Research ArticleDesign of Large-Displacement CompliantMechanisms by Topology Optimization IncorporatingModified Additive Hyperelasticity Technique

Liying Liu,1,2 Jian Xing,3 Qingwei Yang,1 and Yangjun Luo3

1School of River and Ocean Engineering, Chongqing Jiaotong University, Chongqing 400074, China2School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China3State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China

Correspondence should be addressed to Yangjun Luo; yangjunluo@dlut.edu.cn

Received 16 November 2016; Accepted 12 January 2017; Published 9 February 2017

Academic Editor: Giovanni Garcea

Copyright © 2017 Liying Liu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is focused on the topology design of compliant mechanisms undergoing large displacement (over 20% of the structuraldimension). Based on the artificial springmodel and the geometrically nonlinear finite element analysis, the optimization problem isformulated so as tomaximize the output displacement under a givenmaterial volume constraint. Amodified additive hyperelasticitytechnique is proposed to circumvent numerical instabilities that occurred in the low-density or intermediate-density elementsduring the optimization process. Compared to the previous method, the modified technique is very effective and can provide moreaccurate response analysis for the large-displacement compliant mechanism. The whole optimization process is carried out by thegradient-based mathematical programming method. Numerical examples of a force-inverting mechanism and a microgrippingmechanism are presented. The obtained optimal solutions verify the applicability of the proposed numerical techniques and showthe necessity of considering large displacement in the design problem.

1. Introduction

With the advantages of fast fabrication and easy miniatur-ization, compliance mechanisms have been widely used inmodern drivetrain systems, aerospace structures, electronicequipment, bioengineering, biomedical devices, and manyother applications. The optimal layout of compliant mecha-nisms can be typically found by using the topology optimiza-tion methods. Such a design problem can be viewed as atrade-off between the flexibility to deliver required motionsand the rigidity to overcome external resistance.

Early attention to the topology optimization of compliantmechanisms was focused on the mathematical modeling ofthe design problem. Among others, Ananthasuresh et al. [1]and Saxena and Ananthasuresh [2] used the combination ofoutput displacement and strain energy as the objective func-tion. In this formulation, an actuation force was applied at theinput port and a spring was added to the output port. In order

to make the optimal mechanisms-like structure adaptedto the stiffness of the workpiece, Sigmund [3] presentedan optimization formulation to maximize the mechanicaladvantage (MA) of compliant mechanisms. Hetrick and Kota[4] proposed an energy efficiency criterion, which is the ratiobetween the net energy transferred at the output and theinput energy. In addition, an artificial spring formulation [5]has also been introduced to address this problem, in whichthe actuator’s stiffness was modeled by an input spring andonly the output displacement was taken as the objectivefunction. Recently, a comparative study by Deepak et al.[6] revealed that all above formulations yield almost similartopologies, though they show different convergence ability.Based on these optimization formulations and the linearfinite element analysis, topology optimization of small-displacement compliant mechanisms has been successfullyrealized by using some familiar approaches including thesolid isotropic material with penalization (SIMP) approach

HindawiMathematical Problems in EngineeringVolume 2017, Article ID 4679746, 11 pageshttps://doi.org/10.1155/2017/4679746

2 Mathematical Problems in Engineering

[7], the evolutionary structural optimization (ESO) method[8, 9], the level set-based topology optimization method [10,11], and some nodal variable-based methods [12, 13].

In many circumstances, compliant mechanisms undergolarge displacement during operation. To obtain a meaningfuloptimal topology, it is necessary to accurately describe thenonlinear large-displacement structural behavior of com-pliant mechanisms in the optimization process. Therefore,many works have been published in the field of topologyoptimization of compliant mechanisms by incorporating thegeometrically nonlinear finite element analysis. Based on theSIMP approach and the total Lagrangian finite element for-mulation, Pedersen et al. [14] proposed a synthesis tool for thedesign of large-displacement and path-generating compliantmechanisms. Their results showed that the practical outputdisplacement obtained by nonlinear topology optimizationwould be up to 2.5 times the output displacement obtained bythe linearmethod. Sigmund [15] further extended the nonlin-ear method to the topology optimization of thermoelectricalmicro actuators. Bruns and Tortorelli [16] investigated thetopology optimization problem of compliant mechanismsconsidering geometrical and material nonlinearities. Theysuggested an element removal and reintroduction strategy torelieve the instability associated with mesh distortion in low-density elements [17]. Maute and Frangopol [18] designedmicroelectromechanical systems (MEMS) by geometricallynonlinear topology optimization accounting for stochasticuncertainties. Luo and Tong [19] developed a parameteriza-tion level set method for shape and topology optimization ofcompliant mechanisms involving large deformation. In thesestudies, the motion of compliant mechanisms is typicallylimited by about 10% of the length scale of the design domain.

