multi-layer, pseudo 3d thermal topology optimization … · stokes (e.g. [5,6]) and kinetic theory...

Proceedings of the ASME 2012 International Mechanical Engineering Congress & ExpositionIMECE 2012

November 9-15, 2012, Houston, Texas, United States

IMECE2012-93093

MULTI-LAYER, PSEUDO 3D THERMAL TOPOLOGY OPTIMIZATION OF HEAT SINKS

Caroline McConnell∗Department of Engineering

Union UniversityJackson, Tennessee 38305

Email: [email protected]

Georg PingenDepartment of Engineering

Union UniversityJackson, Tennessee 38305Email: [email protected]

ABSTRACTIn order to overcome the high computational costs of 3D

fluid-thermal topology optimization we introduce a pseudo 3Dmodel consisting of a conductive base-layer and a coupled fluid-thermal design layer. While developed with heat sink designs inmind, the resulting topology optimization approach can be ap-plied to other fluid-thermal design problems where 2D flow ef-fects dominate. We introduce the coupled fluid-thermal solverbased on the lattice Boltzmann method, develop and validate thesensitivity analysis for the model, and illustrate the resulting op-timization framework for a heat sink design problem.

INTRODUCTIONIn component and system cooling applications the design of

flow channels and surfaces to promote maximum heat transferis of great importance. In recent years, particular focus has beenplaced on the utilization of micro-scale heat transfer effects, lead-ing to multi-scale and multi-physical design problems that are of-ten nonlinear and non-intuitive. To enhance the solution processfor such coupled thermo-fluidic design problems, the authors in-troduce a multi-layer, pseudo 3D thermal topology optimizationalgorithm. While the specific focus of this work is on the optimaldesign of heat sinks, the resulting framework is a generic topol-ogy optimization framework that can be applied to a wide varietyof coupled thermo-fluidic design problems.

Microchannel heat sink optimization is of considerable in-terest for a wide variety of applications. For example, Liu andGarimella [1] consider analytical models with closed form so-

∗Address all correspondence to this author.

lutions, Chen et al. [2] consider the optimal design of strip-finheat sinks, and Balagangadhar and Roy [3] use computationalshape optimization to optimize fin and duct designs for coupledfluid-thermal applications, requiring close-to-optimal initial de-signs as illustrated in Figure 1b. In recent years, increased fo-

FIGURE 1. Shape vs. Topology Optimization

cus has been placed on topology optimization, which does notrequire close-to-optimal initial designs and can generate new de-sign features as shown in Figure 1c. Starting with the pioneeringwork of Borrvall and Petersson [4], gradient-based flow topologyoptimization approaches have been developed using both Navier-Stokes (e.g. [5, 6]) and kinetic theory based flow solvers [7], andhave found application in commercial software packages suchas COMSOL Multiphysics. These flow topology optimization

1 Copyright c© 2012 by ASME

FIGURE 2. Illustration of a 2D thermal optimization, not permitting heated fins surrounded by fluid.

FIGURE 3. Illustration of a pseudo-3D thermal optimization with a conductive base-layer and a design layer allowing for both heat and masstransport.

methodologies have been developed for pure fluid flows and ex-tended to various multi-physical flow phenomena, including 2Dcoupled heat and mass transfer problems (e.g. [8–11]). Whilethese approaches have shown great promise and an extension ofthe basic concept from 2D to 3D is straightforward, the compu-tational cost can quickly become prohibitive as shown by Dede[12,13] who use a ’flattened’ thin 3D structure in their recent 3Dthermal fluidic topology optimization work. On the other hand,3D effects are essential for many heat transfer applications andcannot be neglected as illustrated in Figure 2 which shows that2D optimizations do not permit the generation of heated fins sur-rounded by a fluid as they are ’free-floating’ and disconnectedfrom the remaining structures. To permit the solution of a subsetof 3D thermo-fluidic topology optimization problems at close to2D computational cost, we introduce the use of a thermal baselayer that models 3-dimensional conduction effects as illustratedin Figure 3. Thus, while the fluid flow is treated as a 2D prob-lem, heat transfer can occur in 3 dimensions, permitting topologyoptimization of e.g. lab-on-a-chip devices.

