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Composite Structures journal homepage: www.elsevier.com/locate/compstruct

Two-scale optimal design of structures with thermal insulation materials Xiaolei Yan a,b, Xiaodong Huang b,c,⇑, Guangyong Sun c, Yi Min Xie b a

School of Mechanical and Automobile Engineering, Fujian University of Technology, Fuzhou 350108, China Centre for Innovative Structures and Materials, School of Civil, Environmental and Chemical Engineering, RMIT University, GPO Box 2476, Melbourne 3001, Australia c State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, China b

a r t i c l e

i n f o

Article history: Available online 22 October 2014 Keywords: Topology optimization Two-scale Concurrent design Bi-directional evolutionary structural optimization (BESO)

a b s t r a c t This paper introduces a two-scale topology optimization approach by integrating optimized structures with the design of their materials. The optimization aims to ﬁnd a multifunctional structure composed of homogeneous porous material. Driven by the multi-objective functions, macrostructural stiffness and material thermal conductivity, stiff but lightweight structures composed of thermal insulation materials can be achieved through optimizing the topologies of the macrostructures and their material microstructure simultaneously. For such a two-scale optimization problem, the effective properties of materials derived from the homogenization method are applied to the analysis of macrostructure. Meanwhile, the displacement ﬁeld of the macrostructure under given boundary conditions is used for the sensitivity analysis of the material microstructure. Then, the bi-directional evolutionary structural optimization (BESO) method is employed to iteratively update the macrostructures and material microstructures by ranking elemental sensitivity numbers at the both scales. Finally, some 2D and 3D numerical examples are presented to demonstrate the effectiveness of the proposed optimization algorithm. A variety of optimal macrostructures and their optimal material microstructures are obtained. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction In engineering applications, material selection is a long and complex process which involves not only material properties but also the service conditions such as structural conﬁguration, applied loads and boundary conditions for a speciﬁc product. Integrating designs of structure and material will be of great signiﬁcance to the ﬁelds of structural engineering and materials engineering [1]. Porous materials are receiving increasing attention because of their ability to combine good properties with light-weight. They have been used in diverse areas of applications such as thermal insulation, energy absorption, buoyancy and soundprooﬁng, and they also exist as biomaterials [2–3]. To meet multifunctional requirements of engineering structures, this paper will investigate the concurrent design of structure and its porous material so that the resulting structure has multifunctional properties. Structural topology optimization aims to ﬁnd optimal topology to maximize structural performance while satisfying various constraints such as a given amount of material. Compared with size ⇑ Corresponding author at: Centre for Innovative Structures and Materials, School of Civil, Environmental and Chemical Engineering, RMIT University, GPO Box 2476, Melbourne 3001, Australia. Tel.: +61 3 99253320; fax: +61 3 96390138. E-mail address: [email protected] (X. Huang). http://dx.doi.org/10.1016/j.compstruct.2014.10.013 0263-8223/Ó 2014 Elsevier Ltd. All rights reserved.

and shape optimization, topology optimization provides much more freedom and allows the designer to create totally novel and highly efﬁcient conceptual designs for structures. Over the last two decades, various topology optimization algorithms, e.g. homogenization method [4], solid isotropic material with penalization (SIMP) [5,6], evolutionary structural optimization (ESO) [7,8], and level set technique [9] have been developed. Unlike the continuous density-based topology methods, the ESO/BESO method represents the structural topology and shape with discrete design variables (solid or void) with a clear structural boundary [10,11]. It was originally developed based upon a simple concept of gradually removing redundant or inefﬁcient material from a structure so that the resulting topology evolves towards an optimum [7]. The later version of the ESO method, namely bi-directional ESO (BESO), allows not only to remove elements from the least efﬁcient regions, but also to add elements in the most efﬁcient regions simultaneously [12–15]. It has been demonstrated that the BESO method is capable of generating reliable and practical topologies for various types of structures with high computational efﬁciency. To date, topology optimization techniques are mainly used to solve one-scale design problems either for the macrostructures to improve their structural performance or for the materials to develop new microstructures with prescribed or extreme properties