As the literature survey reveals, topology optimizationwith geometrically nonlinear analysis provides an attractiveway for the design of compliant mechanisms. However,few studies address the topology optimization of compliantmechanisms undergoing very large displacement (e.g., over20% of the structural dimension), partly because of theextreme distortion of mesh and serious “element instabilityphenomenon” exhibited in the optimization process. In thiscase, the tangent stiffness matrix of distorted elements willbecome negative definite, which leads to serous nonconver-gence of the nonlinear finite element analysis as well as thetopology optimization process.

In this paper, we aim to develop an effective method ofthe topology optimization for large-displacement compliantmechanisms.The artificial spring formulation associatedwiththematerial interpolation scheme is adopted in the optimiza-tion model. The additive hyperelasticity technique reportedin our previous work [20] is extended and improved tonot only eliminate the element instability phenomenon thatoften appears in problems with large deformation, but alsoprovide more accurate response analysis. After obtaining thesensitivity information, the topology optimization problem issolved by using the Method of Moving Asymptotes (MMA)[21]. Finally, twonumerical examples are presented to validatethe proposed technique and to investigate the effects oflarge displacement on the optimal topology of compliantmechanisms.

Design domain

Fin

kin kout

uout

VoidSolid

Figure 1: Design problem of compliant mechanisms.

2. Topology Optimization Formulation ofCompliant Mechanisms

We consider a general design problem of compliant mech-anisms for maximizing specified output displacement oroutput force under a given input force. As mentioned in theIntroduction, several optimization formulations have beenproposed for such a problem.Here, the artificial springmodelproposed by Bendsøe and Sigmund [5] is employed. To thisend, two artificial springs are attached to the design domain.As shown in Figure 1, one spring is applied at the output portto simulate the resistance from a workpiece, and another isput at the loading point to exert a certain control on the netinput work.

By taking the material relative densities as design vari-ables, the topology optimization of compliant mechanisms isformulated as

max𝜌

𝑢out (𝜌)s.t. R (u (𝜌) ,𝜌) = 0,

𝑁∑𝑒=1

𝜌𝑒𝑉𝑒 ≤ 𝑉∗,0 < 𝜌min ≤ 𝜌𝑒 ≤ 1, (𝑒 = 1, 2, . . . , 𝑁) ,

(1)

where 𝑢out(𝜌) denotes the resulting output displacement, 𝜌 =[𝜌1, 𝜌2, . . . , 𝜌𝑁]𝑇 is the vector of elemental relative densities,𝑁 is the total finite element number of the discretized designdomain, and u and R are the global displacement vectorand the global residual force vector, respectively. Note thatR depends nonlinearly on the elemental material properties,which are modeled as a power-law function of the elementalrelative densities by the SIMP approach. The equilibriumstate of the large-displacement compliant mechanism isrepresented by R(u(𝜌),𝜌) = 0 and should be solved byan iterative procedure, for example, the Newton-Raphsonmethod. 𝑉𝑒 is the volume of the eth element and 𝑉∗ is theupper bound on material volume. 𝜌min = 0.001 is the lowerbound of material relative density.

Mathematical Problems in Engineering 3

3. Solution Strategy

3.1. Modified Additive Hyperelasticity Technique. During thecourse of material density-based topology optimization, low-density elements are inevitable, especially when a sensitivityfilter or density filter technique is applied. For problemsinvolving large deformation, these low-density elements maycause severe instability due to local buckling or mesh distor-tions. The local instability induced by low-density areas maycause serious convergence issue in the geometrically nonlin-ear finite element analysis. Indeed, this phenomenon is oneof the main difficulties hindering the topology optimizationof large-displacement compliant mechanisms.

Two simple and intuitive treatments, the “convergencecriterion relaxation” [22] and the “element removal” [17],have been proposed to overcome local instability in theminimum-density elements. The former method excludesthe nodes surrounded by minimum-density elements fromthe convergence criterion, while the latter method removesminimum-density elements from the finite element mesh.However, both methods cannot completely overcome thisdifficulty in the design of compliant mechanisms undergoinga very large deformation due to the fact that local instabil-ity may also occur in intermediate-density elements (e.g.,𝜌𝑒 = 0.01 to 0.3). Other more complex methods, such asthe element connectivity parameterization method [23, 24]and the Levenberg-Marquardt method [25], have also beenintroduced to stabilize the optimization process. In addition,Wang et al. [26] proposed a new energy interpolation schemeto alleviate the numerical instability in the low stiffnessregion. In this study, amodifiedmethod based on the additivehyperelasticity technique proposed by Luo et al. [20] isemployed.