In the following sections, we will first introduce the overalldesign problem and problem formulation, followed by an analyt-ical motivation, a discussion of the lattice Boltzmann flow solver,and a derivation and validation of the corresponding analyticalsensitivity analysis. We will illustrate the topology optimizationapproach by presenting preliminary results for the optimal designof a heat sink using: a) general topology optimization, b) the op-

timization of a single and triple fin, and c) the optimization of amulti-fin heat sink. As part of this work, we will show that thenonlinear interactions between thermal and fluidic effects leadto poor convergence behavior of the optimizer with a preferencefor undesired intermediate design variables. By employing dif-ferent functional dependencies between the design variables andfluid/thermal properties as well as utilizing penalty formulations,we begin to explore options to improve the desired convergenceof the optimization algorithm to final designs consisting of purefluid and pure solid material.

DESIGN PROBLEM FORMULATIONThe goal of the present work is the development of a topol-

ogy optimization algorithm for heat sink optimizations as illus-trated in Figure 3b. Having a heated wall, thermal base-layerpermitting conduction, and a fluid-thermal design layer, we wantto determine the geometry of a design that promotes the maxi-mum heat transfer from the heated walls. We can formulate thefollowing generic unconstrained topology optimization problem:

mins

F (s, f (s),g(s)),

s.t.{

f ,g, solves the governing equations for a given s,(1)

where F is a particular performance (objective) functional (neg-ative heat transfer in this case), f and g are the respective corre-


sponding state variables for mass and heat transfer, and s is thevector of design variables. To solve for the mass and heat transferstate variables, we utilize the lattice Boltzmann method as dis-cussed later. The thermal and hydrodynamic lattice Boltzmannequations are then augmented to enable a continuous transitionfrom fluid to solid as needed for gradient-based topology opti-mization. The resulting thermal optimization problem presentedin equation (1) is solved with a gradient-based optimization algo-rithm, the Globally Convergent Method of Moving Asymptotes(GCMMA) presented by Svanberg [14].

In order to reduce computational cost, the resulting analy-sis and optimization framework will not be fully-3D, but insteadconsist of a thermally conductive base-layer and thermo-fluidicdesign layer as illustrated in Figure 3. This approach is motivatedby the common use of layer-deposition techniques in the fabri-cation of micro-fluidic devices and has been used by Kreissl etal. [15] for the optimal design of flexible micro-fluidic devices.This approximation will be most accurate when the flow featuresare relatively high compared to the flow-channel width, such thatthe fluid flow is accurately represented by the 2D model.

Analytical Solution for Optimal Number of FinsIn order to show that one would indeed expect the formation

of fins for the design problem shown in Figure 3b under typicalflow conditions, a brief analytical analysis is performed. Thethermal energy balance for fluid flowing between heated walls isgiven as:

Q = n f hA∆Tlm = ṁc∆Tm, (2)

where Q is the heat flux, n f is the number of flow channels, h isthe convection coefficient, A is the surface area, ∆Tlm is the log-mean temperature difference (∆Tlm = ∆Te−∆Tiln(∆Te/∆Ti) , where ∆Te =Ts −Te and ∆Ti = Ts −Ti with e, s, and i denoting exit, surface,and inlet respectively), ṁ is the mass flowrate, c is the specificheat, and ∆Tm = Te−Ti is the mean temperature difference of thefluid.

