Level-set based Topology Optimisation of Convectively Cooled Heatsinks

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1 Level-set based Topology Optimisation of Convectively Cooled Heatsinks M. Santanakrisnan* 1, T. Tilford 1, C. Bailey 1 1.Department of Matematical Sciences, Te University of Greenwic, London, UK *m.santanakrisnan@greenwic.ac.uk Abstract: Te level set based topology optimisation tecnique is applied to optimise a 2D and 3D convectively cooled eatsink. Te approac is used to determine te design providing minimal termal compliance and minimal viscous dissipation. Te optimisation as been performed utilising te COMSOL multipysics for solution of pysics and sensitivity analysis and te MATLAB Livelink functionality for level set advection and reinitialisation. Tis paper describes te details of implementation, te topology optimisation model and presents results obtained using tis formulation. Keywords: Level set, Topology optimisation, Heat sink, Forced convection, Re-initialisation 1. Introduction A eat sink is a passive cooling device wic transfers te eat received from te eat source to adjacent fluid medium troug forced or natural convection. Many electronic devices, suc as CPU, GPU, power transistors and LEDs generate significant levels of eat energy wic must be efficiently dissipated in order to ensure reliable operation. Topological optimisation (TO) is a matematical approac tat optimises material layout witin a given design space, for a given set of constraints suc tat te resulting layout meets a prescribed set of performance objectives [1]. Te tecnique as primarily been used for structural optimisation problems but it is been applied to various pysical problems. Te two most prevalent TO approaces are density based metod and Level-set metods (LSM) [2], wit te latter preferred due to te ability to sarply capture inter-material interfaces. termal conductivity ratios for te objectives of minimum termal compliance (TC) and viscous dissipation (VD). Te rest of te paper details te LSM TO formulation in Section 2, Numerical Implementation in Section 3 and Computational details in Section 4. Results of te study and discussion are given in Section 5 and Conclusions are given in Section Level set based TO formulation Te seminal papers on LSM TO were written by Allaire [3] and Wang [4], were tey applied tis tecnique for te optimisation of structural members. Callis and Guest [5] applied tis tecnique for te optimisation of Stokes flow problems. Subsequently tis tecnique as been extended by many researcers to various fields. In tis tecnique, Signed Distance Function (SDF) is used as level set function (LSF). Positive SDF (ψ) is considered to represent solid and negative SDF is considered to represent fluid (Figure 1). Tis is enforced by te ersatz projection approac [3], using Heaviside function. = x Ω (boundary) ψ = { > x Ω + (Solid region) < x Ω (Fluid region) (1) COMSOL Multipysics software as been successfully used to apply LSM TO for design optimisation in a variety of pysical problems. In tis study we use COMSOL Multipysics solver combined wit MATLAB Livelink for performing te LSM TO wit re-initialisation of te level set functions at regular iteration intervals to enance accuracy. In tis study Two & Tree dimensional eat sinks will be developed for two different solid to fluid Figure 1 Design domain and level set function Brinkman s porosity term ( ) is used to differentiate solid and liquid and it is modelled as below. = ( max - min)*h + min (2) Were, H is Heaviside function, wic take unit value wen LSF is positive and takes zero value Excerpt from te Proceedings of te 217 COMSOL Conference in Rotterdam