The basic idea of the additive hyperelasticity technique isto add a specified soft hyperelasticmaterial to the low-densityelements thatmay become unstable.The additive hyperelasticmaterial for the eth element has the following strain energyfunction according to the Yeoh model [27]:

Ψadd𝑒 (𝐼1) = (1 − 𝜌𝑒𝑝) (𝑐𝑒1 (𝐼1 − 3) + 𝑐𝑒2 (𝐼1 − 3)2) , (2)

where 𝐼1 = tr(C) is the first invariant of the right Cauchy-Green strain tensorC, 𝑐𝑒1 > 0 and 𝑐𝑒2 > 0 arematerial constantsof the additive hyperelastic material for the eth element, 𝜌𝑒 isthe elemental density, and 𝑝 is the penalization factor usedin the SIMP approach. It is seen that the shearing capacityof the additive hyperelastic material is gradually increased asa result of increasing strain. Thus, the stiffness of low-densityelement is enhanced under compression to prevent itself frominstability. In order to achieve this, two issues need to befurther addressed.

The first issue is how to identify the low-density elementsthat tend to be unstable during the optimization process. Asimple criterion is the ratio between the element strain in thekth optimization step and the specified threshold value 𝜀∗;that is,

𝜂(𝑘)𝑒 = 𝜀(𝑘)𝑒,von𝜀∗ , (3)

where 𝜀𝑒,von is the average vonMises strain of the eth element.From our numerical experiments, we set the threshold valueas 𝜀∗ = 1.0 in the compliant mechanism topology optimiza-tion problem. If 𝜂(𝑘)𝑒 ≥ 1, this means that the deformation ofthe eth element is close to the instability point. Otherwise,there is little possibility for the eth element to becomeunstable in the next iteration step.

The second issue is the choice of parameters 𝑐𝑒1 and 𝑐𝑒2 forthe additive hyperelastic material. Under small strain, localinstability is not an issue and thus initial Young’s modulus(𝐸ini = 6𝑐𝑒1) of the additive hyperelastic material should besmall enough. Therefore, we set the minimum value 𝑐𝑒1 =𝜌min𝑝𝐸0/6, where 𝐸0 is Young’s modulus of the solid elastic

material. Under large strain, the stiffness of the additivehyperelastic material highly depends on the parameter 𝑐𝑒2and a large value of 𝑐𝑒2 can ensure effective elimination ofinstability. However, it may also lead to a large predictiondiscrepancy of the actual behavior of the mechanism. Areasonable value of 𝑐𝑒2 for the additive material in the ethelement can be updated by an adaptive strategy, which isexpressed by

𝑐𝑒,(𝑘+1)2 = {{{𝑐𝑒,(𝑘)2 ⋅ √𝜂(𝑘)𝑒 , if 𝜂(𝑘)𝑒 < 1𝑐𝑒,(𝑘)2 ⋅ (𝜂(𝑘)𝑒 )3 , if 𝜂(𝑘)𝑒 ≥ 1, (4)

where the superscripts (𝑘 + 1) and (𝑘) denote the iterationsteps of optimization.

Comparedwith our previous work [20], which uses a uni-form value of additive material parameters for all elements,this study proposed a modified strategy to eliminate theelement instability by selecting a more reasonable value of 𝑐𝑒2for each element. As will be illustrated in Section 4.1, by usingthe proposed modified algorithm, the discrepancy betweenthe responses of the remodeled structure and the actualstructure for large-displacement compliant mechanisms willbe substantially reduced.

3.2. Nonlinear Finite Element Analysis Procedures. We nowperform the finite element analysis of the remodeled compli-antmechanism undergoing large displacement and rotations.The remodeled mechanism is composed of original linearelastic material and additive hyperelastic material. Here,the additive hyperelastic material is invoked to describenonlinearmaterial behaviors under large strains. Using linearHooke’s law, we express the second Piola-Kirchhoff stress Soriin the original elastic material as follows:

Sori = Dori𝜀, (5)

where 𝜀 is the Green-Lagrange strain andDori is the constitu-tive elasticity tensor of the original elastic material. Using theSIMP model, for each element, the elasticity tensor Dori

𝑒 is afunction of the elemental relative density 𝜌𝑒:

Dori𝑒 = 𝜌𝑝𝑒 D0 (𝑒 = 1, 2, . . . , 𝑁) , (6)

where D0 is the elasticity tensor of the fully solid materialand 𝑝 > 1 is the penalization factor. In this study, we set

4 Mathematical Problems in Engineering

𝑝 = 3 for penalizing intermediate-density values during theoptimization process.

For the additive hyperelastic part, the second Piola-Kirchhoff stress Sadd can be obtained from the derivative ofthe strain energy function Ψadd; that is,

Sadd = 𝜕Ψadd

𝜕𝜀 = 2𝜕Ψadd

𝜕C = 2 3∑𝑖=1

(𝜕Ψadd

𝜕𝐼𝑖𝜕𝐼𝑖𝜕C) , (7)

where 𝐼1, 𝐼2, and 𝐼3 are the three invariants of C. Then,according to (2), the second Piola-Kirchhoff stress in theadditive hyperelastic material for each element is computedby

Sadd𝑒 = 2 (1 − 𝜌𝑒𝑝) (𝑐𝑒1 + 2𝑐𝑒2 (𝐼1 − 3)) 𝜕𝐼1𝜕C(𝑒 = 1, 2, . . . , 𝑁) .

