This paper proposes an optimum design method for a two-dimensional microchannel heat sink under a laminar flow assumption that simultaneously provides maximal heat exchange and minimal pressure drop, based on a topology optimization method incorporating Pareto front exploration. First, the formulation of governing equations for the coupled thermal-fluid problem and a level set-based topology optimization method are briefly discussed. Next, an optimum design problem for a microchannel heat sink is formulated as a bi-objective optimization problem. An algorithm for Pareto front exploration is then constructed, based on a scheme that adaptively determines weighting coefficients by solving a linear programming problem. Finally, in the numerical example, the proposed method yields a Pareto front approximation and enables the analysis of the trade-off relationship between heat exchange and pressure drop, confirming the utility of the proposed method.
Issue Section:
Design Automation
References
1.
Tuckerman
, D. B.
, and Pease
, R.
, 1981
, “High-Performance Heat Sinking for VLSI
,” IEEE Electron Device Lett.
, 2
(5
), pp. 126
–129
.2.
Qu
, W.
, and Mudawar
, I.
, 2002
, “Experimental and Numerical Study of Pressure Drop and Heat Transfer in a Single-Phase Micro-Channel Heat Sink
,” Int. J. Heat Mass Transfer
, 45
(12
), pp. 2549
–2565
.3.
Morini
, G. L.
, 2004
, “Single-Phase Convective Heat Transfer in Microchannels: A Review of Experimental Results
,” Int. J. Therm. Sci.
, 43
(7
), pp. 631
–651
.4.
Mahalingam
, R.
, and Glezer
, A.
, 2005
, “Design and Thermal Characteristics of a Synthetic Jet Ejector Heat Sink
,” ASME J. Electron. Packag.
, 127
(2
), pp. 172
–177
.5.
Maveety
, J.
, and Jung
, H.
, 2000
, “Design of an Optimal Pin-Fin Heat Sink With Air Impingement Cooling
,” Int. Commun. Heat Mass Transfer
, 27
(2
), pp. 229
–240
.6.
Yu
, S.-H.
, Lee
, K.-S.
, and Yook
, S.-J.
, 2011
, “Optimum Design of a Radial Heat Sink Under Natural Convection
,” Int. J. Heat Mass Transfer
, 54
(11
), pp. 2499
–2505
.7.
Herrmann-Priesnitz
, B.
, Calderón-Muñoz
, W. R.
, Valencia
, A.
, and Soto
, R.
, 2016
, “Thermal Design Exploration of a Swirl Flow Microchannel Heat Sink for High Heat Flux Applications Based on Numerical Simulations
,” Appl. Therm. Eng.
, 109
(Pt. A), pp. 22
–34
.8.
Kim
, S. J.
, 2004
, “Methods for Thermal Optimization of Microchannel Heat Sinks
,” Heat Transfer Eng.
, 25
(1
), pp. 37
–49
.9.
Bello-Ochende
, T.
, Meyer
, J.
, and Ighalo
, F.
, 2010
, “Combined Numerical Optimization and Constructal Theory for the Design of Microchannel Heat Sinks
,” Numer. Heat Transfer Part A: Appl.
, 58
(11
), pp. 882
–899
.10.
Hung
, T.-C.
, Yan
, W.-M.
, Wang
, X.-D.
, and Huang
, Y.-X.
, 2012
, “Optimal Design of Geometric Parameters of Double-Layered Microchannel Heat Sinks
,” Int. J. Heat Mass Transfer
, 55
(11
), pp. 3262
–3272
.11.
Prager
, W.
, 1974
, “A Note on Discretized Michell Structures
,” Comput. Methods Appl. Mech. Eng.
, 3
(3
), pp. 349
–355
.12.
Svanberg
, K.
, 1981
, “Optimization of Geometry in Truss Design
,” Comput. Methods Appl. Mech. Eng.
, 28
(1
), pp. 63
–80
.13.
Pironneau
, O.
, 1984
, Optimal Shape Design for Elliptic Systems
, Springer, Berlin.14.
Sokolowski
, J.
, and Zolesio
, J.-P.
, 1992
, Introduction to Shape Optimization
, Springer
, Berlin
.15.
Bendsøe
, M. P.
, and Kikuchi
, N.
, 1988
, “Generating Optimal Topologies in Structural Design Using a Homogenization Method
,” Comput. Methods Appl. Mech. Eng.
, 71
(2
), pp. 197
–224
.16.
Bendsøe
, M. P.
, and Sigmund
, O.
, 2003
, Topology Optimization Theory, Methods, and Applications
, Springer
, Berlin
.17.
Lee
, H.-A.
, and Park
, G.-J.
, 2012
, “Topology Optimization for Structures With Nonlinear Behavior Using the Equivalent Static Loads Method
,” ASME J. Mech. Des.
, 134
(3
), p. 031004
.18.
Zhu
, B.
, Zhang
, X.
, and Fatikow
, S.
, 2014
, “Level Set-Based Topology Optimization of Hinge-Free Compliant Mechanisms Using a Two-Step Elastic Modeling Method
,” ASME J. Mech. Des.