We can now utilize the illustration in Figure 4 for constanttemperature fins with fully developed laminar flow to analyti-cally determine a relation for the optimum number of fins neededto achieve maximum heat transfer. Considering the solution forflow between parallel plates, a relation for the average channelvelocity Vavg can be found for a given pressure drop ∆P= P2−P1as:

Vavg =∆Pw2

12µL, (3)

where w is the individual channel width defined by the ratio ofoverall width W to the number of flow channels n f (w = Wn f ), µ

FIGURE 4. Notation for Analytical Fin Optimization

is the dynamic viscosity, and L is the channel length. The massflowrate can then be determined as:

ṁ = ρVavgA = ρVavgWH =∆PW 3H12n2f νL

, (4)

where ρ is the fluid density, H is the height of the channels,and ν is the kinematic viscosity. We can now use these rela-tions to express the right side of Eq. 2 in terms of the numberof flow channels n f , fluid temperatures, and known geometri-cal and fluid properties. The left side of Eq. 2 can be modifiedthrough the use of the Nusselt number. For fully developed lam-inar flow through channels with constant surface temperaturesand an infinite height to width ratio inherent to our pseudo 3Dmodel, a Nusselt number of Nu = 7.54 = hDhk is commonly givenin literature [16], where h is the convection coefficient, Dh = 2 Wn fis the hydraulic diameter, and k is the conduction coefficient ofthe fluid. Further, replacing the surface area of the channel withA = 2n f LH leads to the following heat flux balance with the onlyunknowns being the number of flow channels n f and the fluid exittemperature Te:

Q =n2f NukLH∆Tlm

2W=

∆PW 3Hc∆Tm12n2f νL

. (5)

Solving this equation for the fluid exit temperature Te and plug-ging back in leads to the following relation for the overall heatflux as a function of the number of flow channels n f :

Q =∆PW 3Hc12n2f νL

[Ts − (Ts −Ti)e

−6NuαL2µn4f

∆PW4 −Ti], (6)

where α is the thermal diffusivity (α = kρc ). We can now deter-mine the optimal number of fins by setting dQdn f = 0 and solving


for n f . Alternatively we can plot Eq. 6 as done for typical latticeBoltzmann fluid and geometry values in Figure 5, which showsthat having approximately 6 flow channels represents the opti-mal fin/channel configuration for a particular choice of parame-ters. At this point, it should be pointed out that the thickness ofthe fins was neglected during the present analytical analysis andwould lead to a decrease in the optimal number of flow channels.However, the presented analytical derivation shows that there isa specific number of fins that lead to optimum heat transfer. It isthis realization that motivates the development of a generalizedtopology optimization algorithm for heat sink design.

FIGURE 5. Relation between number of flow channels n f and heatflux Q in Watts.

Thermal Lattice Boltzmann MethodSince its introduction, the Lattice Boltzmann method (LBM)

has become a viable alternative to Navier-Stokes based meth-ods for many flow scenarios [17,18]. For coupled thermo-fluidicproblems three main methods - the two distribution approach, thehybrid passive-scalar approach, and the multiple-relaxation-timesingle distribution approach - have been developed and appliedto a variety of problems including forced and natural convection(e.g. [19, 20]). In the present work we use the two distributionapproach due to its simplicity and ability to represent varyingPrandtl numbers.

The two distribution function thermal LBM as used in thiswork closely follows the models presented by Kao et al. [21] andYan and Zu [20] except for the use of three dimensional thermaldistribution functions. The thermal hydrodynamic lattice Boltz-mann method approximates the Navier-Stokes and energy equa-tions and results in a two-step computational process for fluidicf and thermal components g. The key difference between thefluid and thermal distribution functions is the definition of the

equilibrium distribution function as illustrated in the followingequations:

Collision − C :

f̃α(xi, t) = fα(xi, t)−1τ f

[ fα(xi, t)− f eqα (xi, t)], (7)

g̃α(xi, t) = gα(xi, t)−1τt[gα(xi, t)−geqα (xi, t)], (8)

Propagation − P :fα(xi +δ teα , t +δ t) = f̃α(xi, t), (9)gα(xi +δ teα , t +δ t) = g̃α(xi, t). (10)

