ADE664489 1 10

ADE664489 1 10 Research Article Advances in Mechanical Engineering 2016, Vol 8(8) 1–10 � The Author(s) 2016 DOI 10 1177/1687814016664489 aime sagepub com An augmented formulation of distributed compli[.]

Research Article An augmented formulation of distributed compliant mechanism optimization using a level set method Advances in Mechanical Engineering 2016, Vol 8(8) 1–10 Ó The Author(s) 2016 DOI: 10.1177/1687814016664489 aime.sagepub.com Ruixian Li1 and Benliang Zhu2 Abstract Topology optimization has emerged as one of the key approaches to design compliant mechanisms However, one of the main difficulties is that the resulted compliant mechanisms often have de facto hinges For this reason, a simple yet efficient formulation for designing hinge-free compliant mechanisms is developed and examined within a level set–based topology optimization framework First, the conventional objective function is augmented using an output stiffness Second, the proposed formulation is solved using a level set method for designing some benchmark problems in the literature It is shown that the proposed augmented objective function can prevent the de facto hinges in the obtained compliant mechanisms Finally, some concluding remarks and future work are put forward Keywords Topology optimization, compliant mechanisms, level set method, lumped compliance, distributed compliance Date received: June 2016; accepted: 21 July 2016 Academic Editor: Fakher Chaari Introduction A compliant mechanism is regarded as a mechanism that gains its mobility from the deflection of its flexible members.1,2 Over the past decades, compliant mechanisms have been extensively studied Several approaches have been developed for the design of compliant mechanisms Generally, these approaches can be categorized into two types The first one is the kinematics synthesis approach.1 In this approach, the compliant mechanism is derived from a known rigid-body mechanism.1,3 Although the method has been successfully used in designing compliant mechanisms for precision applications, it requires a good deal of designers’ intuition and involvement The second approach is derived from the structural topology optimization approach.2,4–7 As one of the most challenging tasks in the optimization design, topology optimization has been deeply explored and applied to a variety of design problems, for example, the minimum mean compliance problem,8 the vehicle component design problems,9 and the top-down structural assembly synthesis problem.10 During the last decades, several methods have been developed, such as the ground structure method,11–13 the solid isotropic material with penalization (SIMP) method,8,14 and the level set method.15–17 At the very beginning, the level set method was developed for numerically tracking fronts and free boundaries.18,19 After Sethian and Wiegmann20 first introduced it to design structural boundaries, Osher and Santosa21 extended the method by introducing the School of Automation Science and Engineering, South China University of Technology, Guangzhou, P.R China School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou, P.R China Corresponding author: Benliang Zhu, School of Mechanical and Automotive Engineering, South China University of Technology, Building 10, 381 Wushan Road, Tianhe District, Guangzhou 510641, P.R China Email: meblzhu@scut.edu.cn Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage) 2 shape sensitivity analysis into the framework In level set–based topology optimization methods, the structural boundary is treated as the design parameter and is implicitly embedded into a scalar function and updated by solving the level set equation.22 Therefore, topology changes, for example, merging and splitting, can be easily handled Furthermore, several numerical instabilities that usually occurred in the density-based topology optimization approaches, such as checkerboard patterns and gray scales, can be eliminated23–25 as well When using continuum topology optimization methods to design compliant mechanisms, one of the significant challenges is their strong tendency to result in de facto hinges.4 In the context of material distribution– based topology optimization methods, several methods have been developed to eliminate the de facto hinges in the design of compliant mechanisms It is surely possible to redesign the de facto hinge regions as continuous material bridges.26 However, the obtained mechanisms will deviate from the original mechanism Alternatively, one may use the procedures derived from the image processing method,27 such as the