Robust topology optimization for continuum structures ...

Engineering Computations Robust topology optimization for continuum structures with random loads Jie Liu, Guilin Wen, Qixiang Qing, Fangyi Li, Yi Min Xie,

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Article information: To cite this document: Jie Liu, Guilin Wen, Qixiang Qing, Fangyi Li, Yi Min Xie, (2018) "Robust topology optimization for continuum structures with random loads", Engineering Computations, Vol. 35 Issue: 2, pp.710-732, https://doi.org/10.1108/EC-10-2016-0369 Permanent link to this document: https://doi.org/10.1108/EC-10-2016-0369 Downloaded on: 02 May 2018, At: 17:52 (PT) References: this document contains references to 57 other documents. To copy this document: [email protected] The fulltext of this document has been downloaded 30 times since 2018* Access to this document was granted through an Emerald subscription provided by Token:Eprints:ZRDXZK4NGPQ3CIR9FN6M:

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Robust topology optimization for continuum structures with random loads Jie Liu

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Received 28 October 2016 Revised 11 May 2017 7 August 2017 9 August 2017 Accepted 9 August 2017

School of Mechanical and Electric Engineering, Guangzhou University, Guangzhou, China, State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, China and Centre for Innovative Structures and Materials, RMIT University, Melbourne, Australia

Guilin Wen School of Mechanical and Electric Engineering, Guangzhou University, Guangzhou, China, State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, China and Jupiter Original Ecology Science and Technology (Hunan) Co., Ltd., Changsha, China

Qixiang Qing State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, China

Fangyi Li School of Automotive and Mechanical Engineering, Changsha University of Science and Technology, Changsha, China, and

Yi Min Xie Centre for Innovative Structures and Materials, School of Engineering, RMIT University, Melbourne, Australia and XIE Archi-Structure Design (Shanghai) Co., Ltd., Shanghai, China

Abstract Purpose – This paper aims to tackle the challenge topic of continuum structural layout in the presence of random loads and to develop an efficient robust method. Design/methodology/approach – An innovative robust topology optimization approach for continuum structures with random applied loads is reported. Simultaneous minimization of the expectation and the variance of the structural compliance is performed. Uncertain load vectors are dealt with by using additional uncertain pseudo random load vectors. The sensitivity information of the robust objective function is

Engineering Computations Vol. 35 No. 2, 2018 pp. 710-732 © Emerald Publishing Limited 0264-4401 DOI 10.1108/EC-10-2016-0369

The first author is partially supported by scholarship No. 201506130053 of China Scholarship Council. This work was supported jointly by the National Science Fund for Distinguished Young Scholars in China (No. 11225212) and the National Natural Science Foundation of China (No. 11072074, 11302033). The authors also would like to thank the Collaborative Innovation Center of Intelligent New Energy Vehicle, the Hunan Collaborative Innovation Center for Green Car and the “Chair Professor of Lotus Scholars Program” in Hunan province. The authors are grateful to the reviewers for their comments on the paper.

obtained approximately by using the Taylor expansion technique. The design problem is solved using bidirectional evolutionary structural optimization method with the derived sensitivity numbers. Findings – The numerical examples show the significant topological changes of the robust solutions compared with the equivalent deterministic solutions. Originality/value – A simple yet efficient robust topology optimization approach for continuum structures with random applied loads is developed. The computational time scales linearly with the number of applied loads with uncertainty, which is very efficient when compared with Monte Carlo-based optimization method.

Keywords Topology optimization, Robust design, BESO method, Uncertain applied load