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[16–20]. An ideal design should be that structures with different boundary conditions and geometries have their own optimal macrostructural topologies, meanwhile are composed of tailored materials with optimal microstructures. It is understandable that the concurrent design of structures and materials could be more effective since it provides more design freedoms on multiple scales to achieve an efﬁcient solution. However, the research on topology optimization considering structure and material at both scales simultaneously is still limited. Rodrigues et al. [21,22] proposed a hierarchical computational procedure by integrating macrostructures with local material microstructures. This methodology treats the material design in a point wise manner within the macrostructure. Therefore the result brings up extra difﬁculties in manufacturing. Coelho et al. [23] extended this hierarchical procedure to three-dimensional elastic structures. Another type of the approach for concurrent design at two scales assumes that material is uniformly distributed in the macrostructure such as in Liu et al. [24] and Yan et al. [25]. Similarly, Deng et al. [26] studied the design of a thermoelastic structure by minimizing its structural compliance and thermal expansion simultaneously. This paper attempts to propose a multi-objective and multiscale topology optimization approach based on the BESO method. The optimization objective is to design the structure with multifunctional applications that require not only maximum structural stiffness but also superior thermal insulation capabilities. As we know, most engineering structures are expected to function under different physical conditions. The design just considering one single criterion is often unable to satisfy the multifunctional requirement for engineering structures [27,28]. To achieve the multifunctional needs, here we adopt a multi-objective and multi-scale optimization method, where two conﬂicting design objectives are taken into account to maximize structural stiffness at the macro level and minimize material thermal conductivity at the micro level. In this paper, it is assumed that a macrostructure is uniformly composed of lightweight porous material and the optimization needs to ﬁnd the optimal topologies for the macrostructure and for the microstructure of the porous material. The homogenization theory will be used to calculate the effective properties of porous material so as to establish a link between the macro level and the micro level. The BESO algorithm with binary design variables is employed to obtain clear solid-void solutions for the macrostructure and the microstructure simultaneously. Finally, 2D and 3D examples are presented to validate the proposed design method. Following this introduction section, the paper presents a multiobjective concurrent optimization model, and conducts its twoscale sensitivity analysis based on ﬁnite element method in Section 3. The concurrent optimization procedure based on the BESO method is detailed in Section 3. Finally, several typical 2D and 3D examples are discussed in Section 4 and some conclusions are drawn in Section 5. 2. Multi-objective concurrent optimization models 2.1. Optimization formulation In the concurrent optimization, macro structural design and micro material design are combined into one system by using the homogenization theory and optimization is conducted at both levels simultaneously. Consider a structure with known boundary conditions and external forces as illustrated by Fig. 1. The macrostructure is composed of uniformly distributed porous material with microstructures periodically repeated by the unit cell. In the unit cell of the porous material, the base material is represented with black and the void with white. Two types of ﬁnite element

Fig. 1. Illustration of two-scale design for a structure composed of a porous material.

models, namely a macro FE model and micro FE model are employed to represent the macrostructure and material microstructure respectively. The macro and micro relative densities, xi and xj, are considered to be the macro and micro design variables, respectively. In this paper, the optimization objective is to seek for a lightweight structure which is not only with maximum structural stiffness (or minimum compliance) but also composed of porous material with minimum thermal conductivity so that the resulting structure has multiple functions, i.e. best load-bearing and superior thermal insulation capabilities simultaneously. The macrostructural mean compliance can be expressed as

f 1 ðxi ; xj Þ ¼

1 T F U 2

, M 1X C0 ¼ U Ti K i ðxi ; xj ÞU i C 0 2 i¼1

ð1Þ

where F and U represent the external force vector and the nodal displacement vector of the macrostructure, respectively. Ui is the displacement vector of the ith element in the macrostructure. Ki is the stiffness matrix of the ith macrostructural element. M is the total number of ﬁnite elements in the macro FE model. C0 is the mean compliance of the full design. xi is the binary design variables for the macro FE model. xi = 1 means element i is porous or solid and xi = 0 means element i is void. The summation of the diagonal elements in the effective thermal conductivity matrix can be used to measure the overall conduction capability for the orthotropic material design [29,30], as