filtering method,28 to eliminate the de facto hinges These methods can insure the absence of the one-node connected hinges However, the lumped compliance, which is caused by the de facto hinges, is sometimes inevitable Researchers also tried to develop new design models, such as Rahmatalla and Swan4 developed a new springbased method, to prevent de facto hinges Another method can be regarded as a so-called hybrid discretization method.29,30 In this method, during each optimization iteration, each design element will be subdivided for finite element analysis This often leads to a low computational efficiency although the de facto hinges can be eliminated For using the level set method, a logarithmic barrier penalty term has been utilized to ensure topological connection of the structural components.31 However, this cannot eliminate the de facto hinges A possible strategy is to control the geometric width of structural components using a quadratic energy functional, as stated in Luo et al.32 and Chen et al.33 In addition, an intrinsic characteristic stiffness method is developed by Wang and Chen.34 Deepak et al.35 indicate that this method can also lead to point flexures when a large objective geometrical advantage (GA) is needed Zhu and Zhang6 developed two alternative formulations for developing hinge-free compliant mechanisms using level set method This work has been further extended for designing compliant mechanisms with multiple outputs.7 The fundamental reason for the occurrence of the de facto hinges lies in the mathematical formulation of the design problem.36 For this reason, in this study, we were trying to propose a new objective function for the design of hinge-free compliant mechanisms and verified Advances in Mechanical Engineering its validity by designing several two-dimensional (2D) numerical examples that are widely studied in the literature of compliant mechanism optimization Level set method In level set method, the structural boundary ∂O is implicitly embedded in a scalar function f18,19 as its zero level set Therefore < f(x, t).0 f(x, t) = : f(x, t)\0 if x O if x ∂O if x DnO ð1Þ where D is the reference domain to contain all permissible shapes of the design domain O, ∂O is the interface of the structure, and DnO represents the void area The optimization is achieved by solving the following Hamilton–Jacobi equation ∂f + Vn jrfj = ∂t ð2Þ where t is the time and velocity Vn determines the motion of the interface.15,16 Optimal synthesis of compliant mechanisms and the challenge of de facto hinges Conventional mathematical formulation of the optimization problem Synthesis of compliant mechanisms has been formulated in many different ways under two main groups The first of which is to establish the objective function by maximizing a mechanical measurement, such as the mechanical advantage (MA),2 the GA,6,37 and the mechanical efficiency (ME).32 The second of which is formulated by considering both flexibility and compliance to meet the function and strength requirements.11 A comparison study of these formulations can be found in Deepak et al.35 For designing compliant mechanisms with single input–output behavior, the design domain can be illustrated in Figure where Gd indicates the Dirichlet boundary An input force fin is applied at the input port i uin and uout indicate the displacements occurred at the input port i and the output port o due to fin , respectively A spring with stiffness kout is attached to the output port to imitate the reaction force from the workpiece by fout = kout uout In this study, in selecting the stiffness of kout , the bounding spring value method4 is employed The value of kout is set to be 104 kb where kb is computed by applying a unit load at the input port of the reference domain D when D is fully occupied by the structural material Li and Zhu maximization of GA is equivalent to minimization of GA The energy bilinear functional a(u, v, f) and the load linear functional l(v, f) are, respectively, expressed as follows ð a(u, v, f) = Eijkl eij (u)ekl (u)H(f)dO ð4Þ D ð l(v, f) = fvd(f)jrfjdO ð5Þ D where Eijkl and eij are the material property tensor and strain tensor, respectively Since the design condition considered in this article does not concern the body force, only the boundary traction f is considered in the above equations Using the dummy load method,2 the displacements uout and uin can be expressed by superposition of two loading cases applied at the input and output ports Therefore Figure The design domain of topology optimization of compliant mechanisms GA = Figure A compliant displacement