Struct Multidisc OptimDOI 10.1007/s00158-013-1003-9

RESEARCH PAPER

A multi-objective method of hinge-free compliantmechanism optimization

Benliang Zhu · Xianmin Zhang · Sergej Fatikow

Received: 10 May 2013 / Revised: 23 July 2013 / Accepted: 14 August 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract A new multi-objective formulation for topologysynthesis of hinge-free compliant mechanisms is presentedbased on the SIMP method. A weighted sum formed objec-tive function is developed by taking into considerationthe input and output mean compliances. The weightingfactors are set based on the information that is obtainedfrom the previous iteration and automatically updated witheach optimization iteration step. Shape sensitivity analysisis addressed. Some numerical examples are presented toillustrate the validity of the proposed method.

Keywords Topology optimization · Compliantmechanisms · Hinge-free · SIMP · Multi-objective

1 Introduction

A compliant mechanism gains at least some of its mobil-ity from the deflection of its flexible members rather thanmovable joints only (Howell 2001). Compliant mecha-nisms have undergone considerable development due tosome intrinsic advantages, e.g., ease in assembly, no lubri-cation etc. These advantages have produced a growing

B. Zhu · X. Zhang (�)Guangdong Province Key Laboratory of Precision Equipmentand Manufacturing Technology, South ChinaUniversity of Technology, Guangzhou,Guangdong 510640 Chinae-mail: zhangxm@scut.edu.cn

B. Zhue-mail: lllang123@163.com

S. FatikowDivision Microrobotics Department of Computing Science,University of Oldenburg, Uhlhornsweg 84,A1, 26111 Oldenburg, Germany

interest in the compliant mechanisms synthesis method.The developed approaches typically follow one of twoprincipal approaches, i.e., the kinematic-based approach(Howell 2001) and the structural topology optimization-based approach (Frecker et al. 1999; Bendsøe and Sigmund2003; Wang et al. 2003).

In the kinematic-based approach, a compliant mechanismis designed using a rigid-body replacement method startingfrom rigid-synthesis results (Howell 2001). In the structuraltopology optimization based approach, the design problemis recast as an optimal material distribution problem to findthe best layout within a given design domain (Bendsøeand Sigmund 2003). Compared with the kinematic-basedapproach, the topology optimization-based approach doesnot need a basic configuration of the design and has beena mature tool for designing the compliant mechanismsproblem (Bendsøe and Sigmund 2003; Deepak et al. 2009).

In designing of compliant mechanisms by using the con-tinuum topology optimization methods (such as the SIMPmethod (Sigmund 1997) and the level set method (Chen2007)), one of the significant challenges is their typi-cally results in compliant mechanisms with de facto hinges(Wang 2009; Rahmatalla and Swan 2005). These hingesmake the compliant mechanisms not truly compliant butfunction essentially as the rigid-body mechanisms (Fig. 1).When a hinged mechanism is loaded, only the materialaround the hinge areas will contribute to the deflection.Besides, the hinged mechanisms are very difficult to befabricated especially in the micro scale (Rahmatalla andSwan 2005).

For eliminating this kind of hinges, one notable approachis to use the second stage design strategy (redesign the hingeareas). However, this may cause the obtained mechanismdeviating from the original design (Rahmatalla and Swan2005; Chen 2007). Other approaches primarily focus on

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Fig. 1 De facto hinge of structure (a) and the revolute joint of rigidbody (b) (Wang 2009)

employing the filtering methods (Sigmund 2007), extra con-straints ( such as the stress constraint (Duysinx and Bendsøe1998; Wang and Zhang 2012), the MOLE method (Poulsen2003), energy functional (Luo et al. 2008)), Hybrid Dis-cretization method (Kim et al. 2005; Zhou 2010) or usinga wavelet-based topology optimization formulation (Yoonet al. 2004) etc. In addition, via two different sets of arti-ficial spring models, Rahmatalla and Swan (Rahmatallaand Swan 2005) proposed a new method for designinghinge-free compliant mechanisms. Zhu and Zhang (Zhu andZhang 2012) proposed a new multi-objective method fortopology optimization of distributed compliant mechanismsbased on the level set method and have extended the methodfor designing compliant mechanisms with multiple outputs(Zhu et al. 2013). However, the weighting factors have to bepreset manually.