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Paper type Technical paper

1. Introduction Topology optimization of structures as a rapidly growing field of particular interest to the automotive and aerospace industries has been extensively investigated in past three decades. Generally, the type of topology optimization may be classified into two categories, i.e. topology optimization of discrete structures (Hajela and Lee, 1995; Rozvany, 1996) and topology optimization of continuum structures (Bendsøe and Kikuchi, 1988; Suzuki and Kikuchi, 1991; Xie and Steven, 1993, 1994, 1997; Andreassen et al., 2011; Huang and Xie, 2010; Zhu et al., 2015; Wang et al., 2003; Guest, 2015; Özkal and Uysal, 2016; Maleki and Shariat, 2015; Cai et al., 2014; Bochenek and Tajs-Zielinska, 2013). The former is to search for the optimal spatial order and connectivity of the bars. The latter aims at finding the optimal candidates by determining the best locations and geometries of cavities in the design domain, which usually leads to great a reduction in weight or improvement of structural behavior such as stiffness, strength or dynamic response. For a more extensive overview of the field, the reader is referred to the review articles (Deaton and Grandhi, 2014; Rozvany, 2009) and the references therein. Note that topology optimization of continuum structures is considered in this work. Although topology optimization methods can offer efficient means to provide effective candidates, uncertainties are usually ignored in the engineering design process. Uncertainty is the inherent characteristic of engineering structure; in other words, the uncertain disperse of structural parameters about their nominal values is ineluctable in a practical design problem because of incomplete information, manufacturing imperfections, observation errors, etc. (Guo et al., 2013). Some previous studies have shown that uncertain variations of applied loads, material properties and geometry dimensions may have considerable effects on the optimal designs. Therefore, it is significantly important to consider uncertainties in structural optimization design. Normally, there are two ways to include uncertainties in the structural optimization process. One way named reliability-based design optimization (RBDO) (Du et al., 2015; Youn and Choi, 2004; Tu et al., 1999) incorporates uncertainties as a quantified probability of failure and optimizes structural performance metrics, maintaining design constraint satisfaction at an expected reliability level. Robust design optimization (RDO) (Beyer and Sendhoff, 2007; Jung and Lee, 2002; Du and Chen, 2002; Huang and Du, 2007; Li et al., 2013, 2015) as the other way aims to minimize the influence of variations and uncertainties on the objective. The main difference between RDO and RBDO, as noted by Asadpoure et al. (2011), is that RDO largely cares about lower-order statistical properties of the uncertainties being in presence (i.e. mean and variance), while RBDO is more concerned with the higher-order statistical properties such as the tails of the probability density function. RBDO has been widely studied and applied to solving various practical and multidisciplinary problems. However, it was not until the mid-2000s that these statistical

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and probabilistic design methods were introduced in topology optimization, known as reliability-based topology optimization (RBTO). Topology optimization, even in a deterministic situation, consists of a large number of design variables, so including uncertainties results in significant challenges to topology optimization. The first study in RBTO problem was published by Kharmanda and Olhoff (2001). They tackled structural compliance minimization problems with a probability of failure as a constraint and regarded the geometry and the applied loads as random parameters. Later on, more and more researchers have explored this field, and a mass of success works have been accomplished (Liu et al., 2016a; Patel and Choi, 2012; Jung and Cho, 2004; Kharmanda et al., 2004; Yoo et al., 2011; Chen et al., 2010; Maute and Frangopol, 2003; Kang and Luo, 2009; Xu et al., 2016). However, in contrast to RBTO, the application of RDO techniques combined with topology optimization, known as RTO, has been relatively less studied at the early stage. Ben-Tal and Nemirovski (1997) seemed to be the first to study RTO problems. They proposed a robust topology optimization method to design truss structures where the loading uncertainties were considered. Kanno and Guo (2010) reported a mixed integer programing formulation for robust topology optimization of trusses subjected to the stress constraints in the presence of uncertain loads. Asadpoure et al. (2011) put forward an efficient strategy for robust structural topology by including uncertainties in material properties and structural geometry. A robust topology optimization method for the design of trusses with random geometric imperfections due to fabrication errors was developed by Jalalpour et al. (2011). Typical works of RTO in the preceding discussion in the previous paragraph are related to truss design. RTO for designing continuum structures is, to some extent, more valuable and challenging. This area is classified into two categories in this work by the authors for convenience, namely, RTO including loading uncertainties and RTO including other kinds of uncertainties, such as material properties or/and geometry dimensions. For the latter kind, the readers are referred to previous studies by Tootkaboni et al. (2012), Jansen et al. (2013), Amir et al. (2012), Chen et al. (2010) and Wang et al. (2011). Formal incorporation of loading uncertainty into the design optimization framework has recently become a strong focus of the topology optimization community (Liu et al., 2016b). Several researches have proposed RTO algorithms for continuum structures in the presence of loading uncertainties. Kogiso et al. (2008) proposed a new robust topology optimization method for compliant mechanisms. They took the variation of the input load direction into account. However, the algorithm in their study seems can only solve the design problems having only one uncertain applied load. To solve the computationally intractable characteristic of the RTO problem, Dunning et al. (2011), Dunning and Kim (2013) derived analytical formulations for RTO problems in the presence of uncertainties in applied loads. However, the aforementioned studies have two main disadvantages, namely, computational complexity and low computational efficiency. Thus, this study aims to propose a simple yet highly efficient robust algorithm. For example, Dunning et al. (2011) proposed an algorithm that requires (3l þ 1) load cases to accurately solve the robust design problem, while we only need (l þ 1) load cases, demonstrating the high efficiency of our proposed algorithm. Note that l is the number of the uncertain loads. In the present work, the uncertain applied load is modeled by using additional uncertain pseudo random load vectors, which can avoid the complex mathematical computation. The use of truncating the Taylor series of the robust compliance and the technique of modeling the uncertain load approximately transform the RTO problem to deterministic topology optimization problem with a few additional load cases. The deterministic design problem is solved using the bi-directional evolutionary structural optimization (BESO) method with the derived