inverter mechanism suffers from the de facto hinges, which are marked with dashed line circle u1, o u1, i  kout u1, i u2, o + kout u1, o u2, i where u1, i , u1, o , u2, i and u2, o are the displacements included in the displacement field u1 and u2 , respectively u1 is the displacement field which is obtained by applied a unit load f1 at the input port of the design domain u2 is the displacement field which is obtained by applied a unit load f2 at the output port of the design domain u1, i and u1, o are the displacements at the input and the output ports, respectively, caused by f1 u2, i and u2, o are the displacements at the input and the output ports, respectively, caused by f2 For more details, please refer to Sigmund2 and Chen.37 H(f) is the Heaviside function defined as follows  and the output port is restrained, see Rahmatalla and Swan.4 Here, we choose GA to quantify the performance of the compliant mechanisms Incorporating with the level set method, a conventional formulation for topology optimization of the compliant mechanisms can be formulated as follows37 : s:t: : GA(u, f) =  uuout in max Ð uin  uin Vol = H(f)dO  Volmax ð3Þ D a(u, v, f) = l(v, f)8v U where uin is constrained by an upper limit umax in for indirectly controlling the maximum stress level.2Vol denotes the total material usage and is constrained by an upper limit Volmax v denotes the virtual displacement fields in space U Note that a minus is used in equation (3) since ð6Þ H(f) = if if f0 f\0 ð7Þ and d(f) is the one-dimensional delta function defined as follows d(f) = dH(f) df ð8Þ De facto hinge problem A hinged compliant inverter which is obtained using equation (3) is shown in Figure These kinds of hinges are not needed since they make the obtained compliant mechanisms very difficult to fabricate, especially in the micro-scale.4 Since the flexibility of the mechanism is only provided in localized areas (hinge areas), the stress in the hinge areas would approach very high and the mechanism would break 4 Advances in Mechanical Engineering The reason for de facto hinges lies behind the objective formulations.36 In fact, for designing compliant mechanism, many developed formulations specify two main purposes, that is, maximizing the elastic deformation at the output port, meanwhile minimizing the overall compliance This makes the true optimum of the problem a rigid-body linkage with revolute joints The reason is that it can generate the largest output motion and has the minimum strain energy From this point of view, the de facto hinges are inevitable There are some other formulations that try to avoid these two main purposes to avoid the de facto hinges, such as the characteristic stiffness formulation.37 A comparative review of those formulations can be found in Deepak et al.35 The output stiffness and a new formulation For topological synthesis of a compliant mechanism, although the optimum mechanism could not be known in advance, all the three regions (input region, output region, and fixed region) must be connected to one another in order to form a meaningful structure.5 Here, we proposed an output compliance that can be built in the conventional formulation (equation (3)) The idea of designing the hinge-free compliant mechanisms by augmenting the conventional optimization problem with additional energy functionals is not new For example, Luo et al.32 proposed an augmented objective function using a quadratic energy functional to control the geometric width of the mechanism However, it is difficult to implement this formulation to other topology optimization methods, such as the SIMP method or evolutionary structural optimization method.38 As shown in Figure 3, the output stiffness Eout is determined based on the case that an external unit force is only applied at the output port o while keeping the input port i as a free boundary (unfixed) Therefore, Eout can be illustrated as follows ð ð9Þ Eout = f2 u2 dG Figure Schematic for determining Eout Figure Schematic of a (a) hinged and (b) hinge-free compliant mechanism Figure 4(b), when the mechanism is loaded with a force at the output port, the corresponding displacement at the output port will become smaller, that is ð10Þ ∂D uh uhf and this makes the proposed formulation very easy to use since no extra finite element analysis needs to be addressed The reason for introducing the energy function to prevent de facto hinges can be stated as follows As shown in Figure 4(a), for a compliant mechanism that suffers de facto hinges, when the mechanism is loaded with a force at the output port, the corresponding displacement