More recently, it has been recognized that the funda-mental reason of the hinges lie behind the mathematicalrepresentation of the designing problem (Wang 2009). Formathematically modeling the compliant mechanisms topol-ogy optimization problem, several available formulationshave been developed and these formulations can be studiedunder two main groups. The first one is established by maxi-mizing some kind of mechanical measurements (such as theoutput displacement (Bendsøe and Sigmund 2003), the geo-metrical advantage (Chen 2007), the mechanical advantage(Sigmund 1997) and the mechanical efficiency (Luo et al.2008)) while the second one is established by consideringboth the function requirement and the strength requirementof a compliant mechanism (Frecker et al. 1997; Nishiwakiet al. 2001; Saxena and Ananthasuresh 2000). In reference(Deepak et al. 2009), five different design formulations areexamined and it shows that all the formulations lead tohinges.

Ideally, compliant mechanisms should have the com-pliance uniformly distributed throughout the mechanismrather than limiting it to a few hinges. In this study, weare mainly concerned with developing an adequate mathe-matical formulation to eliminate the hinges in the obtainedcompliant mechanisms and acquire designs with distributed

compliance. The remainder of the paper is organized asfollows. In Section 2, the optimization problems of design-ing hinge-free compliant mechanisms are proposed basedon the SIMP method. Sensitivity analysis is evaluated.In Section 3, several numerical examples are presentedto demonstrate the effectiveness of the proposed method.Finally, conclusions are addressed.

2 Problem setting

The design domain for topology optimization of compliantmechanisms with single input-output behaviour can be illus-trated in Fig. 2. At the input port i, an input force Fin isapplied, with its displacement magnitude uin. At the outputport o, it is desired to have an output displacement in magni-tude uout . As mentioned before, various formulations havebeen developed. Yet, there is no single universally acceptedformulation (Deepak et al. 2009).

An alternative design model is proposed in this paperwhich can prevent hinges in the obtained mechanisms. In themethod, in addition to maximize desired displacement uout

in the specified direction, the input compliance (displace-ment at input location due to force at input location) andthe output compliance (displacement at output location dueto force at output location) are added to the objective func-tion in order to ensure that the resulting design is reasonablystiff, automatically implying in the absence of hinges.

2.1 Input and output compliances

As shown in Fig. 3, the input compliance Cin and the out-put compliance Cout are respectively determined from thefollowing equations:

Cin =∫

D

fin · UCind� (1)

Cout =∫

D

fout · UCout d� (2)

Fig. 2 The design domain of compliant mechanisms

A multi-objective method of hinge-free compliant mechanism optimization

Fig. 3 Loading conditions fordetermining: a inputcompliance, and b outputcompliance (Zhu et al. 2013)

where fin and fout are unit loads that are applied at theinput and output ports, respectively. UCin and UCout are thedisplacement vectors due to fin and fout , respectively.

The optimization problem actually is transferred into amulti-objective optimization problem since three objectives(uout , Cin and Cout ) need to be taken into consideration. Themaximization of uout is to meet the function requirementof the compliant mechanism while the minimization of Cin

and Cout is to meet the stiffness requirement.

2.2 An objective function and the weighting factor settingscheme

One of the most commonly used approaches to deal withthe multi-objective optimization problem is the weightingsum method. The optimization problem can be formulatedas follows

Minimize : J = −uout + αCin + βCout (3)

Subject to : V ≤ Vmax (4)

where α and β are the weighting factors for Cin and Cout ,respectively. V is the material usage of the mechanismconstrained with upper limit Vmax . Note that a minus isused for uout since minimizing −uout is equivalent withmaximizing uout .

The major problem in the sum objective formulation ischoosing the values of α and β. Different design problemsmay need different α and β.

Since the best definition for α and β is problem-dependent, we therefore propose an adaptive way to assignthese values, based on the values of uout , Cin and Cout ofthe previous iteration.

Therefore, the objective function J can be rewritten as

J = −ukout + αkCin + βkCout (5)

where αk and βk are weighting factors which change witheach iteration k of the optimization algorithm. They areupdated using the scheme

αk =⎧⎨⎩

uk−1out

Ck−1in

γ

k ≥ 2k = 1

(6)

βk ={

uk−1out

Ck−1out

γ

k ≥ 2k = 1

(7)

where uk−1out , Ck−1

in and Ck−1out denote the values of uout , Cin

and Cout in the (k − 1) − th iteration, respectively. γ ≥ 0 isan arbitrary constant. For simplicity, γ is set to be 0.