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sensitivity information, and filtering schemes are implemented to overcome the checkerboard patterns and mesh-dependency problems. The layout of the paper is as follows. After the introduction, the deterministic topology optimization for continuum structures is briefly discussed in Section 2. The robust topology optimization formulation and the sensitivity analysis are presented in Section 3. The BESO procedure and filtering schemes are introduced in Section 4. A series of representative examples is presented in Section 5 to demonstrate the effectiveness of the proposed approach. Finally, Section 6 presents concluding remarks. 2. Topology optimization of continuum structures In this section, the deterministic topology optimization problem is first discussed without including uncertainties as a prerequisite. In a topology optimization problem, the goal is to obtain a structure with the best performance by determining the material distribution, which normally minimizes compliance for a structure with given loads and supports, subject to a prescribed volume. The compliance or the strain energy minimization problem can be formulated as: Find xj ; j ¼ 1; . . . ; n

(1a)

to minimize C ¼ f T u

(1b)

subject to

n X

Vj xj # vol

(1c)

j¼1

K ðx Þu ¼ f

(1d)

xj ¼ xmin > 0 or 1

(1e)

where xj is the density variable, xmin is a small positive value (e.g. 0.001) to prevent the stiffness matrix from singularity, u and f are the global displacement vector and global force vector, respectively, K is the global stiffness matrix, Vj is the material volume and vol is the allowable volume fraction. Using the solid isotropic material with penalization approach (Bendsøe and Sigmund, 1999), the sensitivity of the objective function and the constraint function with respect to the design variable can be obtained, respectively, as: @kj @C ¼ uTj uj ¼ pxp1 uTj k0j uj j @xj @xj

(2)

@g @V ¼ ¼1 @xj @xj

(3)

Now, the optimization problem can be solved using the BESO algorithm with the derived sensitivity information in equations (2) and (3). For more details about the BESO algorithm, the readers are referred to papers by Huang and Xie (2009, 2011).

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3. Robust topology optimization considering uncertain applied loads Uncertainty in applied loads is an important factor that often needs to be considered in the design of engineering structures. Thus, the variation of the applied loads is taken into account in the robust topology optimization procedure in this paper. The main idea and procedures of an efficient robust topology optimization approach for continuum structures with uncertain applied loads are presented in this section.

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3.1 Description of the uncertain applied loads To describe the variation of the applied load, an uncertain pseudo load vector s whose direction is perpendicular to the applied loads is introduced, as depicted in Figure 1. From Figure 1, the following two equations can be obtained: f ¼ fc þ s

(4)

fc  s ¼ 0

(5)

where f c is the deterministic load vector whose mean and standard deviation are lf c ¼ f c and s f c ¼ 0, respectively, and f is the uncertain load vector. Without loss of generality, the observations of the uncertain load vector s are assumed such that the load can be modeled by a Gaussian distribution with zero mean, i.e. m s = 0 and a standard deviation of s . 3.2 Construction of robust optimization objectives Because the unit of the variance of the function is the square of that of the expectation in the general robust objective, the non-dimensional factors are introduced to the objectives (Dunning and Kim, 2013). Then, the following robust objective is constructed as:     l 1l H ðC Þ ¼ VarðC Þ (6) EðC Þ þ x x2 where l is a weighting factor for the two parts of the objective whose value is between 0 and 1, E(·) and Var(·) denote the expectation and the variance, respectively, and x is the nondimensional factor. x can be defined utilizing Young’s modulus Em and the deterministic displacement vector uc (Dunning and Kim, 2013), as: fc s