f 2 ðxj Þ ¼

2X or 3

, H mm ðxj Þ

j

js 0

ð2Þ

m¼1

where jHmm is the mth diagonal element in the effective thermal conductivity matrix of the porous material. js0 represents the summation of the diagonal elements in the base material thermal conductivity matrix. xj is the binary design variables for the micro FE model. xj = 1 means element j is solid and xj = 0 means element j is void. Thus, to achieve the multifunctional designs for the structure and material, a multi-objective design problem can be formulated in terms of a weight average as

Minimize : f ðxi ; xj Þ ¼ gf 1 ðxi ; xj Þ þ ð1 gÞf 2 ðxj Þ

ð3-aÞ

Subject to : Kðxi ; xj ÞU ¼ F

ð3-bÞ

PN PM xi V i j¼1 xj V j V f ¼ Pi¼1 ¼ V mac V mic ¼ V f PN f f M V V i j i¼1 j¼1

ð3-cÞ

xi ; xj ¼ 0 or 1;

ð3-dÞ

ði ¼ 1; 2; . . . ; M; j ¼ 1; 2; . . . ; NÞ

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where g is the weight factor which is chosen to control the proportion or emphasis between the above two objectives and 0 6 g 6 1. K is the stiffness matrix of the macrostructure which can be assemR bled by the elemental stiffness matrix K i ¼ V i BT DMA BdV i (B is the MA strain–displacement matrix, D is the elasticity matrix of the ith element). N is the total number of ﬁnite elements in the micro FE model. Vi is the volume of element i in the macro FE model, and Vj is the volume of element j in the micro FE model. V mac and V mic f f are the volume fractions in the macro and micro FE models, respec tively. Vf is the total volume fraction of the structure and V f is the prescribed volume fraction of the structure. From Eq. (3), we can be seen that the objective function depends on both the macro variable xi and the micro variable xj. It is a typical two-scale topology optimization problem where the topologies of the macrostructure and material unit cell should be determined simultaneously. A series of optimal designs with different emphasis on the load-bearing capability and material thermal insulation capability can be achieved by adjusting the weight factor from 0 to 1. The material interpolation schemes should be adopted at both scales to obtain clear topologies. At the micro scale, the elemental elasticity matrix can be expressed as [31] MI

D

¼

xpj D0

ð4Þ

where p is the exponent of penalization (p = 3 is used in this paper). D0 denotes the elasticity matrix of the base material. Similarly, the thermal conductivity matrix of the jth element at the material unit cell is deﬁned by

xpj

MI

j ¼ j0

ð5Þ

where j0 denotes the thermal conductivity matrix of the base material.At the macro scale, the elasticity matrix of the ith element can be expressed as MA

D

¼

xpi DH

ð6Þ

H

where D is the effective elasticity matrix which can also be computed from the micro-scale analysis of the material unit cell through the homogenization method.

Because the thermal conductivity is independent of the macro design variable xi, the second term on the right-hand side of Eq. (9) is zero. Substituting Eq. (6) into the elemental stiffness matrix in Eq. (9), the above equation can be rewritten as

@f p g p1 T ¼ x Ui @xi 2 C0 i

Z

BT DH BdV i U i

ð10Þ

Vi

In the proposed optimization framework, it is assumed that the porous material is uniformly distributed in the macrostructure. Therefore, the design variable, xj in the material unit cell is related to all elements in the structure. The sensitivity of the objective against the micro design variable, xj, can be expressed as M 2or3 @f 1 gX @K i ðxi ; xj Þ 1 gX @ jHmm ðxj Þ ¼ U Ti Ui þ @xj 2 C 0 i¼1 @xj js0 m¼1 @xj

ð11Þ

With the help of the material interpolation scheme in Eq. (4), the derivation of DH with respect to micro design variable xj can be obtained as [35] p1 @DH pxj ¼ @xj jYj

