Jikai Liu1

Department of Mechanical Engineering,

University of Alberta,

Edmonton, AB T6G 1H9, Canada

e-mail: jikai@ualberta.ca

Yongsheng MaDepartment of Mechanical Engineering,

University of Alberta,

Edmonton, AB T6G 1H9, Canada

Sustainable Design-OrientedLevel Set Topology OptimizationThis paper presents a novel sustainable design-oriented level set topology optimizationmethod. It addresses the sustainability issue in product family design, which means anend-of-life (EoL) product can be remanufactured through subtractive machining intoanother lower-level model within the product family. In this way, the EoL product isrecycled in an environmental-friendly and energy-saving manner. Technically, a sustain-ability constraint is proposed that the different product models employ the containmentrelationship, which is a necessary condition for the subtractive remanufacturing. A novellevel set-based product family representation is proposed to realize the containment rela-tionship, and the related topology optimization problem is formulated and solved. Inaddition, spatial arrangement of the input design domains is explored to prevent highlystressed material regions from reusing. Feature-based level set concept for sustainabilityis then used. The novelty of the proposed method is that, for the first time, the product life-cycle issue of sustainability is addressed by a topology optimization method. The effec-tiveness of the proposed method is proved through a few numerical examples.[DOI: 10.1115/1.4035052]

Keywords: level set, topology optimization, sustainability, remanufacturing

1 Introduction

In recent years, level set topology optimization (LSTO) hasgained the popularity as a computational product design tech-nique. It has demonstrated the adaptability subjected to differentengineering disciplines. For instance, Wang et al. [1] and Allaireet al. [2] solved the compliance-minimization problems; and later,many other researchers investigated the stress-minimization andstress-constrained problems [3–8]; in Refs. [9–11], the multimate-rial design problems were addressed. Other than the solid mechan-ics problems, LSTO has also been applied to design the fluid flowchannels for energy dissipation minimization, and different flowtypes have been considered including Darcy flow, Stokes flow,and Navier–Stokes flow [12–15]; heat conduction problems werealso addressed [16,17]. For more details, two comprehensivereviews can be found in Refs. [18,19].

As revealed by the literature survey that the authors conducted,so far, LSTO pursues the extreme structural performances; how-ever, little attention was paid to other lifecycle aspects, includingmanufacture, operation/use, service/maintenance, and disposal/recycling [20]. It has always been a headache for engineers tomanually post-treat the topology optimization solution to addressthese lifecycle considerations. Many researchers have realizedthis issue and several manufacture-oriented LSTO methods[21–24] have been developed to enhance the manufacturability.However, in general, many lifecycle aspects still lack solutions,which is also the case for other topology optimizationmethods [19].

Therefore, this paper presents a sustainable design-orientedLSTO method, which addresses the lifecycle issue of recycling.Sustainable design is popular and sometimes compulsive becauseit can reduce the negative environmental impact. An importantstrategy is to reuse the end-of-life (EoL) product, e.g., remanufac-turing, and thus, to create a close-loop product lifecycle. Com-pared to the traditional landfilling, this strategy reduces theeconomic and environmental costs greatly. A few approaches ofEoL product reuse are demonstrated in Fig. 1.

Among the approaches shown in Fig. 1, material recyclinginvolves procedures of collecting, sorting, and processing the EoLproduct, through which raw materials are recycled and can befreely used in new production. However, both recycling andreproduction are energy-costly [25], and designers are reluctant touse the recycled materials because of the quality uncertainty [26].

Remanufacturing through additive manufacturing is an emerg-ing and very promising approach, especially given the rapiddevelopment of additive manufacturing technology. Thisapproach has been proven effective [27,28], which added newmaterials to an EoL product, either to repair the defects such asoverwear, or to create new features to produce a different productmodel. However, there are limitations of this approach that theproduct quality processed through the currently available additivemanufacturing methods is still unstable, e.g., property weakening[29], and very likely may not meet the expectations [30].