Figure 1. Depiction of the uncertain applied load

Ω

f

T

x ¼ ðuc Þ uc =Em

(7)

Now, the robust optimization objectives are to be calculated. As discussed above, l is deterministic, E(C) and Var(C) should be computed if the robust objective is expected to be explicitly expressed. The computation procedure is detailed below. The structural compliance C is divided into two parts, namely, CD and CU. CD is caused by the deterministic loading f c and CU is caused by the random loading s. Then, the following equation holds:

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C ¼ CD þ CU

(8)

It should be noted that the robust objective is H(C). Here, we use CU and CD to easily obtain the sensitivity information of the robust objective, as discussed below. As the random loading has a zero mean, the expectation of the structural compliance caused by the random loading is zero, namely, E(CU) = 0. Thus: EðC Þ ¼ EðC D þ C U Þ ¼ EðC D Þ þ EðC U Þ ¼ EðC D Þ ¼ C D ¼ f c uc

(9)

At the same time, the variance of the structural compliance caused by the deterministic loading is zero as well. So, the variance of the structural compliance can be written as: VarðC Þ ¼ VarðC D þ C U Þ ¼ VarðC U Þ

(10)

To compute the variance of the structural compliance, Var(CU) should be known. The structural compliance CU caused by the random loading can be expressed by a relation f of a set of random variables: C U ¼ f ðs1 ; s2 ; . . . ; sl Þ

(11)

where l is the number of the uncertain applied loads and (s1,s2,. . .,sl) is the realization of the uncertain random vector s. Using a Taylor series expansion at the mean value of random variables, the structural compliance CU can be formulated as: C U ¼ f ð m s1 ; m s2 ; . . . ; m sl Þ þ 1 þ 2

i

i¼1

l X l X i¼1

l X @f ðsi  m si Þ @s

@ f ðsi  m si Þðsj  m sj Þ @s @s þ . . . i j j¼1 2

(12)

Because the use of the first-order variance is considered adequate for most actual engineering applications, the following approximated variance of the structural compliance is used in the robust topology optimization. The first-order variance of CU, denoted as Var(CU), can be shown as: VarðC

UÞ



 l  X @f i¼1

@si

2

Varðsi Þ

(13)

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It should be noted that equation (13) is obtained given that si and sj in equation (12) are uncorrelated. To acquire the second-order mean and the first-order variance of the structural compliance, the first-order partial derivatives of f with respect to si should be computed:     T @ ðs i ÞT K 1 ðs i Þ @ f i @ s i ui ¼ ¼ ¼ 2usi (14) @si @si @si

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where usi refers to the displacement vectors caused by the ith random load vector. Substituting equation (14) into equation (13), the first-order variance of CU can be expressed as: VarðC U Þ 

 l  X @f i¼1

@si

2

Varðsi Þ ¼ 4

2 2 l  l  X X  s T rTi usi ¼4 ui K usi i¼1

(15)

i¼1

where usi denotes the displacement vector caused by the dummy force vector s i. It should be noted that we use s i as a dummy force vector to solve the above equation. Thus, the variance of the structural compliance now is obtained as: VarðC Þ ¼ VarðC U Þ ¼ 4

l  X 

usi

T

K usi

2 (16)

i¼1

3.3 Robust topology optimization Based on the analysis above, the robust topology optimization problem such that the objective is minimized subject to an upper limit on structural volume is shown below:     l 1l VarðC Þ EðC Þ þ Minmize H ðC Þ  x x2 (17) N X subject to gðx Þ ¼ xj V j  V * # 0 j¼1

Using equations (9) and (15), the robust objective function can be reformulated as:     l 1l EðC Þ þ H ðC Þ  VarðC Þ x x2   l    2  s T l 1l X ðu c ÞT K u c þ 4 ¼ ui K usi 2 x x i¼1

(18)

The sensitivity of the objective function with respect to the design variable will be calculated in detail. For the sake of being easy to describe, the following equations are introduced:  X l l  c T u K uci (19) GðC Þ ¼ x i¼1 i