The fluid and thermal equilibrium distribution functions aregiven respectively as:

f eqα = wα, f ρ[

1+3(eα ·u)+92(eα ·u)2 −

32

u2], (11)

geqα = wα,T T [1+3(eα ·u)] , (12)

where ρ is the macroscopic density, T is the macroscopic temper-ature, u is the macroscopic velocity, and wα are lattice weightsthat depend on the lattice geometry. The lattice geometry differsfor the velocity and thermal distributions. The velocity distri-bution is defined by a two dimensional, nine velocity (D2Q9)model (see Figure 6a), allowing for the recovery of the macro-scopic density, velocity and pressure. The thermal distribution isdefined as a three dimensional, 7 velocity (D3Q7) model [22](see Figure 6b), allowing for the heat transfer from the ther-mal base layer to the design layer and the determination of themacroscopic temperature values. Due to the difference in lat-tice geometry the weights for the velocity distribution are wα, f =[ 49 ,

19 ,

19 ,

19 ,

19 ,

136 ,

136 ,

136 ,

136 ] and the weights for the thermal distri-

bution are wα,T = [ 14 ,18 ,

18 ,

18 ,

18 ,

18 ,

18 ].

FIGURE 6. Directions for D2Q9 and D3Q7 LBM lattices.

Viscosity (ν) and thermal diffusivity (α) can be shown tobe related to their respective relaxation times τg and τ f by a


Chapman-Enskog expansion of the Lattice Boltzmann equations[23, 24]. The relationships are given by

ν = (τ f −1/2)c2s δ t (13)α = (τg −1/2)c2s δ t. (14)

For a given fluid, ν and α are related by the dimensionlessPrandtl number

Pr =να. (15)

In order to continuously transition from fluid to solid andvice-versa as required for gradient-based topology optimization,the following physical properties/variables must vary as a func-tion of the mapped design variables p(x) = p(s(x)): velocity,thermal diffusivity and base-layer to design-layer conductivity.Here it should be noted that the ultimate goal is not the accuraterepresentation of intermediate material as the final design shouldconsist primarily of pure fluid and pure solid material. Pingenet al [7] have shown that the porosity model of Spaid and Phe-lan [25] can be used to continuously scale the velocity of thefluid:

ũ(t,x) = (1−p(x)κv) u(t,x), (16)

where p(x) is substituted into the equilibrium distribution func-tions (11) and (12) instead of u and a shaping factor of κv = 3 hasbeen introduced in order to improve the convergence propertiesof the design optimization process.

Pingen and Meyer [8] have introduced a similar scaling totransition from the thermal diffusivity of a fluid α f to that of asolid αs using the shaping factor κT :

α̃(t,x) = α f +(αs −α f )p(x)κT . (17)

Finally, in order to transition from an insulated (fluid) to aconductive (solid) boundary between the base-layer and design-layer, a partial bounce-back boundary condition - analogous tothe work by Pingen et al [26] for fluids - is applied to the thermaldistribution function in place of the propagation step:

g̃25(t,x) = g26(1−p(x)κp)+g15(p(x))κp (18)

g̃16(t,x) = g15(1−p(x)κp)+g26(p(x))κp . (19)

Here, the shaping factor is defined by κp and the boundary in-teraction is illustrated in Figure 7 for one time-step. The result

FIGURE 7. Scaling of conductivity between base-layer and design-layer.

is an interface between the layers that transitions from being a’zero-flux’ insulator when p = 0 (reflective to thermal fluxes) topermitting full heat flux through the boundary in the case of asolid p = 1.

It is important to note that the developed thermal topologyoptimization framework is currently applied to steady-state flowsapproximated through the following fixed-point formulation:

R( f ,g,p) = M( f ,g,p)−

{f

g

}= 0. (20)

Here R denotes the residual vector and the operator M performsone collision and propagation step to advance the flow and tem-perature to the next time step.

