On honeycomb representation and SIGMOID material assignment in optimal topology synthesis of...

Finite Elements in Analysis and Design 43 (2007) 1082–1098www.elsevier.com/locate/finel

On honeycomb representation and SIGMOID material assignment in optimaltopology synthesis of compliant mechanisms

Rajat Saxena, Anupam Saxena∗,1

Mechanical Engineering, Indian Institute of Technology, Kanpur 208016, India

Received 24 October 2006; received in revised form 26 July 2007; accepted 20 August 2007Available online 1 October 2007

Abstract

This paper introduces a new honeycomb based domain representation and SIGMOID material function to model continuum topologyoptimization domains with fixed grids. The two-point connectivity ensured with hexagonal cells is shown to circumvent singularities related tocheckerboard patterns and point flexures that are typically observed with one-point connected rectangular cells. The novel SIGMOID functionis developed to assign the ‘solid’ material for very low values of design variables (probabilities) and ‘void’ material for those further lower thanthe threshold, thus encouraging the binary material assignment. The performance of the SIGMOID function is compared with the previouslyproposed SIMP and PEAK material assignment functions. It is shown that both SIMP and PEAK functions can be over-restrictive in that thematerial is assigned only for probabilities close to one. The proposed modeling is suitable for topology optimization objectives wherein thenumber of constraints is large and gradient-based algorithms are chosen. Numerous examples on material layout determination for compliantmechanisms are solved with flexibility-stiffness and flexibility-strength multi-criteria formulations to illustrate the essence of this paper.� 2007 Elsevier B.V. All rights reserved.

Keywords: Topology optimization; Honeycomb parameterization; Compliant mechanisms; Flexibility-stiffness; Flexibility-strength; SIMP; PEAK; SIGMOID

1. Introduction

A topology design problem entails determining the optimalmaterial distribution or layout in a design region which can berepresented using a fixed array of sub-regions (unit cells), whichin turn are approximated using finite elements. Each cell (or fi-nite element) is assigned a design variable that models its mate-rial state. Design variables are related to the physical attributesof a cell or element, like the cross-section or elastic modulus.A zero value of a design variable represents no-material (void)state while a unit value symbolizes the full-material state. Thematerial states are determined using an optimization methodin conjunction with the finite element analysis that evaluatesthe functional objective (e.g., maximizing stiffness or compli-ance) of the design. Ideally, an optimal topology should solelyconsist of solid and void states in the so called “black andwhite” design, implying that “integer” values of only 0 or 1

∗ Corresponding author.E-mail address: [email protected] (A. Saxena).

1 Currently, visiting faculty in Cornell University.

0168-874X/$ - see front matter � 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.finel.2007.08.004

are sought for design variables. However, if one is restrictedto using gradient-based optimization, for instance in problemswhere the number of constraints are very large and need tobe satisfied in each sub-problem, to facilitate the computationof objective and constraints’ gradients, the design variables arerelaxed as continuous functions bounded between “0” (voidstate) and “1” (solid state).

Parameterization of a design region with a finite grid in-volves its representation in two levels: (a) physical or geomet-ric level that entails discretizing a domain into sub-regions(cells) of known shapes and (b) mathematical that requiresmodeling the material state of each cell as a function of itsdesign variable. Any parameterization scheme should ensurethat (i) at any step in optimization, the global finite elementstiffness matrix should always be non-singular for objectiveand constraint values and/or their gradients to be unique, (ii)the variables or sub-regions span the entire design region toensure the participation of all sites and (iii) the optimal topol-ogy should be manufacturable. In view of (i), singularity ofthe stiffness matrix may be partly avoided by using a small butpositive parameter ∈ as the lower bound for design variables.

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R. Saxena, A. Saxena / Finite Elements in Analysis and Design 43 (2007) 1082–1098 1083

Also, the sub-regions (or finite elements) should be well con-nected. The requirement (ii) may not be essential in that theoptimal topologies may be determined using a relatively coarsemesh wherein some sites in the domain may not participate intopology optimization (e.g., when using truss/frame elements).However, mesh dependency may exist and also, the search maybe limited to a few of many possible topologies. Finally, fromthe manufacturing viewpoint (iii), appropriate functions shouldbe used to model the relationship between the design variablesand respective material states such that the optimal topology isrealizable, ideally, without requiring user interpretation.