QðC Þ ¼ 4

  l  2  s T 1l X s u K u i i x2 i¼1

(20)

Robust topology optimization

(21)

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Then, equation (21) holds: H ðC Þ  GðC Þ þ QðC Þ

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It is easy to obtain the sensitivity of the first part of the objective, G(C), which is the object of the deterministic topology optimization in equation (1):   @GðC Þ pl p1  s T 0 s ¼ uj kj uj xj @xj x

(22)

However, the sensitivity of Q(C) cannot be computed directly. To obtain this sensitivity, the augmented Lagrangian function of Q(C) is introduced as: LðC Þ ¼ 4

  l  2 X l    s T 1l X s u K u þ h Ti K usi  ri i i 2 x i¼1 i¼1

(23)

where h Ti is the Lagrangian multiplier, K usi  ri ¼ 0 is the equilibrium equation of the static structure in finite element analysis and s i is the variance of the ith uncertain load vector si. The sensitivity of the modified Q(C) can be expressed as: !   l   d us T  s T  s T dusi  s T dK s dLðC Þ 1l X ð iÞ s s ¼8 ui K ui K ui þ ui K þ ui u dxj x2 dxj i dxj dxj i¼1 l X  d h Ti  dus dri dK s þ Kusi  ri þ h Ti ui þ K i  dxj dxj dxj dxj i¼1

!! (24)

On account of the equilibrium equation, the following equation holds:  d h Ti  K usi  ri ¼ 0 dxj

(25)

In addition, it is assumed that the variation of an element has no effect on the applied load vector, and therefore: dri ¼0 dxj

(26)

Considering equations (25) and (26), the sensitivity of the modified Q(C) can be rewritten as:

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  s  l   dui dLðC Þ X 1  l  s T s s T ui Kui Kui þ h i K ¼ 16 2 dxj dxj x i¼1   X l l  s T dK s X 1  l  s T dK s s u þ8 K u u u þ h Ti u i i i i x2 dx dxj i j i¼1 i¼1

(27)

dus

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To remove the unknown dxij from the sensitivity expression, the Lagrangian multiplier is determined by the following equation:    1  l  s T s s T Ku u 16 i i Ku i þ h i K ¼ 0 x2

(28)

Solving the above equation, the Lagrangian multiplier can be expressed as:    1  l  s T ui K usi usi h i ¼ 16 2 x

(29)

Substituting the above Lagrangian multiplier into the equation (27), the sensitivity of the modified Q(C) becomes:   l    s T dLðC Þ 1l X s s dK s ¼ 8 u K u u i i ui dxj x2 dxj i i¼1

(30)

Using the interpolation scheme (Bendsøe and Sigmund, 1999), equation (30) can be reformulated as:   l  X  s T 0 s 2 dLðC Þ 1l 2p1 ¼ 8 uij kj uij px j dxj x2 i¼1

(31)

Thus, the sensitivity of the objective function can be expressed as: @H ðC Þ @GðC Þ @QðC Þ ¼ þ @xj @xj @xj     l  X  s T 0 s 2 pl p1  c T 0 c 1l 2p1 px ¼ uj kj uj  8 uij kj uij xj j x x2 i¼1

(32)

4. Bi-directional evolutionary structural optimization procedure Using the BESO method to solve the robust topology optimization problem, the sensitivity number is defined by relatively ranking of the sensitivity of an individual element as:

aej ¼ 

¼

x @H ðC Þ p @xj

Robust topology optimization

8   l  >  s T 0 s  2  c T 0 c 1l X > > > l u k u þ 8 uij kj uij > j j j > x < i¼1

when xj ¼ 1

  > l  > >  s T 0 s 2 1  l 2p1 X p1  c T 0 c > > l x u k u þ 8 uij kj uij x > j j j min min : x i¼1

