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Topology optimization of hinge-freecompliant mechanisms using level setmethodsBenliang Zhu a , Xianmin Zhang a , Nianfeng Wang a & SergejFatikow a ba Key Laboratory of Precision Equipment and ManufacturingTechnology of Guangdong Province, School of Mechanical andAutomotive Engineering , South China University of Technology ,Guangzhou , Guangdong , 510640 , PR Chinab Division Microrobotics, Department of Computing Science ,University of Oldenburg , Uhlhornsweg 84, A1, 26111 , Oldenburg ,GermanyPublished online: 10 May 2013.

To cite this article: Benliang Zhu , Xianmin Zhang , Nianfeng Wang & Sergej Fatikow , EngineeringOptimization (2013): Topology optimization of hinge-free compliant mechanisms using level setmethods, Engineering Optimization, DOI: 10.1080/0305215X.2013.786065

To link to this article: http://dx.doi.org/10.1080/0305215X.2013.786065

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Engineering Optimization, 2013http://dx.doi.org/10.1080/0305215X.2013.786065

Topology optimization of hinge-free compliant mechanismsusing level set methods

Benliang Zhua, Xianmin Zhanga*, Nianfeng Wanga and Sergej Fatikowa,b

aKey Laboratory of Precision Equipment and Manufacturing Technology of Guangdong Province, Schoolof Mechanical and Automotive Engineering, South China University of Technology, Guangzhou,

Guangdong, 510640, PR China; bDivision Microrobotics, Department of Computing Science, Universityof Oldenburg, Uhlhornsweg 84, A1, 26111 Oldenburg, Germany

(Received 4 September 2012; final version received 5 March 2013)

A new level set-based multi-objective optimization method is proposed for topological design of hinge-free compliant mechanisms. Firstly, the flexibility requirement of compliant mechanisms is formulated byusing the mutual energy. Two types of mean compliance are developed to meet the stiffness requirement.Secondly, several objective functions are developed for designing hinge-free compliant mechanisms basedon the weighting method in which a new scheme for determining weighting factors is used. Thirdly, severalnumerical examples are performed to demonstrate the validity of the proposed method. It is shown thatthe resulting compliant mechanism configurations contain only strip-like members which are suitable forgenerating distributed compliance and decreasing stress concentration.

Keywords: compliant mechanism; distributed compliance; level set method; weighted sum; de factohinges

1. Introduction

A compliant mechanism gains at least some of its mobility from the deflection of its flexiblemembers (Howell 2001; Sigmund 1997). The many advantages of compliant mechanisms haveproduced a growing interest in the synthesis method of compliant mechanisms. These approachescan be categorized into two types: the kinematics synthesis approach and the topology optimiza-tion synthesis approach (Rahmatalla and Swan 2005). In the kinematics synthesis approach, thecompliant mechanism is derived from a corresponding rigid-body mechanism (Howell 2001)while in the topology optimization approach, it does not need a known rigid mechanism and theobtained design has the optimum force displacement relationship.

Ever since Bendsøe and Kikuchi (1988) came up with the homogenization design method,structural topology optimization has been paid more attention and several methods havebeen developed and applied to certain problems such as structural flexibility minimization(Bendsøe and Sigmund 2003; Sigmund 2001), thermal diffusivity maximization problems(Yamada, Izui, and Nishiwaki 2011), and the design of compliant mechanisms (Luo, Tong, andWang 2007; Zhu and Zhang 2011) during the past decades. The black-and-white design is a verypowerful tool for determining the geometric representation of a structure even though optimal

*Corresponding author. Email: zhangxm@scut.edu.cn

© 2013 Taylor & Francis

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2 B. Zhu et al.

design problems are often ill-posed (Bendsøe and Sigmund 1999). Hence, some element basedmethods, including the homogenization design method (Bendsøe and Kikuchi 1988) and the SIMPmethod (Bendsøe and Sigmund 2003) have been developed to relax the black-and-white optimaldesign problem and make it well-posed. Substantial research has been done for extending thistheory (Bendsøe and Sigmund 2003; Fuchs, Jiny, and Peleg 2005; Sigmund 2007) and a criticalreview can be found in Rozvany (2009).

Level set (Osher and Fedkiw 2002; Sethain 1999) based topology optimization methods haveemerged as an attractive alternative method for topology optimization. In level set based methods,the structural boundary is treated as the design parameter and implicitly embedded into one higherdimensional scalar function. This can easily handle topology changes, e.g. merging and splitting,compared to traditional shape optimization. After Sethian and Wiegmann (2000) first introducedlevel set methods to do structural boundary design, Osher and Santosa (2001) introduced shapesensitivity analysis (Sokolowski and Zolesio 1992; Ta’asan 2001) into level set model workas velocity field construction and this work has been further developed by Wang, Wang, andGuo (2003) and Allaire, Jouve, and Toader (2004). Recent successful work using level set-basedtopology optimization includes the solution of multi-material structures (Wang and Wang 2004),structural optimization considering geometrical nonlinearity (Ha and Cho 2008; Van Dijk et al.2010), topology optimization of compliant mechanisms (Chen 2007; Zhu and Zhang 2011), etc.For overcoming some unfavourable features, e.g. the Courant–Friedrichs–Lewy (CFL) conditionand the new hole creating prohibition, in standard level-set-based topology optimization methodssome new methods have been developed such as the parameterization level set method (Luoet al. 2008a; Luo, Tong, and Wang 2007) and the semi-implicit level set method (Luo et al.2008b).

It is well-known that, for topology optimization of the stiffest structures, there is a universallyaccepted formulation (minimizing the mean compliance subject to a constraint on the volumeof material used (Bendsøe and Sigmund 2003). However, a universally accepted formulationdoes not exist for the topology optimization of compliant mechanisms, although several availableformulations have been developed over the past decades (Deepak et al. 2009).

