A direct approach to controlling the topology in structural optimization

Computers and Structures xxx (xxxx) xxx

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Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

A direct approach to controlling the topology in structural optimization Zi-Long Zhao a, Shiwei Zhou a, Kun Cai a, Yi Min Xie a,b,⇑ a b

Centre for Innovative Structures and Materials, School of Engineering, RMIT University, Melbourne 3001, Australia XIE Archi-Structure Design, Shanghai 200092, China

a r t i c l e

i n f o

Article history: Received 25 December 2018 Accepted 24 October 2019 Available online xxxx Keywords: Structural optimization Structural complexity control Interior holes Diverse and competitive solutions

a b s t r a c t Structural shape and topology optimization has undergone tremendous developments in recent years due to its important applications in many fields. However, effectively controlling the structural complexity of the optimization result remains a challenging issue. The structural complexity is usually characterized by the distribution and geometries of interior holes. In this work, a new approach is developed based on the graph theory and the set theory to control the number and size of interior holes of the optimized structures. The minimum distance between the edges of any two neighboring holes can also be constrained. The structural performance and the effect of the structural complexity control are well balanced by using this approach. We use three typical numerical examples to verify the effectiveness of the developed approach. The optimized structures with and without constraints on the structural complexity are quantitatively compared and analyzed. The present methodology not only enables the designer to have a direct control over the topology of the optimized structures, but also provides diverse and competitive solutions. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Structural shape and topology optimization techniques are widely used as powerful tools to maximize the performance of a structure by optimizing its material layout. Several optimization techniques have been developed in the past three decades, including the solid isotropic material with penalization (SIMP) [1–3], the level set [4–6], and the bi-directional evolutionary structural optimization (BESO) [7–11]. These techniques have made significant contributions to the areas of, e.g., advanced structures and materials, mechanical and civil engineering, architecture industry, and aerospace and automotive industry [12–16]. The techniques have also provided an excellent platform for the biomechanical morphogenesis of living systems with hierarchical structures [17] and the design of micro and nano systems such as photonic crystals, waveguides, resonators, filters, and plasmonics [18–20]. Although a number of remarkable achievements have been made in the field of structural topology optimization, technologically challenging and practically important issues still exist. One of them is the structural complexity control (SCC) in topology optimization. Due to their free forms, the optimized structures emerging from the optimization techniques may be difficult to fabricate [21]. In

⇑ Corresponding author at: Centre for Innovative Structures and Materials, School of Engineering, RMIT University, Melbourne 3001, Australia. E-mail address: [email protected] (Y. M. Xie).

recent years, much effort has been directly towards improving the manufacturability and mechanical robustness of the results of structural optimization. A series of manufacturing constraints have been formulated and integrated into existing optimization techniques. For instance, the filtering techniques are used to process the elemental sensitivities with the aim of avoiding patterns of checkerboards and diagonal chains in the optimized topologies [22–24]. Several approaches have been developed based on, e.g., the Heaviside projection [25,26] and the concept of structural skeleton [27–29], to control the maximum and minimum length scales of structural components. By selecting ground structures with different complexities, the constraint on the structural complexity is imposed in the optimal design of truss systems [30]. By iteratively adding or removing bars and nodes, the cost for constructing the fully connected ground structure can be reduced [31]. By using the regularized step function to estimate the structural complexity, the constraint on the maximum allowed number of truss members is imposed [32]. However, most of existing studies about SCC focus on discrete systems [33,34], where the presence or absence of structural members is determined in a fixed ground structure. The structural complexity of a continuum system is usually characterized by the distribution and geometries of its interior holes. Four examples are used to illustrate the structures with different complexities (Fig. 1). A small part is removed from an edge of the solid cube in Fig. 1a. Such a structure without any enclosed

https://doi.org/10.1016/j.compstruc.2019.106141 0045-7949/Ó 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. Structures with different numbers of interior holes, where the opaque (left), semi-transparent (middle), and half (right) versions of each structure are used for visualization. There are no interior holes in (a) and (b), one hole in (c), and two holes in (d).

cavities is simply connected. In Fig. 1b, the hollow cylinder makes the cube non-simply connected although there is no cavity enclosed in the structure. The cubes in Fig. 1c and d are both multiply-connected, and they have one and two enclosed cavities, respectively. On the one hand, the enclosed cavities should be sometimes avoided as they may result in difficulties in fabrication. For example, the powder or support structures need to be removed once the products have been built by additive manufacturing technologies such as the selective laser sintering [35], fused deposition modeling [36], and 3D printing. On the other hand, generating enclosed cavities in an optimized structure is sometimes necessary and of practical importance. For instance, the numbers of the rooms, windows, and/or holes, need to be prescribed in the optimal design of buildings, bridges, and their components for both functional and aesthetic consideration. The volume and shape of the enclosed cavity need to be constrained in the design of a container.