(8)

Under large displacement, the Green-Lagrange strain isdefined by

𝜀𝑖𝑗 = 12 (𝑢𝑖,𝑗 + 𝑢𝑗,𝑖 + 𝑢𝑘,𝑖𝑢𝑘,𝑗) (9)

and can be rewritten after discretization as

d𝜀 = B (u) du = (B0 + B𝐿) du, (10)

where B0 is the linear part of the strain-displacement matrixand B𝐿 represents the second-order term.

The Green-Lagrange strain is work-conjugate to the sec-ond Piola-Kirchhoff stress. Therefore, the static equilibriumstate of the remodeled mechanism is expressed by

R (u,𝜌) = ∫Ω0

B (u)𝑇 (Sori + Sadd) dΩ + Kspringu − P

= 0,(11)

where R denotes the residual force, Ω0 is the undeformedvolume of the design domain, Kspring is the stiffness matrixobtained by assembling the contributions of two artificialsprings, and P is the vector of prescribed input force.The static equilibrium (11) for the compliant mechanism isnonlinear and can be typically solved by using the Newton-Raphson procedure.

3.3. Design Sensitivity Analysis. In this section, the adjointvariable method for sensitivities calculation of the objectivefunction 𝑢out(𝜌) is presented. Direct differentiation of thestatic equilibrium function (11) with respect to the designvariable 𝜌𝑒 leads todR (u,𝜌)

d𝜌𝑒 = 𝜕R (u,𝜌)𝜕𝜌𝑒 + 𝜕R𝑇

𝜕u𝜕u𝜕𝜌𝑒 = 0,

(𝑒 = 1, 2, . . . , 𝑁) .(12)

Then, the gradient 𝜕𝑢out/𝜕𝜌 is computed by

𝜕𝑢out𝜕𝜌𝑒 = (𝜕𝑢out𝜕u )𝑇 𝜕u𝜕𝜌𝑒= −(𝜕𝑢out𝜕u )𝑇(𝜕R𝑇𝜕u )

−1 𝜕R (u,𝜌)𝜕𝜌𝑒 ,(𝑒 = 1, 2, . . . , 𝑁) ,

(13)

where 𝜕R/𝜕u = K𝑇 is the global tangent stiffness matrix atthe equilibrium state.

To calculate (13), we first introduce an adjoint vector 𝜆,which can be easily obtained as the solution to the linearadjoint equation

K𝑇𝜆 = 𝜕𝑢out𝜕u . (14)

The partial derivative of the global residual with respectto the eth design variable is

𝜕R (u,𝜌)𝜕𝜌𝑒 = ∫

Ω0

B (u)𝑇(𝜕Sori𝜕𝜌𝑒 +𝜕Sadd𝜕𝜌𝑒 ) dΩ. (15)

Here, the partial derivatives of the second Piola-Kirchhoffstresses Sori and Sadd with respect to the design variable areobtained from the differentiation of (5) and (8).

Submitting (14) and (15) into (13), it yields that

𝜕𝑢out𝜕𝜌𝑒 = −𝜆𝑇∫Ω0

B (u)𝑇(𝜕Sori𝜕𝜌𝑒 +𝜕Sadd𝜕𝜌𝑒 ) dΩ,(𝑒 = 1, 2, . . . , 𝑁) .

(16)

In the finite element implementation, the 4-node planestress elements are used. It is well known that the use of low-order finite elements in topology optimization may lead tocheckerboard patterns and mesh dependency. In this study,the sensitivity filter technique [28] is employed to circumventthis problem; that is,

𝜕𝑢out𝜕𝜌𝑒 = 1𝜌𝑒∑𝑁𝑚=1𝐻𝑒,𝑚

𝑁∑𝑚=1

(𝐻𝑒,𝑚𝜌𝑚 𝜕𝑢out𝜕𝜌𝑚 )(𝑒 = 1, 2, . . . , 𝑁) ,

(17)

where 𝐻𝑒,𝑚 = max{0, 𝑟min − Δ(𝑒,𝑚)/𝐿} is the weight factor,Δ(𝑒,𝑚) represents the distance between the centroids ofelement 𝑒 and element𝑚, 𝐿 is the side length of the elements,and 𝑟min is the filter radius.

3.4. Algorithm. The detailed topology optimization proce-dures for the design of a compliant mechanism using theabove formulation and the modified additive hyperelasticitytechnique are described in Figure 2. With the proposedtechnique, the element instability in low-density elementscan be completely overcome without sacrificing the accu-racy of structural response analysis. Note that the main

Mathematical Problems in Engineering 5

Update elementalconstitutive matrix

by equations (6) and (8)

Stop

Converge?