Physical representation of a two-dimensional design re-gion via a fixed finite element mesh may involve the use ofeither discrete (truss/frame) elements or continuum (triangu-lar/quadrilateral) elements. With the former, full or partialground structures are often used. A full ground structure [1] isan assemblage wherein every node is connected via a truss orframe element to every other node in the domain. In view of(ii) above, attempt is made to map all sites in the design regioninto the finite element mesh. A drawback, however, consider-ing (iii) above, is that the single-piece optimal topologies maybe difficult to manufacture as some overlapping elements mayexist. A partial truss ground structure, on the other hand, en-sures no element overlap but at the expense of losing numeroussites in the finite element mesh and thus numerous possibletopologies. Discrete elements that assume non-existing states(their cross-sections near the lower bound ∈) at convergencestill contribute, though insignificantly (depending on howsmall ∈ is), to the structural stiffness matrix. Complete extrac-tion of such elements from the optimal topology may causethe resultant global stiffness to become singular. This may bethe case when using truss elements as their ends are modeledas pin joints. The optimal topology resulting from the phys-ical removal of mathematically non-existing elements mayretain some dangling trusses with only one end connected.Frame elements in a partial ground structure, on the otherhand, are rigidly connected with non-zero torsional stiffnessat both their ends, and thus do not pose the aforementioneddifficulty. With frame elements in a partial ground structure,the extracted optimal topologies can be manufactured so longas the cross-sections (as design variables) are larger than thepermissible manufacturing limit. Thus, limited only by the in-ability to explore a comprehensive set of possible topologies,material layout design of compliant mechanisms with partialframe ground structures has been accomplished by Saxena andAnanthasuresh [2], Canfield and Frecker [3] and others withlinear deformation analysis and Saxena and Ananthasuresh [4]using geometrically large deformation to achieve prescribednonlinear force-displacement characteristics and curved out-put paths. Some recent works on topology optimization ofcompliant mechanisms using geometrically nonlinear frameelements are by Saxena [5,6], Rai et al. [7] and Zhou andTing [8,9].

Continuum sub-region models have additional advantagesover the discrete ones in that the former span the designregion more comprehensively satisfying criterion (ii) of thedesign region parameterization above and shape and topology

optimization get performed simultaneously. Initially conceivedby BendsZe and Kikuchi [10] who used homogenization tonumerically approximate the stiffness–properties of a unit rect-angular cell with a hole, continuum models have evolved fromthe use of layered microstructures [11] to that of SIMP (solidisotropic material with penalization) models [12–15]. In ho-mogenization, a hole in a unit cell can either be rectangular orelliptical in shape. The design variables that model the lengthand width of a rectangular hole, or the axes of an ellipticalone (say �1 and �2 in both cases, called the void parameters)represent the material state of the cell. With �1 and �2 both1, the hole occupies the cell area, and the cell is in its no-material state. When �1 = �2 = 0, the cell assumes the solidstate. Stiffness of the cell is computed numerically in terms ofthe void parameters. The second type of continuum models,the layered microstructures are obtained in a repetitive processin each step of which, a new layering of given direction isadded to the microstructure. The number of times the steps areperformed depends on the rank of the microstructure. Rank 1microstructure is obtained by arranging alternately thin layersof stiff and relatively flexible materials in the proportion, �1and 1 − �1 in the direction n1. Rank 2 microstructure is ob-tained likewise using layers of stiff and Rank 1 materials inthe ratio �2 and 1 – �2 in the direction n2. Material stiffnessof such microstructures is determined as functions of �i usingsmear-out or quasi-convexification techniques. A detailed re-view of topology optimization with fixed grids and continuumelements is provided by Eschenauer and Olhoff [16].

More commonly used continuum models in recent times arethe rectangular SIMP cells, advantages over the hole-in-cell andlayered microstructures being (i) the reduced number of designvariables per cell and (ii) avoidance of numerical procedures orquasi-convexification methods to estimate the stiffness proper-ties. Note that for �1 = �2 = �, the number of design variablesin homogenization and SIMP parameterization is the same. De-sign sub-regions as rectangular cells are directly approximatedusing four-node finite elements wherein the elasticity tensor Ei

for each cell i is given as

Ei = x�i E0, (1)

where E0 is the tensor for the solid element, xi is the cell de-sign variable (∈ �xi �1) representing the material state and� is the user-specified penalization exponent. The SIMP pa-rameterization has been applied to a range of topology op-timization problems. For minimum mean compliance of stiffstructures, BendsZe [12], Zhou and Rozvany [13], Mlejnik andSchirrmacher [14], and others have used linear deformationanalysis while Buhl et al. [17] have employed nonlinear de-formation model. For compliant mechanisms’ design, Sigmund[18] has used linear deformation to maximize the geometricadvantage. Pedersen et al. [19] have employed geometricallynonlinear analysis to achieve prescribed force displacement re-lationship. Sigmund [38,39] has extended the use of SIMP pa-rameterization to topology optimization of compliant mecha-nisms for applications with multiple output ports and materialsusing multi-physics analysis. As opposed to discrete elementswherein the variables representing the respective cross-sections

1084 R. Saxena, A. Saxena / Finite Elements in Analysis and Design 43 (2007) 1082–1098

Fig. 1. (a) A checkerboard pattern; (b) a continuum segment with zero stiffnesspoint flexure sites.

are physically interpretable, with the rectangular SIMP cell, in-termediate elastic moduli (x�

i E0 between ∈� E0 and E0) arehard to construe from the manufacturing viewpoint. BendsZeand Sigmund [15] show that a SIMP cell with �� 3 satisfies theHashin-Shtrikman upper bound [15] and can be physically in-terpreted as a composite cell that has an elasticity tensor equalto that of the SIMP cell.