The third approach is to remanufacture the EoL product throughthe traditional subtractive machining. Product family is a widelyaccepted concept in industrial production, where a group of prod-uct models are produced through a common platform, and employthe similar function but different levels of performance. Forinstance, Fig. 2 shows a family of four-jaw support structures,each of which works under the same boundary condition, butdemonstrates at a different level of load-bearing capacity. Fromthe perspective of sustainability, it would be ideal that the high-level model coming to the EoL can be remanufactured throughsubtractive machining into another lower-level model (lower vol-ume fraction). The overwear or corroded surfaces and the dam-aged areas could be removed to make it new. Advantages of this

Fig. 1 Approaches of EoL product reuse

1Corresponding author.Contributed by the Design Automation Committee of ASME for publication in

the JOURNAL OF MECHANICAL DESIGN. Manuscript received January 15, 2016; finalmanuscript received October 9, 2016; published online November 11, 2016. Assoc.Editor: James K. Guest.

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approach are evident that a remanufacturing process only con-sumes around 20–25% of the energy of the initial production [31],and it would not consume any energy to recycle the raw materials.

In summary, just as claimed in Ref. [25], remanufacturingthrough subtractive machining is the best among the three EoLproduct-reuse approaches simply because of the fact that, thisapproach is the most environmental-friendly and reliable for thecurrent practice. To implement this approach, satisfying the con-tainment relationship is mandatory, i.e., a product model shouldspatially be a subset of any higher-level model. This containmentrelationship is mathematically formulated in the following equa-tion, which is named the sustainability constraint:

Xi > Xiþ1 (1)

where Xi represents the material domain of model i, and thesmaller number of i means a higher level.

Then, we explore the strategies of product family designthrough topology optimization. Figures 3(a) and 3(b) show twoexisting strategies. Subject to the compliance-minimization crite-ria, strategy 1 is simple and no adaption is required to the existingtopology optimization algorithms. Specifically, the product familyis composed of three models, which are sequentially designedthrough three mutually dependent topology optimization proc-esses. That means the optimization result of model i is used as theinput design domain of model iþ 1. With this strategy, the sus-tainability constraint is strictly satisfied. However, optimality ofthe low-level models is severely sacrificed because of therestricted design space. Strategy 2 is even simpler (see Fig. 3(b)),where all the topology optimization processes use the raw part asthe input design domain and thus, all the models can be optimallydesigned given the fully explored design space. However, the sus-tainability constraints are very likely violated.

In this work, a novel strategy is proposed which concurrentlyaddresses the overall design optimality and the sustainability

constraints (Fig. 3(c)). To be specific, a level set-based productfamily representation is proposed which employs multiple levelset functions to simultaneously represent the multiple productmodels. Accordingly, a multi-objective topology optimizationproblem is formulated through a weighted sum of structural per-formances of the multiple product models. During the solutionprocess, the different product models are simultaneously opti-mized and interactively influenced to improve the overall optimal-ity. A novel multilevel set interpolation is proposed tocompulsively satisfy the sustainability constraints.

Research innovation is evident that this is the first time the life-cycle issue of recycling is addressed by a topology optimizationmethod. It is worth noticing that this method is much more com-plex than the product family design performed by Torstenfelt andKlarbring [32,33] which only realized the partial material domainoverlap.

The following contents are organized as: Section 2 presents theconventional level set geometry modeling methods, includingboth the single-material and multimaterial schemes; Sec. 3 intro-duces the proposed level set-based product family representation,and the related topology optimization problem formulation; Sensi-tivity analysis is performed in Sec. 4; and two numerical examplesare studied in Sec. 5; Sec. 6 explores design enhancement throughspatial arrangement of the input design domains; Sec. 7 demon-strates the feature-based level set method for sustainability; Aconclusion is given in Sec. 8.