In case of γ = 0, since there are no constraints on theCin and Cout , a weak structure with a very large uout and avery small Cin and Cout will be obtained. However, after thefirst iteration, α and β will become very large to enhancethe weights of Cin and Cout . Therefore, a well-posed designstill can be obtained.

From the above analysis one can see that there is no needfor manually operating the functions because the weight-ing factors can actually self-adjust based on the informationobtained from the previous iteration.

The reason of setting α and β based on the informationat the previous step is that it can avoid uout , Cin or Cout

going insanely large. Based on the weighting factor settingscheme, if uout becomes larger, then α and β will becomelarger. The weights of Cin and Cout are increased and thusthey make the created mechanism stiffer. Conversely, if uout

becomes smaller, then α and β will become smaller. Thiswill decrease the weights of Cin or Cout to make the createdmechanism more flexible.

2.3 SIMP method

The use of the SIMP method for topology optimization ofdistributed compliant mechanisms allows us to convert the

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optimal topology problem into a sizing problem. It relatesthe Young’s modulus E to the density x as

E(x) = xpE0 (8)

where x denotes density which is the design function. In thisstudy, the density of each finite element is treated as an opti-mization variable. E0 is the stiffness for the given isotropicmaterial. p is a free parameter to penalize intermediate den-sities and lead approximately to a 0 − 1 design. Usually,p ≥ 3 is required.

Incorporating with the SIMP method, the optimizationproblem stated as in (3) can be rewritten by using the FEMform as follows:

Minimizex

: J = −LT U + αfTinUCin + βfToutUCout (9)

Subject to :N∑

e=1

vexe ≤ Vmax (10)

KU = Fin (11)

KUCin = fin (12)

KUCout = fout (13)

0 < xmin ≤ xe ≤ 1 (14)

e = 1, 2, ...N (15)

where L is a unit load vector consisting of zero except forposition o with an entry of Lo = 1, and U is the dis-placement vector generated by the load vector Fin appliedat the input port. UCin and UCout are displacement vectorsgenerated by the load vector fin and fout , respectively. Thestiffness matrix K depends on the stiffness Ee in element e

which can be written in the form

K =N∑e

Ke(Ee) (16)

where Ke is the element stiffness matrix in the global level.

2.4 Sensitivity analysis

The gradients with respect to the design variables are calleddesign sensitivities. The total derivative of J in (9) withrespect to xe can be written as:

∂J

∂xe

= − ∂(LT U

)∂xe

+ α∂

(fTinUCin

)∂xe

+ ∂α

∂xe

fTinUCin

+ β∂

(fToutUCout

)∂xe

+ ∂β

∂xe

fToutUCout

(17)

The sensitivities of the three parts (i.e., LT ∂U∂xe

,∂(fTinUCin

)∂xe

and∂(fToutUCout

)∂xe

) can be easily obtained by using the

adjoint method (Bendsøe and Sigmund 2003), which can beexpressed as

∂uout

∂xe

= −UTCout

∂K∂xe

U = −pxp−1e UT

CoutKeU (18)

∂(fTinUCin

)∂xe

= pxp−1e UT

CinKeUCin (19)

∂(fToutUCout

)∂xe

= pxp−1e UT

CoutKeUCout (20)

For solving the sensitivities of α and β, since they arerestricted to be constant in each individual optimization iter-ation and only uout , Cin and Cout are treated to be variables,the following two states can be obtained.

∂α

∂xe

= 0 (21)

∂β

∂xe

= 0 (22)

2.5 Algorithm

There are several mathematical methods which canbe employed to solve the topology optimization prob-lem, e.g., the sequential linear programming method,the sequential convex programming method (Barthelemyand Haftka 1993). Optimality criteria (OC) methods areespecially efficient for problems with many variablesand few constraints (Sigmund 2001). Since the opti-mization problem stated in (9) has just one constraintin addition to a simple constraint giving upper andlower limits on the density variable, a standard optimal-ity criteria (OC) method (Sigmund 2001) is employedto solve the proposed optimization problem in thispaper.