A certain number of holes with specific sizes and shapes are required in the optimal design of systems with embedding components for further assembly of functional equipment [37,38]. Several approaches were recently developed to realize the SCC in the structural topology optimization of a continuum system. By introducing a modification of the evolutionary structural optimization technique, an approach to controlling the number of enclosed cavities was proposed [39]. This approach has the additional benefit of eliminating the formation of checkerboard patterns, but it can only realize the SCC in an indirect manner. By introducing a new design variable field, a penalty method was developed for retaining a fixed-area interior hole in a compliance minimization problem [37]. The method allows the hole to be flexibly reshaped and repositioned. An identification method for enclosed voids restriction was established by using an equivalent description of simply-connected constraint [40]. It was assumed

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that the void and solid areas in the structure are filled with virtual materials with different heat conductivities. Existence of enclosed voids is equivalent to that the structure will have a very high temperature when it is heated. By employing the concept of structural skeleton, a SCC approach was proposed under the framework of level-set representation, which is able to control the minimum sizes of the interior holes [41]. Most recently, based on the moving morphable component (MMC) method, an approach was developed to control the structural complexity in the 2-dimensional optimization problems [42]. Currently, there is still a lack of an explicit and efficient approach with simple concept and concise formulae to directly controlling the number and geometries of interior holes in structural optimization. Motivated by the practical importance and potential applications of the SCC, a new approach is here developed based on the fundamental graph theory and the set theory to control the structural complexity in topology optimization. The approach can be easily integrated into the BESO-based computational framework and the algorithm is easy-to-understand. The effectiveness of the developed approach is verified by three typical numerical examples. The new approach enables the designer to explicitly control the number and size of the interior holes of both 2-dimensional and 3-dimensional structures. The structural performance and the effect of the SCC can be well balanced by using this approach. The present methodology can effectively enhance the manufacturability of the optimized designs generated by structural optimization and provide the designer with diverse and competitive solutions [43]. This work holds great promise in advanced manufacturing and computational morphogenesis. The paper is organized as follows. The BESO-based computational framework is first described in Section 2. The new approach to controlling the structural complexity is developed and integrated into the BESO technique in Section 3. Three typical numerical examples of compliance minimization problems are given in Section 4. Advantages and potential applications of the developed approach are discussed in Section 5. Main conclusions from this study are drawn in Section 6.

x

n 1 T 1X U KU ¼ x p u T ki u i ; 2 2 i¼1 i i

F ¼ KU and v f ðxÞ ¼

subject to :

n X i¼1

v i xi =

ð1Þ n X

v i ¼ v^ f ;

ð2Þ

i¼1

where C, x ¼ fxi ji ¼ 1; 2; . . . ; ng, U, K, and F are the structural compliance, design variables, global displacement vector, global stiffness matrix, and global force vector, respectively. ui and ki , obtained from the finite element analysis, are the local displacement vector and stiffness matrix for element i, respectively. p is the penalty exponent [45]. The design variables are updated according to the efficiency of each element, which is assessed through the elemental sensitivities. In order to calculate the elemental sensitivities, the gradient of the compliance C with respect to the design variable xi is derived by using the adjoint method:

@C 1 ¼  pxp1 uTi ki ui : @xi 2 i

ð3Þ

The sensitivity of element i is defined as

ai ¼ 

1 @C : pv i @xi

ð4Þ

In order to achieve a mesh-independent solution, the filtering technique is used to process the raw sensitivity [46–48]: n P

a^ i ¼

wij aj

j¼1 n P

;