2 Level Set Geometry Modeling

Osher and Sethian [34] proposed the level set function as a nat-ural way of closed boundary representation. Specifically, let D 2Rn ðn ¼ 2 or 3Þ be the initial design domain, X 2 Rn ðn ¼2 or 3Þ represent the material domain, and @X be the material/void interface. The level set function, UðXÞ : Rn 7!R; is definedas

UðXÞ > 0; X 2 X=@XUðXÞ ¼ 0; X 2 @XUðXÞ < 0; X 2 D=X

8<: (2)

Normally, the Heaviside function (see Eq. (3)) and the Diracdelta function (see Eq. (4)) are employed to support the numericalgeometry modeling

HðUÞ ¼ 1; U � 0

HðUÞ ¼ 0; U < 0

�(3)

dðUÞ ¼ @HðUÞ=@U (4)

Then, the interior and boundary of the material area can be rep-resented by

X ¼ fX j HðUðXÞÞ ¼ 1g (5)

@X ¼ fX j dðUðXÞÞ > 0g (6)

If applied to multimaterial geometry modeling, multiple levelset functions are required. Generally, there are two approaches:the “color” level set [9] and the multimaterial level set (MMLS)[35]. Especially for the color level set, it has been widely acceptedby multimaterial topology optimization implementations[9,10,36–38]. Other multimaterial level set models, such as piece-wise constant level set [39–41] and reconciled level set [42], arealso effective in supporting multimaterial topology optimization,which will not be specified in this paper.

The color level set approach was developed in Refs. [9,10],and it only requires m level set functions to represent n ¼ 2m

material phases. As presented in Fig. 4(a), the two level setfunctions UiðXÞ ði ¼ 1; 2Þ could represent four material

Fig. 2 Product family of the cube structure: (a) model 1 (vol-ume ratio 5 0.4), (b) model 2 (volume ratio 5 0.3), and (c) model3 (volume ratio 5 0.2)

Fig. 3 Strategies of product family design: (a) strategy 1, (b)strategy 2, and (c) new strategy

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phases: xj¼fX j H¼ ½HðU1ðXÞÞ;HðU2ðXÞÞ� ¼ constant vectorgðj¼ 1; 2; 3; 4Þ. The specific multimaterial interpolation is real-ized through the following equation:

DðXÞ ¼ HðU1ðXÞÞHðU2ðXÞÞD1 þ HðU1ðXÞÞ½1� HðU2ðXÞÞ�D2

þ ½1� HðU1ðXÞÞ�HðU2ðXÞÞD3 þ ½1� HðU1ðXÞÞ�� ½1� HðU2ðXÞÞ�D4

(7)

in which D1, D2, D3, and D4 are the elasticity tensor of materialphase 1, 2, 3, and 4, respectively.

The MMLS is a different approach that mþ 1 material phasesare represented by m level set functions. As presented in Fig. 4(b),the two level set functions UiðXÞ ði ¼ 1; 2Þ represent three mate-rial phases: xj ðj ¼ 1; 2; 3Þ. Therefore, the multimaterial inter-polation is formulated through

DðXÞ ¼ HðU1ðXÞÞ½1� HðU2ðXÞÞ�D1 þ HðU1ðXÞÞHðU2ðXÞÞD2

þ ½1� HðU1ðXÞÞ�D3

(8)

By using both approaches, the completeness D ¼ [nk¼1xk and

uniqueness xk\xl ¼1 ðk 6¼ lÞ are satisfied.

3 Level Set-Based Product Family Modeling

Inspired by the MMLS, a novel approach is developed to sup-port the product family modeling while satisfying the sustainabil-ity constraints.

As shown in Fig. 5, it requires m level set functions to representm product models. The ith product model is defined byxi ¼ fXjU1ðXÞ > 0;…;UiðXÞ > 0g. It is obvious that the sus-tainability constraints are compulsively satisfied. The related mul-tilevel set interpolation would be

D1ðXÞ ¼ HðU1ðXÞÞDD2ðXÞ ¼ HðU1ðXÞÞHðU2ðXÞÞDD3ðXÞ ¼ HðU1ðXÞÞHðU2ðXÞÞHðU3ðXÞÞD

………

(9)

where DiðXÞ is the interpolated elasticity tensor of the productmodel i, and D is the elasticity tensor of the solid material. It is

noted that only one solid material type is involved because weassume the product to be homogeneous, which reflects the com-mon case. The weak material properties for the void [1,2] areignored in the expression but will be applied during numericalimplementation. Accordingly, the compliance-minimizationtopology optimization problem for product family design is for-mulated as follows:

Min: Jðu1;u2;U1;U2Þ

¼ w1

ðD

Deðu1Þeðu1ÞHðU1ÞdXþ w2

ðD

Deðu2Þeðu2ÞHðU1Þ

� HðU2ÞdX

s:t: aðu1; v1;U1Þ ¼ lðv1Þaðu2; v2;U1;U2Þ ¼ lðv2Þ 8v1; v2 2 Uadð

D

HðU1ÞdX � V1ðD

HðU1ÞHðU2ÞdX � V2

aðu1; v1;U1Þ ¼ð

D

Deðu1Þeðv1ÞHðU1ÞdX

aðu2; v2;U1;U2Þ ¼ð

D

Deðu2Þeðv2ÞHðU1ÞHðU2ÞdX

lðv1Þ ¼ð

D

pv1dXþð

D

sv1dS

lðv2Þ ¼ð

D

pv2dXþð

D

sv2dS ð10Þ

in which aðÞ is the energy bilinear form and lðÞ is the load linearform. u is the deformation vector, v is the test vector, and eðuÞ isthe strain. U ¼ fv 2 H1ðXÞdjv ¼ 0 on CDg is the space of kine-matically admissible displacement field, p is the body force, and sis the boundary traction force, which are assumed not spatiallyvarying. V1 and V2 are the upper bounds of the material volumefor product model 1 and 2, respectively. w1 and w2 are theweighting factors, which satisfy w1þ w2 ¼ 1.

It is noted that this problem formulation is defined by assumingonly two product models and it would be trivial to make theextensions.

We can see from Eq. (10) that this is a multi-objective optimi-zation problem and the objective function is a weighted sum ofthe structural compliances of the two product models. Two finiteelement analyses are required in each iteration to separately evalu-ate the status variables u1 and u2.

4 Sensitivity Analysis

In order to solve the optimization problem, the level set func-tions U1 and U2 are used as design variables. The sensitivity anal-ysis is needed to derive the velocity fields Vn1 and Vn2 to properlyevolve the level set contours in the steepest descent direction.Generally, the Lagrange multiplier method is applied to solve theoptimization problem; material derivative and the adjoint methodare employed for shape sensitivity analysis. The shape sensitivityanalysis for multimaterial compliance-minimization problemunder the level set framework is well known, which have beenpresented in previous publications [35,37]. Therefore, the detailedprocess will not be repeated while only the major results are pre-sented; see below

L0 ¼ð

D

R1dðU1ÞVn1jrU1jdXþð

D

R2dðU2ÞVn2jrU2jdX

R1 ¼ k1 þ k2HðU2Þ � w1� Deðu1Þeðu1Þ � w2� Deðu2Þeðu2Þ� HðU2Þ

R2 ¼ k2HðU1Þ � w2� Deðu2Þeðu2ÞHðU1Þ ð11Þ

Fig. 4 Multimaterial level set geometry modeling

Fig. 5 Level set-based product family representation

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where L is the Lagrangian; k1 and k2 are the Lagrange multi-pliers; R1 and R2 are the shape sensitivity densities.

Then, by following equation:

Vn1 ¼ �R1

Vn2 ¼ �R2(12)

L can be guaranteed to change in the descent direction, as shownin the following equation:

L0 ¼ð

D

�R21dðU1ÞjrU1jdXþ

ðD

�R22dðU2ÞjrU2jdX (13)

Guided by the calculated velocity fields, update of the level setcontours can be performed by solving the Hamilton–Jacobi equa-tion. This is well established and details will not be repeated here.Interested readers can refer to Refs. [1,2].

5 Numerical Example

In this section, the benchmark cantilever example is studied toprove the effectiveness of the proposed method. We assume thattwo models form the product family and the related boundary con-ditions are presented in Fig. 6. The same homogeneous elasticmaterial is applied to both models which employs the Young’smodulus of 1.3 and the Poisson’s ratio of 0.4. The optimizationproblem is to minimize the structural compliance under the maxi-mum volume fraction of 0.5 for model 1 and 0.4 for model 2.