Check the termination criterion

Yes

No

Perform FE analysis by Newton-Raphson iterations

Define and discretizeinitial design domain

Sensitivity analysis

Start

Evaluate output displacement Evaluate element strains

k = 0; set initial design value 𝜌(0) = 0.5

and initial parameter value c(0)2 = 0.0001E0

Update design variables𝜌(k+1) by MMA

Update additive material parameters c(k+1)2 by equation (4)

k = k + 1

Figure 2: Flowchart of the topology optimization incorporating modified additive hyperelasticity technique.

computational cost is the Newton-Raphson iterations forsolving the nonlinear equilibrium equations. The MMAalgorithm proposed by Svanberg [21] is used as an optimizerfor updating design variables. The termination criterion ofthe optimization processes is the following: the maximaldifference between the design variables of two adjacentiterations is less than 0.01.

4. Numerical Examples

The proposed topology optimization methodology is appliedto the design of two compliant mechanisms undergoing largedisplacement.The solid material properties for the compliantmechanisms are given as follows: Young’s modulus is 𝐸0 =100MPa and Poisson’s ratio is ] = 0.3. The filter radius fordesign sensitivities is 𝑟min = 1.8.4.1. Design of an Inverter Mechanism. The design domainand the boundary conditions of the displacement inverter aresketched in Figure 3. The top left corner and the bottom leftcorner of the design domain are fixed and an external force𝐹in is applied in the middle (point A) of the left side. Thedomain size is 100mm × 100mm. Two artificial springs with

Fin

kin kout

uoutBA

10 mm

10 mm

100 mm

Figure 3: Design domain of the inverter mechanism.

stiffness values 𝑘in = 1.0N/mm and 𝑘out = 0.2N/mm are putat point A and point B, respectively. The design problem is tofind a force-inverting mechanism that maximizes the outputdisplacement𝑢out at point B.Thematerial volume is restrictedto 25% of the total volume of the design domain.

In view of the symmetry, one half of the design domainis discretized by 5000 (100 × 50) 4-node plane elements. The

6 Mathematical Problems in Engineering

(a) (b) (c)

Figure 4: Comparison of optimal topologies for the inverter mechanism considering different inputs. (a) 𝐹in = 2N. (b) 𝐹in = 18N. (c)𝐹in = 36N.

Table 1: Solution details for the inverter mechanism by the proposed method.

Number ofiterations

Maximum value of 𝑐𝑒2(×𝐸0) 𝑢in (mm) Input work𝑊in

(Nmm) 𝑢out (mm) Output work𝑊out (Nmm) 𝑊out/𝑊in

𝐹in = 2N 65 1.0000𝐸 − 8 0.68 0.449 1.49 0.222 49.44%𝐹in = 18N 76 1.0000𝐸 − 8 6.01 36.030 11.98 14.352 39.83%𝐹in = 36N 61 2.5615𝐸 − 6 13.16 150.287 23.15 53.592 35.66%

relative density of each element is treated as a design variable.Three different values of𝐹in = 2N,𝐹in = 18N, and𝐹in = 36Nare considered.

The optimization results considering geometrical nonlin-earity for different input forces are compared in Figures 4(a)–4(c).The black region indicates the fully solid material, whilethe white region represents the low-density or void elements.It is noted that the optimal topologies change as the inputforce increases. For a small input 𝐹in = 2N, the optimummaterial distribution is shown in Figure 4(a) with smalloutput displacement of 𝑢out = 1.49mm. When 𝐹in = 36N,the topology optimization procedure yields a rather differentdesign as depicted in Figure 4(c). The main difference is thelength and the incline angle of the right bars connected to theoutput point. In this case, the maximum output displacementis predicted to be 𝑢out = 23.15mm.

Figure 5 plots the iteration histories of the proposedoptimization processes. It can be seen that the optimizationobjective for each case converges to a stable value within 80iterations.The corresponding details of solutions, such as thenumber of iterations, themaximumvalue of parameter 𝑐𝑒2 , theinput displacement 𝑢in, the input work𝑊in = (1/2)(𝐹in𝑢in −𝑘in𝑢2in), the output displacement 𝑢out, the output work𝑊out =(1/2)𝑘out𝑢2out, and the work transmission ratio𝑊out/𝑊in, arelisted in Table 1. It is observed that the topology in Figure 4(a)gains the most energy-efficient way to transfer input tooutput. However, it cannot generate large displacement at theoutput point.

The deformed shapes of the optimized topologies forthree cases of input force are shown in Figures 6(a)–6(c).

−10

−5

0

5

10

15

20

25

30

Number of iterations

Fin = 2N

Fin = 18N

Fin = 36N

Obj

ectiv

euout

(mm

)

0 10 20 30 40 50 60 70 80

Figure 5: Optimization histories for the inverter mechanism.