ð5Þ

wij

j¼1

  where wij ¼ max r f  dij ; 0 is the weight function, r f the filter radius, and dij the distance between the centers of elements i and j. In order to achieve a convergent solution, the sensitivity of the current iteration k (k P 2) is averaged by that of the previous iteration [49]:

ae i ¼ 2. BESO-based computational framework The SCC approach is here developed under the BESO-based computational framework, which has become a widely adopted design methodology for both academic research and engineering applications due to its efficiency and robustness [44]. In this section, a brief description of the BESO-based computational framework is provided. In the structural optimization, the material is first modeled as a solid assemblage of n elements that are uniformly distributed in the design domain X. During the evolutionary structural optimization, less efficient elements are removed iteratively from the design domain, and the volume fraction v f of material decreases ^ f . By using the BESO technique, the gradually to a target value v removed elements can be re-admitted in later evolutions if their efficiency is sufficiently high. Finite element analysis and design variable update are involved in each iteration. Here we consider the compliance minimization problems as examples to investigate the SCC. The governing equations are derived based on an assumption of static, linear, and elastic behavior. The volume and relative density of element i are denoted as v i and xi (xi ¼ 1 or r), respectively, where 0 < r1. xi ¼ 1 and xi ¼ r indicate the presence (solid) and absence (void) of element i, respectively. In the structural form-finding process, the contribution of the void elements to, e.g., the structural compliance and volume, can be neglected as 0 < r1. The optimization problem is described as

C ðxÞ ¼

min :

 1  ðk Þ a^ i þ a^ ði k1Þ : 2

ð6Þ

The optimization process is convergent and will be terminated if

  X Z  Z  X  C ðktþ1Þ  C ðkZtþ1Þ  6 s C ðktþ1Þ ;  t¼1 t¼1

v ðfkÞ ¼ v^ f and 

ð7Þ

where k is the current iteration number, s an allowable convergent

error, and v f and C ðkÞ the volume fraction and the structural compliance in the k th iteration. The integer number Z is selected to be 5 which implies that the change in the mean compliance over the last 10 iterations is acceptably small. The following parameters are used ðkÞ

in the numerical examples:

r ¼ 1  103 , p ¼ 3, and s ¼ 1  103 .

3. Structural complexity control The BESO-based computational framework is introduced in Section 2. In this section, the SCC approach will be developed and integrated into the computational framework. The graph theory [50– 52], the set theory, and the finite element based structure definition are here combined to investigate the connectivity of a structure. The idea of graph theory based topology optimization is to represent a design as a mathematical graph [53]. A series of graph models have been established for transforming the connectivity properties of finite element meshes into the topological properties of their graphs [54,55]. The graph-based description of topologies

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is suitable for matrix analysis, which has been successfully applied to the computer-aided structural analysis and optimal design in recent years [56]. The set theory is used, in combination with the graph-based description of topologies, to control the complexity of structures. The new approach developed in the present work allows the designer to directly control the structural complexity. By using this approach, the constraints on the number and size of the interior holes can be easily imposed. The minimum distance between the edges of any two neighboring holes can be explicitly controlled. The structural optimization with SCC will start from an initial design with a certain number of interior holes. In each iteration, the boundary elements of the structure is first identified. Only boundary elements will be involved in the design variable update, where the elemental sensitivities in Eq. (6) will be used. In Section 3.1, the method of identifying the boundary elements of a finite element based structure is described. In Section 3.2, the method of how to update the design variables is established for realizing the SCC. 3.1. Identification of the boundary elements At the beginning of each iteration step, the set B of boundary elements needs to be determined. The neighboring set of element i is defined as

  N E i ¼ jjN i \ N j –£ ;

ð8Þ

where N i and N j denote the node sets of element i and element j, respectively. For any element i, it belongs to the set B of boundary elements if

N i \ N @ X –£ or

    max xj – min xj ; j2N ei

j2N ei

ð9Þ

where N @ X denotes the set of the nodes that are located at the edge of the design domain X. A few examples are given in Fig. 2 to illustrate how to identify the boundary elements by using Eq. (9). Elements 1 and 2 belong to B because both solid and void elements are included in the neighboring sets N E 1 and N E 2 . Element 3 belongs to B because two of its nodes are located at the edge of the design domain (i.e., N 3 \ N @ X –£). Both elements 4 and 5 do