A concern with the use of rectangular SIMP cells (in factboth continuum representation schemes mentioned above) is theappearance of checkerboard and/or stiffness singularity (pointflexure) regions. Checkerboard patterns have high numericalstiffness [20] and comprise alternate arrangement of solid andvoid cells (Fig. 1(a)). They appear mostly in stiffness designor related problems wherein minimization of mean compli-ance is posed as an objective. Regions with point flexures(Fig. 1(b)) often appear in compliant topology design prob-lems wherein it is required to maximize the output displace-ment in a prescribed direction (flexibility requirement) whilesimultaneously minimizing the strain energy of the contin-uum (stiffness requirement). Yin and Ananthasuresh [21] ex-plain that such kineto-statically singular regions with zero tor-sional stiffness are bound to appear with flexibility-stiffnessmulti-criteria formulations. This is because a rigid-linkage is atrue optimum for a flexible-stiff compliant topology for whichthe output deformation is large and the strain energy is zero.Both checkerboard and point flexure discrepancies are unde-sired as, after physical removal of void rectangular cells fromthe optimal topology for further analysis or fabrication, notonly does the stiffness matrix becomes singular but also it be-comes difficult to manufacture the continua with sub-regions inpoint contact.

With rectangular cells, formation of checkerboard patternsmay be avoided by (a) imposing a perimeter constraint on thestructure [22], (b) imposing a pointwise constraint on the de-sign variable [11], (c) by using a material redistribution scheme[23], (d) by using higher order finite elements [24] or (e) byemploying the image processing inspired filtering scheme [18].Liang and Steven [25] have suggested a performance based op-timization method to avoid this problem. In optimal complianttopology synthesis, to discourage point flexures to achieve dis-tributed compliance, Yin and Ananthasuresh [21] suggest andpenalize excessive relative rotation at all nodes in the domain.

The schemes mentioned above, in some manner, work towardssmoothing of gradients, thereby, disallowing abrupt variationin the variable values over adjacent cells. As a result, the voidsites in checkerboard patterns and around point-flexures getinfused with some material. Such supplementary schemes ef-fectively fillet the point contacts although they pose additionalcomputational costs on topology optimization. However, cellswith intermediate material states (between void and solid) doresult in optimal topologies.

As evident in checkerboard and point flexure regions, the di-agonally placed rectangular cells appear in point contact whilethe neighboring cells remain in void state. This is one of thecardinal causes for stiffness singularities. If the possibility ofpoint contact between any two neighboring cells is eliminatedin the geometrical discretization, one may not observe checker-board or point flexure regions in resultant topologies. Conse-quently, gradient smoothing schemes will not be needed. A wayto do so is to ensure at least a two-point or an edge contactbetween any two neighboring cells in the domain. This paperexplores the use of honeycomb discretization of a design regionwherein each cell is hexagonal in shape and shares at least anedge with its neighboring cell. The paper also explores the ob-jective of achieving the “black and white” optimal topologiesusing the PEAK and SIGMOID dependence of material stateson respective variables. Layout examples involve the optimaltopology synthesis of compliant mechanisms with single outputport using the flexibility-stiffness and flexibility-strength multi-criteria formulations with linear deformation analysis. Thoughgenetic algorithms have been used extensively in designing op-timal compliant continua using large deformation procedures[5–9], in this paper, because of the presence of a large num-ber of (stress) constraints in the flexibility strength formula-tion, a gradient based optimization approach, that emphasizesfeasibility of the intermediate solution within each iteration,is preferred which necessitates modeling topology optimiza-tion using continuous variables. Notwithstanding that accuratedisplacements can be captured using large deformation anal-ysis, in this paper, the primary goals are to illustrate the useof a honeycomb based design representation and performanceof two previously used and one newly proposed material as-signment functions. The following section describes the newlyproposed honeycomb based design representation and sensitiv-ity analysis of objectives and constraints. Section 3 briefs theflexibility-stiffness and flexibility-strength multi-criteria objec-tives, previously used in literature, for synthesis of compliantmechanisms. The performance of SIMP material parameteriza-tion is illustrated in Section 4. PEAK and newly proposed SIG-MOID material representations are discussed in Sections 5 and6, respectively. Section 7 discusses the limitations of this workand some methods to overcome them, after which conclusionsare drawn.

