Topology optimization of conduction path in laminated metals composite materials

International Journal of Thermal Sciences 96 (2015) 183e190

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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts

Topology optimization of conduction path in laminated metals composite materials Chin-Hsiang Cheng*, Yen-Fei Chen Institute of Aeronautics and Astronautics, National Cheng Kung University, No.1, University Road, Tainan 70101, Taiwan, ROC

a r t i c l e i n f o

a b s t r a c t

Article history: Received 17 April 2014 Received in revised form 16 May 2015 Accepted 16 May 2015 Available online xxx

A topology optimization method that can be used to optimize the conduction path in laminated metallic materials between unequal isothermal surfaces is proposed in this study. The volume-of-solid (VOS) method presented by Cheng and Chen [20] for homogeneous and isotopic materials shape design has been firstly applied to deal with the composite materials. The materials used to make the laminate largely determine the properties, costs, and thereby its suitability for different applications. In this study, three-layer laminated metallic composite materials are considered in the test problems. These metallic layers are made of copper, aluminum, stainless steel or iron. Two possible orientations of the composite materials, vertical and horizontal, are investigated. Optimal shapes of the thermal conduction path between a higher- and a lower-temperature isothermal surfaces are determined in order to maximize three different objective functions, namely Q_ =m; Q_ =V and Q_ =USD. By using the present approach, optimal thermal conduction paths leading to maximum heat transfer rate per unit mass, per unit volume, or per unit cost can be readily yielded. © 2015 Elsevier Masson SAS. All rights reserved.

Keywords: Topology optimization Volume-of-solid method Laminated metals composite materials Heat conduction path

1. Introduction Topology optimization is a computational approach which optimizes material layout for given design space, material mass, loads or boundary conditions, such that the designed layout may meet the desired performance requirement. Using topology optimization, engineers can find the best conceptual design at a lower cost and in a shorter period of time. Recently, rapid growth in the rapid prototyping or 3D printing technology really helps advance the manufacturing of the complex shapes. With the help of 3D printing, engineers are able to present novel ideas of design that could not be manufactured before. In the past several decades, the topological design concept is widely applied to structural optimization [1e3], mechanical design [4,5], and thermal problems [6e8]. The studies of structural optimization problems were focused on maximization of stiffness, shape or eigenfrequency [9,10]. As for the thermal optimization problems, it is typically to find an optimal shape of an object that leads to maximum heat transfer performance [6e8]. Nowadays, there have been a number of topology

* Corresponding author. Tel.: þ886 6 2757575x63627. E-mail address: [email protected] (C.-H. Cheng). http://dx.doi.org/10.1016/j.ijthermalsci.2015.05.005 1290-0729/© 2015 Elsevier Masson SAS. All rights reserved.

optimization methods proposed, such as solid isotropic material with penalization (SIMP) method [11], evolutionary structural optimization (ESO) method [12], and constructal method [13], and so on. In the field of heat conduction structures, the progress of the optimization methods has also been advanced recently. For example, Bejan and coworkers [14,15] developed a tree network for heat transfer based on the constructal theory. Guo, Cheng, and Xia [16] performed heat conduction structure optimization based on the least dissipation principle of heat transport potential capacity for several practical examples. Cheng and Chang [17,18] designed the shapes for cylinders and sliders to meet different required loading conditions by means of the simplified conjugate-gradient method (SCGM) proposed by the same group of authors [19]. Most recently, Cheng and Chen [20] presented a novel approach based on the solution of the volumeof-solid function equation (VOS equation) in a non-constrained formulation. The VOS method was successfully applied for topology design of heat conductive solid paths made of homogeneous materials. The materials used to make the laminates largely determine the properties, costs, and thereby its suitability for different applications. For example, copper conducts heat evenly and quickly due to its high conductivity; however, cost of it is relatively high compared to other metals like stainless steel and iron. It might be

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Nomenclature keff ks L m n N Q_ S T Ts V x, y USD

effective thermal conductivity, W m1 K1 thermal conductivity of pure solid, W m1 K1 side length of original design domain, m total mass of solid materials in use, kg power constant in effective thermal conductivity equation normal coordinate to boundary of original design domain heat transfer rate, W coefficient of Equation (3), W m1 temperature in original design domain, K pure solid temperature in heat conduction path, K total volume per unit length, m2 rectangular coordinates, m total cost, in US dollar