not belong to B as the neighboring sets N E 4 and N E 5 contain only void and solid elements, respectively. In Fig. 2, the design domain is discretized by 4-node rectangular elements, and a 3  3 scheme is used to identify the boundary elements. Schemes with different dimensions and shapes may be used if the design domain has a complex mesh. Nevertheless, Eq. (9) is generally valid for identifying the boundary elements, regardless of the complexity of the mesh. 3.2. Update of the design variables The design domain X is divided into several parts, including the structure (solid), the holes enclosed in the structure (void), and the region outside the structure (void). By introducing a new design variable field, these parts are labeled and distinguished, and their boundary elements are categorized into different groups. This method allows us to control the geometric features of the interior holes when we update the relative density of the boundary elements. New labels will be assigned to the elements once their relative density is changed. The l th interior hole of the structure, denoted by Hl (l ¼ 1; 2; . . . ; N), consists of a certain number of connected void elements, where N is the number of the interior holes in the structure. The set of solid elements is defined as H0 , and the assemblage of the void elements outside the structure, designated by HNþ1 , is taken as a generalized hole. The void elements of HNþ1 can be either connected or disconnected. The design domain is the union of the above sets:

X¼

Nþ1 [

Hl :

As shown in Fig. 3a, there are two interior holes H1 and H2 in the design domain, therefore X ¼ H0 [ H1 [ H2 [ H3 . By using Eq. (9), one can readily obtain the set B of boundary elements. The set B is the union of the following sets:

B¼

Nþ1 [

@Hl ;

ð11Þ

l¼1

where @Hl consists of (i) the boundary elements of Hl , (ii) the solid elements i (i 2 H0 ) if N E i \ Hl –£, and (iii) the solid elements which have nodes in N @ X if l ¼ N þ 1. As shown in Fig. 3b, the elements that belong to @H1 , @H2 , and @H3 are highlighted in green, red, and blue, respectively. In order to impose geometric constraints (e.g., size) on the interior holes, the elements that belong to different regions are distinguished by different labels. The label hi of element i is defined as

8 Nþ1 > < 0; if i 2 H and i R S @H 0 l hi ¼ : l¼1 > : l; if l P 1 and i 2 Hl [ @Hl

Fig. 2. Illustration of how to identify the boundary elements in the design domain.

ð10Þ

l¼0

ð12Þ

For the elements of B, their relative densities (design variables) will be updated in the current iteration. First we change all the boundary elements to solid by setting xi ¼ 1 if i 2 B (Fig. 3c), and then remove a certain amount of elements from B in two steps by using the following method. In the first step, the element removal is carried out in the neighborhood @Hl of the interior holes (l ¼ 1; 2; . . . ; N). The constraints on the size of the holes and the minimum distance between the edges of any two neighboring holes will be imposed. It is required that after removal (i) all interior holes should not be smaller than a ^ h , and (ii) the distance between any two holes prescribed volume v (including the generalized hole HNþ1 ) should not be smaller than a minimum allowable value t h . As highlighted in yellow in Fig. 3c, a spherical test region with a radius of t h is introduced for any

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Fig. 3. Illustration of the optimization technique with SCC. (a) Two interior holes are included in the design domain. (b) The boundary elements of different regions are highlighted in different colors. (c) The boundary elements with less structural efficiency are removed while the SCC is achieved. (d) The new design is obtained after the update of the design variables.

element i of @Hl (l ¼ 1; 2; . . . ; N). In both the first and second step, element i will not be removed if the elements within the test region i have different non-zero h values, i.e.,

    max hj – min hj P 1; j2T

j2T

i

ð13Þ

i

where the set T i is defined as



 T i ¼ jjk rj  ri k 6 t h and hj –0 ;

ð14Þ

and ri (rj ) represents the position vector of the centroid of element i (j). Neighboring interior holes might be merged in the degenerated case of t h ¼ 0. In the simulations, we set t h ¼ r f . If the condition Eq. (13) is not satisfied, the element can be removed from @Hl (l ¼ 1; 2; . . . ; N), and the elements with a low sensitivity (structural efficiency) will be first removed. The update of the design variables is performed iteratively in the first step, where the iteration variable l increases from 1 to N. Each iteration will be terminated if

v hl ¼

X

v i ð1  xi Þ P v^ h

ðl ¼ 1; 2; . . . ; NÞ;