The solution of only running model 1 is demonstrated in Fig. 7.It is observed that the loading point of model 2 is in status of void,which means the sequential procedure, as shown in Fig. 3(a), can-not be applied. Therefore, among the three strategies as presentedin Fig. 3, the newly proposed strategy is appropriate to solve thissustainability-constrained problem.

To demonstrate the effectiveness of the proposed method, a fewnumerical implementations are performed under different weightfactors. The optimization results are presented in Fig. 8.

Through analysis of the optimization results, it can be con-cluded that the proposed method can effectively design the prod-uct family, and the sustainability constraints are always satisfied.In addition, as indicated by Fig. 9, result optimality of the twomodels is balanced by the weight factor.

In Fig. 8, the derived structural topologies of the different prod-uct models are always identical, while it should not be the case ifthe volume fraction difference is enlarged. Therefore, this exam-ple is further studied subjected to maximum material fraction of0.5 for model 1 and 0.25 for model 2. The related optimizationresults are demonstrated in Fig. 10. We can see from the results

that, by increasing the volume fraction difference, the derivedstructural topologies are not always consistent.

6 Spatial Arrangement of the Design Domains

In Sec. 5, spatial arrangement of the input design domains israndomly determined and thus, there is still room to furtherenhance the overall design effect by carefully exploring the spatialarrangement. Principally, it is preferable to prevent the highlystressed material regions from reusing. In addition, the derivedstructural performances are tightly related to the spatial arrange-ment and it is of our interest to investigate how the structural per-formances will be affected.

6.1 The Cantilever Structure Problem. For the cantileverstructure, material areas around the left edge are highly stressed.Therefore, it is preferable to remove these areas when reusing thepart, which can be achieved by repositioning the boundary condi-tion of model 2; see Fig. 11.

Correspondingly, the optimization results are demonstrated inFig. 12. It can be foreseen that the results of model 2 after spatialarrangement are more reliable because the reused materials areless stressed in history.

Fig. 6 Boundary conditions of (a) model 1 and (b) model 2

Fig. 7 The independent solution of model 1

Fig. 8 Optimization results under different weight factors. (a)Results of model 1 (compliance 5 51.68), model 2 (compli-ance 5 34.52), and the spatial overlap under the weight factorw1 5 0.9; (b) results of model 1 (compliance 5 51.99), model 2(compliance 5 32.42), and the spatial overlap under the weightfactor w1 5 0.8; (c) results of model 1 (compliance 5 52.09),model 2 (compliance 5 32.08), and the spatial overlap under theweight factor w1 5 0.7; (d) results of model 1 (compli-ance 5 53.40), model 2 (compliance 5 30.81), and the spatialoverlap under the weight factor w1 5 0.6; (e) results of model 1(compliance 5 54.76), model 2 (compliance 5 30.90), and thespatial overlap under the weight factor w1 5 0.5; and (f) resultsof model 1 (compliance 5 56.21), model 2 (compliance 5 30.70),and the spatial overlap under the weight factor w1 5 0.4.

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6.2 The L-Bracket Problem. In this example, the samehomogeneous elastic material is applied to both models whichemploys the Young’s modulus of 1.3 and the Poisson’s ratio of0.4. The optimization problem is to minimize the structural com-pliance under the maximum material volume fraction of 0.4 formodel 1 and 0.2 for model 2.

The boundary condition of model 1 is shown in Fig. 13(a) andtwo sets of boundary conditions are applied to model 2; seeFigs. 13(b) and 13(c), which correspond to the two sets of spatialarrangements.

The optimization results are presented in Fig. 14 given the spa-tial arrangement 1 and in Fig. 15 given the spatial arrangement 2.

Generally, the most stressed area of the L-bracket locates at thereentrant corner. Therefore, it is observed from Figs. 14 and 15that the optimization results subjected to the spatial arrangement 2effectively prevent most of the highly stressed areas from reusing.In addition, the derived structural performances subjected to thespatial arrangement 2 are obviously better than those of the spatialarrangement 1.

7 Feature-Based Product Family Design

Geometric features are widely applied in mechanical engineer-ing domain, especially for design and manufacturing activities.Therefore, this section presents the feature-based level set methodsubjected to the sustainability consideration.