Greek symbols boundary of constant-temperature object volume-of-solid function optimal cut-off value of VOS function

a z zc

Subscripts i index of constant-temperature object j index of layer

a benefit to have a mix of materials to form a multiple-layers structure to take advantage of the better properties of each material and avoid properties that may have a negative effect. The point is that there should be a balance between the function and the cost of the materials. Thermal behavior of the laminated metallic materials is one of the interesting issues which has been widely investigated [21e25]. However, the previous studies of the laminated metallic materials were majorly focused on mechanical or thermal properties like effective thermal conductivity. To the authors' knowledge, there is no published article relevant to shape design or topology optimization for the laminated metallic materials. For the laminated metallic materials, an essential issue is how to design the structures with a rational distribution of the shapes of the metals layers so as to increase the heat transfer rate but reduce the mass or cost of metals utilized. However, the determination of the heat transfer path in the laminated metallic materials is a relatively complicated task because it involves different metallic layers. It definitely cannot be carried out merely based on the experience of the designers or by costly experiments. Under these circumstances, in this study computation optimization of the laminated metallic materials is attempted. The concept of the VOS method proposed by Cheng and Chen [20] is firstly employed to optimize the conduction path in the laminated metallic materials between unequal isothermal objects. In this study, three-layer laminated metallic composite materials are considered in the test problems. These metallic layers are made of copper, aluminum, stainless steel or iron. Copper and aluminum are relatively high conductive metals for heat conduction; however, prices of them are high. On the other hand, conductivities of stainless steel and iron is relatively low compared to copper and aluminum, whereas their prices are also low. Densities and thermal conductivities [26] and prices of the selected metals [27] are provided in Table 1. Note that the prices of the metals were part of the

Table 1 Thermal conductivities and prices of selected metals. Metal

Densities [26] [kg/m3]

Thermal conductivity [26] [W/m K]

LME official price [27] [USD/kg]

Copper Aluminum Stainless steel (AISI 304) Iron

8933 2702 7900

401 237 14.9

7.375 1.815 0.30

7870

80.2

e

statistics published by London Metal Exchange (LME) website on December 27, 2013 [27]. Two possible orientations of the metallic layers of the composite materials, vertical and horizontal, are taken into consideration. Table 2 shows the labeled cases with their orientations and metals used. Fig. 1 shows the schematic of the physical models. The design is carried out in a design domain having an area of 10 cm  10 cm. Sizes of the hot and the cold objects are set to be 10/3 cm and 5 cm, and temperatures of the hot and the cold objects are maintained at 80  C and 30  C, respectively. In this study, the locations of the hot and the cold objects are fixed. Conventionally, the optimization tasks were performed by building a mathematical model which minimizes an objective function subject to some certain constraints. A suitable thermal performance index must be selected as the objective function. In some certain existing studies, for example, Ref. [28], the constraint of the optimization of heat conduction structure is given with a fixed amount of mass m. Using this constraint, one may obtain some interesting results. However, firstly, this constraint can only fix the amount of solid mass in use rather than increase the utilization efficiency of the solid mass. In real engineering applications, fixing mass is not a practical constraint because a designer is not able to know the amount of mass before he starts designing. If the designer does not know the amount of mass, fixing mass is not meaningful. Secondly, because the metals used in the laminate have different densities and even prices. Thus, for the laminated materials composed of different metals it does not make any sense to design the structure simply by fixing the total mass m. Therefore, in this study the optimal shapes of the conduction path in the laminated metallic materials between a higher- and a lower-temperature objects are determined based on three objective functions, Q_ =m; Q_ =V and Q_ =USD, where m, V and USD represent total mass, total volume and total cost, in US dollar, of the metals used in the laminate. The topology optimization is actually non-constrained by directly maximizing the magnitudes of the individual objective functions, not just fixing m. Based on the present concept, optimal thermal conduction paths leading to maximum heat transfer rate per unit mass, per unit volume, or per unit cost can be pursued for improving the utilization efficiency of the solid materials.