Thermal Sensitivity Analysis for Topology Optimiza-tion

In order to solve the optimization problem (1) using agradient-based optimization algorithm, the design sensitivities ofthe performance functional with respect to the design variablesdFds j

are required. Due to the large number of design variables,an adjoint formulation is traditionally used and was derived forhydrodynamic and thermal lattice Boltzmann based topology op-timization by Pingen and co-workers [7, 8] as:

dFds j

=∂F∂ s j

−[

J−T∂F

∂ [f,g]

]T ∂R∂ s j

, (21)

where f and g are the steady-state fluid and thermal distributionfunctions approximated through the fixed-point formulation (20)and J = ∂R∂ [f,g] =

∂M∂ [f,g] − I is the Jacobian of the fixed-point prob-

lem.


The evaluation of the partial derivatives ∂F∂ s j ,∂F

∂ [f,g] ,∂R∂ s j

, and∂M

∂ [f,g] is necessary to solve the adjoint sensitivity equation (21).The evaluation of these derivatives based on analytically derivedexpressions is discussed in detail by Pingen and co-workers [7,8]and the derivatives of the residual with respect to the design vari-ables ∂R/∂ s j follow a procedure similar to the Jacobian. Thus,the present work specifically focuses on the differences betweenthe analytical 2D thermal LBM Jacobian derived by Pingen andMeyer [8] and the pseudo 3D Jacobian required for the presentwork. Those differences are due to the use of the D3Q7 ther-mal model and the thermal interaction between base-layer anddesign-layer as shown in equation (18).

For the Jacobian, the derivative of the operator M can beseparated into a collision and a propagation step:

(∂M

∂ [f,g]

)T=

(∂P

∂ [f,g]

)T( ∂C∂ [f,g]

)T. (22)

Collision Jacobian: The sensitivities of the collisionoperator with respect to the distribution functions, (∂C/∂ [f,g])Tare determined by differentiating the fluid and thermal collisionsteps (7) and (8) leading to the following localized collision stepJacobian at each lattice node:

∂C∂ [f,g]

=∂ [ f̃ , g̃]α∂ [ f ,g]β

=∂ [ f ,g]α∂ [ f ,g]β

− 1τ

[∂ [ f ,g]α∂ [ f ,g]β

− ∂ [ f ,g]eqα

∂ [ f ,g]β

]. (23)

Here, α and β represent the 9 discretizations of velocity spacefor the fluid distribution function f and the 7 discretizations ofvelocity space for the thermal distribution function g. The partialderivatives ∂ fα/∂ fβ = 1 and ∂gα/∂gβ = 1 when α = β andzero otherwise.

Thus, only the derivatives of the equilibrium distributionfunctions (11) and (12) are non-trivial. As a one-way couplingis present between fluid and thermal state variables (no buoy-ancy effects are considered), the velocity equilibrium distributionfunction f eq depends only on the mass transfer variables ρ and u,but the thermal equilibrium distribution function geg is coupledand depends on both mass and thermal transfer through the vari-ables u and T. Following the detailed derivation by Pingen andMeyer [8] for 2D problems, this leads to a block diagonal globalJacobian and a highly populated, 16×16 (pseudo 3D), local Ja-cobian of the collision operation Ci at each lattice node i shownin Figure 8:

(∂Ci

∂[

fβ ,gβ])T = (∂

[f̃α , g̃α

]∂[

fβ ,gβ] )T =

∂ f̃0∂ f0

. . . ∂ g̃6∂ f0...

. . ....