2. Honeycomb representation

A honeycomb representation shown in Fig. 2 is a staggeredarrangement of hexagonal cells such that each cell shares anedge with the neighboring one. Consequently, after void cells


Fig. 2. Honeycomb parameterization of the design region.

Fig. 3. (a) Vertically split, (b) left-split and (c) right-split hexagonal cells.

are physically removed, optimal topologies with honeycombdiscretization will have well-connected sub-regions, therebyeliminating stiffness singularities like those in diagonally con-nected rectangular cells. Such topologies shall further be man-ufacturable provided that all variable values are very close, ifnot equal, to their lower or upper bounds at convergence andthat no cell has an intermediate material state. This may pos-sibly be accomplished using an appropriate material modelingfunction with gradient based optimization.

When using hexagonal cells, analysis becomes easier when acell is further divided into two four-node finite elements. Finiteelement algorithms for quadrilateral elements can readily beused for the analysis and gradient computation with hexagonalcells. These quadrilateral elements inherit the same variablevalue or material properties as the parent hexagonal cell. Ahexagonal cell may be split into two four-node elements in threepossible ways shown in Fig. 3. Based on the direction of thesplit line drawn from the upper vertex to a diagonally oppositelower one, the three cells can be termed as vertically split,left-split or right-split cells. A honeycomb finite element meshcan have the cells with same or different split line directions.The arrangement can be uniform, periodic or random. Uniformarrangement may comprise all cells with the same split-linedirections, and in a periodic mesh, cells with different split-lines may follow a periodic trend.

Assuming the sensitivities are known for component quadri-lateral elements i and j of a cell c, the sensitivities for the cellcan be determined using the chain rule. That is

��

�xc

= ��

�xi

�xi

�xc

+ ��

�xj

�xj

�xc

. (2)

Fig. 4. Design domain and problem specification.

Here xc is the variable for the hexagonal cell, and xi and xj arevariables for the component elements. � represents either theobjective or a constraint. Noting that xc = xi = xj , the abovebecomes

��

�xc

= ��

�xi

+ ��

�xj

. (3)

3. Compliant mechanisms

Topology synthesis examples are presented for compliantmechanisms in this paper to assess the implementation of (i)honeycomb representation of a design domain and (ii) differentmaterial modeling functions. Compliant mechanisms are singlepiece devices designed for motion, force or energy transferusing the elastic deformation of constituting members. Re-duction of the rigid body joints offer advantages like reducedfriction, wear and tear, part count, ease in assembly, reducedbacklash, increased precision, reduction in noise and vibrationand many others over their rigid body counterparts [26]. Besidesthe kinematics-based approach to designing compliant mech-anisms [26], significant research effort early on has gone intothe topology design approach (for example [27,28,18,2,29]).Two multi-criteria formulations have mainly resulted. Thefirst is the flexibility-stiffness formulation wherein the outputdeformation is maximized while minimizing the strain energysimultaneously. The contention is that maximization of the out-put deformation renders adequate flexibility to the complianttopology while minimization of strain energy provides it theload bearing capability. The second is the flexibility-strengthformulation wherein, in addition to maximizing the outputdeformation, the failure criterion is modeled directly usinglocal stress constraints. For input–output specifications, givena design region with a single output port shown in Fig. 4, theoutput deformation �out is computed using the unit dummyload method [30] as

�out = VTKU, (4)

where V is the displacement response due to the dummy unitload along the direction of the output deformation specified,and U is the same for input loads only. Here, K is the globalstiffness matrix. The strain energy, SE, is expressed as

SE = 12 UTKU. (5)


In the flexibility-stiffness formulation [28], the objective is to

Minimize − �out

SE

Subject to V =N∑

i=1

xi �V ∗

∈ �xi �1. (P1)

Here, V is the continuum volume with V ∗ as its upper limit,xi is the design variable representing the material for the ithhexagonal cell, ∈ is a small positive number used to avoidstiffness singularities and N is the number of hexagonal cellsin the discretization. Sensitivities of the output deforma-tion and strain energy can be computed for a quadrilateralelement j as

��out

�xj

= −vTj

�kj

�xj

uj (6)

and

�SE

�xj

= −1

2uT

j

�kj

�xj

uj , (7)

where uj and vj are displacement responses of the jth quadri-lateral element for input and dummy loads respectively and kj

is the element stiffness matrix. Eqs. (6) and (7) can be employedwith Eq. (3) to compute the function and gradient sensitivitiesfor the hexagonal cell.