Some numerical instabilities, e.g., checkerboards andmesh-dependent, will occur during the optimization. Thisstudy employs the filtering method (Bendsøe and Sigmund2003) to smooth the sensitivities information to overcomesuch instabilities.

Here, we summarize the optimization procedure for thetopology optimization of distributed compliant mechanismsas follows:

Step 1. Make an initial design by distributing the materialhomogeneously.

Step 2. Do the finite element analysis to obtain uout , Cin

and Cout .Step 3. Calculate the weighting factors α and β by

using (6) and (7) based on the information that

A multi-objective method of hinge-free compliant mechanism optimization

is obtained from the previous step. For the firstiteration step, αk and βk can be simply set tobe 0.

Step 4. Calculate the sensitivity of the optimization prob-lem with respect to design changes. Filter thesensitivities.

Step 5. Update the design variables using a standard Opti-mality Criteria (OC) method.

Step 6. Check for convergence. If not convergence, returnto Step 2. The convergence check scheme isdirectly employed in reference (Sigmund 2001).

3 Numerical examples

In this section, the proposed method is applied to the designof two distributed compliant mechanisms. The artificialmaterial properties are described as: Young’s modulus forsolid material is E = 1 and Poisson’s ratio υ = 0.3. Theinput load Fin is set to be unity. For numerical reasons, thedensities of the design variables are restricted by a lowerbound xmin = 10−3.

3.1 Push gripper

The design of push gripper is firstly considered. The designdomain and boundary conditions are shown in Fig. 4. Thedesign domain is assumed to be rectangular and discretizedby square finite elements. The goal of the design is tomaximize the output displacement uout in o.

Equation (3) is developed for designing compliant mech-anisms with single input-output behavior. Although thedesign problem dealing in this section is a multi-outputproblem, only half of the design domain is taken into consid-eration due to symmetry. Therefore, the actually consideredoptimization process is a single output problem. In order toshow a clear material distribution, the topology of the wholedesign domain is plotted.

Finuout

uout

i

o

o

Fig. 4 The design domain of the push gripper

Fig. 5 The push gripper is obtained by using the proposed formula-tion without filtering the sensitivities: a optimized configuration, andb deformed configuration

3.1.1 The effect of the proposed method on preventingde facto hinges: with or without filtering the sensitivities

The maximum material usage is restricted to 30 % of themass of the design domain. The design domain is discretizedby 80 × 80 finite elements for the elastic analysis. The voidarea(gap size) is set to be 40 × 20 finite elements.

Firstly, the formulation is employed without using thefiltering method for smoothing the sensitivities. The opti-mal configuration is shown in Fig. 5a. Note that the finaltopology does not contain any of the de facto hinges whichconfirms the validity of the proposed formulation on sup-pressing the lumped compliance problem. The correspond-ing deformed configuration of the optimized configurationis shown in Fig. 5b. It is shown that the mechanism isfunctioning with well-distributed elastic deformation.

The final topology is suffering from the checkerboardpatterns. Secondly, in order to overcome the checkerboardpatterns and show a smooth material distribution, the designproblem is resolved by filtering the sensitivities. The fil-tering method has been one of the most popular methodsfor mitigating checkerboard patterns and mesh-dependencyin stiffness topology optimization (Sigmund 2001; Bendsøeand Sigmund 2003). Filtering methods can be divided into

Fig. 6 The optimal configuration of the push gripper obtained byusing the proposed method with filtering: a final topology, and b localenergy density

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Fig. 7 The convergencehistories of uout , Cin and Cout

density-based and sensitivity-based methods. In the density-based methods, each element density is redefined as aweighted average of the densities in a neighborhood ofthe element (Bourdin 2001; Guest J et al. 2004; Sigmund2007). In the sensitivity-based methods, after calculatingthe sensitivities, they are modified as weighted averagesof the sensitivities in the neighborhoods (Borrvall 2001;Wang and Wang 2005). The filtering methods have shownsome success at preventing the point flexure by present-ing some extra elements at the hinged areas (Luo Z et al.2005; Yin and Ananthasuresh 2003). However, it is still acase of lumped compliance. The sensitivity filtering methodthat is used in Sigmund (2001) is employed to preventthe checkerboard patterns due to its simplicity and easeof implementation.