Z

ðe0 ej ÞT D0 ðe0 ej ÞdV j

ð12Þ

Vj

Similarly, the derivation of able xj is p1 @ jHmm pxj ¼ @xj jYj

Z Vj

jHmm with respect to micro design vari-

T

m m m ðvm 0 vj Þ j0 ðv0 vj ÞdV j

ð13Þ

By the virtue of Eqs. (12) and (13), Eq. (11) can be rewritten as

@f g ¼ @xj 2C 0 M X pxp1 j xp U T jYj i¼1 i i

þ

(Z

p1 or 3 1 g pxj 2X js0 jYj m¼1

T

B

Vi

Z Vj

"Z

#

)

T

ðe0 ej Þ D0 ðe0 ej ÞdV j BdV i U i

Vj T

m m m ðvm 0 vj Þ j0 ðv0 vj ÞdV j

ð14Þ 2.2. Homogenization and sensitivity analysis When the size of the material unit cell is very small compared with the size of the structure body, the homogenization theory can be applied for obtaining the effective elasticity matrix of the material. In the micro FE model, the homogenized elasticity matrix can be expressed by [32,33]

DH ¼

1 jYj

Z

DMI ðe0 eÞdY

ð7Þ

1 jYj

3. The concurrent BESO procedure

Y

where e0 is unit test strain ﬁeld and e is the induced strain ﬁeld. |Y| is the total area or volume of the material unit cell. In the same way, the effective thermal conductivity is expressed as

jH ¼

The above section indicates that the optimal designs of macrostructure and material unit cell are coupled, i.e. the material properties from the homogenization of micro base cell are used in the ﬁnite element analysis of the macrostructure while the sensitivities for the micro design variables need the displacement ﬁeld of the macrostructure.

Z Y

jMI ðv0 vÞdY

ð8Þ

where v0 is unit temperature gradient ﬁeld and v is the induced temperature gradient ﬁeld.To implement the BESO optimization technique, sensitivity analysis is necessary for guiding the search direction in the iteration process. With the help of Eqs. (1) and (2), the derivation of the objective with respect to the design variable xi can be expressed using the adjoint variable method [34], as or 3 @f 1 g T @K i ðxi ; xj Þ 1 g 2X @ jHmm ðxj Þ ¼ Ui Ui þ @xi 2 C0 @xi js0 m¼1 @xi

ð9Þ

In the BESO method, the sensitivity numbers which denote the relative ranking of the elemental sensitivities will be used to update the design variables xi and xj. In the concurrent designs of macrostructure and material unit cell, elemental sensitivity numbers for the macro and micro models should be put together and compared. Therefore, elemental sensitivities of both models should be normalized by the variations of their corresponding volumes so that they can be treated at the same level. When removing or adding an element in the macrostructure, the change of the volume of P the base material is Nj¼1 xj V j . Therefore, the sensitivity number of the ith element in the macrostructure for minimizing the objective under the prescribed volume constraint can be deﬁned as

ai ¼

@f @xi

X N xj V j j¼1

ð15Þ

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Because all elements in the macrostructure have the same material unit cell, the sensitivity number of the jth element in the material unit cell can be deﬁned as

@f @xj

aj ¼

X M xi V j

ð16Þ

i¼1

Using Eqs. (10), (14), (15) and (16), the sensitivity number ai at the macro level and sensitivity number aj at the micro level can be easily calculated. To realize the concurrent designs of structure and material, sensitivity numbers ai and aj will be sorted together from the highest to the lowest. Because the design variables xi and xj are restricted to be discrete values either 0 or 1, we can devise a simple scheme for updating the design variables xi, xj = 0 for elements with the lowest sensitivity numbers and xi, xj = 1 for elements with highest sensitivity numbers. In the numerical implementation, a prescribed value small as 106 can be used for void elements to avoid any possible numerical singularities [8].To avoid checkerboard pattern and mesh-dependency which are common phenomenon in the topology optimization [36]. Here, a mesh-independent ﬁlter for discrete design variables [8] has to be applied for the ith elemental sensitivity number at the macro level as