These formulations can be studied under two main groups, the first of which is established bymaximizing some kind of mechanical measurements. Sigmund (1997) proposed a formulationin which the mechanical advantage (MA) is employed as the objective function. In his method,the maximum input displacement is also limited for indirectly controlling the maximum stresslevel. Some available formulations created based on other mechanical measurements, such asgeometrical advantage (GA) (Chen 2007; Zhu and Zhang 2012), mechanical efficiency (ME)(Luo et al. 2008a) and output displacement (Bendsøe and Sigmund 2003; Luo and Tong 2008)have also been developed. A comparative study of different objective functions for topologyoptimization of compliant mechanisms can be found in Deepak et al. (2009). The second of whichgroups is formulated by treating the design as a multi-objective problem where flexibility andcompliance are both considered to meet the function requirement and the strength requirement ofa compliant mechanism (Nishiwaki et al. 1998).Ananthasuresh (1994) proposed a weighted linearcombination formulation to accomplish simultaneously the two requirements of both flexibility(maximizing mutual potential energy) and compliance (minimizing strain energy). However,choosing the value of the normalized weight in the global sense is rather difficult. To overcomethis limitation, Frecker et al. (1997) proposed an alternative multi-criteria objective of maximizingthe ratio of the flexibility requirement and the mean compliance. Nishiwaki et al. (1998) consideredmultiple objectives in the optimization of compliant mechanisms by using the homogenizationmethod, but found that the use of the homogenization method is somewhat troublesome. Saxenaand Ananthasuresh (1998) proposed an energy based method to accomplish two requirements ofthe compliant mechanisms topology optimization by maximizing the flexibility, minimizing thecompliance and maximizing the mechanical advantage of the mechanism simultaneously. Luo

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Engineering Optimization 3

et al. (2005) proposed a new multi-objective method for the topology optimization of compliantmechanisms, in which a hybrid-filtering scheme is developed for solving numerical instabilities. Itwas shown that the scheme can also prevent one node connected hinges in the design of compliantmechanisms.

Compliant mechanisms can be classified into two categories: lumped compliant mechanismsand distributed compliant mechanisms (Howell 2001). As described by Ananthasuresh (1994),the flexibility of a lumped compliant mechanism is provided in localized areas (hinge areas) whilethe flexibility of a distributed compliant mechanism is distributed through the whole mechanism.

One significant challenge when applying the continuum topology optimization method to com-pliant mechanism designs is de facto hinges (Bendsøe and Sigmund 2003), i.e. the resultingcompliant mechanisms always have lumped compliance. Accordingly, a variety of strategies havebeen investigated for dealing with this problem in the continuous topology design of compliantmechanisms. Filter methods and specific constraints methods have been developed based on SIMPand the homogenization design method (Bendsøe and Sigmund 2003; Yoon et al. 2004; Sigmund2007). Poulsen (2003) proposed a MOnotonicity based minimum LEength(MOLE) scale methodto force the generated compliant mechanisms to have distributed flexibility. However, this methodcan only eliminate de facto hinges and cannot completely eliminate flexible hinge ports. Sigmund(2007) introduced a new class of morphology-based restriction schemes that work as densityfilters during the process of optimization to generate manufacturable solutions. By using twodistinctly different sets of springs, Rahmatalla and Swan (2005) proposed a formulation for thedesign of hinge-free compliant mechanisms. In terms of level set based methods, specific energyfunctionals are always built into the traditional optimization model to force the geometric widthof structural components in the created mechanisms. Using a quadratic energy functional, Wangand Luo (2011), Luo et al. (2008a), Chen, Wang, and Liu (2008) and Chen (2007) proposed anew objective function to generate hinge-free compliant mechanisms. The geometric width ofstructural components in the created mechanisms is constant, therefore it can greatly improvethe restructurability of the design. Wang and Chen (2009) and Chen (2007) proposed an intrin-sic characteristic stiffness method to design distributed compliant mechanisms. In their method,springs attached to the input and output ports are no longer needed. This method still suffers flex-ible hinges when a large objective geometrical advantage is set (Deepak et al. 2009). Using thegeometrical advantage as the objective function incorporating two types of mean compliance, Zhuand Zhang (2012) proposed a level set based method for the topology optimization of distributedcompliant mechanisms. De facto hinges can be successfully prevented.

In this study, a new multi-criteria level set based method for the topology optimization ofhinge-free compliant mechanisms is proposed. Three available formulations are developed basedon the weighted sum method by incorporating flexibility and two types of mean compliance.A self-adjusting scheme is developed for determining the weighting factors that are used in theformulations. During each optimal iteration step, the weighting factors are updated based on theresults in the previous optimization step. Certain optimization algorithms, e.g. the shape derivativeand gradient method, are employed to determine the final layout of the mechanisms. Numericalresults show that the resulting compliant mechanisms are completely free of de facto hinges andhave distributed compliance that favours decreasing stress concentration and possible fatiguebreakage.

The remainder of the article is organized as follows. In Section 2, the basic concepts of theproposed method are introduced. The structural topology optimization model is set. Three func-tions for the topology optimization of hinge-free compliant mechanisms are proposed. Necessaryoptimization algorithms are put forward. The shape sensitivity derivative is used as well as thegradient method for obtaining the optimal solution of the optimization problem. In Section 3, sev-eral numerical examples are presented to demonstrate the effectiveness of the proposed method.Finally, conclusions and a discussion for further work are developed.

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4 B. Zhu et al.

2. Optimization problem

2.1. De facto hinges and lumped compliance

As mentioned before, in the context of designing compliant mechanisms using continuum topologyoptimization methods, one difficulty is the tendency for de facto hinges to appear in the compliantmechanisms obtained. Compliant mechanisms containing de facto hinges should belong to thelumped compliant mechanisms since the flexibility of the mechanisms is provided only in the defacto hinge regions, as shown in Figure 1.

The causes of de facto hinges have been studied from the viewpoint of maximizing energytransfer, as in Luo et al. (2008a) and Chen (2007). Compliant mechanisms containing de factohinges are not truly compliant but rather they evolve to rigid-link mechanisms (Bendsøe andSigmund 2003). One of the biggest shortcomings is that the stress in the sharp hinge regions couldapproach infinity and the structure would break in the area of the de facto hinges. Moreover, itis very difficult to fabricate such de facto hinges for micro-scale mechanical systems. Therefore,for most real-world applications, hinge-free compliant mechanisms are preferable.