ð15Þ

i2Hl [@Hl

where vhl is the volume of the lth interior hole. In the second step, a certain number of solid elements with less structural efficiency will be further removed from B. The volume of the material of B before and after the design variable update are calculated respectively as

v ðBk1Þ ¼

X

v i xði k1Þ

and

v ðBkÞ ¼

i2B ðk1Þ

X

v i xði kÞ ;

ð16Þ

i2B ðkÞ

where xi (xi ) indicate the design variables in the ðk  1Þ th (k th) iteration. The solid elements with the smallest sensitivity will be iteratively removed from B until

v ðBkÞ 6 v ðBk1Þ  v ðfk1Þ Vd

or

 X ðkÞ 1  xi P jBjw;

ð17Þ

i2B

where V is the volume of the design domain, d the prescribed evolutionary rate (typically 1% or 2%), jBj the cardinality of B, and w the maximum required volume fraction (typically between 60% to 90%) of the voids in B. At the end of the k th iteration, the design variable update is completed and a new structure is obtained (Fig. 3d). According to the label hi and updated design variable xi , the sets Hl (l ¼ 1; 2; . . . ; N þ 1) are redefined as

H0 ¼ fijxi ¼ 1g; Hl ¼ fijxi ¼ 0 and hi ¼ lg;

ð18Þ ðl ¼ 1; 2; . . . ; N þ 1Þ;

ð19Þ

which will be used in the next iteration. The above approach enables the designer to directly control the number and size of the interior holes in the structural optimization. The optimization algorithm is developed on the basis of the BESO technique, implemented by using Matlab and Python, and

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linked to Abaqus. The basic arithmetic functions of the Python language are not optimized and may therefore lead to poor performance when a large amount of computation (e.g., determining the weight functions wij in Eq. (5)) is involved [49]. Once the finite element model is created by Abaqus, wij is calculated by using the Matlab code, in which the matrix mathematics are expressed directly instead of through generalized programming constructs. The weight functions are independent of the elemental sensitivity and only need to be calculated once. We divide the design domain into multiple subdomains such that wij in different subdomains can be calculated separately and in parallel. The finite element analysis, filtering of sensitivity, and update of design variables are performed by using the Python script, which is developed based on the Abaqus environment. The convergence and meshindependency of the BESO-based computational framework has been analyzed and verified [9]. 4. Numerical examples In this section, three typical optimization problems are used to verify the effectiveness of the developed approach for SCC. The optimized designs of the structures with and without SCC are quantitatively compared and analyzed. The influence of the number, size, and initial distribution of the interior holes on the optimized design is revealed. For illustration purpose, the material properties, external forces, and geometric parameters are chosen as dimensionless. All materials are assumed to be homogenous, isotropic, and linearly elastic, where the Young’s modulus and Poisson’s ratio are set as E ¼ 1 and m ¼ 0:3, respectively. The Python code we develop here for the SCC is generally valid for 3dimensional topology optimization. 3-dimensional models are established in all numerical examples, while the design domains in the first two examples have only one layer of brick elements. Eight-node hexahedral elements are used for finite element discretization.

shown in Fig. 4. The concentrated force that acts on the free end of the beam is set as F ¼ 1. The design domain is discretized by a 240  180  1 finite element mesh. The filter radius rf ¼ 12 and ^ f ¼ 30%. the target volume fraction v The structural optimization without SCC is first investigated (Fig. 5). Fig. 5a shows the optimized design of the cantilever beam, where there are three interior holes naturally generated in the structure. The optimized beam structure is gradually tapered from the clamped end to the free end. The external force is transferred in the reversed direction through the V-shaped frame, which is much thicker than the inner X-shaped member. The usage of large amount of material for the primary load bearing component (i.e., the V-shaped frame) ensures the high stiffness of the structure. The evolutionary histories of the volume fraction v f and structural 

compliance C are plotted in Fig. 5b, where the compliance is normalized by its initial value. When the volume fraction decreases from 100% to 30%, the structural compliance increases to 2:45 times of its initial value. The final structure converges to a stable topology after about 80 iterations. The structural optimization with SCC is further investigated (Figs. 6–8). The initial design (left), optimized design (middle), 

and evolutionary histories (right) of v f and C of the cantilever beam are plotted in each line of Fig. 6, where there are one, two, three, and four interior holes in the cases of 6a–6c, 6d–6f, 6g–6i, 6j–6l, respectively. In the structural optimization, each hole is required to be not smaller than 2% of the volume V of the design domain. The optimized designs with (Fig. 6b, e, h, and k) and without SCC (Fig. 5a) are distinctly different. In the optimized structures with SCC, the V-shaped frame also serves as an important load-bearing component. The thin members inside the V-shaped frame contribute to stiffening the beam by creating more load paths. Fig. 6c shows that at the beginning of the structural optimiza