when xj ¼ bmin

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(33) It is worth pointing out that the robust topology optimization formulation derived in this article only requires (l þ 1) load cases to approximately calculate the robust objective and the required sensitivity numbers. As vast amount of computational time is wasted on the finite element analysis in the topology optimization problem, the computational cost of the algorithm can be regarded as being linearly related to the number of uncertain applied loads. This is seen as much more efficient than the Monte Carlo-based optimization (MCBO) methods. The concept of the BESO procedure is simple. The materials are iteratively removed or added according to the sensitivity number of the element to obtain the best structural robust compliance. However, when a continuum structure is discretized utilizing low-order bilinear of trilinear finite elements, the sensitivity number could discontinuous across elements boundaries. This leads to checkerboard pattern in the final topologies. In addition, there is always mesh-dependency problem in the topology optimization. To circumvent these two problems, in the proposed robust topology optimization algorithm, the commonly used numerical filter technique (Sigmund and Petersson, 1998; Li et al., 2001; Zuo and Xie, 2015) is used to modify the elemental sensitivity number as: X v ðrjp Þanp p aej ¼ X (34) v ðrjp Þ p

where rjp is the distance between the center of the jth element and the pth node; v is a weighting function for averaging the sensitivity numbers; and v (rjp) = max(0, rmin – rjp), where rmin denotes the filter radius and anp represents nodal sensitivity number, which is defined as:

anp ¼

M X

g j aej

(35)

j¼1

where M is the total number Pof the elements connected to the pth node; g j is the weight factor of the jth element and M j¼1 g j ¼ 1; g j is defined as: 0 1 g jp 1 (36) g j ¼ B1  M MB X C C @ g jp A j¼1

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An averaging technique of element sensitivity number with its historical information is introduced to stabilize the evolutionary process (Huang and Xie, 2009, 2011; Zuo and Xie, 2015), which is given as:

ae_average ¼ j

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720

e;kþ1 ae;k j þ aj 2

(37)

e;kþ1 where k is the current iterative number and ae;k are the filtered elemental j and aj sensitivity numbers at k and k þ 1 iterations, respectively. The robust topology optimization procedure is terminated until the predefined volume is reached, and the following convergence criterion is satisfied:   X R      j¼1 Hkjþ1  HkRjþ1  #t (38) error ¼ XR H kjþ1 j¼1

where t is an allowable convergence tolerance and R is an integer number. Based on the above mentioned, the flow chart of the proposed robust topology optimization procedure is summarized in Figure 2. 5. Numerical examples In this section, four well-studied examples are discussed to illustrate the validity of the proposed approach, as well as the importance of considering load uncertainties in structural topology optimization problems. Table I lists the general settings that apply to every example unless noted otherwise. The relationship between the deterministic loading and its corresponding random loading is defined as s i ¼ cvi fic , and cvi is a coefficient. 5.1 Robust design of a simple column subjected to a concentrated force As a first test case, a simple column is designed with design domain of 100 mm in length and 100 mm in height, as shown in Figure 3. The design domain is clamped on the bottom side. The available material volume will cover 30 per cent of the design domain. In other words, the volume fraction of final design will be 30 per cent. The coefficient cv1 = 0.3. The proposed algorithm is now applied to design the column for various combination weights l . The optimal design outcomes are shown in Figure 4. When l = 1, the robust topology optimization problem is degenerated to a deterministic one without considering the influence of the random loading. It can be seen that the deterministic solution is a straight column [Figure 4(a)]. This is not surprising, as this solution does not take the variance into account during the optimization process. However, this optimum may be in unstable equilibrium and may fail under non-perpendicular perturbation in load. When the value of combination weight is a non-zero number, the corresponding solutions are robust ones such that the variances of the structural compliance are considered. It is interesting to find that the vertical bar becomes two bars after the variance is considered. This change is of great meaning, as it allows withstanding the possible loading in the non-perpendicular direction. It is also interesting to notice that in this specific example, the distance between two bars increases gradually as the combination weight value increases. This is easy to understand. The larger is the combination weight value, the greater is the influence of the variance on the final layout. From another point of view, it may be safe to say that it is very important to

Define initial design domain, related BESO parameters and volume constraint

Compute element sensitivity number using equation (33)

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Filter element sensitivity number according to equation (34)

Average the filtered element sensitivity number utilizing equation (36)

Update the design variable according to the sensitivity numbers and current target volume

No

Is volume constraint satisfied?

Yes

No

Is convergence criterion satisfied?