This study aims at developing an alternative method of designing hinge-free compliant mech-anisms based on constructing a specific trade-off between flexibility and stiffness during theoptimization process. In the proposed method, this trade-off can be kept stable at each optimiza-tion iteration to avoid generating de facto hinges. Note that the small linear deformation assumptionis used in this study since the goal is to design compliant mechanisms which qualitatively deformin the desired direction.

2.2. Flexibility formulation

As shown in Figure 2, consider a linear elastic body occupying a 2D domain D and fixed atboundary �d . Further, consider two cases, Case (a) and Case (b). In Case (a), the design domainis subjected to a traction tin at boundary �tin while in Case (b) the design domain is subjected toa traction tout at boundary �tout as shown, respectively, in Figure 2. Suppose that tin is the appliedinput force and the corresponding displacement field is uin, and tout is a unit dummy load and thecorresponding displacement field is uout. Now the mutual mean compliance CM can be used as ameasure of the flexibility of the designed compliant mechanism and defined as

CM = Lout(uin) = a(uout, uin), (1)

(a) (b)

Figure 1. A lumped displacement inverter compliant mechanism (the total volume of material used isVol = 0.2,Young’smodulus for the solid material is E = 1, the void area is assumed to have aYoung’s modulus of E = 0.001 and a Poisson’sratio of υ = 0.3).

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Engineering Optimization 5

(a) (b)

Figure 2. Schematic for calculating mutual mean compliance.

where

Lout(uin) =∫

�

toutuin d� (2)

a(uout, uin) =∫

DEijklεkl(uout)εij(uin) d�, (3)

where Eijkl and ε are the elasticity tensor and the linearized strain tensor, respectively.The mutual mean compliance CM is interpreted as the deformation of boundary �tout when tin

is applied at �tin . The larger the CM is, the more flexible(compliant) the designed mechanism is.Therefore, to accomplish the requirement for flexibility of the topology optimization of compliantmechanisms, CM should be maximized.

2.3. Compliance formulations: new determination cases

The idea of incorporating stiffness into a compliant mechanism topology design to ensure ameaningful design is not new. One widely used single loading condition for determining thestiffness is that the boundary of the applied load of the loading condition is considered to be fixedwhile a virtual load is applied at the output boundary (Frecker, Kikuchi, and Kota 1999; Nishiwakiet al. 1998; Luo et al. 2005). Another loading condition for determining the stiffness that needs tobe maximized is that only the input load is applied at the input boundary and an artificial springis attached to the output boundary (Saxena and Ananthasuresh 1998; Saxena and Ananthasuresh2000). Nishiwaki et al. (2001) proposed two loading conditions for determining the stiffness. Thefirst is that applying load at the input boundary and the output boundary is fixed, and the secondis that the boundary of the applied load of the loading condition is considered to be fixed while avirtual load is applied at the output boundary.

In this study, two new types of mean compliance that can be incorporated into the optimizationmodel are developed. Since the two proposed loading conditions do not involve changing theinput or output boundaries into fixed boundaries and, additionally, the mutual mean compliancecan also be determined based on the displacement field obtained from the two proposed loadingconditions, this can simplify the computational process of finite element analysis.

Suppose that the original design domain D for topology design of compliant mechanisms isfixed at �d and subjected to an input load Fin as shown in Figure 3(a). The first type of meancompliance is determined based on the case that an external unit force is only applied at the inputport i while keeping the output port o as a free boundary (unfixed) as shown in Figure 4(b), denotedas Cin. The second type of mean compliance is determined based on the case that an external unitforce is only applied at the output port o while keeping the input port i as a free boundary (unfixed)as shown in Figure 4(c), denoted as Cout.

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6 B. Zhu et al.

(a) (b) (c)

Figure 3. Schematics for a compliant mechanism design.

Figure 4. Flowchart of the proposed optimization procedure (A0 is used).

Therefore, using the finite element method, Cin and Cout can be simply written in discreteform as

Cin = fTinuin = uT

inKuin (4)

Cout = fToutuout = uT

outKuout, (5)

where fin is a vector consisting of zero except for the input port position, where its value is finas shown in Figure 3(b) and uin is the displacement field due to fin, respectively. fout is a vectorconsisting of zeros except for the output port position, where its value is fout as shown in Figure 3(c)and uout is the displacement field due to fout, respectively. K is the stiffness matrix at the global level.

Based on the above analysis, the mutual mean compliance CM can be written as

CM = FinuToutKuin. (6)

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Engineering Optimization 7

2.4. Optimization model for topology optimization of hinge-free compliant mechanisms

Towards the optimization of compliant mechanisms, the created mechanisms should satisfykinematic and stiffness requirements. The optimization problem for topology optimization of com-pliant mechanisms incorporating above ideas can be transferred into maximizing a monotonicallyincreasing function J(CM, Cin, Cout) of CM, Cin and Cout.

Two widely used methods dealing with multi-objective optimization problems are the weightedsum method and the ratio method (Nishiwaki et al. 1998; Nishiwaki et al. 2001; Saxena andAnanthasuresh 2000; Luo et al. 2005) which can be, respectively, written as follows:

J = �CM − (1 − �)[ωCin + (1 − ω)Cout] (7)

J = CM

ωCin + (1 − ω)Cout, (8)

where � and ω are the weighting factors.Traditionally, the values of � and ω used in Equations (7) or (8) need to be pre-set. Constants

are always employed through the whole optimization process (Nishiwaki et al. 2001; Luo et al.2005) and different values will lead to different topologies. It is difficult to find suitable valuesmathematically. Furthermore, choosing the values of � and ω in a global sense is rather difficult.Different design problems may need different weighting factors (Saxena andAnanthasuresh 2000).

A new scheme for updating the values of the weighting factors is developed in this paper.The underlying idea of the proposed method is to prevent the stiffness or the flexibility of aresulting compliant mechanism approaching values that are too high, i.e. the proposed methodcan guarantee a good trade-off between flexibility and stiffness. Therefore, de facto hinges can besuccessfully prevented. The optimization model A0, which is composed of an objective functionand a maximum material usage constraint, can therefore be written as follows:

A0 Minimize: J = −CM + αCin + βCout (9)

Subject to: Vol ≤ Volmax, (10)

where α and β are weighting factors for Cin and Cout, respectively. Volmax is the maximumallowable material usage.