Table 2 Labeled cases and their specifications. Case Case Case Case Case Case Case

V1 V2 V3 H1 H2 H3

Orientation

Layer 1

Layer 2

Layer 3

Vertical

Copper Copper Copper Copper Copper Copper

Aluminum Stainless steel Iron Aluminum Stainless steel Iron

Stainless steel Aluminum Aluminum Stainless steel Aluminum Aluminum

Horizontal

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vTj v k vx eff vx



 þ

vTj v k vy eff vy

185

 ¼0

for layer j

(3)

Effective thermal conductivity

keff =ksj ¼ f ðzÞ

for layer j

(4a)

where ksj is already given for each of the metallic layers in Table 1, and

f ðzÞ ¼ zn

(4b)

The basic assumption of the algorithm is that the solid material bridging the hot and the cold objects ought to be ‘paved’ basically along the direction of the heat flow; meanwhile, thermal boundary conditions on the isothermal surfaces of the objects must be felt. It means that the effects of the objects' temperatures located at all boundaries must be included. The VOS function equation is referred to as the transport equation of the conserved VOS function (z) in the heat flux field. The diffusion term on the right-hand side of Equation (2) is added to include the effects of all boundaries. In the original design domain, each computation grid cell contains solid material of volume fraction z and void of (1- z). As suggested by Cheng and Chen [20], the function f(z) is expressed by the Bruggeman model with n ¼ 2.0. It is noted that the present approach can be adapted to any expressions of the effective thermal conductivity, not restricted to the Bruggeman model. In addition, effects of the magnitudes of s and n have been discussed by Cheng and Chen [20]. It appeared that a change in s or n does not remarkably alter final results of the optimal shape. Herein the value of s is suggested to be 300. Boundary and Interface Conditions. The boundary conditions for VOS function and temperature are described as follows:

ð ¼ 80 CÞ on a1

ð1Þ z ¼ 1 and T3 ¼ TH

ðat high  temperature object0 s boundaryÞ ð2Þ z ¼ 1 and T1 ¼ TL

ðat low  temperature object0 s boundaryÞ

Fig. 1. Schematic of physical model.

2. Topology optimization method

ð3Þ z ¼ 0 and

The problem seeking for heat conductive solid paths can be stated mathematically in the VOS method as:

Find : zðx; yÞ; with z2½0; 1 Maximize : Q_ =m; Q_ =V or Q_ =USD Subject to : T ¼ Ti on ai :

(1)

The VOS method is an approach based on a VOS function, z, which represents the volume fraction of solid material in the grid cells and solid material distribution as well. The optimal design of the heat conduction path is performed by determining the solid material distribution z(x, y) leading to maximum Q_ =m; Q_ =V or Q_ =USD. For solid material region, z ¼ 1 and for void region, z ¼ 0. The solution of z(x, y) is obtained in two steps: Step 1 Solving VOS Function and Heat Conduction Equations in Original Design Domain VOS function equation

keff VTj $Vz ¼ V$ðSVzÞ

for layer j

ð ¼ 30 CÞ on a2

(2)

where S ¼ skeff, in W m1, and keff represents the effective thermal conductivity, in W m1 K1. Heat conduction equation

vTj ¼0 vN

j ¼ 1; 2; 3

ðelsewhereÞ

(5a)

(5b)

(5c)

where N denotes normal coordinate to the boundary of the original design domain. At interface between layers j and j þ 1, the conditions of local equilibrium and energy conservation are applied as

ð1Þ Tj ¼ Tjþ1 ð2Þ  kj

j ¼ 1 and 2

vTj vTjþ1 ¼ kjþ1 vn vn

(6a) j ¼ 1 and 2

(6b)

The contact resistances at the interfaces are neglected in the present model. The solutions of z(x, y) and Tj(x,y) in layer j are obtained simultaneously from Equations (2) and (3) by means of finite-difference method. Step 2 Selecting Optimal Cut-Off Value of VOS Function to Determine Shape Outline In Step 2, the solution of z(x, y) is further used to determine the optimal shape of the conduction path. The constant-z contours of z(x, y) are generated and then serve as a family of candidate shapes to select. The optimal shape of the heat

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conduction path is the best among the family of candidate shapes, which leads to maximum of Q_ =m; Q_ =V or Q_ =USD. That is, the optimization criterion is

  v Q_ =f vz

¼ 0 at z ¼ zc

(7)

where f ¼ m, V or USD, and zc is the optimal cut-off value of VOS function. The optimal shape of the conduction path is yielded by cutting off the region of z < zc. The remaining part of z > zc is regarded as the heat conduction path. By using this criterion, one can maximize the heat transfer rate per unit mass, per unit space, or per dollar spent. In order to calculate Q_ , the temperature distributions in the metallic layers, Tsj(x,y) (j ¼ 1, 2, 3), in the tentative heat conduction path are solved from heat conduction equations for pure solids