2 wen LSF is negative and it as smoot transition between te two levels in order to enable differentiability. Derivative of Heaviside function is function wose expression is also given below. H(ψ) = (ψ ) 5 8 (ψ ) (ψ )5 (3) δ(ψ) = (1 (ψ )2 ) 2 (4) Were max =1e4 and min =.1 Evolution of te LSF represents evolution of sape and tis evolution wit respect to time is governed by Hamilton-Jacobi (HJ) equation, given in (5). Tis equation is marced in time to convect te LSF in te decreasing direction of objective value. Tis is done by taking te velocity of convection equal to sape sensitivity. G( ) i,j = { a = D x i,j = i,j i 1,j b = D x + i,j = i+1,j i,j c = D y i,j = i,j i,j 1 d = D y + i,j = i,j+1 i,j max ((a + ) 2, (b ) 2 ) + max ((c + ) 2, (d ) 2 ) 1 if i,j > max ((a ) 2, (b + ) 2 ) + max ((c ) 2, (d + ) 2 ) 1 if i,j < oterwise HJ equation: ψ t = V n ψ (5) N+1 i,j = N i,j Δt S ( i,j ) G( N i,j ) (8) Tis equation is solved using an explicit first order upwind sceme. Te time step for te marcing sould satisfy te CFL criterion for stability [3]. Every time te pysical problem is solved, te HJ equation is marced in time several time steps in order to obtain new sape or new level set function. Te MATLAB code (TOPLSM) written by Wang [6] demonstrates te various steps involved in level set based topology optimisation for simple structural mecanics problems. Te formulation of te approac requires te gradient of te LSF as to be unity. During LSM boundary convection tis gradient may vary from unity and result in in-accuracy of boundary definition. In order to overcome tis, te Eikonal equation (6) is solved to re-initialise te LSF [3]. Te unsteady equation is time marced till steady state is obtained, te steady state ensures te gradient of level set equals 1. ψ t + w. ψ = S(ψo) w = S(ψo) ψ ψ (6) Were S is smooted sign function S( )={ 1 1 i,j S( ) i,j = (7) 2 i,j +mes 2 Gradients calculated troug forward and backward difference formulas are used to solve te equation. Te difference formula used for time marcing is given below. 3. Numerical Implementation Te topology optimisation of eatsinks using Level set metod wit re-initialisation is implemented following te works of Liu [7], Kawamoto [8] and Deng [9]. Te works of Liu and Kawamoto describe te coupled LSM TO formulation witin COMSOL multipysics, wereas Deng as used COMSOL Multipysics for solving te pysics and MATLAB Livelink for solving te HJ equation and for reinitialisation. Te paper follows te approac of Deng [9] but applied to te callenge of eat sink design. Some of te notable works on eat sink design using topology optimisation are by Yoon [1], Alexanderson [11] using Density metod and works of Yaji [12] and Coffin [13] using Level set metod. Yaji designed liquid cannel cooled eat sinks and Coffin designed convectively cooled eat sinks using level set wit Extended Finite Element Metod geometry mapping. Te various steps involved in te process are given in Figure-2, in wic steps enclosed witin te dased line box are carried out in COMSOL Multipysics and rest of te steps are carried out in MATLAB Livelink. First, te design domain is initialised wit some initial guess of LSF in MATLAB and it is exported to COMSOL Multipysics. For te eat sink design, te points witin te design domain are differentiated into solid or fluid by interpolating te properties viz, termal conductivity, Cp, density and material impermeability. Tis in turn depends on te Excerpt from te Proceedings of te 217 COMSOL Conference in Rotterdam

3 sign of level set function and Heaviside function as given in Table-1. function is evolving, no accuracy will be lost due to interpolation. By solving te pysical problem in COMSOL, te sape sensitivity is calculated and ten it is retrieved in MATLAB Livelink using command mpeval, te syntax is given below. mpeval(model, 'sapesens','dataset','dset2','selection',3) were sapesens is te variable name, dset2 is dataset number and 3 is te design domain number. No-slip is imposed by initialising velocity equal to zero in solid regions i.e., were H equal to 1. For TC objective, te velocity term in te HJ equation (5) is equal to, Vn = [Kgam (( T x )2 + ( T y )2 + ( T z )2 ) + λ + (Volume difference)] (15) Figure 2 Level set topology optimisation procedure Name Kgam Cpgam gam Expression (Ks- Kf)*H + Kf (Cps- Cpf)*H + Cpf ( s- f)*h + f Table 1 Interpolation of termal properties Te eat sink optimisation problem for two different objectives are stated below. Objective TC: min Ω kgam ( T) 2 dω Objective VD: min μ ( u i ) 2 dω (9) Subjected to, Ω x j Cp(u. T) =. (k T) + Q (1) (. u) = (11) (u. u) = p +. {µ{ u + ( u) T }} u (12) H( )u= (13) Volume constraint =.4*Design volume (14) In order to import te LS in COMSOL from MATLAB, Interpolation Function is used. Since te mes remains same and only te level set Were te first term on te rigt and side denotes te sape sensitivity for te termal compliance objective, is te Lagrangian multiplier and is te area penalty factor. For VD objective, sape sensitivity is, 2 )) x i Vn =.5 μ (( u i ) + ( u j + λ + x j (Volume difference) (16) Te Lagrangian multiplier and Area penalty factor are updated as follows. λ k = λ k 1 Λ k 1 (Voume Difference) (17) Λ k = 1 β Λ k 1 (18) Te value of is.9, and te same value is used for all simulations irrespective of te objective of te study but value of Lagrangian multiplier, and te area penalty factor, are cosen differently for different objectives. Te reason for coosing different value is te difference in magnitude of te objective values. Suitable value of tese factors are cosen by trial and error metod. Te level sets are re-initialised at regular intervals (every 4 t iteration or 5 t iteration) to maintain teir slope. It is noted tat due to re-initialisation, te mean line of te boundary is sligtly moved or tere may be a pase lag. Level set distribution before and after re-initialisation at one instant is sown in Figure-3. Topology gradient term is not implemented in te algoritm so no new oles are nucleated in te design domain. Excerpt from te Proceedings of te 217 COMSOL Conference in Rotterdam