The flexibility-strength objective [29] is considered betterthan the flexibility-stiffness formulation for topology opti-mization of compliant mechanisms as in the former, the localfailure criterion is formulated more directly. Also, optimaltopologies with the flexibility-stiffness objective can be overlystiff as minimization of strain energy restricts the overall de-formation field. The flexibility-strength formulation can beposed as

Minimize: − �out

Subject to: g(xi)= (�VM)i

�a− ∈r

f (xi)n − 1�0, i = 1, . . . , N

V

V ∗ − 1�0, (P2)

where (�VM)i is the von Mises stress averaged at gauss pointsin a hexagonal cell, �a is the allowable stress limit, ∈r is the re-laxation parameter, f (xi) is the material interpolation functionand n is a chosen exponent greater than 1. As xi approaches itslower bound, the material function f (xi) approaches zero forwhich the effective upper bound on the mean von Mises stress,that is, �a(∈r/f (xi)

n + 1) approaches infinity. In other words,stress constraints are not applied at sites approaching their non-existing or void material state. Constraint sensitivity calcula-tions and constraint handling in gradient-based optimization arecomputationally expensive and therefore, it is desired to have aminimum number of constraints. A stress constraint is local innature in that, there exists a constraint for every hexagonal cellin the mesh. However, all stress constraints are not “active”,i.e., in many cells in the domain, the stress values may be be-low the safe limit. Only those elements are critical from failure

point of view, in which the stresses exceed the allowable limit.Inclusion of all stress constraints in the optimization procedureunnecessarily increases the computational load. Duysinx andBendsZe [31] suggested an active constraints strategy by usingonly active constraints and their sensitivities for optimization.This may be done by first calculating the stresses in each finiteelement and then calculating the sensitivities for only those ele-ments in which the stress constraints are violated. In the begin-ning of the optimization process, the set of active constraintsmay be large and unstable. However, near convergence, this setis stable.

In what follows, topology design of compliant mechanismsis attempted with three material interpolation functions us-ing the hexagonal representation of the design region. Synthe-sis examples are first presented with SIMP material function(Eq. (1)) wherein it is found, and noted elsewhere as well, thatcells with intermediate material states do appear in optimaltopologies. It is argued that the penalization exponent can begradually increased (from 3 onwards) with iterations to closelyapproximate the binary “black” and “white” material assign-ment. A similar function, the PEAK material model is testednext for which the standard deviation is reduced with itera-tions to a very small value, which results in optimal solutionswith disconnected sub-regions. However, proper connectivityis regained when the standard deviation value is relaxed. BothSIMP and PEAK models are over-restrictive in that they tendto assign material only for large values of design variables.A design variable may be considered as a measure of probabil-ity with which material can be assigned to the correspondingcell. Thus, both SIMP and PEAK models assign material only ifthe probability is very large. A SIGMOID material assignmentis proposed as the third alternative which, in contrast, assignsmaterial if the corresponding design variable, larger than thethreshold, indicates even the slightest probability of materialassignment.

4. Synthesis with SIMP material function

The first synthesis example is inspired by the optimal lay-out design of microstructures for negative Poisson’s ratio inLarsen et al [32]. The Poisson’s ratio for solid isotropic ma-terial is confined in the interval 0–0.5 and it may approach avalue of −1 for porous materials. By suitably designing themicrostructure of the material, the latter can be made to ex-pand transversely when subjected to elongation, i.e., to have anegative Poisson’s ratio. Note, however, that the result in thispaper cannot be used directly for the purpose because of dif-ferent boundary conditions used. The input–output specifica-tions and boundary conditions of the upper symmetric half ofthe design region are shown in Fig. 5. Boundary conditionsinvolve a roller support at the bottom face in addition to afixed lower region of the left edge. A 20 × 10 hexagonal cellmesh is used to approximate the domain. The gradient basedoptimization algorithm used is the Method of Moving Asymp-totes [33]. Volume constraint is 30% of the maximum possible.Two input forces (Fin) of 10 N each are applied to the designdomain. The length of a side of the hexagonal cell is 2 mm,


Fig. 5. Design domain of the symmetric half of a compliant mechanism.

Fig. 6. Optimal topologies for example in Fig. 5 for (a) vertically split cells;(b) right split cells; (c) left split cells and (d) alternate right and left split cells.

which makes the domain dimensions as 69 mm × 31 mm. Outof plane thickness is 1 mm. The SIMP material model in Eq.(1) is employed with the penalization exponent � = 4. No gra-dient filtering method to avoid checkerboard or point-flexurepatterns, suggested before for rectangular cells, is used in thisexample.

The optimal topologies obtained with meshes having hexag-onal cells with (a) vertical, (b) right, (c) left, and (d) alternateright and left split-line directions are shown in Fig. 6 with theirdeformed configurations. For the first case (Fig. 6(a)), the out-put deformation is 24 mm and the input displacement is 10 mm.The objective (P1) is minimized to −0.23. For the second re-sult (Fig. 6(b)), the output and input displacements are 6 and3 mm, respectively. The objective function value at optimum is−0.19. For third and fourth cases (Figs. 6(c) and (d)), the outputdisplacements are 12 and 7 mm while the input displacementsare 10 and 8 mm, respectively. The result in Fig. 7 is obtainedusing a structured array of rectangular cells with no sensitivityfiltering scheme employed.