4.1. Cantilever beam

tion, the compliance C decreases with decreasing the volume fraction v f . This is because the structural performance is not maximized in the prescribed initial design, which includes a large interior hole. When the distribution and geometries of the hole

In this example, the classical short beam problem is considered. The loading and boundary conditions of the cantilever beam are

evolve to be reasonable, the compliance C will increase with the further reduction of the material. By reducing the initial size of





the interior holes (Fig. 6d, g, and j), the compliance C increases with decreasing the volume fraction v f from the beginning (Fig. 6f, i, and l). In comparison to the globally optimized structure in Fig. 5a, the 

Fig. 4. Loading and boundary conditions of a cantilever beam.

increments of C due to SCC are only 4%  6% for the designs in Fig. 6b, e, h, and k. The requirements that there should exist a certain number of holes in the structure and each of them must not be smaller than V=50 only result in a slight loss of structural performance. It is worth noting that for a symmetric optimization problem the optimal topology is in general unique and symmetric (e.g., Fig. 5a). However, there may exist multiple non-symmetric optimal solutions in some cases. The symmetry and non-uniqueness in topology optimization have been systematically studied [57,58]. Here unsymmetrical solutions are generated in symmetric problems (Fig. 6). This is because some geometric constraints are imposed on the interior holes during the form-finding process. For example, a symmetric design with two identical holes H1 and H2 is achieved and needs to be further optimized. In this design, solid elements i and j with the same sensitivity are located symmetrically on the boundaries of H1 and H2 , respectively. Consider that element i is removed first, and the distance between the edges of H1 and H2 will be smaller than the allowable value th if element j is also removed. To constrain the minimum distance between the edges of any two neighboring holes, element j must

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Fig. 5. (a) Optimized design of the cantilever beam without SCC. (b) Evolution histories of the volume fraction

vf



and structural compliance C .



Fig. 6. Influence of the number of the holes on the optimized design of the cantilever beam. The initial design, optimized design, and evolution histories of v f and C of the beam are plotted from left to right, where there are one, two, three, and four holes in the cases of (a)–(c), (d)–(f), (g)–(i), (j)–(l), respectively. Each interior hole is required to be not smaller than V=50 in the optimization.

be retained and the symmetry of the design will be broken. It should be mentioned that the mirror of an unsymmetrical optimal structure is also the optimum. The influence of the hole size on the structural optimization is further investigated (Fig. 7). Fig. 7a shows that two holes are included in the initial design. Before the volume fraction of material reaches the target value, each hole is required to be not smaller than 0 for Fig. 7b and V=50 for Fig. 7c and d. At the end of the formfinding process, each hole is required to be not smaller than 0 for Fig. 7b, V=20 for Fig. 7c, and V=10 for Fig. 7d. Since there is no constraint on the minimum size of the holes, one hole is degenerated in the optimized design (Fig. 7b) such that a higher structural effi-

ciency can be achieved. The structural stiffness is reduced when the interior holes are required to have a larger size. The compliance of the optimized structure, in comparison to Fig. 5a, increases 3% for Fig. 7b (v h =V P 0), 6% for Fig. 7c (v h =V P 1=20), and 8% for Fig. 7d (v h =V P 1=10). Although the optimized design is substantially changed by imposing different constraints, the structural efficiency is guaranteed. It suggests that the proposed approach is able to provide diverse solutions of high structural performance. The influence of the initial design on the optimized design of the cantilever beam is shown in Fig. 8. The initial design in Fig. 8a evolves into the optimized structure in Fig. 8b, while the design in Fig. 8c evolves into the structure in Fig. 8d. In both exam-

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ples, there exit three interior holes in the beam and each of the holes are required to be not smaller than V=50. In the first example, the optimized structure (Fig. 8b) has the similar material layout with that in Fig. 5a, where the interior holes of the former are larger than that of the latter. In comparison to Fig. 5a, the increments of structural compliance are only 1% for Fig. 8b and 5% for Fig. 8d, suggesting that both the designs with SCC are rather competitive. The evenly distributed member size can effectively enhance the manufacturability and mechanical robustness of the products designed by structural optimization techniques. 4.2. MBB example

Fig. 7. Influence of the hole size on the optimized design of the cantilever beam. (a) Two interior holes are included in the initial design. In the optimization, each hole is required to be not smaller than 0 for (b), V=20 for (c), and V=10 for (d).

