36 150.17208.00206.00204.00 ;;-202.00 S200.00 ~198.00 .i196.00 ::G>194.00 Q192.00 I100.00 8.16800• 186.006 7 8 9MFigure5: Power density distributionlilid 20, Bil. 3 (Disember 2008)350.00300.00I.i 250.00! 200.00~ 150.00E' 100.00iii~ 50.00000: 0.0. 12 4321 MN 34 5 6 7 B glB765Figure 6: Prediction of temperature behaviorJumal Teknologi Maklumat1460.20.8l,~-fD.b.s0.4Mal: L206e-H.10. 111.2Mh: 1,02Qe-12Figure 7: Visualization of temperature using COMSOL Multiphysics softwareThe on-chip temperature distribution is dependent on the value of power density given to thechip. The advantages of the temperature prediction are to reduce the numbers of rejected full-chip. In chip industry, the process in visualizing the power and temperature distribution onmultilayered chip involves large scale of computational complexity. Therefore, we developthe algorithm introduced in [2] using C language on Linux platform. Development of thealgorithm in C language makes it possible for future advancement of the algorithm indistributed computing environment. Distributed computing will reduced the visualizingprocess complexity where the task is distributed to several processors and produce the resultsin less time compared to running the code on a single processor. In addition, using Linux asoperating system platform will ensure the low-cost implementation, since it is from an opensource environment where no licensing procedure required.Table 3: Comparison of Numerical AnalysisMethod Gauss Seidel Gauss Seidel Red BlackAnalvsisExecution time (microsecond) 626 624Iteration 9000 200--Computational complexity:Add 27000 600Divide 9000 200_.Jilid 20, Bi!. 3 (Disember 2008) Jumal Teknologi MaklumatJ'tfTable 3 shows the comparison of numerical analysis between the calculation using Gauss-Seidel and Red-Black Gauss Seidel methods. The comparison is done in terms of executiontime, number of iterations and computational complexity. The analysis shows that Red-BlackGauss Seidel is significantly better than Gauss-Seidel method. This is because, Red-BlackGauss Seidel results in small value of every term that is to be analyzed.6. CONCLUSIONThe novelty of this research is the two-dimensional visualization of power density andtemperature distribution for multilayered full-chip structure governing by elliptic equation.This research will be potentially enhanced by implementing parallel algorithm of the powerdensity and temperature visualization on distributed parallel computer system. Developmentof the algorithm in C language on Linux operating system is an added value and brightpotential for further advancement in parallel and grid computing environment.ACKNOWLEDGMENTThe authors acknowledge the Institute of Ibnu Sina, Fundamental Science Studies, UTM andMinistry of Science, Technology and Innovation Malaysia (MOSTI) for the fmancial supportunder vote 79217.REFERENCES[1] B.Wang and P. Mazumder, ""Fast Thermal Analysis for VLSI Circuits VIa Semi-analytical Green's Function in Multi-layer Materials," iEEE international Symposium onCircuits and Systems, ppA09-412, May 2004.[2] Y. Zhan and S. S. Sapatnekar, "A High Efficiency Full-Chip Thermal SimulationAlgorithm," Proceedings ofthe 2005 iEEE/ACM international conference on Computer-aided design, pp. 635 - 638,2005.[3] R.P. Canale, S.C Chapra,"Numerical Method for Engineers," Mc Graw-Hill, 1998.[4] L. Lapidus, G.F Pinder,"Numerical Solution of Partial Differential Equation in Scienceand Engineering," Wiley-Interscience, 1982.Jilid 20, Bi!. 3 (Disember 2008) Jumal Teknologi Maklumat148[5] Evans, D.J. "Group explicit methods for the numerical solution of partial differentialequations, Gordon and Breach Science Publishers, 1997.[6] R.L. Burden and J.D. Faires,''Numerical Analysis,"PWS-KENT PublishingCompany, I989.[7] A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers andScientists, Chapman & Hall/CRC Press, Boca Raton, 2002.[8] Arfken, G. Mathematical Methods for Physicists, 3rd ed.T Orlando, FL: Academic Press,pp. 485-486,905, and 912, 1985.Jilid 20, Bi!. 3 (Disember 2008) Jumal Teknologi Maklumat

The visualization of power density and temperature distribution of electronic system using elliptic PDE solver

http://eprints.utm.my/8494/1/NormaAlias2008_TheVisualizationofPowerDensity.pdf

The visualization of power density and temperature distribution of electronic system using elliptic PDE solver

Abstract

Similar works

Full text

Available Versions

Universiti Teknologi Malaysia Institutional Repository