7.1 Feature-Based Level Set Geometry Modeling. First,individual geometric features can be represented by parametriclevel set functions. For instance, a circle feature can be modeledby

Uf ðXÞ ¼ Rc � sqrtððx� x0Þ2 þ ðy� y0Þ2Þ (14)

and a square feature by

Uf Xð Þ ¼minHs

2� x� x0ð Þ;

Hs

2þ x� x0ð Þ;

Hs

2� y� y0ð Þ;

Hs

2

�

þ y� y0ð Þ�

(15)

in which (x0; y0) is the feature primitive center coordinates; Rc isthe circle radius; and Hs is the square length.

Then, a complex geometry can be constructed through Booleanoperations of the individual feature primitives; see the followingequation for the Boolean operators:

Uf 1 [ Uf 2 ¼ maxðUf 1; Uf 2ÞUf 1 \ Uf 2 ¼ minðUf 1; Uf 2Þ

Uf 1nUf 2 ¼ minðUf 1;�Uf 2Þ(16)

Finally, the geometry is constructed through the following equation:

UðXÞ ¼ CðUf 1;Uf 2;Uf 3;…Þ (17)

where C is the set of Boolean operations. It is worth noting that,level set topology optimization on constructive solid geometry(CSG) model was originally studied in Ref. [43]. About Eqs.(15)–(17), they are C1 discontinuous, which are nondifferentiable.This issue can be solved by employing differentiable geometryrepresentations by smoothing the sharp corners [44–46] and theanalytical descriptions of the Boolean operators [47]. On the otherhand, in many topology optimization implementations involvingdiscrete geometric elements, Eqs. (15)–(17) are directly applied tocalculate the piecewise continuous sensitivity result, which hasbeen proven effective by many numerical implementations[44,45,47].

7.2 A Torque Arm Example. As depicted in Figs. 16(a) and16(b), the two torque arm product models are both geometric fea-ture based, and the common hierarchical structure of Booleanoperations is demonstrated in Fig. 16(c) and in the followingequation:

U1 ¼ ðUf 1 [ Uf 2[ Uf 3Þ=ðUf 4 [ Uf 5[ Uf 6Þ ðmodel 1ÞU2 ¼ ðUF1 [ UF2[ UF3Þ=ðUF4 [ UF5[ UF6Þ ðmodel 2Þ

(18)

Fig. 9 Influence of the weight factor

Fig. 10 Optimization results after changing the volume frac-tions: (a) results of model 1 (compliance 5 52.66), model 2(compliance 5 49.41), and the spatial overlap under the weightfactor w1 5 0.8; (b) results of model 1 (compliance 5 54.99),model 2 (compliance 5 47.18), and the spatial overlap under theweight factor w1 5 0.4

Fig. 11 Rearrangement of the boundary condition of model 2

Fig. 12 Optimization results after spatial arrangement: (a)results of model 1 (compliance 5 51.88), model 2 (compli-ance 5 31.74), and the spatial overlap under the weight factorw1 5 0.5; (b) results of model 1 (compliance 5 56.44), model 2(compliance 5 30.83), and the spatial overlap under the weightfactor w1 5 0.2

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We can see that the initial design domains are not identical.Therefore, a preparation step is necessary to unify the separateddesign domains by overlapping the fixed circular voids, and theunified design domain is represented by Eq. (19).

U ¼ U1 [ U2 (19)

The next step is to distinguish the design domain into designa-ble and nondesignable subdomains because some geometric fea-tures are employed for assembly purpose and therefore, theirshape characteristics should be resevered. Hence, in this example,Uf 1 ðUF1Þ, Uf 3 ðUF3Þ, Uf 4 ðUF4Þ, and Uf 6 ðUF6Þ are nondesigna-ble, while only Uf 2 ðUF2Þ and Uf 5 ðUF5Þ are freely designablesubdomains.

Equation (10) is still applied as the optimization problem for-mulation. The maximum material volume fraction is 0.4 for model1 and 0.3 for model 2. Again, a few numerical implementationsare conducted under different weight factors, and the optimizationresults are shown in Fig. 17.