Fig. 7. Optimal topology with a regular mesh of rectangular elements obtainedwith the design specifications in Fig. 5; checkerboard and point flexure regionsare present at many sites.

The following observations can be made from the resultsabove: (a) Comparing the solutions in Fig. 6 with that in Fig. 7,both checkerboard and point flexure patterns are not observedwith the hexagonal scheme. (b) Cells with intermediate mate-rial states do appear in optimal topologies with the SIMP ma-terial model. Such cells have the effective elastic moduli lessthan a solid cell and thus are more flexible. Thus, as suggestedby Yin and Ananthasuresh [21] that flexibility-stiffness objec-tives prefer local flexures, in case with hexagonal cells, theydo appear. However, hexagonal flexures have some finite (non-zero) torsional stiffness unlike point flexures in Fig. 1(b). Be-sides, as mentioned earlier, hexagonal cells with intermediatevalues of design variables (between 0 and 1) are physicallyinterpretable for � � 3 as they satisfy the Hashin–Shtrikmanupper bound [15]. (c) Different topologies are obtained whenusing hexagonal cells with different split line directions sug-gesting the dependence of the optimal solutions on the orienta-tion of the split lines. We conjecture that different orientationsgive rise to slightly different numerical stiffnesses (and the de-sign sensitivities) for the hexagonal cells leading to differenttopological results.

Feasibility interpretation of optimal topologies becomes anissue from the mathematical, and more so from the manufac-turing standpoint when some cells in optimal solutions haveintermediate “gray” material states. For manufacturing such so-lutions, a way may be to restore the material in such cells to thesolid state at the cost of the optimality of topological solutions.Another way is to replace such a cell by a “solid” thin beamlike sub-region that exhibits the same respective deformation


Fig. 8. Behavior of (a) SIMP function with variations in penalization exponent; (b) PEAK model for variation in standard deviation.

characteristics (or has the same torsional stiffness) as the graycell, which may prove tedious. The third alternative, within thedomain of topology optimization, is to use an appropriate ma-terial model that would ensure the cells to have the materialstates very close to “void” or “solid”.

It is mandatory for any material interpolation f (xi) like theone in Eq. (1) to be at least differentiable in ∈ �xi � 1 so thatthe objective and constraints’ sensitivities can be computed.Also, f (xi) should be close to but not zero for xi= ∈, andf (1) = 1. Further, f (xi) for ∈ �xi � 1 is expected to approx-imate the “black and white” schema near convergence. That is,f (xi) should either be close to “1” or “0” for a design vari-able xi . The SIMP material model satisfies most of the criteriaabove except that for smaller values of the penalization coef-ficient �, the binary representation is not approximated well.Fig. 8(a) shows the plot of SIMP material interpolation fordifferent values of the penalization coefficient. Notice that forany � > 1 (usually ��3 is chosen), f (xi) < xi which impliesthat the SIMP function has a tendency to force the materialstates towards the lower bound. Only for xis close to 1 is whenthe material state is close to “solid”. This becomes apparentfor larger values of the penalization exponent. The penaliza-tion coefficient is usually held constant during topology opti-mization. However, if � is increased gradually over iterations,one would expect to observe less number of gray cells in theconverged solution and the latter would appear almost like abinary design.

5. Synthesis with PEAK material function

The second material model, the PEAK function, was pro-posed by Yin and Ananthasuresh [34] and was used to syn-thesize compliant mechanisms with multiple materials. For atwo-phase (void-solid) material interpolation, the PEAK func-

tion is expressed as

E = E0

{exp

[− (xi − �)2

2�2

]+ �

}= E0f (xi). (8)

Here � (a small positive number) is the ratio of the elasticmoduli of void and solid material states, and � and � are meanand standard deviation, respectively with �=1 representing thesolid state. Further, with small enough standard deviation, themodeling relation in Eq. (8) approximates a delta function withthe peak at xi = � = 1. Thus, a treatment similar to that for thepenalizing coefficient in SIMP material model can be renderedto the standard deviation. That is, the standard deviation can bedecreased with the increase in the number of iterations to ensurematerial states only around the peak as suggested in Fig. 8(b).