The obtained optimized configuration and its corre-sponding local energy density are shown in Fig. 6. De factohinges do not appear in this example. It should also be saidthat lumped results as designed in the previous work usingthe external spring model (Bendsøe and Sigmund 2003)which tend to resemble rigid-body linkages of the equiv-alent kinematic functions do not appear either. The localenergy density shows when the mechanism is loaded, nearly

all parts of the mechanism are contributing to the deflec-tion at the output port (although the fixed points that nearthe upper and lower corners and also the input force appliedposition have a relatively higher compliance). Hence, it isin favour of decreasing stress concentration and possiblefatigue breakage.

The convergence histories of uout , Cin and Cout areshown in Fig. 7. The self-adjusting of the weighting fac-tors can be more intuitively seen from this figure. Note that

Fig. 8 Push gripper obtained with gap size 50 × 40 and volume ratio0.2: a the optimized configuration, and b its deformed configuration

A multi-objective method of hinge-free compliant mechanism optimization

Fig. 9 The optimalconfiguration of the pushgripper obtained by using thespring method: a final topology,and b local energy density. TheMatlab code can be directlyfound in reference (Bendsøe andSigmund 2003)

since the values of α and β can be easily obtained fromFig. 7, the corresponding curves are no longer performedhere. Oscillations occur at the initial 100 iterations. Afterα and β achieve a more stable level, the objective functionuses 230 iterations to converge to a minimum.

The proposed formulation (9) is a summation of threeterms with the same units. In some problems they canassume significantly different magnitudes, which can beseen from Fig. 7 where the values of uout , Cin and Cout are69.5830, 62.2639 and 309.8125, respectively. However, Itseems that this magnitude difference will not cause difficul-ties in the convergence and de facto hinges prevention. Inimplementation of (9), if the value of Cin(or Cout ) is large, itmeans that a small value of α (or β) will give a great weightof Cin(or Cout ).

To further demonstrate this, we here perform anotherexample. The push gripper design problem is resolved. Thedesign domain is discretized by 80 × 80 finite elements andthe gap size is set to be 50 × 40 finite elements. The maxi-mum material usage is restricted to 20 % of the mass of thedesign domain. The obtained configuration and its deformedconfiguration are shown in Fig. 8. It is shown that theobtained configuration is completely free of de facto hingesand the deformed configuration shows that the compliance

is well-distributed throughout the mechanism. In this exam-ple, the values of uout , Cin and Cout are 251.5652, 116.1999and 1868.2397, respectively. The input-output behaviourdue to a unit load is 116.1999-251.5652.

3.1.2 Comparison study

The obtained push gripper (Fig. 6a) has the input-outputbehaviour due to a unit load Fin as follows: uin is 62.2639and uout is 69.5830. In order to show the advantage ofthis method compared with the use of springs (Bendsøe andSigmund 2003), we here perform a push gripper obtained byusing the spring method that has close performances (input-output behaviour due to a unit load) with the mechanism inFig. 6a.

For using the spring method, the material usage is alsorestricted to be 30 % and the design domain is also dis-cretized by using 80 × 80 finite elements. Both of themathematical model and the matlab code can be foundin reference (Bendsøe and Sigmund 2003) (Page 96 andPage 269). In order to obtain a mechanism that has compa-rable performances with the mechanism shown in Fig. 6a,the stiffness of the input spring and the output spring are setto be 0.012 and 0.0016, respectively.

Fig. 10 The optimalconfigurations of the pushgripper obtained with differentmesh refinement: a 40 × 40,b 120 × 120, and c 360 × 360

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The obtained push gripper and its corresponding localenergy density are shown in Fig. 9. The resulting input-output behaviour due to a unit load is: uin (62.7823) anduout (68.9976).

The optimal configuration obtained by using the springmethod is clearly suffering form the hinge problem. Thelocal energy density distribution confirms that the flexibil-ity of a hinged mechanism is mainly provided in the localareas (hinged areas) which makes it suffer from the stressconcentration problem and hard to be manufactured in thesmall scale.

3.1.3 Effect of the mesh refinement

In most material density distribution-based methods, a finemesh will lead to a more topologically complex structureand this phenomena is called mesh-dependent.

In this section, to study mesh-independence, the designdomain is discretized by using 40 × 40, 120 × 120, and360 × 360 elements, respectively. The maximum materialusage is restricted to be 30 %.