PMi

ai ¼ Pr¼1 M

wir ar

r¼1 wir

ð17Þ

where Mi is the set of elements around element i for which the center to center distance D(i, r) is smaller than the ﬁlter radius Rmin. wir is the ﬁlter weight factor given as

wir ¼ max½0; Rmin Dði; rÞ

ð18Þ

The jth elemental sensitivity number at the micro level will be ﬁltered in the same way. The sensitivity number can be further modiﬁed by averaging with its historical information to improve the convergence of the solution [8], as

ð19Þ

where k is the current iteration number.A threshold of sensitivity number can be easily determined by the target volume for the next iteration and the ranking of sensitivity numbers ai and aj. The design variables xi, xj are changed from solid to void for elements with lowest sensitivity numbers, and from void to solid for these with highest sensitivity numbers. The concurrent evolutionary design process stops when the objective material volume is satisﬁed and the objective function converges. The whole BESO procedure for concurrently designing a macrostructure and its material microstructure can be outlined in Fig. 2. More details about the numerical implementation of the BESO method can also refer to [10,25,37]. 4. Numerical examples and discussions To demonstrate the capability of the proposed optimization approach, 2D and 3D numerical examples are presented in this section to illustrate the concurrent design of structure and material. It is assumed that the base material is isotropic, its Young’s modulus is E0 = 1.0, Poisson’s ratio l0 = 0.3 and thermal conductivity j0 = 1. The material unit cell that represents the microstructure of the material is discretized into 81 81 4-node quadrilateral elements for 2D case and 30 30 30 8-node brick elements for 3D case, respectively. For simplicity, a uniform meshes with element size 1 1 and 1 1 1 are assigned to all 2D and 3D macrostructures, respectively. The used BESO parameters are evolution rate ER = 2% mic and ﬁlter radii Rmac min ¼ Rmin ¼ 3 .

Fig. 2. Flow chart of the concurrent two-scale BESO optimization.

4.1. 2D cantilever Fig. 3 shows a cantilever with length L = 120 and height H = 60 undergoing a concentrated vertical force at the center of the right edge. When the weight factor is set to be g = 0.3 and the material

F= 1

H=60

1 2

ai ¼ ðai;k þ ai;k1 Þ

L=120 Fig. 3. A cantilever with length L and high H.

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Fig. 4. Optimized macrostructure and its material microstructure for the cantilever with weight factor g = 0.3 and material volume fraction V f ¼ 30%: (a) topology of the cantilever; (b) topology of the material unit cell; (c) topology of 2 2 unit cells; (d) effective elasticity matrix DH; (e) effective thermal conductivity matrix jH.

Fig. 5. Evolution histories of the objective and volume fractions when g = 0.3 and V f ¼ 30% .

volume fraction V f ¼ 30%, the optimized macrostructure, material unit cell and the effective material properties are given in Fig. 4. It can be seen that the resulting material has higher stiffness in x-direction than that in y-direction to resisting the bending of the cantilever.

Fig. 5 plots the evolution histories of the objective function and material volume fractions. BESO starts from the initial full design and gradually decreases the total volume fraction to the prescribed value 30% and then keeps constant. At the early stage of the optimization, the macrostructure almost keeps as the full design and the decrease of the total volume fraction mainly attributes to the decrease of the micro volume fraction. After 25 iterations, the topologies of the macrostructure and the material unit cell change simultaneously. At the last stage of the optimization, objective function and volume fractions are convergent to stable solutions and the resulting optimized topologies are achieved as shown in Fig. 4. Table 1 lists the optimized results for the cantilever with various weight factors but the material volume fraction V f is kept to be 30%. It is interesting to observe how the structural compliance objective f1 and the material conductivity objective f2 affect the optimal topologies of macrostructures and material unit cell. As the weight factor g increases, a heavier emphasis is given to minimize the mean compliance of the structure and more and more material is shifted from the micro scale to the macro scale. When g reaches 0.8, the material unit cell becomes solid and the lightweight of the structure attributes to the removal of materials at