If CM is much larger than Cin and Cout, a very weak and fragile structure will be obtained.Conversely, if Cin and Cout are much larger than CM, a very stiff structure will be obtained thatcannot accomplish the kinematic requirement of a compliant mechanism. Therefore, a simpleway to choose the values of the weighting factors α and β is that they can make the orders ofmagnitude of CM, αCin and βCout be of the same order.

Define a Lagrangian to convert the constrained optimization problem A0 into an unconstrainedproblem written as follows:

L = −CM + αkCin + βkCout + λ(Vol + χ2 − Volmax), (11)

where χ is a slack variable to convert the inequality mass constraint into an equality one. αk andβk are weighting factors changing with each iteration k of the optimization algorithm and theyare updated using the scheme

αk+1 ={

CkM/Ck

in k ≥ 1

0 k = 0(12)

βk+1 ={

CkM/Ck

out k ≥ 1

0 k = 0,(13)

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8 B. Zhu et al.

where CkM, Ck

in and Ckout denote the values of CM, Cin and Cout in the kth iteration, respectively.

From Equations (11), (12) and (13), the following equation can be obtained:

Ck+1M : αk+1Ck+1

in : βk+1Ck+1out = (Ck

M + �CM) :

(Ck

M + CkM

Ckin

�Cin

):

(Ck

M + CkM

Ckout

�Cout

),

(14)

where �CM, �Cin and �Cout are increments of CM, Cin and Cout from iteration k to k + 1,respectively.

Since compared to CM, Cin and Cout, �CM, �Cin and �Cout are small, especially when theoptimization process is close to convergence, the following equation can be obtained:

Ck+1M : αCk+1

in : βCk+1out ≈ Ck

M : CkM : Ck

M = 1 : 1 : 1. (15)

One can see that, during iteration k + 1, if CkM is larger than Ck

in or Ckout, then αk+1 > 1 or

βk+1 > 1. This will increase the weight of Cin or Cout to make the created mechanism much stiffer.Conversely, if Ck

M is smaller than Ckin or Ck

out, then αk+1 < 1 or βk+1 < 1. This will decrease theweight of Cin or Cout to make the created mechanism more flexible.

From the above analysis one can see that an n step optimization problem has n objectivefunctions. However, there is no need for manually operating the functions because the weightingfactors can actually self-adjust based on the information obtained from the previous iteration.

In the first iteration, e.g. k = 1, since α and β are set to be 0, the optimization problem A0 isactually reduced to

Minimizeφ

: J = −CM (16)

Subject to: Vol ≤ Volmax. (17)

This will lead to a very large uout and, at the same time, very small Cin and Cout (one can also seethis from the numerical examples in Section 3). Therefore, a very weak structure can be obtainedthat cannot perform kinematic outputs. However, after the first iteration, α and β will becomevery large to enhance the weighting of Cin and Cout. Therefore, a well-posed design can still beobtained.

2.5. Alternative models

Based on the weighting factors setting scheme, it is easy to see that there are two alternativefunctions, the first of which can be stated as follows:

A1 Minimize: J = −αA1CM + Cin + βA1Cout (18)

Subject to: Vol ≤ Volmax, (19)

where αA1 and βA1 are the weighting factors for CM and Cout, respectively. The scheme for updatingthe αA1 and βA1 can be stated as follows:

αk+1A1 =

{Ck

in/CkM k ≥ 1

0 k = 0(20)

βk+1A1 =

{Ck

in/Ckout k ≥ 1

0 k = 0,(21)

where CkM, Ck

in and Ckout denote the values of CM, Cin and Cout in the kth iteration, respectively.

αk+1A1 and βk+1

A1 are the values of αA1 and βA1 in the (k + 1)th step, respectively. One can see that

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Engineering Optimization 9

the following results can be easily obtained by following the method proposed in Equation (14):

αA1Ck+1M : Ck+1

in : βA1Ck+1out ≈ Ck

in : Ckin : Ck

in = 1 : 1 : 1. (22)

The second alternative function can be expressed as follows:

A2 Minimize: J = −αA2CM + βA2Cin + Cout (23)

Subject to: Vol ≤ Volmax, (24)

where αA2 and βA2 are the weighting factors for CM and Cin, respectively. The scheme for updatingthe αA2 and βA2 can be stated as follows:

αk+1A2 =

{Ck

out/CkM k ≥ 1

0 k = 0(25)

βk+1A2 =

{Ck

out/Ckin k ≥ 1

0 k = 0,(26)

where CkM, Ck

in and Ckout denote the values of CM, Cin and Cout in the kth iteration, respectively.

αk+1A2 and βk+1

A2 are the values of αA2 and βA2 in the (k + 1)th step, respectively. Similarly, thefollowing results can be easily obtained by following the method proposed in Equation (14):

αA2Ck+1M : βA2Ck+1

in : Ck+1out ≈ Ck

out : Ckout : Ck

out = 1 : 1 : 1. (27)

By using the proposed optimization model A0 (or A1 and A2), there is no need to set theweighting factors artificially. The resulting compliant mechanisms in this work are completelyfree of de facto hinges that favour decreasing stress concentration and possible fatigue breakage(as shown in Section 3).

The proposed method is developed based on a new design model and a self-adjusting weightingfactors choosing scheme without special requirements. Therefore, the implementation of theproposed method is extremely easy since no extra constraints need to be considered. Althoughthe weighting factors continue to change during the optimization process, their derivatives withrespect to the design variables are zero. Therefore the sensitivities of the proposed objectivefunction are also cheap to compute.

2.6. Level set based optimization method

Suppose that D is the design domain that completely contains the material domain �, and anotherdomain D \ � represents the void area. The underlying idea behind the level set method is torepresent the structural boundaries ∂� as the zero level set of one higher dimensional function φ

(Osher and Fedkiw 2002; Sethain 1999), therefore

φ(x, t) > 0 if x ∈ �

φ(x, t) = 0 if x ∈ �

φ(x, t) < 0 if x ∈ D\�,

⎫⎪⎬⎪⎭ (28)

where � indicates the structural boundary.The optimization process can be transferred into the evolution of the level set function φ, which

can be stated as∂φ

∂t+ Vn|∇φ| = 0, (29)

where Vn determines the motion of the interface, which can be derived from the shape sensitivityanalysis.