The performance with PEAK material function is tested us-ing the example in Fig. 5 for a resource constraint of 20%. Thestandard deviation is decreased gradually from 1 to 0.01 in 150iterations. The upper symmetric half of the optimal topologyis shown in Fig. 9(a). As can be seen, the optimal continuumhas many gray cells (intermediate material states) and appearsdisconnected at certain regions. The observation is confirmedin Fig. 9(b) that shows the plot of the material states (‘�’) withrespect to the variable values. For standard deviation of 0.01,certain design variables with values greater than 0.9 have ma-terial states close to “void”. In other words, even when somevariables close to 1 exhibit strong inclination towards “solid”material assignment, smaller values of the standard deviationseem restrictive. For the same optimal design variable vector,if the standard deviation is relaxed to 0.1, the PEAK materialstates (‘o’) approach the solid state and the optimal solution ap-pears well defined (Fig. 9c). With standard deviation increasedfurther to 0.2, the material states (‘*’) appear even closer to thesolid state.


Fig. 9. Optimal topology of example in Fig. 5 using the PEAK function material model. (a) Optimal topology with � = 0.01; (b) change in PEAK materialstates for increasing � values; (c) optimal result for � = 0.2; (d) objective gradients at optimum and (e) deformed profile of the full topology.

The behavior of the optimal solution with change in the stan-dard deviation value can be investigated. From the Taylor seriesexpansion, one has

�(� + ��) = �(�) + ��

��

= �(�) + ��N∑

i=1

��

�f (xi)

�f (xi)

��, (9a)

where � is the flexibility-stiffness objective, �� is the changein the standard deviation value and f (xi) is the PEAK materialfunction in Eq. (8). Note that the function gradient ��/�xi foreach cell is known during optimization and thus ��/�f (xi) canbe computed using chain rule as

��

�f (xi)=

[�f (xi)

�xi

]−1 ��

�xi

. (9b)

With the analytical dependence of f (xi) on xi and � also knownfrom Eq. (8), the variation in the objective value �� from Eq.(9a) can be written as

�� = −��

�

N∑i=1

��

�xi

(xi − �). (10)

For the optimal solution in Fig. 9a, the gradients of theflexibility-stiffness objective are plotted in Fig. 9(d) whichare either negative or zero. For increased standard deviation(�� > 0) and noting that ∈ �xi �1, the product (��/�xi)

(xi − �) is always greater than or equal to zero implying that�� 0. In other words, an increase in standard deviationresults in further decrease of the objective. In this exam-ple, �� = −0.28 for � increased to from 0.01 to 0.1 and�� = −0.59 for � = 0.2. With this standard deviation value,the full topology with deformed profile is shown in Fig. 9(e).


Fig. 10. Optimal topology of pliers using the PEAK function material model. (a) Design specifications; (b) optimal topology with � = 0.0025; (c) change inmaterial states for increasing � values (d) optimal result for � = 0.1; (e) objective gradients at optimum and (f) deformed profile of the full topology.

As can be observed, the optimal result appears well connectedwith all retained cells close to the solid state and thus manu-facturable.

The second synthesis example with PEAK material interpola-tion is the design of compliant pliers. The design region (uppersymmetric half) of length 173 mm and width 61 mm is shown


Fig. 11. Optimal topology of compliant Atkinson’s mechanism using the PEAK function material model. (a) Schematic of the mechanism; (b) design domainwith loading and boundary conditions; (c) optimal topology with � = 0.01 (d) change in material states for increasing � values; (e) optimal result for � = 0.2,(f) objective gradients at optimum and (g) deformed profile with the kinematic setup.


Fig. 12. Optimal continuum for the specification in Fig. 5 using the flexibility-strength objective with PEAK function. (a) Optimal solution with � = 0.01;(b) PEAK material states; (c) optimal solution with � relaxed to 0.2; maximum stress values exceed the allowable limit.

with boundary conditions in Fig. 10(a) and is discretized using872 hexagonal cells. Three input forces of magnitude 10 N eachare applied vertically downward at the top left corner and theoutput displacement is sought along the vertically downwarddirection shown around the bottom right corner. The objectivein (P1) is minimized with design variables between 10−5 and1, out-of plane thickness as 2 mm and E0 as 2000 N mm−2.Fig. 10(b) shows the optimal solution for the resource con-straint of 40% obtained by gradually decreasing the standarddeviation from 1 to 0.0025 over 300 iterations. The flexibility-stiffness objective is minimized to −0.87. The optimal solutionappears disconnected as suggested by the material states (‘�’)shown in Fig. 10(c). Also, a few cells appear in the gray state.As before, note that some design variables at optimum are veryclose to 1 that are assigned the void states for a very low valueof standard deviation. Fig. 10(d) shows the topology after thestandard deviation is increased to 0.1, which appears well con-nected and also, the material cells (‘*’) are very close to thesolid state. The gradients of the flexibility-stiffness objectiveare shown in Fig. 10(e). Using Eq. (10) above, the change in theobjective value �� is recorded as −43.6 which implies that thesolution obtained with increased standard deviation has the ob-jective value smaller than the original solution. The deformedconfiguration of the full topology is shown in Fig. 10(f).