Table 1 The optimized macrostructures and their material microstructures for the cantilever with various weight factors when V f ¼ 30%. Weight factor

Material volume fraction

g = 0.8

V mac ¼ 30:0% f V mic ¼ 100:0% f

g = 0.6

V mac ¼ 31:06% f V mic ¼ 96:62% f

g = 0.4

V mac ¼ 42:50% f V mic ¼ 70:55% f

g = 0.2

V mac ¼ 72:86% f V mic ¼ 41:17% f

Macro structure

Micro unit cell (2 2)

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decrease with the decrease of the total volume fraction V f . However, the decrease of the total volume fraction mainly attributes to the signiﬁcant change of the optimized topologies of macrostructures. The optimized topologies of material base cells have almost no change. This indicates that, in this case g = 0.3, more emphasis assigned to the minimization of thermal conductivity results in removing material mainly at the micro-scale level to a certain extend. However, the further material removal may lead to the signiﬁcant increase of the mean compliance and thus the optimization prefers to remove material at the macro-scale level. Therefore, material removal at the macro-scale or micro-scale level totally depends on the deﬁned optimization problem and is automatically realized in the proposed optimization algorithm.

the macro level. This demonstrates that solid material is preferable for minimizing the mean compliance of the structure while porous material is preferable for maximizing the heat-insulating capability of the structure. From Table 1, we can also see how the two conﬂicting objectives compete through the re-distribution of materials in the admissible design spaces at both scales when the weight factor varies. The variations of the multi-objective functions for weight factors from 0.1 to 0.7 are plotted in Fig. 6. It can be seen that the mean compliance of the structure increases while the thermal conductivity of the material decreases with the increase of the weight factor. Values of the two conﬂicting objectives generally form an approximate convex curve in the admissible design space. Therefore, the optimal design of stiff structures with thermal insulation porous materials is a matter of making a trade-off decision from a set of compromising designs. Table 2 lists the optimized topologies of macrostructures and their material microstructures when the weight factor g is set to be 0.3 but with various volume fraction constraints. From the table, it can be seen that the ﬁnal macro and micro volume fractions

4.2. 3D beam In this example, we will conduct the two-scale multifunctional designs for a 3D beam. The geometry and boundary conditions of the beam are shown in Fig. 7. A vertical unit line load applies on the top surface. Table 3 shows the resulting optimized topologies of the macrostructures and the material microstructures for the beam with two different volume constraints when the weight factor g = 0.3. Similar to 2D cases, the total volume optimally allocates to the volumes at the macro level and the micro level. But different from 2D cases, the ﬁnal volume at the micro level generally increases from 69.63%

F=1 y

H=20

Fig. 6. The variations of multi-objective functions for various weight factors when V f is set to be 30%.

z x L=100

Fig. 7. Geometrical model and boundary conditions for a 3D beam.

Table 2 The optimized macrostructures and their porous microstructures for the cantilever with different volume fractions when g = 0.3. Material volume fraction V f ¼ 70% ¼ 96:69% V mic ¼ 72:35%) (V mac f f

V f ¼ 50% (V mac ¼ 80:36% V mic ¼ 62:23%) f f

V f ¼ 30% (V mac ¼ 51:69% V mic ¼ 58:09%) f f

V f ¼ 10% (V mac ¼ 17:75% V mic ¼ 56:41%) f f

Macro structure

Micro unit cell (2 2)

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Table 3 The optimized macrostructures and their porous microstructures for the beam with different volume fractions when g = 0.3. Material volume fraction V f

Macro structure

Micro unit cell (2 2 2)