at the output port will approach very high since the surrounding materials of the de facto hinges undergo essentially rigid-body rotations This is also fairly intuitive as shown in Figure However, when the mechanism is completely free of de facto hinges as is shown in This means a hinged compliant mechanism corresponds to a large displacement while a hinge-free compliant mechanism corresponds to a small one Conversely, by minimizing the displacement at the output port due to the unit load (which is equivalent to minimize the output stiffness Eout ), one can obtain a compliant mechanism which is free of the de facto hinges Based on the above analysis, the objective function can be set by minimizing GA meanwhile minimizing Eout Incorporating the level set model, a new mathematical formulation of the optimization problem can be rewritten as follows Li and Zhu : s:t: : GA(u, f) + aEout (u, f) max Ð uin (f)  uin max H(f)dO  Vol ð11Þ D a(u, v, f) = l(v, f) 8u U where SGA , SEout , Suin , and SVol can be derived from h∂GA=∂f, ui, h∂Eout =∂f, ui, h∂uin =∂f, ui, and h∂Vol=∂f, ui, respectively And zuin , zVol can be determined from the mentioned Kuhn–Tucker optimally condition as follows where a is the weighting factor of Eout The new objective function derived from the original function (equation (3)) is augmented with the energy function Eout The minimization of Eout is to make the mechanism structure stiffer, while the maximization of GA is to make the mechanism more flexible Shape sensitivity analysis In order to obtain the sensitivity of the objective function with respect to the boundary perturbations, the shape derivative method39,40 is employed The optimization problem represented in equation (11) is reformulated using Lagrange’s method of undetermined multipliers as follows J = GA + aEout + luin (uin  umax in + uuin ) + lVol (Vol  Volmax + u2Vol ) ð12Þ      ∂J ∂GA ∂Eout ,u ,u = ,u +a ∂f ∂f ∂f     ∂uin ∂Vol , u + lVol ,u =0 + luin ∂f ∂f  ð13Þ ∂J = uin  umax in + uuin = ∂luin ð14Þ ∂J = Vol  Volmax + u2Vol = ∂lVol ð15Þ ∂J = 2luin uuin = ∂uuin ð16Þ ∂J = 2lVol uVol = ∂uVol ð17Þ luin  0, lVol  ð18Þ where h∂J =∂f, ui indicates the Fre´chet derivative of J with respect to f in the direction of u Equation (13) can be expressed as   ð ∂J SGA + aSEout d(f)udO ð19Þ ,u = + zuin Suin + zVol SVol ∂f D  zVol = luin lVol if if uin umax in uin  umax in if Vol.Volmax if Vol  Volmax ð20Þ ð21Þ In order to calculate the shape sensitivity of J, we need to calculate the shape sensitivities of the individual functions, that is, h∂GA=∂f, ui, h∂Eout =∂f, ui, h∂uin =∂f, ui, and h∂Vol=∂f, ui Taking Fre´chet derivative of the GA with respect to f in the direction of u leads to       ∂GA ∂GA ∂u1, i ∂GA ∂u1, o ,u + ,u ,u = ∂f ∂u1, i ∂f ∂u1, o ∂f     ∂GA ∂u2, i ∂GA ∂u2, o ,u + ,u + ∂u2, i ∂f ∂u2, o ∂f ð22Þ where uuin , uVol are slack variables to convert the inequality constraint into the equality one and luin , lVol are Lagrange multipliers Applying the Kuhn–Tucker conditions of J leads to   zuin = Furthermore, taking Fre´chet derivative of the uin with respect to f in the direction of u leads to       ∂uin ∂uin ∂u1, i ∂uin ∂u1, o ,u = ,u + ,u ∂f ∂u1, i ∂f ∂u1, o ∂f     ∂uin ∂u2, i ∂uin ∂u2, o ,u + ,u + ∂u2, i ∂f ∂u2, o ∂f ð23Þ Since ∂GA=∂u1, i , ∂GA=∂u1, o , ∂GA=∂u2, i , ∂GA=∂u2, o , ∂uin =∂u1, i , ∂uin =∂u1, o , ∂uin =∂u2, i , and ∂uin =∂u2, o can be directly obtained from equation (6), only h∂u1, i =∂f, ui, h∂u1, o =∂f, ui, h∂u2, i =∂f, ui, and h∂u2, o =∂f, ui need to be solved The Fre´chet derivative of u1, i with respect to f in the direction of u can be written as D ∂u1, i ∂f E ,u = Ð D   u1 ) 2½kf1 u1 + ∂(f∂n  d(f)udO Eijkl eij (u1 )ekl (u1 ) ð24Þ Ð d(f) ∂f + G jrfj ∂n f1 u1 udG where k is the mean curvature of the structural boundary defined by k = div  n, where n is the normal vector of the structural boundary For design cases where the external force is applied at a point, equation (24) is reduced to   ð ∂u1, i , u =  Eijkl eij (u1 )ekl (u1 )d(f)udO ∂f D ð25Þ Advances in Mechanical Engineering Similarly, the Fre´chet derivatives of u1, o , u2, i , Eout , and u2, o can be written as     ∂u1, o ∂u2, i ,u = ,u ∂f ∂f ð = Eijkl eij (u1 )ekl (u2 )d(f)udO ð26Þ D     ∂Eout ∂u2, o ,u = ,u ∂f ∂f ð =  Eijkl eij (u2 )ekl (u2 )d(f)udO ð27Þ  d(f) = 3  3e D  fD3 if others if D  f  D ð32Þ Numerical examples In this section, in order to demonstrate the validity of the proposed formulation, we present several examples The used artificial material properties are as follows: Young’s modulus for solid material is E = and Poisson’s ratio is y = 0:3 The void area is assumed with