Yes Output the robust outcome

consider the variance when designing a structure. The histories of the robust objective and volume with different combination weights are depicted in Figure 5. It can be found that all the solutions converge well and have a predefined material volume. Regarding the computational efficiency, the robust topology optimization of the simple column only requires two load cases to obtain the final layout, which is very efficient when compared with MCBO methods. In this specific example, we try to investigate the effects of the material volume constraint and coefficient cv1 on the optimal results (l = 0.45). Figure 6(a)-(c) shows the robust designs when the material volume constraints are 0.1, 0.4 and 0.8, respectively. It can

Figure 2. Flow chart of the proposed robust topology optimization algorithm procedure

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Setting/value

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Element type Mesh Thickness Young’s modulus for solid element Young’s modulus for void element Poisson’s ratio Density filter Evolutionary ration Allowable convergence tolerance Integer number R

Q4 Fixed regular mesh with 1  1 mm size 1 mm E0 = 1 Emin = 109E0 y = 0.3 rmin = 3.0 er = 0.02 t = 0.01% R = 10

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Table I. General settings

f1c = 10 kN

Figure 3. The initial design domain, support and applied load of the simple column for Example 1

Figure 4. Robust designs for Example 1 and different values of weighting factor l

100 mm

100 mm

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Figure 5. Corresponding histories of the robust objective and volume for Example 1

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be found that different material volume constraints yield evidently different results. Particularly, there is no hole in the structure when vol = 0.8. Figure 7(a)-(c) depicts the robust designs when the coefficients (cv1) are 0.01, 0.3 and 1.0, respectively, showing that the coefficient affects the robust design significantly. It means that the fluctuation in the applied load is minor when cv1 is 0.01, and the robust design is expected to be similar to the deterministic design [Figure 4(a)]. cv1 = 1.0 denotes that the level of the uncertainty in the applied load is large. It can be seen that there is a big distance between the two bars in the robust design when cv1 = 1.0. 5.2 Robust design of a simple carrier plate In this example, the proposed algorithm is applied to design a simple carrier plate (Figure 8). The boundary condition and the initial design domain are the same as those in the first

Figure 6. Influence of the volume constraint on the robust design (l = 0.45)

Figure 7. Influence of the coefficient cv1 on the robust design (l = 0.45)

f1c = 5kN/mm

5 mm

Non-design domain

Figure 8. The initial design domain, support and applied load of the simple column for Example 2

95 mm

100 mm

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example, except that there is a non-design domain, which is 5 mm in height and 100 mm in length. The volume constraint is 55 per cent of the design domain. The coefficient cv1 = 0.1. The deterministic solution, shown in Figure 9(a) with the value of combination weight as 1.0, possesses bars that are mainly aligned to cope with the uniformly distributed loading. Nevertheless, the robust outcomes [Figure 9(b)-(f)] contain more diagonal small bars to resist the uncertainties in applied loads. It is very interesting to note that when l = 0.05, four additional new small bars appear at the bottom of the solution, which is very different from other solutions. The reason behind this may be that when larger uncertainties are involved, more diagonal bars are needed to withstand the uncertainty loading. Figure 10 depicts the corresponding convergence histories of the solutions, demonstrating that all the robust solutions converge well within 70 iterations. In this specific example, two times of finite element analyses at each iteration are only needed to compute the robust object. Thus, high computational efficiency of the proposed algorithm is highlighted.

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5.3 Robust design of a square plate In the third numerical example, a square plate is optimized to verify the proposed algorithm. The design domain has a width of 80 mm and a length of 80 mm, which is clamped at its four corners (Figure 11). The design domain is modeled by 80  80 elements. One concentrate force is applied at the center of the design domain. The volume constraint is 50 per cent of the design domain. The coefficient cv1 = 0.8. The deterministic solution, without considering the variance, owns members mainly in its top part to bear the vertical loads, as depicted in Figure 12(a). Correspondingly, the convergence history of the deterministic solution is displayed in Figure 13(a). From Figure 13(a), the good convergence of our algorithm can be investigated. The robust solution, shown in Figure 12(b), is acquired utilizing only two load cases during per iteration, which demonstrates the high computational efficiency of the presented robust

Figure 9. Robust designs for Example 2 and different values of weighting factor l

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Figure 10. Corresponding histories of the robust objective and volume for Example 2

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algorithm. This robust solution is a very special one because the influence of expectation is ignored and only variance is considered. This particular case does exist in actual engineering structures when the structures are designed without applying any externally load, but they may also have to withstand other uncertain loads such as the wind. It is not surprising to find that the robust solution significantly differs from the deterministic solution. The material spreads to the whole part of the design domain, forming five main bars to cope with the uncertain loads in the horizontal direction. The good convergence of the robust solution is demonstrated in Figure 13(b).