It is seen that some low-density elements in Figure 6(c) forthe case of 𝐹in = 36N undergo very large strains. However,these elements do not suffer from instability because the

Mathematical Problems in Engineering 7

(a) (b) (c)

Figure 6: Deformed shapes of optimized structures for the inverter considering different inputs. (a)𝐹in = 2N, 𝑢out = 1.49mm. (b)𝐹in = 18N,𝑢out = 11.98mm. (c) 𝐹in = 36N, 𝑢out = 23.15mm.

additive hyperelastic material is added. The final contour ofthe parameter 𝑐𝑒2 for the case of 𝐹in = 36N is plotted inFigure 7. In most parts of the design domain, the parameter𝑐𝑒2 is extremely low. Therefore, the additive material onlyhas slight effects on the overall response of the compliantmechanism. In order to show this, the optimal topology inFigure 4(c) was also analyzed by the nonlinear finite elementmethod incorporating the element removal technique, inwhich the low-density elements with 𝜌𝑒 ≤ 0.01 are removedfrom the finite element model, as shown in Figure 8. Thecurves of output displacement versus input work predictedby the modified additive material model are compared withthe element removal model in Figure 9. In this large-displacement case, the maximum discrepancy between thetwo sets of results is no more than 1.0%. On the contrary, ifone adopted the previous method as suggested in [20] thatuses a uniform value of 𝑐2 = 2.57 × 10−6𝐸0 for all ele-ments, the maximum discrepancy becomes 4.7%. Therefore,the modified additive hyperelasticity technique is capableof eliminating the element instability and providing moreaccurate response analysis than the previous method duringthe topology optimization of large-displacement compliantmechanisms.

4.2. Design of a Gripper Mechanism. The second benchmarkexample considered in this study is the topology optimizationof a gripper mechanism. The geometrical dimension of thedesign domain and the boundary condition are given inFigure 10. The spring parameters are set as 𝑘in = 0.4N/mmand 𝑘out = 0.1N/mm. Here, the output displacement is tobe maximized under two different input force values 𝐹in =2N and 𝐹in = 20N. The allowable material volume ratiois specified as 25%. In the finite element model, only thesymmetric upper design domain is discretized by 4839 nodesand 4688 four-node finite elements by using an element sizeof 1mm.

The optimized solutions obtained by the proposedmethod for the two input force values are presented in

2.57

2.25

1.93

1.61

1.29

0.97

0.65

0.33

0.01

(×10−6E0)

Figure 7: Contour of the parameter 𝑐𝑒2 for the inverter mechanismin the case of 𝐹in = 36N.

Figure 8: Element removal model for the design in Figure 4(c).

8 Mathematical Problems in Engineering

Element removal modelModified additive material modelPrevious method in [18]

Out

put d

ispla

cem

entu

out

(mm

)

Input work Win (Nmm)

25

20

15

10

5

00 20 40 60 80 100 120 140 160

Figure 9: Output displacement versus input work curves for the design in Figure 4(c) by different models.

Finkin

kout

uout

uout

10 mm

10 mm

100 mm

25 m

m

Figure 10: Design domain of the gripper mechanism.

Figures 11(a) and 11(b). The resulting optimal solution inFigure 11(a) has a significant topological difference from thesolution in Figure 11(b) due to the large-displacement effect.The latter design gains output displacement of 12.91mm,while the former gains only 1.44mm. The curves of outputdisplacement versus input work for both designs are com-pared in Figure 12 based on the modified additive materialmodel and the element removal model. The comparisonshows that the relative discrepancy between the additivematerial-based response and the true response is rather small.In addition, even if a large input work (𝑊in = 82Nmm) isimposed, the design in Figure 11(a) can only yield 9.17mmoutput displacement, which is much less than that of thedesign in Figure 11(b). This reveals that it is meaningful to

account for the large-displacement effects in topology designof compliant mechanisms.

The deformed shape of the additivematerial-basedmodelfor the design in Figure 11(b) is shown in Figure 13, fromwhich the effectiveness of using the proposed method toprevent instability in low-density elements can be validated.

It should be noted that, besides the element instability,another main difficulty in the topology optimization ofcompliant mechanisms is that there are some usually smallhinge-like regions in the final topology, for example, inFigures 4 and 6. How to avoid such hinges during thetopology optimization has been extensively investigated up tonow.Many efficient methods, such as the length-scale controlmethod [29], the morphology-based restriction scheme [30],

Mathematical Problems in Engineering 9

(a) (b)

Figure 11: Comparison of optimal layouts of the gripper mechanism with different inputs. (a) 𝐹in = 2N. (b) 𝐹in = 20N.

