Research Papers

Optimization Under Uncertainty Applied to Heat Sink Design

[+] Author and Article Information
Suresh V. Garimella

e-mail: sureshg@purdue.edu
Cooling Technologies Research Center,
NSF IUCRC, School of Mechanical Engineering and Birck Nanotechnology Center,
Purdue University,
West Lafayette, IN 47907-2088

1Corresponding author.

Manuscript received April 1, 2012; final manuscript received June 20, 2012; published online December 6, 2012. Assoc. Editor: Akshai Runchal.

J. Heat Transfer 135(1), 011012 (Dec 06, 2012) (13 pages) Paper No: HT-12-1146; doi: 10.1115/1.4007669 History: Received April 01, 2012; Revised June 20, 2012

Optimization under uncertainty (OUU) is a powerful methodology used in design and optimization to produce robust, reliable designs. Such an optimization methodology, employed when the input quantities of interest are uncertain, yields output uncertainties that help the designer choose appropriate values for input parameters to produce safe designs. Apart from providing basic statistical information, such as mean and standard deviation in the output quantities, uncertainty-based optimization produces auxiliary information, such as local and global sensitivities. The designer may thus decide the input parameter(s) to which the output quantity of interest is most sensitive, and thereby design better experiments based on just the most sensitive input parameter(s). Another critical output of such a methodology is the solution to the inverse problem, i.e., finding the allowable uncertainty (range) in the input parameter(s), given an acceptable uncertainty (range) in the output quantities of interest. We apply optimization under uncertainty to the problem of heat transfer in fin heat sinks with uncertainties in geometry and operating conditions. The analysis methodology is implemented using DAKOTA, an open-source design and analysis kit. A response surface is first generated which captures the dependence of the quantity of interest on inputs. This response surface is then used to perform both deterministic and probabilistic optimization of the heat sink, and the results of the two approaches are compared.