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10 B. Zhu et al.

Based on the above shape representation, the optimization problem (A0 is used as an example)for topology optimization of hinge-free compliant mechanisms can be incorporated in the levelset equation and rewritten as

Minimizeφ

: J(u, φ) = −CM(u, φ) + α(u, φ)Cin(u, φ) + β(u, φ)Cout(u, φ) (30)

Subject to:∫

DH(φ) d� ≤ [Vol]max, (31)

where u is the state variable and H(φ) is the Heaviside function defined as

H(φ) ={

1 if φ ≥ 0

0 if φ < 0.(32)

2.7. Shape sensitivity analysis and velocity field construction

This subsection is devoted to finding out the proper velocity field for solving the optimizationproblem (A0 is used as an example and the results can be directly extended for solving A1 andA2) using the gradient method and shape derivative analysis (Wang, Wang, and Guo 2003). Theprocess of level set based topology optimization can be seen as the level set function φ, changingwith time to minimize the objective function of Equation (9) while satisfying the constraint ofEquation (10).

Applying the Kuhn–Tucher conditions of Equation (11) and incorporating Equation (30)leads to ⟨

dL

dφ, ϕ

⟩=

⟨dJ

dφ, ϕ

⟩+ λ

⟨dVol

dφ, ϕ

⟩

=∫

D(VJ + λ)δ(φ)ϕ d� = 0 (33)

dL

dλ= Vol + χ2 − [Vol]max = 0 (34)

dL

dχ= 2λχ = 0 (35)

λ ≥ 0, (36)

where 〈dL/dφ, ϕ〉 denotes the Fréchet derivative of the regularized Lagrangian function L withrespect to φ in the direction of ϕ. δ(φ) is defined as follows:

δ(φ) = dH(φ)

dφ. (37)

Let

ξ ={

0 if Vol ≤ [Vol]max

λ if Vol > [Vol]max, λ ≥ 0, (38)

then the optimality conditions are reduced to⟨dL

dφ, ϕ

⟩=

∫D(VJ + ξ)δ(φ)ϕ d�. (39)

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Engineering Optimization 11

To ensure the decrease of the objective function L, the velocity field can be simply chosen byletting

Vg = −(VJ + ξ)ϕ. (40)

Then, the following condition can be obtained:⟨dL

dφ, ϕ

⟩= −

∫D(Vl + ξ)2ϕ2δ(φ) d� < 0, (41)

which can guarantee a decrease of L.In order to calculate Equation (40), we need to calculate VJ , i.e. the Fréchet derivative of the

objective function J . Taking the Fréchet derivative of Equation (30) with respect to φ in thedirection of ϕ leads to⟨

dJ

dφ, ϕ

⟩= − ∂J

∂CM

⟨dCM

dφ, ϕ

⟩+ ∂J

∂Cin

⟨dCin

dφ, ϕ

⟩+ ∂J

∂α

⟨∂α

∂φ, ϕ

⟩

+ ∂J

∂Cout

⟨dCout

dφ, ϕ

⟩+ ∂J

∂β

⟨∂β

∂φ, ϕ

⟩.

(42)

Even though α and β are variables during the whole optimization process, their value remainsconstant in each individual optimization step. Therefore, in Equation (42) the following conditionsare satisfied: ⟨

∂α

∂φ, ϕ

⟩= 0 (43)

⟨∂β

∂φ, ϕ

⟩= 0. (44)

Therefore, solving VJ can be transferred into solving the shape derivatives of CM, Cin and Cout.Considering a region � with a smooth boundary, a velocity V is applied along the boundary

for a short dummy time τ , mapping � into �τ . Considering a general cost functional

�(�, z) =∫

�

F(z) d� (45)

depends on the domain � as well as on a function z, which also depends on the domain.According to Sokolowski and Zolesio (1992) and Ta’asan (2001), the shape derivative of

�(�, z) can be written as

d�

dτ=

∫�

F(z)V · n d� +∫

�

Fz(z) d�, (46)

where n is the normal vector of the boundary. � is the boundary of �.Similarly, in terms of boundary functionals

�(�, z) =∫

�

F(z) d�. (47)

The shape derivative of �(�, z) can be written as follows:

d�

dτ=

∫�

(∇F(z) · n + κF(z))V · n d� +∫

�

Fz(z) d�, (48)

where κ is the mean curvature of the boundary �.

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By using Equations (46) and (46), the shape derivative of CM can be obtained directly as follows:⟨dCM

dφ, ϕ

⟩=

∫�N1

[∂(Finuout)

∂n+ κFinuout

]V · n d�

+∫

�2

[∂(foutUin)

∂n+ κfoutUin

]V · n d�

−∫

�

Eijklεij(Uin)εkl(uout)V · n d�,

(49)

where ε is the strain field. Uin is the displacement field due to Fin which equals to Finuin. Notethat, in the cases studied in this work, the Neumann boundary is zero since the force is applied ata point, so only the last term is considered in Equation (49).

Similarly, the shape sensitivity of Cin and Cout can be expressed as⟨dCin

dφ, ϕ

⟩=

∫�

Eijklε(uin)ε(uin)V · n d� (50)

⟨dCout

dφ, ϕ

⟩=

∫�

Eijklε(uout)ε(uout)V · n d�. (51)

For more details of sensitivity analysis, please refer to Allaire, Jouve, and Toader (2004), Choiand Kim (2005), Ta’asan (2001) and Sokolowski and Zolesio (1992).

2.8. Optimization algorithm

A flowchart of the proposed optimization method is shown in Figure 4. The optimization proce-dure for the topology optimization of hinge-free compliant mechanisms can be summarized asfollows.

Step 1. Initialize the level set function φ to represent the original structural boundary.Step 2. Do finite element analysis to obtain the displacement fields, Ck

M, Ckin and Ck

out.Step 3. Calculate the weighting factors α and β that are used in the next step. If k = 1, α and β

can be simply set to be 0.Step 4. Calculate the velocity field which defines the speed of propagation of the level set function

φ.Step 5. Solve the level set equation using ENO2 (Osher and Fedkiw 2002) to update the level set

function φ.Step 6. Check for convergence. Convergence is achieved if the volume is within 0.02 of the

required value Volmax and the values of CM, Cin and Cout in previous three steps are alsowithin 5% tolerance of the values in current step. If no convergence, return to Step 2.