Through analysis, similar conclusions can be drawn regardingthe cantilever example: the sustainability constraints are alwayssatisfied and the weight factor can well balance the design qualitybetween the two models; see Fig. 18.

8 Conclusion

This paper contributes toward a novel sustainable design-oriented LSTO method. It addresses the sustainability issue ofproduct family design, for the first time, through topology optimi-zation. The proposed method enables EoL product reuse; this

Fig. 13 Boundary conditions of (a) model 1, (b) model 2 (given the spatial arrangement 1),and (c) model 2 (given the spatial arrangement 2)

Fig. 14 Optimization results subject to the spatial arrangement1: (a) results of model 1 (compliance 5 80.51), model 2 (compli-ance 5 53.57), and the spatial overlap under the weight factorw1 5 0.5; (b) results of model 1 (compliance 5 89.90), model 2(compliance 5 50.03), and the spatial overlap under the weightfactor w1 5 0.2

Fig. 15 Optimization results subject to the spatial arrangement2: (a) results of model 1 (compliance 5 77.39), model 2 (compli-ance 5 20.27), and the spatial overlap under the weight factorw1 5 0.5; (b) results of model 1 (compliance 5 84.67), model 2(compliance 5 18.93), and the spatial overlap under the weightfactor w1 5 0.2

Fig. 16 The torque arm example: (a) the design domain andboundary condition of model 1; (b) the design domain andboundary condition of model 2; and (c) construction of thegeometry through R-functions

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function is of engineering significance to reduce production-related environmental and energy issues through remanufacturing.

Technically, a sustainability constraint is proposed that the dif-ferent product models employ the containment relationship. Thissustainability constraint is addressed through a novel multilevelset interpolation, which is inspired by the previous MMLS model-ing. The overall design optimality is guaranteed by the concurrentmultiproduct model optimization.

Spatial arrangement of the input design domains is explored toprevent highly stressed material regions from reusing becausethese regions are prone to accumulating fatigue damages. Throughtwo numerical studies, it is proven that the highly stressed areascan be prevented from reusing through spatial arrangement, andthe rearranged design domains do not evidently sacrifice thederived structural performances. Hence, carefully positioning theinput design domains is an effective approach to enhancing reus-ability of the product models.

On the other hand, a more stringent way to consider the fatiguedamage is to add local fatigue constraints when formulating the

optimization problem [48]. However, fatigue-constrained topol-ogy optimization falls out of scope of this paper, and the mainfocus, as well as the contribution, of this paper is thesustainability-oriented product family design method through anovel multilevel set interpolation. It is practically meaningfulbecause load-bearing parts come to EoL for many possible rea-sons, e.g., overwear and corrosion, instead of only fatigue damageor direct failure. Even so, it is of the authors’ interest to explorethe fatigue-constrained product family design in our future work.

At the end, a few limitations of this work are stated here: (i)subtractive machining constraints are not included in the proposedtopology optimization method, such as tool access, which causesthat the optimized topology design may face challenges for manu-facturing; and (ii) so far, the proposed method is limited to com-pliance minimization, which is monotonic with volume. For otherhighly relevant design criteria, such as stresses, the applicabilityof the proposed method needs more research effort and enhance-ment. Hence, overcoming these limitations will be the main focusof our future work.

Acknowledgment

The authors would like to acknowledge the support from ChinaScholarship Council (CSC) and Natural Sciences and EngineeringResearch Council of Canada (NSERC).

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Fig. 17 Optimization results under different weight factors: (a)model 1 (w1 5 0.9), (b) model 2 (w1 5 0.9), (c) model 1 (w1 5 0.8),(d) model 2 (w1 5 0.8), (e) model 1 (w1 5 0.7), (f) model 2(w1 5 0.7), (g) model 1 (w1 5 0.6), (h) model 2 (w1 5 0.6), (i)model 1 (w1 5 0.5), (j) model 2 (w1 5 0.5), (k) model 1 (w1 5 0.4),and (l) model 2 (w1 5 0.4)

Fig. 18 Influence of the weight factor

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