a Young’s modulus E = 0:001 and the same Poisson’s ratio y = 0:3 D The derivative of allowable material usage Vol with respect to f in the direction u can be written as follows   ð ð ∂Vol dH(f) ,u = udO = d(f)udO ∂f df D ð28Þ D For more details, please refer to previous studies.16,39–41 Numerical implementation For the implementation of the proposed method, a number of numerical issues need to be addressed here The second-order accurate essentially non-oscillatory (ENO2)18 is employed for solving equation (2) In order to ensure stability, the following reinitialization equation42 is used ∂f + S(f0 )(jrfj  1) = ∂t ð29Þ where S(f0 ) is a sign function taken as in O, 1 in DnO, and on the interface In numerical implementation, S(f) is approximated by f0 S(f0 ) = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f20 + (Dx)2 Displacement inverter The designing of the displacement inverter is first considered The design domain is shown in Figure The length and height of the design domain have the same size The left top corner and the left bottom corner are fixed A single horizontal load F = is applied at the center point of the left side of the design domain An inverse displacement is expected at the center point of the right side of the design domain Due to symmetry, only half the design domain is taken into consideration 80 40 finite elements are used for discretization Effect of a on the optimum topology This section is focused on examining the effectiveness of Eout on preventing the de facto hinges The effect of a on the optimal topology is examined A volume ratio Volmax = 0:3 is considered For all studied cases, umax in is set to 50 Five cases are studied, in which a is set to 0.001, 0.002, 0.003, 0.004, and 0.006 The initial design as well as the corresponding final designs are shown in Figure For all studied five cases, the created compliant mechanisms not suffer de facto hinges When large a is used, the elastic hinges can also be eliminated ð30Þ where Dx is set to the mesh size In order to avoid the numerical difficulties, we adopted the method proposed in Wang et al.15 in which the Heaviside function is approximated using equation H(f) = < e :  f 3e D  f3 3D3 + 1+e if f\  D if D  f  D if f.D ð31Þ where e is small value to ensure the numerical stiffness nonsingular In our case, e is set to be 103 D is set to be 0:75Dx Using equations (31) and (8), d(f) can be obtained directly as Figure The design domain of the displacement inverter topology optimization problem Li and Zhu Figure The intermediate designs of the displacement problem with a = 0:002: (a) step 1, (b) step 25, (c) step 50, (d) step 100, (e) step 150, and (f) step 190 Figure Topology optimization of the displacement inverter using equation (11) with different a: (a) the initial configuration, (b) a = 0:001, (c) a = 0:002, (d) a = 0:003, (e) a = 0:004, and (f) a = 0:006 Figure The convergence histories of the displacement inverter problem with a = 0:002 Figure The effect of a on the geometric advantages and the iterations of the obtained mechanisms and compliant mechanism that only contain strip-like members is obtained which is in favor of generating distributed compliance such as case of a = 0:006 The value of a has direct impact on the GA of the created mechanism, see Figure In fact, one can find that a larger a surely will lead to a small GA A proper selection of a depends on applications, and no universal optimal setting may exist However, to reduce the difficulty of selection, one may rephrase the objection function in (equation (11)) using the normalization method so that the value of a lies between and In fact, equation (11) can prevent de facto hinges not only in the final topology but also during the optimization process This can be seen from some intermediate designs of the displacement inverter with a = 0:002 in Figure It should be pointed out that the present method cannot generate new holes freely inside the design domain Therefore, for using the proposed method, a good initial guess of the topology (with a certain number of holes) is needed for obtaining a reasonable solution The convergence histories of using the proposed method with the case of a = 0:002 are shown in Figure It needs more than 190 iterations before convergence Future works will be focused on reducing Figure 10 The design domain of the push gripper topology optimization problem Advances in Mechanical Engineering Figure 12 Level set surfaces of the optimal topologies of the push gripper mechanism (only half of the design domain is plotted) The final designs obtained using the proposed formulation