4 Te evolved level sets are again feedback to te COMSOL Multipysics and tis procedure is repeated till convergence Pi Pi.6.15 Pi.8 Pi and a eat flux of 7W/m 2 is specified as eat source in te bottom wall and zero pressure boundary condition is applied at te outlet. TO is carried out for two different material sets i.e., fluid to solid conductivity ratios. Te material properties are given in Table-2. Parameter Value K f /K s.1 &.1 f / s 1/892 Cp f /Cp s 4184/385 Table 2 Material properties Computational domain used for 3D study is sown in Figure-5. Te computational domain considered is 1 quadrant of te total domain, making use of symmetry boundary condition on te two sides. Te design domain is a cube of side.1m lengt and it is discretised by 43x43x43 mes elements. Heat flux of 1W/m 2 is applied at te bottom corner of area 1.353e-4 m 2 of te design domain base and a fluid flow of velocity 4e-5m/s is applied at te top surface of te computational domain. Te volume fraction of solid material is constrained at 25% Figure 3 Level set function before and after reinitialisation 4. Computational details Design domain is rectangular in sape, wit eat source at te bottom of te domain and liquid convection injected from te top of computational domain as sown in Figure-4. Te two sides of te computation domain act as outlet. Figure 5 Tree dimensional computational domain Figure 4 Computational domain details Te design domain is discretised wit 15x5 rectangular elements. Te initial level set used for te computation is series of circles. Te level set function is evolved on a grid mes wit gost elements, tese elements surround te 4 sides of design domain. Heat transfer in fluids and Laminar flow modules of COMSOL are used for computation. A liquid flow of velocity.2m/s and temperature 293K is applied at te Inlet. Te inlet velocity corresponds to a Reynolds number of 6 Linear elements or discretisation is used for bot velocity and pressure along wit stream wise diffusion stabilization for finite element solution. In two dimensional case, governing equations are solved in coupled way and for tree dimensional case equations are solved in segregated way. Simulation meeting te area constraint and te objective not varying significantly is considered as converged. 5. Results and Discussion Considering only TC as objective, TO is carried out for 2 fluid to solid conductivity ratios k f/k s=.1 and.1. Since te flow Reynolds numbers are low (Re6), it is expected tat at least for later case, te convective eat transfer will play a comparable role to conduction. Results obtained are given in Section Excerpt from te Proceedings of te 217 COMSOL Conference in Rotterdam

5 5.1 and 5.2 and later section describe te result for VD minimisation case and combination of TC and VD objectives. 5.1 Heat sink of iger solid conductivity k f/k s=.1 case: Te optimized sape for iger solid termal conductivity case, resemble like a tree sape and it is sown in Figure-6. Temperature is uniformly distributed trougout te design domain expect near te periperies. Convergence of Lagrange Multiplier, Area difference and Termal compliance are sown in Figure Heat sink of lower solid conductivity k f/k s=.1 case: Te optimized sape and te temperature distribution witin te design domain are sown in Figure-8. Unlike te ig solid conductivity case, tis doesn t ave many brances but te primary branc connects te eat flux wit te corners of te design domain. Te objective and Maximum temperature observed in te design domain are given in Table-3. Figure 6 Level set and Temperature (K) distribution Figure 8 Optimized sape for termal compliance (Kf/ks=.1) and Temperature (K) distribution Kf/Ks Maximum Termal Temperature Compliance(WK/m) (K) Table 3 Results of minimum termal compliance objective optimisations 5.3 Combined TC and VD For tis case, te objective for te topology optimisation problem is defined as, Objective = F1* TC + F2* VD (19) Figure 7 Convergence History Were F1 and F2 are factors and wen F1 is equal to zero, te optimisation becomes pure VD minimisation problem and wen F2 is zero it becomes TC minimisation problem. Excerpt from te Proceedings of te 217 COMSOL Conference in Rotterdam