V2 Tsj ¼ 0

for layer j

(8)

subject to the boundary conditions

ð1Þ z ¼ 1 and Ts3 ¼ TH

ð ¼ 80 CÞ on a1 0

ðat high  temperature object s boundaryÞ ð2Þz ¼ 1 and Ts1 ¼ TL

ð ¼ 30 CÞ on a2

ðat low  temperature object0 s boundaryÞ ð3Þ z ¼ 0 and

vTsj ¼0 vN

j ¼ 1; 2; 3

ðelsewhereÞ

(9a)

(9b)

(9c)

where N denotes normal coordinate to the boundary of the tentative heat conduction path. In addition, the same interfaces conditions as Equations (6a) and (6b) are introduced in the computation of Tsj(x,y) (j ¼ 1, 2, 3). Once Tsj(x,y) (j ¼ 1, 2, 3) in the heat conduction path is obtained, the heat transfer rate ðQ_ Þ dissipated from the hot object (TH) can be calculated. In the test problems, the heat transfer rate Q_ , total mass m, total volume V, and total cost USD of the tentative heat conduction shape are determined by integrations over the laminated metals of 1 mm thickness. The dependence of Q_ =m; Q_ =V or Q_ =USD on the cut-off value is then calculated, and then the optimal cut-off value, zc, can be selected. The approach is a systematic, efficient, computational process not much more than solving simple partial differential equations.

_ Fig. 2. Dependence of Q_ =m; Q=V and Q_ =USD on z for Case V1.

4.8 W/USD, is reached at zc ¼ 0.4. The optimal shapes of the heat conduction paths at zc ¼ 0.4 and zc ¼ 0.45 can be observed in Fig. 2. Obviously, different objective functions of interests lead to different optimal shapes. If the stainless steel and aluminum layers of the laminate in Case V1 swap their places, it becomes Case V2. Dependence of Q_ =m; Q_ =V and Q_ =USD on z for Case V2 is conveyed in Fig. 3. The optimal shapes leading to maximum Q_ =m of 32.28 W/kg and maximum Q_ =USD of 9.04 W/USD are found at zc ¼ 0.3. In addition, maximum Q_ =V of 209.56 W/m2 is found at zc ¼ 0.35. A comparison between Figs. 2 and 3 shows that Case V2 has higher heat transfer rate than Case V1. This means that the heat transfer rate between the two constant-temperature objects can be elevated simply by swapping the stainless steel and aluminum layers. It appears that the high-conductivity metallic layers should be

3. Results and discussion A comprehensive study of topology optimization of the heat conduction path is performed based on various objective functions, Q_ =m; Q_ =V and Q_ =USD, to cover different design purposes. Fig. 2 shows dependence of Q_ =m; Q_ =V and Q_ =USD on z for Case V1. In Case V1, the laminate consists of three layers made of copper, aluminum, and stainless steel placed vertically. In this figure, it is seen that each one of the three objective functions increases with z, reaches a maximum, and then rapidly descends, as z is increased from zero to unity. One can readily determine the optimal cut-off VOS function zc corresponding to each objective function. In this figure, maximum Q_ =m and maximum Q_ =V are found to be approximately 18.22 W/kg and 112.6 W/m2, respectively, both at zc ¼ 0.45. This implies that the heat conduction path defined by the contour at zc ¼ 0.45 leads to maximum heat transfer performance either in terms of Q_ =m or Q_ =V. On the other hand, maximum Q_ =USD, whose magnitude is

Fig. 3. Dependence of Q_ =m; Q_ =V and Q_ =USD on z for Case V2.

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placed in direct contact with the constant-temperature objects to increase the heat transfer rate. Fig. 4 shows the distributions of TS(x, y) in optimal heat conduction paths for Cases V1 to V3, which are all in vertical arrangement. The objective function considered in this case is Q_ =m. It is found that the optimal shape and temperature distribution are significantly affected by the materials used. The laminated metals composite materials of Case V3 are made of copper, iron, and aluminum in vertical arrangement. As a result, the magnitude of maximum Q_ =m for Case V3 reaches 95.92 W/kg which is the highest value among the three cases of this figure. Figs. 5e7 display the optimization results for the cases in horizontal orientation. Dependence of Q_ =m; Q_ =V and Q_ =USD on z for Case H1 is shown in Fig. 5. For this particular case, the optimal shapes lead to maximum Q_ =m of 33.9 W/kg and maximum Q_ =USD of 10.29 W/USD at zc ¼ 0.3. Furthermore, maximum Q_ =V of 209.16 W/m2 is found at zc ¼ 0.2. A comparison between Figs. 2 and 5 clearly display the effects of orientation of the layers. It appears that the horizontal orientation is accompanied by a higher heat transfer performance for the prescribed thermal boundary conditions. Plotted in Fig. 6 are the variations in the three objective functions with z, for Case H2. In this figure, the optimal shapes are all found at zc ¼ 0.3, independent of the objective function considered. Maximum Q_ =m, maximum Q_ =V, and maximum Q_ =USD are