, 136
(3
), p. 031007
.19.
Yamada
, T.
, Izui
, K.
, and Nishiwaki
, S.
, 2011
, “A Level Set-Based Topology Optimization Method for Maximizing Thermal Diffusivity in Problems Including Design-Dependent Effects
,” ASME J. Mech. Des.
, 133
(3
), p. 031011
.20.
Dede
, E. M.
, Joshi
, S. N.
, and Zhou
, F.
, 2015
, “Topology Optimization, Additive Layer Manufacturing, and Experimental Testing of an Air-Cooled Heat Sink
,” ASME J. Mech. Des.
, 137
(11
), p. 111403
.21.
Borrvall
, T.
, and Petersson
, J.
, 2003
, “Topology Optimization of Fluids in Stokes Flow
,” Int. J. Numer. Methods Fluids
, 41
(1
), pp. 77
–107
.22.
Whitaker
, S.
, 1986
, “Flow in Porous Media—I: A Theoretical Derivation of Darcy's Law
,” Transp. Porous Media
, 1
(1
), pp. 3
–25
.23.
Gersborg-Hansen
, A.
, Sigmund
, O.
, and Haber
, R. B.
, 2005
, “Topology Optimization of Channel Flow Problems
,” Struct. Multidiscip. Optim.
, 30
(3
), pp. 181
–192
.24.
Zhou
, S.
, and Li
, Q.
, 2008
, “A Variational Level Set Method for the Topology Optimization of Steady-State Navier–Stokes Flow
,” J. Comput. Phys.
, 227
(24
), pp. 10178
–10195
.25.
Deng
, Y.
, Liu
, Z.
, and Wu
, Y.
, 2013
, “Topology Optimization of Steady and Unsteady Incompressible Navier–Stokes Flows Driven by Body Forces
,” Struct. Multidiscip. Optim.
, 47
(4
), pp. 555
–570
.26.
Zhang
, B.
, Liu
, X.
, and Sun
, J.
, 2016
, “Topology Optimization Design of Non-Newtonian Roller-Type Viscous Micropumps
,” Struct. Multidiscip. Optim.
, 53
(3
), pp. 409
–424
.27.
Lin
, S.
, Zhao
, L.
, Guest
, J. K.
, Weihs
, T. P.
, and Liu
, Z.
, 2015
, “Topology Optimization of Fixed-Geometry Fluid Diodes
,” ASME J. Mech. Des.
, 137
(8
), p. 081402
.28.
Matsumori
, T.
, Kondoh
, T.
, Kawamoto
, A.
, and Nomura
, T.
, 2013
, “Topology Optimization for Fluid–Thermal Interaction Problems Under Constant Input Power
,” Struct. Multidiscip. Optim.
, 47
(4
), pp. 571
–581
.29.
Yaji
, K.
, Yamada
, T.
, Kubo
, S.
, Izui
, K.
, and Nishiwaki
, S.
, 2015
, “A Topology Optimization Method for a Coupled Thermal–Fluid Problem Using Level Set Boundary Expressions
,” Int. J. Heat Mass Transfer
, 81
, pp. 878
–888
.30.
Dede
, E. M.
, 2012
, “Optimization and Design of a Multipass Branching Microchannel Heat Sink for Electronics Cooling
,” ASME J. Electron. Packag.
, 134
(4
), p. 041001
.31.
Koga
, A. A.
, Lopes
, E. C. C.
, Nova
, H. F. V.
, de Lima
, C. R.
, and Silva
, E. C. N.
, 2013
, “Development of Heat Sink Device by Using Topology Optimization
,” Int. J. Heat Mass Transfer
, 64
, pp. 759
–772
.32.
Alexandersen
, J.
, Aage
, N.
, Andreasen
, C. S.
, and Sigmund
, O.
, 2014
, “Topology Optimisation for Natural Convection Problems
,” Int. J. Numer. Methods Fluids
, 76
(10
), pp. 699
–721
.33.
Alexandersen
, J.
, Sigmund
, O.
, and Aage
, N.
, 2016
, “Large Scale Three-Dimensional Topology Optimisation of Heat Sinks Cooled by Natural Convection
,” Int. J. Heat Mass Transfer
, 100
, pp. 876
–891
.34.
Łaniewski-Wołłk
, Ł.
, and Rokicki
, J.
, 2016
, “Adjoint Lattice Boltzmann for Topology Optimization on Multi-GPU Architecture
,” Comput. Math. Appl.
, 71
(3
), pp. 833
–848
.35.
Yaji
, K.
, Yamada
, T.
, Yoshino
, M.
, Matsumoto
, T.
, Izui
, K.
, and Nishiwaki
, S.
, 2016
, “Topology Optimization in Thermal-Fluid Flow Using the Lattice Boltzmann Method
,” J. Comput. Phys.
, 307
, pp. 355
–377
.36.
Deb
, K.
, Pratap
, A.
, Agarwal
, S.
, and Meyarivan
, T.
, 2002
, “A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II
,” IEEE Trans. Evol. Comput.
, 6
(2
), pp. 182
–197
.37.
Sullivan
, T. A.