During the process of optimization, in order to avoid the level set function φ becoming too flator steep (to decrease the numerical errors and ensure the stability of the optimization process),the level set function φ needs to remain as a signed distance function (SDF), i.e. |∇φ| = 1. In thisstudy, the reinitialization procedure (Osher and Fedkiw 2002) is employed. Therefore, Equation(29) can be reduced to

∂φ

∂t= −Vn. (52)

Therefore, from time to time, Equations (52) can be approximately rewritten as

φk+1 − φk = −�tVn, (53)

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Engineering Optimization 13

Figure 5. Design domain of the displacement inverter.

where Vn is obtained from the shape sensitivity analysis and the steepest descent method isemployed for evolving the level set function in this study. �t should satisfy the CFL condition

�t ≤ min(�x, �y)

max(|Vn|) , (54)

where �x and �y are the grid spaces in the horizontal and vertical directions, respectively.

3. Numerical examples

In this section, the proposed method is applied to the design of two compliant mechanisms.For the following numerical cases, the artificial material properties are described as: Young’smodulus for the solid material is E = 1 while the void area is assumed to have aYoung’s modulusof E = 0.001 and a Poisson’s ratio of υ = 0.3 for both solid and void areas. The second orderaccurate essentially non-oscillatory ENO2 (Osher and Fedkiw 2002) is employed for numericallysolving the level set equation.

3.1. Displacement inverter

Synthesis of the displacement inverter is one of the most well known benchmark problems in thecompliant mechanisms topology optimization field. As shown in Figure 5, the design domain ofthe compliant inverter mechanism has the dimensions 2 mm × 2 mm. The left upper and left lowerparts are fixed as the Dirichlet boundaries. A force Fin = 100 μN is applied horizontally at thecentrepoint of the left side and a reverse displacement is expected at the centrepoint of the rightside. In this paper, A0 is used as the default optimization model, although the results obtainedusing A1 and A2 also testify to the effectiveness of the proposed method.

3.1.1. The displacement inverter design using A0

In order to compare the result with the lumped displacement inverter as shown in Figure 1,the maximal material usage is restricted to 20%. The design domain is discretized with 80 × 80quadrilateral elements. Note that, owing to the symmetry, only the lower half of the design domainis taken into consideration for elastic analysis.

The optimization process was run for 200 iterations. The final topology is shown in Figure 6(a)and the corresponding local energy density is shown in Figure 6(b).

It should be noted that the present formulations can ensure elimination of de facto hinges.Figure 6(a) shows that a continuous, hinge-free compliant displacement inverter can be obtained

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by using A0 without using any extra constraints. It should be also noted that the broken parts thatare shown inside the circles in Figure 1(a) are the de facto hinge regions. The flexibility of thedesign is mainly concentrated in the hinged regions, which can be seen from Figure 1(b).

When the hinge-free displacement inverter is loaded, nearly all parts of the mechanism arecontributing to the deformation as shown in Figure 6(b). When a lumped displacement inverteras shown in Figure 1 is loaded, just material around the hinge regions (as shown in the circles inFigure 1(a)) will contribute to the deflection at the output port. This makes the lumped compliantmechanisms suffer hinge stress concentration. From this point of view, hinge-free compliantmechanisms are preferable to lumped compliant mechanisms in the sense that they decreasestress concentration and possible fatigue breakage.

There is no doubt that for a given structural material and a given material usage constraint,hinged compliant mechanisms generally feature much higher flexibility than the family of hinge-free designs. A quantitative comparison can be seen in the following section. However, it must beborne in mind that the hinge design itself is very fragile. Further, hinged designs are very difficult tofabricate, especially on the micro-scale. Therefore, hinge-free compliant mechanisms such thoseobtained in this work are believed to be superior to compliant mechanisms that contain hinges inthe sense of manufacturability and durability. Further, if a hinge-free compliant mechanism withhigher flexibility is required, the designer can resort to using more compliant structural materialas proposed in Rahmatalla and Swan (2005).

Figures 7(a)–7(f) represent some intermediate designs in their half form. Topology change ismainly concentrated at the first 120 iterations. The proposed method can prevent de facto hingesin the resulting compliant mechanisms not only in the final design but also during the optimizationprocess.

Figures 8(a)–8(f) represent the corresponding level set surfaces of the intermediate designsshown in Figure 7. With the evolving level set function, topology changes such as merging canbe obtained naturally. Note that the re-initialization procedure, which is used for maintainingthe level set function as an SDF, is employed in this study. Therefore, during the optimization,new holes cannot be generated. For using the proposed formulations based on a conventionallevel set method, a certain number of holes should be preset in the initial design. However, theauthors believe that reasonable designs also can be obtained by using the proposed formulationsincorporating other level set methods such as the parameterization level set method.

(a) (b)

Figure 6. Final design of the displacement inverter: (a) final topology; (b) local energy density.

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Engineering

Optim

ization15

(a) (b) (c)

(d) (e) (f)

Figure 7. Intermediate designs of the displacement inverter: (a) step 1; (b) step 30; (c) step 75; (d) step 120; (e) step 150; and (f) step 180.

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Figure 8. Level set surface plots of the intermediate designs of the displacement inverter: (a) step 1; (b) step 30; (c) step 75; (d) step 120; (e) step 150; and (f) step 180.

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Engineering Optimization 17

Table 1. Characteristics comparison between the hinged displacement inverter (as shown in Figure 1(a))and the hinge-free design (as shown in Figure 6(a)).

Type Input load Input displacement Output displacement Maximum stress

Hinged 100 42.1210 −38.9983 765.1055Hinge-free 100 11.0198 −5.8856 7.033

3.1.2. Convergence of the design functions

Figure 9 shows the convergent curves of the CM, volume ratio, two types of mean compliance andthe Lagrange objective function ranging from 0 to 200 iterations. When the CM increases too highdue to the violation of the volume constraint, the CM will change in an opposite direction due tothe violation of two mean compliances to avoid generating a disconnected structure due to highflexibility. Figure 9 shows that the optimization is almost completed at iteration 100. However,from Figure 7 one can see that the last 100 iterations are still necessary to ensure a uniformdistribution of the material.