Fig. 4. Distributions of TS(x,y) in optimal heat conduction paths for Cases V1 to V3.

187

_ Fig. 5. Dependence of Q=m; Q_ =V and Q_ =USD on z for Case H1.

determined to be approximately 33.9 W/kg, 213.51 W/m2 and 10.29 W/USD, respectively. Temperature distributions TS(x,y) in optimal conduction paths, in terms of Q_ =m, for Cases H1 to H3 are shown in Fig. 7. A comparison in maximum Q_ =m between Cases H1 and H2 shows a very small difference. This is because same materials are used in the two cases. However, when iron is used to replace stainless steel in Case H3, the magnitude of maximum Q_ =m is greatly increased to be 91.6 W/kg. Optimal shapes for all the cases tested in this study are summarized in Table 3. It is noticed that Cases V3 and H3 use copper, iron, and aluminum for their laminates, which are materials of relatively higher thermal conductivities; therefore, the optimal shapes for the two cases have much higher heat transfer rates than other cases. In the two cases, the optimal shapes of Case V3 use more mass of copper than Case H3, and hence, it is accompanied by a higher heat transfer performance.

Fig. 6. Dependence of Q_ =m; Q_ =V and Q_ =USD on z for Case H2.

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Fig. 7. Distributions of TS(x,y) in optimal heat conduction paths for Cases H1 to H3.

Furthermore, it is seen again that Case H1 has higher heat transfer rates per unit mass, volume or USD than Case V1 even though the two cases use same materials for the three layers. This may be attributed to a lower thermal resistance with the optimal shape in Case H1 than in Case V1. Actually, it can be observed in Table 3 that among the six cases considered, Case V1 leads to poorest heat transfer rates no matter which objective function is selected. It probably is because the low-conductivity metal, stainless steel, is used and placed directly on the surface of the hightemperature object. In this situation, it forms an ‘insulation layer’ on the surface for heat transfer so that the heat transfer rate is appreciably reduced. It is expected that the size of the original design domain has subtle influence on the optimal shape. Figs. 8e9 are plotted to show the effects of size of the original design domain on the optimal design. The results presented in Fig. 8 for Case V1 clearly reflect this expectation. As a matter of fact, in methods based on fixed design domains, for example, the SIMP method [11] and the VOS method [20], the optimal solution will be influenced by changing the size the original design domain. In Fig. 8, it is naturally found that the optimal design increases with the size of original design domain. However, the maximum value corresponding to the optimal shape is decreased as the size of the original design domain is enlarged. For example, for the original design domain having an area of 10 cm  10 cm, the value of maximum Q_ =m is 18.22 W/kg, whereas when the area is enlarged to 10 cm  15 cm, maximum Q_ =m is reduced to 16.97 W/kg. Fig. 9 shows the detailed information of effects of original design domain size on relationship between Q_ =m and z for Case V1. In this figure, the area is varied from 10 cm  10 cme10 cm  20 cm. It is observed that every curve has a maximum, and as discussed earlier, the maximum Q_ =m is decreased, while the corresponding zc is increased, by increasing the area of original design domain. The original design domain with area of 10 cm  10 cm leads to highest value of maximum Q_ =m. This is why the area of 10 cm  10 cm is suggested and used typically in this study. To illustrate the feasibility of the present approach a case of different configuration from those shown in Fig. 1. A twohorizontal-layer structure made of copper and stainless steel

Table 3 Optimal shapes for different cases based on different objective functions. Case V1 Q_ m