∂ f̃0∂g6

. . . ∂ g̃6∂g6

(24)

FIGURE 8. Local 16×16 Collision Jacobian

Considering the local collision Jacobian depicted in Figure8, the 9 × 9 block in the top left corner represents the hydro-dynamic component of the Jacobian ∂ f̃α∂ fβ and is fully populated.The 9× 7 block in the top right corner represents the couplingbetween thermal and hydrodynamic distribution functions ∂ g̃α∂ fβ .As the stationary distribution function g0 and those moving inthe vertical direction (3rd dimension) g5 and g6 are independentof the flow velocities, those columns are zero. The 7×9 block inthe bottom left corner represents the coupling between hydrody-namic and thermal distribution functions ∂ f̃α∂gβ and is filled with allzeros due to the one-way coupling in the present model. Finally,the 7×7 block in the bottom right corner represents the thermalcomponent of the Jacobian ∂ g̃α∂gβ and is fully populated.

Propagation Jacobian: The second component neededfor the determination of the Jacobian of the lattice Boltzmannsystem is the derivative of the propagation operator with respectto the distribution functions, (∂P/(∂ [f,g]))T . In previous workby Pingen and co-workers [7, 8] this simply led to a shifting ofthe rows from the collision Jacobian as the basic propagationstep leads to a movement of distribution functions to neighbor-ing nodes. In the present model, the partial bounce-back condi-tion (18) modifies the propagation step and leads to the following


augmented propagation Jacobian:

∂P∂ [ f ,g]β

=∂ [ f ,g]α (xi, t)∂ [ f ,g]β (x j, t)

=∂[

f̃ , g̃]

α (xi −δ teα , t −δ t)∂ [ f ,g]β (x j, t)

(25)

=

1−p(xκp) if g, β = [5,6], α = [6,5], x j = xi,p(x)κp if g, β = α = [5,6], x j = xi −δ teα ,1 if β = α , x j = xi −δ teα ,0 otherwise.

From this equation it can be seen that when the Jacobians ofthe collision and propagation operators are combined, the Jaco-bian of the propagation operator primarily leads to a shifting ofrows from the collision Jacobian except for the vertical compo-nents of the thermal distribution function (g5,g6), which lead to ascaling and addition of the corresponding rows from the collisionJacobian.

Objective Function: The objective for the exampleproblems presented in this work is to minimize the negative netheat flux out of the design domain measured between the 2ndand 3rd node at the inlet and the 3rd to last and 2nd to last nodesalong the outlet. Nodes at least 1 removed from the inlet/outletwhere chosen to reduce boundary effects, leading to the follow-ing performance functional:

F = −∆H F net =−(H F out −H F in) (26)

=∫

x=2g1 −

∫x=3

g3 −∫

x=lx−2g1 +

∫x=lx−1

g3.

Sensitivity Validation: In order to ensure that the ana-lytical sensitivity analysis is accurate, a finite difference sensitiv-ity analysis was implemented. The finite difference verificationwas performed by finding the heat flux from the domain witha single, intermediate design variable fin (s = 0.5). The designvariable was then perturbed by a factor ε , leading to a changein the corresponding heat flux. The difference between the twoheat flux values was divided by the perturbation ε and then com-pared to the analytic sensitivity value in order to compute thepercent difference between both. For a small interval around thedesign variable s = 0.5 the sensitivities are expected to behavein a nearly linear manner resulting in good agreement betweenanalytical and finite-difference results. It can be seen from Fig-ure 9 that for epsilon values from 10−7 −10−3 the analytical andfinite-difference sensitivities are in good agreement and the im-plemented analytical sensitivity analysis is accurate. For smallerε values the finite difference results are inaccurate due to numer-ical noise and for larger ε values the linear finite difference ap-

proximation inaccurately approximates the underlying nonlinearproblem.