6 Before optimising te eat sink for combined termal compliance and viscous dissipation, an optimisation is carried out for pure VD minimisation case. Te optimised sape and te velocity field in te design domain are sown in Figure -9. Figure 9 Optimised sape for minimum viscous dissipation and Velocity (m/s) contour Te VD magnitude is many order lower tan TC, as te fluid viscosity (1.2e-3Pa.s) and Re are low. Hence for combined objectives run, F1 is taken as 1e-9 and F2 as 1, so tat bot te objectives will influence te optimization. Optimized sape obtained is sown in Figure-1. In order to allow te smoot flow passage, branced structure as canged into a rectangular block on top of eat source and two islet of solid region acting like a guide vane for te incoming flow. Optimized sape along wit velocity and Temperature contour are given Figure-1. Figure 1 Results of combined TC and VD optimisation, sape, velocity (m/s) and Temperature (K) contour 5.4 Tree dimensional Heat sink A tree dimensional eat sink is optimised for TC objective for conductivity ratio of k f/k s=.1 subjected to laminar forced convection of Re=8. Heat sink sape, velocity and temperature distribution are given in Figure-11, 12. (F1,F2) Termal Compliance (WK/m) Viscous Dissipation (N/s) (, 1) e-8 (1e-9,1) e-8 Table 4 Combined TC and VD optimisation results Excerpt from te Proceedings of te 217 COMSOL Conference in Rotterdam

7 Figure 11 Top view of full (symmetrised) optimised eat sink Figure 12 Velocity (m/s) and Temperature (K) distribution in te design domain 5.5 Discussion Te eat sink sapes obtained for iger solid conductivity case agrees wit te tree like sape reported in literature. Because of te reduced termal conductivity, brances are missing in te low solid conductivity case, wic is understandable. Similarly for te combined TC and VD minimisation case, te optimiser tried to avoid sarp canges in temperature and velocity trougout te domain. Te 3D eat sink as a primary branc extending to te opposite diagonal and oter small brances extend in different directions and te branc tip ends wit bulb like mass. Te structure also as isolated/disconnected regions. Te symmetry boundary condition used during optimisation needs validation as it as not worked well for 2D cases. 6. Conclusions Level set based topology optimisation is applied to te design of 2D and 3D convectively cooled eat sinks for different material sets. In tis formulation, evolution and re-initialization of level set is carried out in MATLAB Livelink wile pysics is solved in COMSOL Multipysics. Tis formulation ensure crisp boundary capture and no-slip condition also applied on te solid boundaries. Heat sink sape obtained for 2D and 3D iger solid conductivity case agree wit tree like/dendritic sape. Te symmetry boundary condition used during 3D optimisation needs validation, tis will be taken as future work. Also, extending tis tecnique to industrial Reynolds number and for various oter application is also planned in future. 7. References 1. Bendsoe MP, Kikuci N, Generating optimal topologies in structural design using a omogenization metod. Comput Metods Appl Mec Eng 71(2): (1988) 2. Sigmund, O. and Maute, K., Topology optimisation approaces: A comparative review, Structural and Multidisciplinary Optimisation, vol 48, pp (213) 3. Allaire G, Jouve F, Toader A-M, Structural optimization using sensitivity analysis and a levelset metod. J Computational Pys 194(1): (24) 4. Wang MY, Wang X, Guo D. A level set metod for structural topology optimization. Computer Metods in Applied Mecanics and Engineering, 192: (23) 5. Callis V, Guest JK, Level set topology optimization of fluids in Stokes flow. Int J Numerical Metods Engg 79(1): (29) 6. TOPLSM199 Matlab code written by MY Wang et.al: ttp://www2.acae.cuk.edu.k/ ~cmdl/download.tm 7. Z. Liu, J.G. Korvink, R. Huang, Structure topology optimization: fully coupled level set metod via FEMLAB, Structural and Multidisciplinary Optimization, Vol 29, pp (25) 8. Kawamoto, T.Matsumori, T.Nomura, T.Kondo, S. Yamasaki and S. Nisiwaki, Topology optimization by a time dependent diffusion equation, Int Jrnal for Numerical metods in Engineering, 93: (213) 9. Yongbo Deng, Z Liu, J Wu, Y Wu, Topology optimization of steady Navier Stokes flow wit body force, Computer metods Applied Mecanical Engineering, 255, (213) 1. Yoon GH. Topological design of eat dissipating structure wit forced convective eat transfer. Journal of Mecanical Science and Tecnology, 24(6): (21) 11. Joe Alexandersen, Niels Aage, Casper Scousboe Andreasen and Ole Sigmund, Topology optimization for natural convection problem, Int. J. for numerical metods in fluids, :1-23 (213) 12. Kentaro Yaji, Takayuki Yamada, Seiji Kubo, Sinji Nisiwaki, A topology optimization metod for a coupled termal-fluid problem using level set boundary expressions, Int Jrnal of Heat and Mass transfer 81,pp , (215) 13. Peter Coffin, Kurt Maute, Level set topology optimization of cooling and eating devices using a simplified convection model, Structural and multidisciplinary optimization, 53(5), pp (216) 8. Acknowledgements Te autor would like to express gratitude for financial support from te University of Greenwic for sponsoring tis work as part of a PD study. Excerpt from te Proceedings of te 217 COMSOL Conference in Rotterdam