are shown in Figure 11 The corresponding level set surface plots are shown in Figure 12 Note that only half of the design domain is plotted in Figure 12 due to the symmetry One can confirm that there are no de facto hinges occurring in the created mechanisms and this can confirm the capability of the proposed formulation (equation (11)) for designing hinge-free compliant mechanisms Conclusion Figure 11 The final designs of the push gripper mechanism the computational effort without affecting the optimization outcomes Push gripper The design of the push gripper has been widely studied previously, and the design domain and boundary conditions are shown in Figure 10 The goal of the design is to achieve a mechanism that when a horizontal force is applied at the input port of the mechanism, the opposing output ports move vertically and therefore it is capable of gripping a workpiece The design domain is discretized by 80 80 finite elements for the elastic analysis The void area (gap size) is set to be 30 30 finite elements Due to symmetry, only half of the design domain is taken into consideration The weighting factor a is set to 0.004 The maximum material usage Volmax is set to 30% and the umax in is set to 60 A simple yet efficient formulation for topology optimization of hinge-free compliant mechanisms is presented by taking into consideration an output stiffness The level set method is used for modeling the optimization problem The proposed method is examined by topology optimization of two benchmark compliant mechanisms, that is, the displacement inverter and the push gripper It is shown that the augmented objective formulation can prevent de facto hinges Although only the level set method is considered, the implementation of the proposed formulation to other topology optimization methods should be straightforward The present method can be used as an alternative method to partially control the de facto hinges of the linear elastic mechanisms Our future research will investigate the validity of the presented method for designing both 2D and three-dimensional (3D) largedisplacement compliant mechanisms Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this Li and Zhu article: This work was supported by the Open Fund of Key Laboratory of Robotics and Intelligent Manufacturing Equipment Technology of Zhejiang Province (RIE2016OSF02), the China Postdoctoral Science Foundation Funded Project (Grant No 2016M590772), and the National Natural Science Foundation of China (Grant numbers 51275174, 51605166) This support is greatly acknowledged References Howell LL Compliant mechanisms Hoboken, NJ: Wiley-Interscience, 2001 Sigmund O On the design of compliant mechanisms using topology optimization Mech Struct Mach 1997; 25: 493–524 Pucheta MA and Cardona A Design of bistable compliant mechanisms using precision–position and rigid-body replacement methods Mech Mach Theory 2010; 45: 304–426 Rahmatalla S and Swan CC Sparse monolithic compliant mechanisms using continuum structural topology optimization Int J Numer Meth Eng 2005; 62: 1579–1605 Wang N and Zhang X Compliant mechanisms design based on pairs of curves Sci Chin Technol Sci 2012; 55: 2099–2106 Zhu B and Zhang X A new level set method for topology optimization of distributed compliant mechanisms Int J Numer Meth Eng 2012; 91: 843–871 Zhu B, Zhang X and Wang N Topology optimization of hinge-free compliant mechanisms with multiple outputs using level set method Struct Multidiscip O 2013; 47: 659–672 Bendsøe MP and Sigmund O Topology optimization: theory, methods and applications Berlin: Springer, 2003 Durgun I_ and Yildiz AR Structural design optimization of vehicle components using cuckoo search algorithm MP Mater Test 2012; 54: 185 10 Yildiz AR and Saitou K Topology synthesis of multicomponent structural assemblies in continuum domains J Mech Design 2011; 133: 01100801–01100809 11 Frecker MI, Ananthasuresh GK, Nishiwaki S, et al Topological synthesis of compliant mechanisms using multicriteria optimization J Mech Design 1997; 119: 238–245 12 Motiee M Development of a novel multi-disciplinary design optimization scheme for micro compliant devices PhD Thesis, University of Waterloo, ON, Canada, 2008 13 Zhan J and Zhang X Topology optimization of compliant mechanisms with geometrical nonlinearities using the ground structure approach Chin J Mech Eng 2011; 24: 257–263 14 Rozvany GIN A critical review of established methods of structural topology optimization Struct Multidiscip O 2009; 37: 217C237 15 Wang M, Wang XM and Guo DM A level set method for structural topology optimization Comput