Figure 11(b), element removal modelFigure 11(b), modified additive material modelFigure 11(a), element removal modelFigure 11(a), modified additive material model

14

12

10

8

6

4

2

00 10 20 30 40 50 60 70 80 90 100

Out

put d

ispla

cem

entu

out

(mm

)

Input work Win (Nmm)

Figure 12: Output displacement versus input work curves for thedesigns in Figure 11.

and the level set-based control method [31], have been pro-posed. Interested readers are referred to some recent publica-tions [7, 32–34].

5. Conclusions

Based on the density penalization model and the modifiedadditive hyperelasticity technique, a topology optimizationmethod for large-displacement compliant mechanisms ispresented. The modified additive hyperelastic material can

Figure 13: Deformed shape of the design in Figure 11(b). 𝐹in = 20N,𝑢out = 12.91mm.

effectively overcome the element instability difficulty, whileexerting only a negligible influence on the structural nonlin-ear response prediction. Compared to the previous method,this modified technique provides more accurate resultsin structural response. The present method is easy to beimplemented, without causing extra computational burden.It can generate a meaningful and well-defined topologywhen the mechanism is required to deliver relatively largedisplacement. Topology optimization of two benchmarkexamples of compliant mechanisms is tested. It is shown thatthe obtained optimal designs with large displacement havedistinct differences in topology and performance comparedwith those with small displacement.

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper.

10 Mathematical Problems in Engineering

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (DUT15RC(3)026).

References

[1] G. K. Ananthasuresh, S. Kota, and Y. Gianchandani, “Amethod-ical approach to the design of compliant micromechanisms,” inProceedings of the Solid-State Sensor and ActuatorWorkshop, pp.189–192, 1994.

[2] A. Saxena and G. K. Ananthasuresh, “On an optimal propertyof compliant topologies,” Structural and Multidisciplinary Opti-mization, vol. 19, no. 1, pp. 36–49, 2000.

[3] O. Sigmund, “On the design of compliant mechanisms usingtopology optimization,” Mechanics of Structures and Machines,vol. 25, no. 4, pp. 493–524, 1997.

[4] J. A.Hetrick and S. Kota, “An energy formulation for parametricsize and shape optimization of compliant mechanisms,” Journalof Mechanical Design, vol. 121, no. 2, pp. 229–234, 1999.

[5] M. P. Bendsøe and O. Sigmund, Topology Optimization: Theory,Methods, and Applications, Springer, Berlin, Germany, 2003.

[6] S. R. Deepak, M. Dinesh, D. K. Sahu, and G. K. Ananthasuresh,“A comparative study of the formulations and benchmark prob-lems for the topology optimization of compliant mechanisms,”Journal of Mechanisms and Robotics, vol. 1, no. 1, Article ID011003, 2009.

[7] E. Lee and H. C. Gea, “A strain based topology optimizationmethod for compliant mechanism design,” Structural and Mul-tidisciplinary Optimization, vol. 49, no. 2, pp. 199–207, 2014.

[8] R. Ansola, E. Veguerıa, J. Canales, and J. A. Tarrago, “A simpleevolutionary topology optimization procedure for compliantmechanism design,” Finite Elements in Analysis and Design, vol.44, no. 1-2, pp. 53–62, 2007.

[9] X. Huang, Y. Li, S. W. Zhou, and Y. M. Xie, “Topology opti-mization of compliant mechanisms with desired structuralstiffness,” Engineering Structures, vol. 79, pp. 13–21, 2014.

[10] B. Zhu andX. Zhang, “A new level setmethod for topology opti-mization of distributed compliant mechanisms,” InternationalJournal for Numerical Methods in Engineering, vol. 91, no. 8, pp.843–871, 2012.

[11] Z. Luo, N. Zhang, J. Ji, and T.Wu, “Ameshfree level-set methodfor topological shape optimization of compliant multiphysicsactuators,” Computer Methods in Applied Mechanics and Engi-neering, vol. 223/224, pp. 133–152, 2012.

[12] Z. Kang and Y. Wang, “A nodal variable method of structuraltopology optimization based on Shepard interpolant,” Interna-tional Journal for Numerical Methods in Engineering, vol. 90, no.3, pp. 329–342, 2012.

[13] Z. Luo,N. Zhang, Y.Wang, andW.Gao, “Topology optimizationof structures using meshless density variable approximants,”International Journal for Numerical Methods in Engineering, vol.93, no. 4, pp. 443–464, 2013.

[14] C. B. W. Pedersen, T. Buhl, and O. Sigmund, “Topology synthe-sis of large-displacement compliant mechanisms,” InternationalJournal for NumericalMethods in Engineering, vol. 50, no. 12, pp.2683–2705, 2001.

[15] O. Sigmund, “Design of multiphysics actuators using topologyoptimization—part I: one-material structures,”ComputerMeth-ods in Applied Mechanics and Engineering, vol. 190, no. 49-50,pp. 6577–6604, 2001.