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5.4 Robust design of a U-shaped structure In the last test, a more complex structure, U-shaped structure, is designed. The design domain, clamped at its left side, is shown in Figure 14. In total, 10,400 elements are used to model the initial design domain. The structure is subjected to two applied loads. The volume constraint is 40 per cent of the design domain. The coefficients are cv1 = 0.01 and cv2 = 0.01. The deterministic solution and the robust solution are depicted in Figure 15(a) and (b), respectively. It can be seen that they are remarkably different from each other. Unlike the deterministic solution, the robust solution owns many diagonal bars to withstand the uncertain loads in the horizontal direction. In this regard, it is greatly meaningful to take the uncertainties into account when designing a structure in actual engineering. In this specific numerical example, only three load cases are involved at each iteration. Again, the efficiency of the proposed algorithm is indicated.

80 mm

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fc1 = 10 kN

Figure 11. Initial design domain of a square plate

Figure 12. Optimal design for a square plate

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Figure 13. (a) Histories of the robust objective and volume for a square plate with l = 1 and (b) histories of the roust objective and volume for a square plate with l = 0

6. Concluding remarks This paper presents an efficient robust topology optimization algorithm for continuum structures considering randomly applied loads. The robust objective is approximately derived using the Taylor expansion technique. The sensitivity information is obtained using the Lagrangian method. An improved BESO method utilizing the derived sensitivity numbers is developed to solve the concerned design problem. Several numerical examples are presented to demonstrate the effectiveness of the proposed algorithm. The numerical examples show the significant topological changes of the robust solutions compared with

80 mm

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729 Figure 14. Initial design for a U-shaped structure

Figure 15. Optimal design for a U-shaped structure

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Tu, J., Choi, K.K. and Park, Y.H. (1999), “A new study on reliability-based design optimization”, Journal of Mechanical Design, Vol. 121 No. 4, pp. 557-564. Wang, M.Y., Wang, X. and Guo, D. (2003), “A level set method for structural topology optimization”, Computer Methods in Applied Mechanics and Engineering, Vol. 192 Nos 1/2, pp. 227-246. Wang, F., Jensen, J.S. and Sigmund, O. (2011), “Robust topology optimization of photonic crystal waveguides with tailored dispersion properties”, Journal of the Optical Society of America B, Vol. 28 No. 3, pp. 387-397. Xie, Y. and Steven, G.P. (1993), “A simple evolutionary procedure for structural optimization”, Computers and Structures, Vol. 49 No. 5, pp. 885-896. Xie, Y.M. and Steven, G.P. (1994), “Optimal design of multiple load case structures using an evolutionary procedure”, Engineering Computations, Vol. 11 No. 4, pp. 295-302. Xie, Y.M. and Steven, G.P. (1997), Evolutionary Structural Optimization, Springer, London. Xu, B., Zhao, L., Li, W., He, J. and Xie, Y.M. (2016), “Dynamic response reliability based topological optimization of continuum structures involving multi-phase materials”, Composite Structures, Vol. 149, pp. 134-144. Yoo, K.S., Eom, Y.S., Park, J.Y., Im, M.G. and Han, S.Y. (2011), “Reliability-based topology optimization using successive standard response surface method”, Finite Elements in Analysis and Design, Vol. 47 No. 7, pp. 843-849. Youn, B.D. and Choi, K.K. (2004), “A new response surface methodology for reliability-based design optimization”, Computers & Structures, Vol. 82 Nos 2/3, pp. 241-256. Zhu, B., Zhang, X. and Fatikow, S. (2015), “Structural topology and shape optimization using a level set method with distance-suppression scheme”, Computer Methods in Applied Mechanics and Engineering, Vol. 283, pp. 1214-1239. Zuo, Z.H. and Xie, Y.M. (2015), “A simple and compact python code for complex 3D topology optimization”, Advances in Engineering Software, Vol. 85, pp. 1-11. Further reading Sigmund, O. (2001), “A 99 line topology optimization code written in matlab”, Structural and Multidisciplinary Optimization, Vol. 21 No. 2, pp. 120-127. Corresponding author Guilin Wen can be contacted at: [email protected]

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