, Van de Ven, J. D., Northrop, W. F., and McCabe, K., 2015
, “Integrated Mechanical and Thermodynamic Optimization of an Engine Linkage Using a Multi-Objective Genetic Algorithm
,” ASME J. Mech. Des.
, 137
(2
), p. 024501
.38.
Bhattacharjee
, K. S.
, Singh
, H. K.
, and Ray
, T.
, 2016
, “Multi-Objective Optimization With Multiple Spatially Distributed Surrogates
,” ASME J. Mech. Des.
, 138
(9
), p. 091401
.39.
Koski
, J.
, 1988
, “Multicriteria Truss Optimization
,” Multicriteria Optimization in Engineering and in the Sciences
, Springer
, New York
, pp. 263
–307
.40.
Haimes
, Y. Y.
, Lasdon
, L. S.
, and Wismer
, D. A.
, 1971
, “On a Bicriterion Formulation of the Problems of Integrated System Identification and System Optimization
,” IEEE Trans. Syst. Man Cybern.
, 1
(3
), pp. 296
–297
.41.
Das
, I.
, and Dennis
, J. E.
, 1998
, “Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems
,” SIAM J. Optim.
, 8
(3
), pp. 631
–657
.42.
Messac
, A.
, Ismail-Yahaya
, A.
, and Mattson
, C. A.
, 2003
, “The Normalized Normal Constraint Method for Generating the Pareto Frontier
,” Struct. Multidiscip. Optim.
, 25
(2
), pp. 86
–98
.43.
Shin
, W. S.
, and Ravindran
, A.
, 1991
, “Interactive Multiple Objective Optimization—Survey I: Continuous Case
,” Comput. Oper. Res.
, 18
(1
), pp. 97
–114
.44.
Nakayama
, H.
, 1992
, “Trade-Off Analysis Using Parametric Optimization Techniques
,” Eur. J. Oper. Res.
, 60
(1
), pp. 87
–98
.45.
Chen
, Y.
, Zhou
, S.
, and Li
, Q.
, 2010
, “Multiobjective Topology Optimization for Finite Periodic Structures
,” Comput. Struct.
, 88
(11
), pp. 806
–811
.46.
Mitchell
, S. L.
, and Ortiz
, M.
, 2016
, “Computational Multiobjective Topology Optimization of Silicon Anode Structures for Lithium-Ion Batteries
,” J. Power Sources
, 326
, pp. 242
–251
.47.
Lee
, J.
, Seo
, J. H.
, and Kikuchi
, N.
, 2010
, “Topology Optimization of Switched Reluctance Motors for the Desired Torque Profile
,” Struct. Multidiscip. Optim.
, 42
(5
), pp. 783
–796
.48.
Izui
, K.
, Yamada
, T.
, Nishiwaki
, S.
, and Tanaka
, K.
, 2015
, “Multiobjective Optimization Using an Aggregative Gradient-Based Method
,” Struct. Multidiscip. Optim.
, 51
(1
), pp. 173
–182
.49.
Sato
, Y.
, Izui
, K.
, Yamada
, T.
, and Nishiwaki
, S.
, 2016
, “Gradient-Based Multiobjective Optimization Using a Distance Constraint Technique and Point Replacement
,” Eng. Optim.
, 48
(7
), pp. 1226
–1250
.50.
Sato
, Y.
, Izui
, K.
, Yamada
, T.
, and Nishiwaki
, S.
, 2017
, “Pareto Frontier Exploration in Multiobjective Topology Optimization Using Adaptive Weighting and Point Selection Schemes
,” Struct. Multidiscip. Optim.
, 55
(2
), pp. 409
–422
.51.
Yamada
, T.
, Izui
, K.
, Nishiwaki
, S.
, and Takezawa
, A.
, 2010
, “A Topology Optimization Method Based on the Level Set Method Incorporating a Fictitious Interface Energy
,” Comput. Methods Appl. Mech. Eng.
, 199
(45
), pp. 2876
–2891
.52.
Koski
, J.
, 1985
, “Defectiveness of Weighting Method in Multicriterion Optimization of Structures
,” Int. J. Numer. Methods Biomed. Eng.
, 1
(6
), pp. 333
–337
.53.
Deb
, K.
, 2001
, Multi-Objective Optimization Using Evolutionary Algorithms
, Vol. 16
, Wiley
, Chichester, UK
.54.
Zitzler
, E.
, and Thiele
, L.
, 1998
, “Multiobjective Optimization Using Evolutionary Algorithms—A Comparative Case Study
,” International Conference on Parallel Problem Solving From Nature
(PPSN
), Edinburgh, UK, Sept. 17–21, pp. 292
–301
.55.
Auger
, A.
, Bader
, J.
, Brockhoff
, D.
, and Zitzler
, E.
, 2009
, “Theory of the Hypervolume Indicator: Optimal μ-Distributions and the Choice of the Reference Point
,” Tenth ACM SIGEVO Workshop on Foundations of Genetic Algorithms
, Orlando, FL, Jan. 9–11, pp. 87
–102
.Copyright © 2018 by ASME
You do not currently have access to this content.