FIGURE 9. Comparison of Analytical and Finite Difference Sensitiv-ities

EXAMPLESThe goal of this work is the development of a topology opti-

mization framework to determine optimal heat-sink designs. Theobjective is to maximize the heat transfer from the thermal base-layer of the design problem illustrated in Figure 10. The bluecolor in the figure represents a prescribed fluid region at the frontand back of the design domain in order to avoid inlet and out-let effects. Similarly, the sides are prescribed as solid material.The temperature of the fluid at the inlet is set to 20 degrees inLBM units and the thermal base-layer side temperature is set to200 degrees in LBM units. The solid material is highly con-ductive, as to model material commonly used in heat sinks, thethermal diffusivity in this problem is αs = 0.5 in LBM units. Thefluid is much less conductive, having a thermal diffusivity set toα f = 0.00211 in LBM units. The scaling factors of this problemare set to κp = 3, κv = 3, and κt = 3. The problem is solved on a40×40 mesh with a design domain of 22×38 where each latticenode is linked to one design variable leading to 836 total designvariables.

With these settings, the optimal design in Figure 11 was ob-tained. The dark blue at the bottom of the spectrum of the poros-ity scale in Figure 11 represents complete fluid, the dark red atthe top of the same scale represents complete solid. The resultsshow a large amount of intermediate porosities throughout thedesign domain, resulting in a non-intuitive, impractical and non-physical design. While the intermediate porosities might suggesta very fine fin structure smaller than permitted by the problemresolution, the ultimately desired result is a design with no inter-mediate porosities in order to produce a realistic heat sink design.


FIGURE 10. Initial Design Domain

FIGURE 11. Optimization of Initial Design Domain on a 40 × 40Mesh

Due to this undesired design outcome we simplify our designsetup in the following section in order to further investigate thebehavior of the optimization algorithm.

Single and Triple Fin ExperimentsInstead of a design domain with design variables corre-

sponding to each lattice node, we simplify the problem to thecreation of one fin in the center of the domain as shown in Figure12. The area around the fin is prescribed fluid and the fin is onlyallowed to change as one design variable. With this new designdomain we keep all remaining variables unchanged.

Using this design domain the results shown in Figure 13 areobtained. The fin converges towards a pure solid but stops at a re-sult of s = 0.9025 instead of becoming completely solid. Havinga fin with this porosity improves the objective value from that ofan empty design domain. With a completely fluid design domainthe objective value of the system is −7.8 and the objective valuewith a fin of porosity s = 0.9025 is −15.2. The design stoppedshort of forming a solid fin because the objective for s = 0.9025is better than that of a completely solid fin as shown in Figure 14.This effect is due to the relative interplay of fluidic and thermaleffects as the design variable changes from fluid to solid and can

FIGURE 12. Single Fin Design Domain

possibly be overcome through the use of different scaling factorsκ .

FIGURE 13. Single Design Variable Fin on a 25×25 Mesh

To create an end result of complete fluid or solid features theporosity vs heat flux graph must be continuously decreasing. Fig-ure 14 shows that objective values at the current design settingsdip lower than the objective value for a complete solid. Optionsto improve this functional behavior are investigated next.

Kappa and Penalty TestingFirst we investigate possible changes to the scaling factors

κv, κt , and κp to change the functional dependency of the hy-drodynamic and thermal parameters on the design variables. Aswe are primarily concerned with designs that consist of completefluid and solid without intermediate porosities, these functionalrelations can be selected in order to obtain a better behaving op-timization problem. The scaling factor kappa allows a change invalues for κv, κt , and κp, without impacting the accuracy of theanalysis for the final converged design.


FIGURE 14. Objective vs. Porosity With All Kappas Equal to 3 for aSingle Design Variable Fin on a 25×25 Mesh

FIGURE 15. Kappa Testing (a) Objective vs. Porosity With κpChanging (b) Objective vs. Porosity With Different κv, κt , and κp Val-ues

A sample of the different testing results is shown in Figure15a and b for different kappa values. Based on the results shownin Figure 15a, increasing κp solved the problem of objective val-ues dipping lower than those with a porosity of one but createdan increase in the objective at low porosities. Since combina-tions of different kappas are not producing the desired continu-ously decreasing result we turn to a penalty formulation. Penaltyapproaches are commonly used in structural topology optimiza-tion and are referred to as SIMP-models (Solid Isotropic Materialwith Penalization) [27]. For our problem we augment the objec-tive through the following penalty formulation:

FP = F +C(p f )(1− p f ), (27)

where C and f are scaling parameters that can be adjusted toimprove the behavior of the optimization problem for interme-diate porosities by penalizing those intermediate values. Usingthe kappa results shown in Figure 15 we can apply this penaltyformulation to see if there is a penalty factor that works for ourdesign problem. Using the present problem with κv = 2, κt = 4,κp = 2, and a penalty factor with C = 2 and f = 2 leads to acontinuously decreasing relation between the design variable andobjective as shown in Figure 16. Applying this result to one de-sign variable fins in the middle of the design domain works forthe 25×25 mesh as shown in Figure 17, leading to a completelyconverged design.

FIGURE 16. Objective vs. Porosity With Penalty Factor Correction

We now want to increase the complexity of the design prob-lem by re-introducing additional design variables. To do so, wesolve a design problem with 3 design variables for the possiblegeneration of three fins. The results for this study are shown inFigure 18 and confirm that the use of κv = 2, κt = 4, κp = 2, and


FIGURE 17. Optimization of One Fin With Penalty Factor for a 25×25 Mesh

a penalty factor with C=2 and f=2 leads to converged optimaldesigns for the current problem set-up.

FIGURE 18. Optimization for Three Fins With Penalty Factor for a25×25 Mesh

Optimization With Seven FinsAs shown in the analytical section of this paper, there is an

optimal number of fins for a given design domain. Having testeda single and triple fin design with one and three design variables,respectively, we now want to investigate a problem with 7 designvariables and therefore a possibility of generating 7 fins in thedomain as shown in Figure 19. The goal is to determine if theoptimizer can find the optimal configuration of fins. In order toaccount for the fact that 7 design variables are now penalized, thescaling factor C has been adjusted to a value of C = 1.30 to fa-cilitate converged designs while all other settings are maintainedfrom the previous problems.

As shown in Figure 20, our optimization framework leadsto a design with three fins, equivalent to that shown in Figure18. Given the design space of seven design variables, this 3 finsolution is indeed optimal. Of interest is also to see how the op-timizer progresses towards the design, which is shown in Figure

FIGURE 19. Initial Design Domain For Seven Fins

20. First the optimizer begins to form 7 fins as shown in Figure20b. Then the optimizer reduces the number of fins to 3 in Fig-ure 20c which become completely solid at different rates. Theprogression of the objective as a function of design iterations isshown in Figure 20a. This example problem shows the poten-tial of the developed thermal topology optimization approach.More work is needed to develop a general penalty approach thatis applicable to the general topology optimization problem whereeach lattice node is linked to one design variable as was the casein the first example problem shown in Figure 11.

CONCLUSIONWe have introduced a lattice Boltzmann method based

pseudo 3D fluid-thermal topology optimization approach andhave developed the corresponding adjoint sensitivity analysis.We have validated the sensitivity analysis and have applied theresulting optimization framework to the optimal design of a heatsink. Differing from past LBM based topology optimization, itwas shown that the non-linear nature and complex interactionof hydrodynamic and thermal effects necessitates the use of apenalty formulation in order to prevent intermediate porosities.The resulting optimization framework was than illustrated for amulti-fin heat sink optimization. Future work will focus on gen-eralizing the penalty formulation such that it is applicable to themore general topology optimization problem where each fluidnode corresponds to a design variable, maximizing the designspace and thus permitting the generation of a wider range of op-timal designs.

ACKNOWLEDGMENTThe authors would like to acknowledge support by Union

University through an Undergraduate Research Grant in supportof the presented work.


FIGURE 20. Iterations of Seven Fins: (a) Optimization vs. Iterations (b) System After 20 Iterations (c) System after 40 Iterations (d) SystemConverged by 70 Iterations

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multi-layer, pseudo 3d thermal topology optimization … · stokes (e.g. [5,6]) and kinetic theory...

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