¼ 18:22 W=kg zC ¼ 0.45

Case H1 Q_ V

Q_ USD

¼ 112:60 W=m2 zC ¼ 0.45

¼ 4:80 W=USD zC ¼ 0.4

Case V2 Q_ m

¼ 32:28 W=kg

zC ¼ 0.3

¼ 95:92 W=kg

zC ¼ 0.4

¼ 33:90 W=kg

zC ¼ 0.3

Q_ V

¼ 209:16 W=m2

Q_ USD

¼ 10:29 W=USD

Q_ V

¼ 213:51 W=m2

Q_ USD

¼ 10:88 W=USD

zC ¼ 0.2

zC ¼ 0.3

Case H2 Q_ V

¼ 209:56

Q_ USD

W=m2

zC ¼ 0.35

¼ 9:04 W=USD

zC ¼ 0.3

Case V3 Q_ m

Q_ m

Q_ m

¼ 33:28 W=kg

zC ¼ 0.3

zC ¼ 0.3

zC ¼ 0.3

Case H3 Q_ V

¼ 611:15

zC ¼ 0.45

W=m2

e e

Q_ m

¼ 91:60 W=kg

zC ¼ 0.35

Q_ V

¼ 566:1 W=m2

zC ¼ 0.4

e e

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189

_ Fig. 9. Effects of original design domain size on relationship between Q=m and z for Case V1.

present approach efficiently leads to an optimal shape for the conduction path in the two-layer laminated material. For this case, the value of the maximum Q_ =m is 28.38 W/kg. It is noted that for the particular cases considered in this study, no holes were created inside the materials of the optimal designs. For the present study which deals with the laminated metallic materials, holes generated inside the laminated metallic materials are not practical due to difficulties in manufacturing. When necessary, the VOS function equation can be slightly modified by adding some source terms to the equation so as to generate holes in the materials. The modification should be open to discussion.

Fig. 8. Effects of original design domain size on optimal heat conduction path and maximum Q_ =m, for Case V1.

subject to different thermal boundary conditions is also tested. Fig. 10(a) shows the schematic of the test case, and the optimal design results are conveyed in Fig. 10(b). For this case, the area of the original design domain is 10 cm  10 cm. It is found that the

Fig. 10. A two-horizontal-layer structure made of copper and stainless steel.

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4. Concluding remarks In this study, topology optimization of heat conduction path in laminated metals composite materials is firstly performed by volume-of-solid method. Optimal shapes of the heat conduction path between a higher- and a lower-temperature isothermal surfaces are determined in order to maximize three different objective functions, namely Q_ =m; Q_ =V and Q_ =USD. By using the present approach, optimal thermal conduction paths leading to maximum heat transfer rate per unit mass, per unit volume, or per unit cost can be readily yielded. Results show that the optimal shape of the heat conduction path is significantly dependent on the objective function, the orientation of the laminates, and the materials of the layers. It appears that the horizontal orientation is accompanied by a higher heat transfer performance under the prescribed thermal boundary conditions. Among the particular tested cases considered in this study, Case V1 leads to poorest heat transfer rates no matter which objective function is selected. It probably is because the low-conductivity metal, stainless steel, is used for one of the layers placed directly on the surface of the high-temperature object. A comparison between Cases V2 and V1 shows that Case V2 has higher heat transfer rate than Case V1. This implies that the heat transfer rate can also be elevated simply by swapping the stainless steel and aluminum layers. Furthermore, different optimal shapes are yielded based on different objective functions of interests. Acknowledgment This research received funding from the Headquarters of University Advancement at National Cheng Kung University, which is sponsored by the Ministry of Education, Taiwan. References [1] M.Y. Wang, X. Wang, D. Guo, A level set method for structural topology optimization, Comput. Methods Appl. Mech. Eng. 192 (2003) 227e246. goire Allaire, François Jouve, Anca-Maria Toader, Structural optimization [2] Gre using sensitivity analysis and a level-set method, J. Comput. Phys. 194 (2004) 363e393. [3] M. Bremicker, P.Y. Papalambros, H.T. Loh, Solution of mixed-discrete structural optimization problems with a new sequential linearization algorithm, Comput. Struct. 37 (1990) 451e461. [4] R.V. Rao, V.J. Savsani, D.P. Vakharia, Teaching-learning-based optimization: a novel method for constrained mechanical design optimization problems, Comput. Aided Des. 43 (3) (2011) 303e315. [5] E. Sandgren, A multi-objective design tree approach for the optimization of mechanisms, Mech. Mach. Theory 25 (1990) 257e272.

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