The multi-objective function proposed in this study actually provides two kinds of velocityfields. The first kind is produced by CM which makes the created mechanism a compliant one.The second kind is produced by the weighting sum of the Cin and Cout which make the createdcompliant mechanism free of de facto hinges.

The main difference among the multi-objective scheme proposed in this study and other multipleobjective functions is that the weighting factors of the objectives does not need to be pre-set andthey can self-adjust during the optimization process. Figure 10 shows the weighting factors curvesof two types of mean compliance. It is easy to note that at the first 80 iterations, due to the topologychange, α and β change by a relatively large amount. And after 80 iterations, α and β basicallyremain unchanged. One can see that, during the whole optimization process, owing to the changeof weighting factors, αCin, βCout and CM remain at the same level.

3.1.3. Comparison with the hinged design

The main contribution of this article is the new development of optimization models in whichde facto hinges and lumped compliance issues can be successfully solved. The standard levelset method is employed for solving the optimization problems in this study, which means theCFL condition and the steepest descent method are also employed. Therefore, the convergenceefficiency does not differ from that of the standard level set method such as in Allaire, Jouve, andToader (2004) and Wang, Wang, and Guo (2003). The comparison study of the results obtainedby using the proposed method and the literature has been added in terms of the flexibility and themaximum Von Mises stress.

The set of computational results presented in Table 1 is intended to make a quantitative com-parison between the hinged displacement inverter and the hinge-free design obtained in this work.The numerical results in the Hinged row are obtained based on the device shown in Figure 1, andthe numerical results in the Hinge-free row are obtained based on the device shown in Figure 6(a).They have the same material usage constraint (20%), the same material properties and the sameinput load. Note that the maximum Von Mises stress of the hinged inverter approach very highvalues because the elements in the hinged areas have very small stiffness and very large deforma-tion. The high stress condition can be improved by replacing the de facto hinges with small-lengthbeams. However, the compliant mechanisms obtained using the replacement method still sufferthe stress concentration problem.

One can see that, by eliminating de facto hinges, the maximum Von Mises stress due to the inputload can be tremendously reduced. Comparing the input displacements of the two mechanisms

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Figure 9. Convergence history of the displacement inverter: (a) CM; (b)volume ratio; (c) Cin and Cout; and (d) objectivefunction.

Figure 10. Weighting factors curve of the displacement inverter.

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Engineering Optimization 19

Table 2. Average CPU time comparison between the proposed two mean compliancesand conventional stiffness determinations.

Method FEA(s) Velocity construction(s) Level set evolution(s)

Equation (9) 5.7649 0.2374 0.0301Conventional 5.8613 0.2378 0.0300

studied for the same input load, one can see that the hinge-free inverter obtained in this work isstiffer.Additionally, the mean compliances introduced clearly have direct impact on the magnitudeof the output displacements.

For obtaining a continuous, hinge-free complaint mechanism that has higher flexibility, onecan impose much more restrictive material usage constraint values. Since the resulting compliantmechanisms do not suffer the hinge problem, a small material usage constraint can be used forachieving higher flexibility. Further, from the results shown in Table 1, one can see that the obtainedhinge-free compliant mechanisms can bear a large input load for achieving a higher kinematicperformance since the stress concentration is no longer an issue.

3.1.4. The effectiveness of the proposed compliance determined cases with respect to the finiteelement analysis

As discussed in Section 2.3, since the two proposed loading conditions for determining the twotypes of mean compliance do not involve changing the input or output boundaries into fixedboundaries, the computational process of the finite element analysis can be simplified. This sectionis devoted to examining the above analysis quantitatively.

The average CPU time of three main numerical steps during one iteration is shown in Table 2.Conventional means that the determination of the compliances involves changing the boundary

condition, see for example Nishiwaki et al. (2001).One finds that the finite element analysis is the most costly step during one iteration. The

proposed method does save CPU time for doing the finite element analysis although very littleCPU time can be saved. More importantly, the mutual mean compliance can also be determinedbased on the displacement field obtained from the two proposed loading conditions, which cansimplify the computational process of the finite element analysis.

3.1.5. Designs obtained by using A1 and A2

There are three possible optimization models for topology optimization of hinge-free compliantmechanisms proposed in this study. The above results have proved the validity of the optimizationmodel A0. This section is devoted to examining the validity of optimization models A1 and A2 bydesigning the displacement inverter. The design domain is discretized using 80 × 80 elements forthe elastic analysis. For both studied cases, the input load is set to be 100 μN and the maximummaterial usage constraint is set to be 20%.

The final designs of the displacement inverter obtained by using A1 and A2 are shown inFigures 11(a) and 11(b), respectively. One can see that results can be obtained that are identicalwith the outcome obtained using A0. The results confirm the proposed self-adjusting weightingfactors scheme for designing hinge-free compliant mechanisms.

There are three objectives in A0 (A1 or A2) that need to be minimized or maximized, i.e. CM,Cin and Cout. The main difference among A0, A1 and A2 is that the reference objective is chosendifferently. For instance, in A0, CM is chosen as the reference objective, while in A1, Cin is chosen.Therefore, during the optimization process, the values of the weighting factors in A0 are different

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(a) (b)

Figure 11. Final displacement inverter designs using different optimization models: (a) A1 and (b) A2.

from the values in A1. Since the same results can be obtained by using any of the three optimizationmodels, the following numerical results in the article are obtained based on A0.

3.1.6. Designs with different meshing refinement

Since the proposed method is independent of the underlying finite element mesh refinement, theoptimal design will also be independent of the mesh refinement in general. Thus, the result of theoptimization problem should not be changed if the finite element mesh is refined. To study meshindependence, three cases are studied.

The design domain is discretized by using 60 × 60 = 3600, 120 × 120 = 14, 400 and 160 ×160 = 25, 600 elements, respectively. The maximal material usage is restricted to 20%. Theoptimum material distributions are shown in Figure 12, respectively.