Method Appl M 2003; 192: 227–246 16 Allaire G, Jouve F and Toader AM Structural optimization using sensitivity analysis and a level set method J Comput Phys 2004; 194: 363–393 17 Coffin P and Maute K Level set topology optimization of cooling and heating devices using a simplified convection model Struct Multidiscip O 2016; 53: 985–1003 18 Sethian JA Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and material science Cambridge: Cambridge University Press, 1999 19 Osher S and Fedkiw R Level set methods and dynamic implicit surfaces New York: Springer, 2002 20 Sethian JA and Wiegmann A Structural boundary design via level set and immersed interface methods J Comput Phys 2000; 163: 489–528 21 Osher S and Santosa F Level set methods for optimization problems involving geometry and constraints: I frequencies of a two-density inhomogeneous drum J Comput Phys 2001; 171: 272–288 22 Yamasaki S, Nishiwaki S, Yamada T, et al A structural optimization method based on the level set method using a new geometry-based re-initialization scheme Int J Numer Meth Eng 2010; 83: 1580–1624 23 Yamada T, Izui K, Nishiwaki S, et al A topology optimization method based on the level set method incorporating a fictitious interface energy Comput Method Appl M 2010; 199: 2876–2891 24 Wang Y, Luo Z, Kang Z, et al A multi-material level setbased topology and shape optimization method Comput Method Appl M 2015; 283: 1570–1586 25 Wang Y, Gao J, Luo Z, et al Level-set topology optimization for multimaterial and multifunctional mechanical metamaterials Eng Optimiz 2016; 1–21 26 Pedersen CB, Buhl T and Sigmund O Topology synthesis of large-displacement compliant mechanisms Int J Numer Meth Eng 2001; 50: 2683–2705 27 Yoon GH, Kin YY, Bendsøe MP, et al Hinge-free topology optimization with embedded translation-invariant differentiable wavelet shrinkage Struct Multidiscip O 2004; 27: 139–150 28 Sigmund O Morphology-based black and white filters for topology optimization Struct Multidiscip O 2007; 33: 401–424 29 Kim JE, Kim YY and Min S A note on hinge-free topology design using the special triangulation of design elements Commun Numer Meth En 2005; 21: 701–710 30 Zhou H Topology optimization of compliant mechanisms using hybrid discretization model J Mech Design 2010; 132: 111003–111010 31 Alexandrov O and Santosa F A topology-preserving level set method for shape optimization J Comput Phys 2005; 204: 121–130 32 Luo J, Luo Z, Chen S, et al A new level set method for systematic design of hinge-free compliant mechanisms Comput Method Appl M 2008; 198: 318–331 33 Chen SK, Wang MY and Liu AQ Shape feature control in structural topology optimization Comput Aided Design 2008; 40: 951–962 34 Wang MY and Chen S Compliant mechanism optimization: analysis and design with intrinsic characteristic stiffness Mech Based Des Struc 2009; 37: 183–200 35 Deepak SR, Dinesh M, Sahu DK, et al A comparative study of the formulations and benchmark problems for 10 the topology optimization of compliant mechanisms J Mech Robot 2009; 1: 1–8 36 Wang MY A kinetoelastic formulation of compliant mechanism optimization J Mech Robot 2009; 1: 021011–021020 37 Chen SK Compliant mechanisms with distributed compliance and characteristic stiffness: a level set method PhD Thesis, The Chinese University of Hong Kong, China, 2007 38 Huang X and Xie YM A further review of ESO type methods for topology optimization Struct Multidiscip O 2010; 41: 671–683 Advances in Mechanical Engineering 39 Sokolowski J and Zolesio JP Introduction to shape optimization: shape sensitivity analysis Heidelberg: Springer, 1992 40 Ta’asan S Introduction to shape design and control, 2001, http://www math cmu edu/shlomo/VKI-Lectures/ lecture1/index html 41 Choi KK and Kim NH Structural sensitivity analysis and optimization 1: linear systems New York: Springer, 2005 42 Sussman M, Smereka P and Osher S A level set approach for computing solutions to incompressible twophase flow J Comput Phys 1994; 114: 146–159  ... Commun Numer Meth En 2005; 21: 7 01? ?? 710 30 Zhou H Topology optimization of compliant mechanisms using hybrid discretization model J Mech Design 2 010 ; 13 2: 11 100 3? ?11 1 010 31 Alexandrov O and Santosa... MP Mater Test 2 012 ; 54: 18 5 10 Yildiz AR and Saitou K Topology synthesis of multicomponent structural assemblies in continuum domains J Mech Design 2 011 ; 13 3: 011 008 01? ?? 011 00809 11 Frecker MI, Ananthasuresh... for 10 the topology optimization of compliant mechanisms J Mech Robot 2009; 1: 1? ??8 36 Wang MY A kinetoelastic formulation of compliant mechanism optimization J Mech Robot 2009; 1: 0 210 11? ??0 210 20