Fig. 8. Influence of the initial design on the optimized design of the cantilever beam with SCC. There are three holes in both the ((a) and (b)) first and ((c) and (d)) second case, where the optimized designs (b) and (d) are evolved from the initial designs (a) and (c), respectively. Each of the interior holes is required to be not smaller than V=50 in the optimization.

In the second example, we consider an MBB problem, where the loading and boundary conditions are shown in Fig. 9a. Due to the symmetry property of the problem, only half of the design domain is considered (Fig. 9b). In the following, the interior holes indicate the ones which are enclosed in the reduced design domain (Fig. 9b). The concentrated force that acts on the beam is set as F ¼ 1. The design domain in Fig. 9b is discretized by a 360  120  1 finite element mesh. The filter radius r f ¼ 6 and ^ f ¼ 35%. the target volume fraction v The structural optimization of the simply supported beam without SCC is first investigated. Fig. 9c shows the optimized design of the MBB beam, where there are five interior holes naturally generated in the structure. The beam evolves into a frame structure which is comprised of a few straight members. The truss-like structures, featured by high stability and stiffness, are widely used by engineers and architects in the lightweight design of bridges and towers. The structural optimization of the MBB beam with SCC is then investigated (Fig. 10). Three and five interior holes are required in the first (Fig. 10a and b) and second example (Fig. 10c and d), respectively. Each of the holes is required to be not smaller than 1=200 of the reduced design domain. The optimized designs in Fig. 10b and d retain the basic feature (i.e., the straight members) of the structure in Fig. 9c. In the MBB beam examples, the SCC is achieved at the cost of slightly compromised structural performance. In comparison to the optimal design without SCC, the increment in structural compliance is only 1% for both Fig. 10b and d. Further, the developed approach is used to optimize the MBB beam, where the merging of interior holes is allowed. Fig. 11 shows

Fig. 9. (a) Loading and boundary conditions of the MBB example. (b) Only half of the design domain is considered due to the symmetry property of the problem. (c) Optimized design of the MBB beam without SCC.

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of the upper right hole vanishes (highlighted in red), seven interior holes are retained in the design domain (Fig. 11b). Next, the two holes highlighted in green and the two highlighted in blue merge successively, and only five interior holes are retained (Fig. 11c and d). The structural complexity of the beam does not change in the following form-finding process (Fig. 11e and f). The difference in structural compliance between Figs. 9c and 11f is less than 0:2%. The developed approach is able to generate highly competitive design if no geometric constraints are imposed on the interior holes. 4.3. 3-dimensional cantilever beam In the third example, we consider a 3-dimensional cantilever beam. The loading and boundary conditions are shown in Fig. 12. Three orthonormal unit basis vectors fi; j; kg are referred. The concentrated force F ¼ 1 that acts at the centroid of the free end is along the opposite direction of i. The design domain is discretized using 1 million elements. The filter radius r f ¼ 5 and the ^ f ¼ 30%. The initial design of the structural target volume fraction v optimization with SCC is shown in Fig. 13, where there is a block

Fig. 10. Influence of the number of the interior holes on the optimized design of the MBB beam. There are three and five holes in the ((a) and (b)) first and ((c) and (d)) second case, respectively. The optimized designs (b) and (d) are evolved from the initial designs (a) and (c), respectively. Each interior hole is required to be not smaller than V=200 in the optimization.

representative topologies of the beam during the structural formfinding process. In this example, the minimum allowable distance t h between the edges of any two interior holes is set as 0. Eight holes are included in the initial design (Fig. 11a). After the wall

Fig. 12. Loading and boundary conditions of a 3-dimensional cantilever beam.