The optimum material distributions are shown in Fig. 10,respectively. Note that the consistent designs can beobtained with the different meshing refinements.

3.1.4 Optimized configurations obtained with small Vmax

This section is denoted to confirm if the proposedmethod is capable for topology optimization of hinge-free compliant mechanisms with a small material usage,since the hinge problem is even a more important issuewhen the maximum material usage is restricted to bevery small.

Two cases are studied. The maximum material usage isrestricted to be 0.1 and 0.2, respectively. The final designsare shown in Fig. 11.

When the total material usage is set to be small, the gen-erated designs are more like the results that are generatedby using the ground structure method (Frecker et al. 1997).Note that no de facto hinges occur even when the maxi-mum material usage is restricted to be very small, e.g., 0.1.However, the small material restriction will result in a largeamount of elements with the intermediate density whichmakes the created mechanisms hardly be converted into lay-outs that can be manufactured. Topology extraction is stillan open topic and a complete investigation is beyond thescope of this paper.

3.2 Displacement inverter

Synthesis of the displacement inverter is one of themost well known benchmark problems in the compli-ant mechanisms topology optimization field. As shown

Fig. 11 The optimal configurations of the push gripper obtained bysetting mesh size 100 × 100 and different volume ratio: a V/Vmax =0.1, and b V/Vmax = 0.2

in Fig. 12, the left upper and left lower parts are fixedas the Dirichlet boundaries. Fin is horizontally appliedat the center point of the left side and a reverse dis-placement is expected at the center point of the rightside.

3.2.1 Optimization process

The whole design domain is discretized by 100 × 100 finiteelements for the elastic analysis. The maximum materialusage is restricted to be 30 %.

The optimization process takes 100 iterations to con-vergence. A number of intermediate solutions are shownin Fig. 13. The initial design is shown in Fig. 13a. Theoptimum material distribution is shown in Fig. 13f.

Obviously no hinges occur in the optimum design ofdisplacement inverter. As shown in Fig. 13f, an even dis-tribution of material is nearly over the whole structure ofthe inverter which confirms the validity of the proposedmethod. More, from the intermediate designs one can seethat the proposed method can prevent hinges not only

Fig. 12 The design domain of the displacement inverter

A multi-objective method of hinge-free compliant mechanism optimization

Fig. 13 The optimizationprocess of topology optimizationof the displacement inverter. aStep 1, b Step 10, c Step 20,d Step 40, e Step 70, f Step 100

in the optimal topology but also during the optimizationprocess.

4 Conclusions

A method for topology synthesis of hinge-free compliantmechanisms is presented based on continuum type topologyoptimization. In addition to maximize the output displace-ment, the input compliance and the output compliance areadded to the objective function in order to ensure thatthe resulting design is reasonably stiff. This automaticallyimplies the absence of hinges.

A new multi-criteria objective function is set by usingthe weighting sum method. The weighting factors are deter-mined based on the information of the output displacement,the input compliance and the output compliance obtainedfrom the previous step. Thus, there is no need for manuallypresetting the weighting factors and they change with eachiteration step.

Numerical examples that are widely studied in the litera-ture of compliant mechanism topology optimization are pre-sented for illustrating the validity of the presented method.It is shown that the proposed method can be used fortopology optimization of hinge-free compliant mechanisms.The obtained hinge-free compliant mechanisms only con-tain strip-like members which can greatly mitigate the stressconcentration and fatigue problem.

Acknowledgments This research was supported by the NationalScience Foundation of China (Grant No. 91223201, 50825504), theUnited Fund of Natural Science Foundation of China and Guangdongprovince (Grant No. U0934004), Project GDUPS (2010), and theFundamental Research Funds for the Central Universities(2012ZP0004). This support is greatly acknowledged.