Note that de facto hinges do not occur in all the designs, even though rather bad discretizationis used (60 × 60). Consistent designs can be obtained with different meshing refinements. Forthe finite element analysis, a fine discretization is surely capable of capturing the spatial partialderivative and the elastic analysis. The examples illustrate that the proposed method does notsuffer from the mesh-dependent problem.

3.1.7. Designs with different maximum material usage

In this section, the effect of the maximum material usage on the final design of the displacementinverter is examined. Two values of the maximum material usage constraint Volmax are examined,corresponding to 10 and 30% of the whole design domain, respectively. The design domainis discretized into 80 × 80 finite elements for the elastic analysis. The final designs and thecorresponding local energy density plots are shown in Figure 13.

The de facto hinges problem could become a more important issue when the maximum materialusage is restricted to be very small (Bendsøe and Sigmund 2003). However, in the proposedmethod, when the total material usage is set to be small, a connected compliant displacementinverter can also be obtained as shown in Figures 13(a) and 13(b). The de facto hinges problemis no longer an issue.

From Figure 13 and the results shown in Figure 6, one can see that hinge-free compliantmechanisms can be obtained no matter whether the maximum material usage is set to be large or

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Engineering Optimization 21

Figure 12. Final displacement inverter designs with different mesh refinement: (a) 60 × 60; (b) 120 × 120; and (c)160 × 160.

small. However, the topology of the final designs is affected by changing the maximum materialusage.

(a) (b)

Figure 13. Final displacement inverter designs with different maximum material usage: (a) Vol = 10% and (b)Vol = 30%.

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Figure 14. The design domain of the push gripper.

3.2. Push gripper

The second example is a push gripper design using the proposed multi-objective scheme. Thesize of the design domain is 2 mm × 2 mm as shown in Figure 14. It is discretized with 80 × 80quadrilateral elements. The left upper and left lower parts are fixed as the Dirichlet boundaries.A force of Fin = 100 μN is applied horizontally at the centrepoint of the left side and the ver-tical displacements of the outer jaws are maximized. The maximal material usage is restrictedto 20%.

3.2.1. Design obtained by using A0

In this section, the size of the gap is set to be 0.4 mm × 0.4 mm and A0 is used. The optimiza-tion process was run for 400 iterations. The final topology is shown in Figure 15(a) and itscorresponding local energy density plot is shown in Figure 15(b).

As shown in Figure 15(b), the obtained push gripper is completely free of de facto hinges.Further, the push gripper configuration obtained only contains strip-like members that favourperforming distributed compliance and decreasing stress concentration.

Figures 16(a)–16(d) show the convergent curves of the CM, volume ratio, two types of meancompliance and the Lagrange objective function ranging from 0 to 400 iterations. At the initial

(a) (b)

Figure 15. The final design of the push gripper: (a) final topology; (b) local energy density plot.

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Engineering Optimization 23

Figure 16. Convergence history of the push gripper: (a) CM; (b)volume ratio; (c) Cin and Cout; and (d) objective function.

Figure 17. Final push gripper designs with the same volume constraint and different gap sizes: (a) 1 mm × 0.1 mm, (b)1 mm × 0.5 mm, and (c) 0.5 mm × 1 mm.

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20 iterations, the objective function increases due to the violation of the volume constraint. Afterthat, owing to the violation of the two mean compliances, the objective function CM changes inthe opposite direction and keeps slightly changing for the remaining iterations. The optimizationis almost completed at iteration 100. However, the last 300 iterations are still necessary to ensurea uniform distribution of the material.

3.2.2. Designs with different sizes of the gap area

This section is devoted to examining the capability of A0 to design the hinge-free push gripperwith different sizes of the gap area. Three cases are studied in which the size of the gap is set tobe 1 mm × 0.1 mm, 1 mm × 0.5 mm and 0.5 mm × 1 mm, respectively. The maximum materialusage constraint is set to be 25% in all cases.

The corresponding final designs of the three studied cases are shown in Figure 17. One cansee that hinge-free mechanisms can be obtained no matter what the size of the gap is. The resultsobtained in this section confirm that the proposed method can be used for the topology optimizationof hinge-free compliant mechanisms.

4. Conclusions

A level set based multi-objective method for the topology optimization of hinge-free compliantmechanisms has been developed in this study.

In the proposed method, there are three objectives that need to be either maximized (CM) or min-imized (Cin and Cout). The proposed formulations are established using the weighted sum methodin which a new self-adjusting weighting factors setting scheme is used. Three possible optimizationmodels are proposed – A0, A1 and A2. Although the weighting factors continue to change duringthe optimization process, their derivatives with respect to the design variables are zero. There-fore, the sensitivities of the proposed objective function are also cheap to compute. The numericalresults show that all three models can be used for the topology optimization of hinge-free compliantmechanisms and that identical results can be obtained using the three optimization models.

The main characteristics of the present method are summarized as follows.

(1) Two types of mean compliance are developed based on the initial design loading conditions.Since the calculation of two mean compliances does not involve changing the input or out-put boundaries into fixed, the computational process of the finite element analysis can besimplified.

(2) It is shown that the compliant mechanisms obtained in this work are completely free ofde facto hinges, which can greatly mitigate the stress concentration and fatigue problem.Further, the compliant mechanism configurations obtained only contain strip-like membersthat are suitable for manufacturing (especially at the micro-scale) and generating distributedcompliance.

(3) The proposed method is very easy to apply since no extra constraints or other filtering schemesneed to be added. It guarantees the existence of an optimal design for the topology optimizationof compliant mechanisms problem and is computationally cheap.

(4) The proposed method incorporates the standard level set method, which means that numericaldifficulties such as the CFL condition and a lack of capability to generate new holes still exist.Therefore, a huge amount of computation (hundreds of iterations) was needed to obtain theoptimal solutions shown in the proposed numerical examples.

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Acknowledgements

This research was supported by the National Science Foundation of China (Grant Nos 91223201 and 50825504), theUnited Fund of the Natural Science Foundation of China and Guangdong Province (Grant No. U0934004), ProjectGDUPS (2010), and the Fundamental Research Funds for the Central Universities (2012ZP0004). This support is greatlyappreciated.

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