[16] T. E. Bruns andD. A. Tortorelli, “Topology optimization of non-linear elastic structures and compliant mechanisms,” ComputerMethods in Applied Mechanics and Engineering, vol. 190, no. 26-27, pp. 3443–3459, 2001.

[17] T. E. Bruns and D. A. Tortorelli, “An element removal and re-introduction strategy for the topology optimization of struc-tures and compliant mechanisms,” International Journal forNumerical Methods in Engineering, vol. 57, no. 10, pp. 1413–1430,2003.

[18] K. Maute and D. M. Frangopol, “Reliability-based design ofMEMS mechanisms by topology optimization,” Computers &Structures, vol. 81, no. 8–11, pp. 813–824, 2003.

[19] Z. Luo and L. Tong, “A level set method for shape and topologyoptimization of large-displacement compliant mechanisms,”International Journal for Numerical Methods in Engineering, vol.76, no. 6, pp. 862–892, 2008.

[20] Y. Luo, M. Y. Wang, and Z. Kang, “Topology optimization ofgeometrically nonlinear structures based on an additive hyper-elasticity technique,” Computer Methods in Applied Mechanicsand Engineering, vol. 286, pp. 422–441, 2015.

[21] K. Svanberg, “The method of moving asymptotes—a newmethod for structural optimization,” International Journal forNumerical Methods in Engineering, vol. 24, no. 2, pp. 359–373,1987.

[22] T. Buhl, C. B.W. Pedersen, and O. Sigmund, “Stiffness design ofgeometrically nonlinear structures using topology optimiza-tion,” Structural and Multidisciplinary Optimization, vol. 19, no.2, pp. 93–104, 2000.

[23] G. H. Yoon and Y. Y. Kim, “Element connectivity parameter-ization for topology optimization of geometrically nonlinearstructures,” International Journal of Solids and Structures, vol.42, no. 7, pp. 1983–2009, 2005.

[24] S. J. Moon and G. H. Yoon, “A newly developed qp-relaxationmethod for element connectivity parameterization to achievestress-based topology optimization for geometrically nonlinearstructures,” Computer Methods in Applied Mechanics and Engi-neering, vol. 265, pp. 226–241, 2013.

[25] A. Kawamoto, “Stabilization of geometrically nonlinear topol-ogy optimization by the Levenberg-marquardt method,” Struc-tural andMultidisciplinary Optimization, vol. 37, no. 4, pp. 429–433, 2009.

[26] F. Wang, B. S. Lazarov, O. Sigmund, and J. S. Jensen, “Interpola-tion scheme for fictitious domain techniques and topology opti-mization of finite strain elastic problems,”Computer Methods inApplied Mechanics and Engineering, vol. 276, pp. 453–472, 2014.

[27] P. A. L. S. Martins, R. M. N. Jorge, and A. J. M. Ferreira, “A com-parative study of several material models for prediction of hyp-erelastic properties: application to silicone-rubber and softtissues,” Strain, vol. 42, no. 3, pp. 135–147, 2006.

[28] O. Sigmund, “A 99 line topology optimization code written inmatlab,” Structural and Multidisciplinary Optimization, vol. 21,no. 2, pp. 120–127, 2001.

[29] T. A. Poulsen, “A new scheme for imposing a minimum len-gth scale in topology optimization,” International Journal forNumerical Methods in Engineering, vol. 57, no. 6, pp. 741–760,2003.

[30] O. Sigmund, “Morphology-based black andwhite filters for top-ology optimization,” Structural andMultidisciplinary Optimiza-tion, vol. 33, no. 4-5, pp. 401–424, 2007.

[31] J. Z. Luo, Z. Luo, S. K. Chen, L. Y. Tong, andM. Y.Wang, “A newlevel set method for systematic design of hinge-free compliant

Mathematical Problems in Engineering 11

mechanisms,” Computer Methods in Applied Mechanics andEngineering, vol. 198, no. 2, pp. 318–331, 2008.

[32] B. S. Lazarov, M. Schevenels, and O. Sigmund, “Topologyoptimization with geometric uncertainties by perturbationtechniques,” International Journal for Numerical Methods inEngineering, vol. 90, no. 11, pp. 1321–1336, 2012.

[33] B. Zhu, X. Zhang, and N. Wang, “Topology optimization ofhinge-free compliant mechanisms with multiple outputs usinglevel set method,” Structural and Multidisciplinary Optimiza-tion, vol. 47, no. 5, pp. 659–672, 2013.

[34] D. M. De Leon, J. Alexandersen, J. S. O. Fonseca, and O. Sig-mund, “Stress-constrained topology optimization for compliantmechanism design,” Structural and Multidisciplinary Optimiza-tion, vol. 52, no. 5, pp. 929–943, 2015.

Submit your manuscripts athttps://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of