Fig. 11. Representative topologies of the MBB beam during the form-finding process, where the merging of interior holes is allowed. There are eight holes in (a) (the initial design), seven holes in (b) and (c), and five holes in (d)–(f).

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highly efficient for compliance minimization. Similar bulblike structures widely exist in natural biological systems, e.g., the venom container of honey bees and scorpions, where the enclosed cavity is used for storing the venom [59,60].

Fig. 13. Initial design of the 3-dimensional cantilever beam for the SCC, where a block cavity with dimensionless sizes 60  60  120 is located at the center of the design domain.

cavity with dimensions 60  60  120 located at the center of the design domain. It is required that the cavity not be smaller than V=10 of the design domain. The optimized designs without (Fig. 14a and b and Supplementary Video S1) and with (Fig. 14c and d and Supplementary Video S2) SCC are compared, where Fig. 14b and d are the rear half of Fig. 14a and c, respectively. The former structure has an open bottom and an X-shaped web near the clamped end. Both its upper and lower flanges shrink gradually from the clamped end to the free end. The X-shaped web is redistributed when the structural complexity constraint is imposed. The structure in Fig. 14c is a bulblike container, in which a large and enclosed cavity is constructed. In spite of the distinct difference in topology, the difference of structural compliance between Fig. 14a and b is less than 0:1%. It suggests that the material distribution of the bulblike structure is

Supplementary Video S1. Optimized design of the 3-dimensional cantilever beam without structural complexity control.

Supplementary Video S2. Optimized design of the 3-dimensional cantilever beam with structural complexity control.

Fig. 14. Optimized designs of the 3-dimensional cantilever beam ((a) and (b)) without and ((c) and (d)) with SCC. The rear halves of the structures in (a) and (c) are shown in (b) and (d), respectively. There is no cavity in the first case, while one cavity in the second case.

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5. Discussion In this section, the advantages and potential applications of the developed SCC approach are discusses. The new approach has the following advantages. It is developed based on the fundamental graph theory and the set theory, in which complex theoretical frameworks or concepts are not involved. The present optimization algorithm is easy-tounderstand. It has been successfully integrated into the BESObased computational framework. Owing to its simple concept, the algorithm can be easily integrated into other computational frameworks, e.g., the SIMP method. The developed approach enables the designer to directly control the structural complexity in topology optimization, which not only guarantees the structural efficiency, but also improves the manufacturability and reduces the manufacturing cost of the products. By changing the initial layout of material or the constraints on the number and size of the interior holes, the developed approach is able to provide a series of solutions of high structural efficiency. The influence of the initial layout on the subsequent form-finding process is induced by the averaging operation of the elemental sensitivities (Eq. (6)). The alternative solutions generated by the SCC approach allow the designer to account for other factors based on the structural optimization. Therefore, the methodology presented here holds great promise in the emerging field of human–computer interactive design. The SCC approach is directly applicable in the optimal design of buildings and bridges, where the number and size of the rooms and/or holes can be controlled in an explicit manner. The new approach will be a powerful mathematic tool to design containers and systems with morphable components, where interior holes with specific sizes and shapes are required in order to meet functional demands. Potential application of the approach might also include the computational morphogenesis of living systems with enclosed cavities for bio-inspired designs. 6. Conclusions In this paper, a new approach is developed to directly control the topology in evolutionary structural optimization. By using this approach, the number and size of the interior holes can be explicitly controlled, and the constraints on the minimum distance between the edges of any two neighboring holes can be easily imposed. The approach has been successfully integrated into the BESO-based computational framework. The effectiveness of the proposed methodology is verified by typical numerical examples, including the compliance minimization of a clamped-free beam, a MMB beam, and a 3-dimensional cantilever beam. The optimized designs with different constraints on the structural complexity are quantitatively compared and analyzed. The results demonstrate that the effect of the structural complexity control and the structural performance can be well balanced. The developed approach can not only improve the manufacturability of the products emerging from the structural optimization, but also provide the designer with diverse and competitive solutions. Acknowledgments The work was supported by the Australian Research Council (DP160101400), the National Natural Science Foundation of China (51778283), and the Fundamental Research Funds for the Central Universities (2014ZD16). References [1] Bendsøe MP. Optimal shape design as a material distribution problem. Struct Optim 1989;1:193–202.

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