¼ 20% ¼ 28:72% V mic ¼ 69:63%) (V mac f f

V f ¼ 10% (V mac ¼ 13:97% V mic ¼ 71:53%) f f

to 71.53% although the volume constraint decreases from 20% to 10%. It also demonstrates that the ﬁnal volumes at the macro level and at the micro level are automatically determined by the optimization algorithm and cannot be predicted beforehand. Therefore, the proposed concurrent topology optimization is hardly replaced by simply performing the macrostructural optimization and the material design separately although the resulting topologies at the macro level have no signiﬁcant difference from that of the sole macrostructural optimization. Table 4 shows the optimized results for the 3D beam with different material volume constraints as g = 0.1. Because of a heavier emphasis on the material thermal insulation design, the optimization automatically allocates larger volume fractions at the macro level and smaller volume fractions at the micro level compared with the examples in Table 3. It can be easily understood that the thermal insulation can be effectively improved by using less base material. The proposed concurrent topology optimization algorithm converges to the trade-off solutions between at the macro and micro levels. Fig. 8 shows the variation of two competitive objective functions when the weight factor changes from 0.1 to 0.5 but the

volume fraction V f is ﬁxed as 20%. The variation of two objective functions can be ﬁtted with a convex curve. Similar to 2D cases, Fig. 8 also indicates that the two-scale optimal design for stiff structures with thermal insulation porous materials is a compromising

Fig. 8. The variations of multi-objective functions for various weight factors when V f is set to be 20%.

Table 4 The optimized macrostructures and their porous microstructures for the beam with different volume fractions when g = 0.1. Material volume fraction V f ¼ 20% (V mac ¼ 49:20% V mic ¼ 40:65%) f f

V f ¼ 10% (V mac ¼ 22:45% V mic ¼ 44:53%) f f

Macro structure

Micro unit cell (2 2 2)

X. Yan et al. / Composite Structures 120 (2015) 358–365

solution, in which a heavier emphasis on the macrostructural stiffness will sacriﬁce the thermal insulation of materials, and vice versa. 5. Concluding remarks In this paper, a multi-objective and two-scale concurrent topology optimization approach is presented based on the BESO method for designing a structure composed of homogenous porous material. The stiff structure with thermal insulation material with a given volume fraction is achieved by minimizing the macrostructural mean compliance and the material thermal conductivity simultaneously. The multi-objective function is deﬁned by the weight scheme where the weight factor is assigned for the importance of each objective. Numerical examples demonstrate that designs at the macro and micro levels interact strongly with each other. Different from the one scale design of structure or material, the proposed concurrent optimization approach optimally allocates volume fractions at two scales automatically and obtains the optimized macrostructures and material microstructures simultaneously. When a larger weight factor for the mean compliance is assigned, the optimization will allocate a smaller volume fraction at the macro level and a larger volume fraction at the micro level. Under extreme cases, the micro unit cell is full of base material and the optimized macrostructure is the same with that from the sole macrostructural topology optimization for minimizing mean compliance. Acknowledgment The authors wish to acknowledge the ﬁnancial support from the Australian Research Council (FT130101094) and Scientiﬁc Research Foundation of Fujian University of Technology (GY-Z14006) for carrying out this work. References [1] McDowell DL, Olson GB. Concurrent design of hierarchical materials and structures. Sci Model Simul SMNS 2008;15:207240. [2] Evans AG, Hutchinson JW, Ashby MF. Multifunctionality of cellular metal systems. Prog Mater Sci 1999;43(3):171–221. [3] Zhang ZZ, Zhang YW, Gao H. On optimal hierarchy of loading-bearing biological materials. Proc R Soc B 2011;278:519–25. [4] Bendsøe MP, Kikuchi N. Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 1988; 71:197–224. [5] Bendsøe MP. Optimal shape design as a material distribution problem. Struct Optim 1989;1:193–202. [6] Bendsøe MP, Sigmund O. Topology optimization: theory, methods and applications. Berlin: Springer-Verlag; 2003. [7] Xie YM, Steven GP. A simple evolutionary procedure for structural optimization. Comput Struct 1993;49:885–96. [8] Huang X, Xie YM. Convergent and mesh-independent solutions for the bidirectional evolutionary structural optimization method. Finite Elem Anal Des 2007;43(14):1039–49. [9] Wang MY, Wang X, Guo D. A level set method for structural topology optimization. Comput Methods Appl Mech Eng 2003;192:227–46.

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