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Eldred, M. S., 2009, “Recent Advances in Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Analysis and Design,” 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Palm Springs, CA, May 4–7, American Institute of Aeronautics and Astronautics Inc.
Eldred, M. S., Giunta, A. A., Wojtkiewicz, S. F., Jr., and Trucano, T. G., 2002, “Formulations for Surrogate-Based Optimization Under Uncertainty,” Proceedings of the 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Buckhead, Atlanta, GA, Sept. 4–6.
Ndao, S., Peles, Y., and Jensen, M. K., 2009, “Multi-Objective Thermal Design Optimization and Comparative Analysis of Electronics Cooling Technologies,” Int. J. Heat Mass Transfer, 52(19–20), pp. 4317–4326. [CrossRef]
Kitajo, S., Takeda, Y., Kurokawa, Y., Ohta, T., and Mizunashi, H., 1992, “Development of a High Performance Air Cooled Heat Sink for Multi-Chip Modules,” 8th Annual IEEE Semiconductor Thermal Measurement and Management Symposium, New York, NY, Feb. 3–5, pp. 119–124. [CrossRef]
Incropera, F. P., and De Witt, D. P., 2002, Fundamentals of Heat and Mass Transfer, John Wiley & Sons, New Jersey.
Bar-Cohen, A., and Rohsenow, W. M., 1984, “Thermally Optimum Spacing of Vertical, Natural Convection Cooled, Parallel Plates,” ASME J. Heat Transfer, 106(1), pp. 116–123. [CrossRef]
Bejan, A., 2004, Convection Heat Transfer, John Wiley & Sons, New York.
Nakayama, W., Matsushima, H., and Goel, P., 1988, “Forced Convective Heat Transfer From Arrays of Finned Packages,” Cooling Technology for Electronic Equipment, Hemisphere, Washington, DC, pp. 195–210.
Bejan, A., and Sciubba, E., 1992, “The Optimal Spacing of Parallel Plates Cooled by Forced Convection,” Int. J. Heat Mass Transfer, 35(12), pp. 3259–3264. [CrossRef]
Ledezma, G., Morega, A. M., and Bejan, A., 1996, “Optimal Spacing Between Pin Fins With Impinging Flow,” ASME J. Heat Transfer, 118, pp. 570–577. [CrossRef]
Kang, H. S., 2010, “Optimization of a Pin Fin With Variable Base Thickness,” ASME J. Heat Transfer, 132(3), p. 034501. [CrossRef]
Khan, W. A., Culham, J. R., and Yovanovich, M. M., 2005, “Optimization of Pin-Fin Heat Sinks Using Entropy Generation Minimization,” IEEE Trans. Compon. Packag. Technol., 28(2), pp. 247–254. [CrossRef]
Najm, H. N., 2009, “Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics,” Annu. Rev. Fluid Mech., 41, pp. 35–52. [CrossRef]
Le Maitre, O. P., Knio, O. M., Najm, H. N., and Ghanem, R. G., 2001, “A Stochastic Projection Method for Fluid Flow,” J. Comput. Phys., 173(2), pp. 481–511. [CrossRef]
Xiu, D., and Karniadakis, G. E., 2002, “The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations,” SIAM J. Sci. Comput. (USA), 24(2), pp. 619–644. [CrossRef]
Sobol’, I. M., and Kucherenko, S., 2009, “Derivative Based Global Sensitivity Measures and Their Link With Global Sensitivity Indices,” Math. Comput. Simul., 79(10), pp. 3009–3017. [CrossRef]
Eldred, M. S., Giunta, A. A., van Bloemen Waanders, B. G., Wojtkiewicz, S. F., Hart, W. E., and Alleva, M. P., 2007, DAKOTA, a Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis: Version 4.1 Reference Manual, Sandia National Laboratories, Albuquerque, NM.
Vanderplaats, G. N., 1973, CONMIN, a FORTRAN Program for Constrained Function Minimization: User's Manual, Ames Research Center and US Army Air Mobility R&D Laboratory, Moffett Field, CA.
Morega, A. M., Bejan, A., and Lee, S. W., 1995, “Free Stream Cooling of a Stack of Parallel Plates,” Int. J. Heat Mass Transfer, 38(3), pp. 519–531. [CrossRef]
Matsushima, H., Yanagida, T., and Kondo, Y., 1992, “Algorithm for Predicting the Thermal Resistance of Finned LSI Packages Mounted on a Circuit Board,” Heat Transfer–Jpn. Res., 21(5), pp. 504–517.
User's Guide for fluent 6.0, 2002, Fluent Inc., Lebanon, NH.
CUBIT 10.0 User's Manual, 2005, Sandia National Laboratories, Albuquerque, NM.
Bodla, K. K., Murthy, J. Y., and Garimella, S. V., 2010, “Microtomography-Based Simulation of Transport Through Open-Cell Metal Foams,” Numer. Heat Transfer, Part A, 58(7), pp. 527–544. [CrossRef]
Lee, P.-S., Garimella, S. V., and Liu, D., 2005, “Investigation of Heat Transfer in Rectangular Microchannels,” Int. J. Heat Mass Transfer, 48(9), pp. 1688–1704. [CrossRef]
Ganapathy, D., and Warner, E. J., 2008, “Defining Thermal Design Power Based on Real-World Usage Models,” 2008 11th IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems ( ITHERM), Orlando, FL, May 28–31, IEEE, pp. 1242–1246. [CrossRef]


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Fig. 1

Nested approach to optimization under uncertainty employed in this work

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Fig. 2

(a) Isometric view with boundary conditions and (b) top view of pin-fin geometry considered

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Fig. 3

Schematic diagram of the heater block design problem

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Fig. 4

Two-dimensional channel flow with uncertain viscosity for the case of Re = 81.24: (a) current results and (b) results from Le Maitre et al. [14]

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Fig. 5

Nusselt number versus nondimensional fin spacing. The results from Ledezma et al. [10] are shown for comparison.

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Fig. 6

PDFs of (a) Nusselt number and (b) pressure drop for the uniformly distributed input parameters in Table 2

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Fig. 7

Response surface plots of Nusselt number as a function of various input parameters

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Fig. 8

Response surface plots of pressure drop as a function of various input parameters

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Fig. 9

Plots of Nusselt number shown for three most sensitive input parameters: (a) fin width, (b) fin height, and (c) length of the base, for OUU of pin-fin heat sinks

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Fig. 10

Nusselt number PDFs shown for the three most sensitive input parameters: (a) fin width, (b) fin height, and (c) length of the base, for OUU of pin-fin heat sinks

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Fig. 11

Convergence history for OUU with uncertain dimensions using (a) deterministic optimization for maximizing Nusselt number and (b) probabilistic optimization for maximizing mean value of Nusselt number

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Fig. 12

Convergence history for OUU with uncertain operating conditions using (a) deterministic optimization for maximizing Nusselt number and (b) probabilistic optimization for maximizing mean of Nusselt number




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