References

Barthelemy JFM, Haftka RT (1993) Approximation concepts foroptimum structural design-a review. Struct Multidiscip Optim5(3):129–144

Bendsøe MP, Sigmund O (2003) Topology optimization: theory meth-ods and applications. Springer

Borrvall T (2001) Topology optimization of elastic continua usingrestriction. Arch Comput Methods Eng 8(4):351–385

Bourdin B (2001) Filters in topology optimization. Int J NumerMethods Eng 50(9):2143–2158

Chen SK (2007) Compliant mechanisms with distributed complianceand characteristic stiffness: a level set method. PhD thesis. TheChinese University of HongKong, China

Deepak SR, Dinesh M, Sahu DK, Ananthasuresh GK (2009) A com-parative study of the formulations and benchmark problems for thetopology optimization of compliant mechanisms. J Mech Robot1:1–8

Duysinx P, Bendsøe MP (1998) Topology optimization of continuumstructures with local stress constraints. Int J Numer Methods Eng43(8):1453–1478

Frecker M, Kikuchi N, Kota S (1999) Topological optimizationof compliant mechanisms with multiple outputs. Struct Optim17(4):269–278

B. Zhu et al.

Frecker MI, Ananthasuresh GK, Nishiwaki S, Kikuchi N, Kota S(1997) Topological synthesis of compliant mechanisms usingmulti-criteria optimization. J Mech Des 119(2):238–245

Guest J, Prevost J, Belytschko T (2004) Achieving minimum lengthscale in topology optimization using nodal design variables andprojection functions. Int J Numer Methods Eng 61(2):238–254

Howell LL (2001) Compliant Mechanisms. Wiley, New YorkKim JE, Kim YY, Min S (2005) A note on hinge-free topology

design using the special triangulation of design elements. Com-mun Nemerical Methods Eng 21(12):701–710

Luo JZ, Luo Z, Chen SK, Tong LY, Wang MY (2008) A newlevel set method for systematic design of hinge-free compliantmechanisms. Comput Methods Appl Mech Engrg 198:318–331

Luo Z, Chen L, Yang J, Zhang Y, Abdel-Malek K (2005) Compli-ant mechanism design using multi-objective topology optimiza-tion scheme of continuum structures. Struct Multidiscip Optim30(13):142–154

Nishiwaki S, Min S, Yoo Jetal (2001) Optimal structural design consid-ering flexibility. Comput Methods Appl Mech Eng 190(34):4457–4504

Poulsen TA (2003) A new scheme for imposing a minimum lengthscale in topology optimization. Int J Numer Anal Methods Eng57(6):741–760

Rahmatalla S, Swan CC (2005) Sparse monolithic compliant mech-anisms using continuum structural topology optimization. Int JNumer Anal Methods Eng 62(12):1579–1605

Saxena A, Ananthasuresh G (2000) On an optimal property of compli-ant topologies. Struct Multidiscip Optim 19(1):36–49

Sigmund O (1997) On the desgn of compliant mechanisms usingtopology optimization. Mech Struct Mach Int J 25(4):493–524

Sigmund O (2001) A 99 line topology optimization code written inmatlab. Struct Multidiscip Optim 21(2):120–127

Sigmund O (2007) Morphology-based black and white filters fortopology optimization. Struct Multidiscip Optim 33(4-5):401–424

Wang M, Wang XM, Guo DM (2003) A level set method for struc-tural topology optimization. Comput Methods Appl Mech Eng 192(1-2):227–246

Wang MY (2009) A kinetoelastic formulation of compliant mechanismoptimization. J Mech Robot 1(2):021,011

Wang MY, Wang S (2005) Bilateral filtering for structural topologyoptimization. Int J Numer Methods Eng 63(13):1911–1938

Wang NF, Zhang XM (2012) Compliant mechanisms designbased on pairs of curves. Sci China Technol Sci 55(8):2099–2106

Yin L, Ananthasuresh GK (2003) Design of distributed compliantmechanisms. Mechan Based Des Struct Mach 31(2):151–179

Yoon GH, Kin YY, Bendsøe MP, Sigmund O (2004) Hinge-freetopology optimization with embedded translation-invariant differ-entiable wavelet shrinkage. Struct Multidiscip Optim 27(3):139–150

Zhou H (2010) Topology optimization of compliant mechanismsusing hybrid discretization model. J Mech Des 132(11):111,003–111,010

Zhu B, Zhang X (2012) A new level set method for topology optimiza-tion of distributed compliant mechanisms. Int J Numer MethodsEng 91(8):843–871

Zhu B, Zhang X, Wang N (2013) Topology optimization of hinge-free compliant mechanisms with multiple outputs using level setmethod. Struct Multidiscip Optim 47(5):659–672