An example of partially compliant Atkinson’s mechanism[35] the kinematic sketch of which is shown in Fig. 11(a) issolved next. The design region of dimensions 60.5 × 52 mm2

with input/output specifications and boundary conditions isshown in Fig. 11(b) which is discretized using 1180 hexago-nal cells. The input force Fin is applied at a height of 9.5 mmfrom the bottom right corner and the component magnitudesare −8 and 2 N along horizontal and vertical directions respec-tively. The output deformation port is at a height of 46.5 mmfrom the bottom left corner. The optimization algorithm usedis the Method of Moving Asymptotes [33] and the standarddeviation is reduced from 1 to 0.01 in 150 iterations. The op-timal topology for a resource constraint of 20% is shown inFig. 11(c) with the objective minimized to −2.96. For standarddeviation of 0.01, even though the cells are very close to the

solid state (‘�’ plot in Fig. 11(d)), the solution appears withdisconnected sub-regions. For design variable vector at opti-mum and standard deviation increased to 0.2, the topology isshown in Fig. 11(e) that is well connected with cells very closeto the solid and void states (‘*’ plot in Fig. 11(c)). The functiongradients are plotted in Fig. 11(f) and application of Eq. (10)results in �� = −99.64 implying that the increase in standarddeviation from 0.01 to 0.2 further results in the lowering of theobjective value. The deformed profile with the kinematic setupis shown in Fig. 11(g).

The honeycomb parameterization with the PEAK functionmaterial model is tested for the flexibility-strength formulation(P2) for compliant mechanisms. The example in Fig. 5 is solvedwith the allowable stress limit of 10 N/mm2. The initial value ofrelaxation parameter ∈r in (P2) is chosen as 0.1 and is decreasedby a factor of 10 after every 50 iterations. The standard deviationin the PEAK function is reduced by a factor of 1.03 after eachiteration. For a standard deviation value of 0.01, the optimalsolution is shown in Fig. 12(a) and the material states (‘�’) withrespect to the design variables is shown in Fig. 12(b). The outputdeformation is maximized to 19.4 mm with the maximum stressvalue as 6.9 N/mm2. The solution appears disconnected, andfor increased standard deviation to 0.2, the resultant solution isshown in Fig. 12(c) which is a well connected “black and white”topology (‘*’ in Fig. 12(b)). However, the maximum stress isrecorded as 42.2 N/mm2 which is higher than the allowablelimit. This suggests that some constraints may get violated ifthe standard deviation in the PEAK material model is relaxedto seek a “binary” solution.

6. Synthesis wih SIGMOID material function

The material functions can be regarded as those that helpin making decisions pertaining to whether material should beplaced within a cell or not based on the probabilities expressedby the corresponding design variables. From what it seems,both SIMP and PEAK models are over-restrictive in that theymake material assignments only if the probability (the designvariable) is very close to 1. In the squeal, performance of the


Fig. 13. SIGMOID material function (a) behavior for increasing � with p = 0.5 and (b) behavior for decreasing p with � = 100.

third material model is explored to achieve the “binary” materialassignment in topology optimization. This SIGMOID functionis developed to have the assignment behavior contrary to SIMPand PEAK models, around xi = p < 1, where p is a parameter,with the intent that f (xi = p) = 1

2 . For ∈ �xi �p, one mayexpress the material assignment function f (xi) as an exponentfunction below.

f (xi) = c0 + c1 exp[−�(xi + c2)], ∈ �xi �p < 1. (11)

Here, � is a parameter such that �(xi + c2)�0, and c0, c1 andc2 are unknowns. It is expected that as � approaches +∞ andxi approaches ∈, f (xi) approaches 0 representing the “void”state. For c2 �= 0, the above requirement results in c0 = 0.Further, for f (xi = p) = 1

2 , one gets

12 = c1 exp[−�(p + c2)]. (12)

For Eq. (12) to be true for any �, a solution is c2 = −p whichimplies c1 = 1

2 . Eq. (10) then becomes

f (xi) = 12 exp[−�(xi − p)], ∈ �xi �p and � < 0. (13)

If � is chosen a positive parameter a priori, Eq. (13) can berevised as

f (xi) = 12 exp[−�(p − xi)], ∈ �xi �p. (14)

For p < xi �1, the material interpolation can be expressed sim-ilar to Eq. (11) as

f (xi) = d0 + d1 exp[−�(xi + d2)], p < xi �1, (15)

with �(xi +d2)� 0 and d0, d1 and d2 are unknowns which maybe determined as follows. As before, as � approaches +∞ andxi approaches 1, f (xi) is expected to approach 1 representingthe “solid” state. This implies that d0 = 1 for �(1 + d2) �= 0.Further, since the material value should be 1

2 at xi = p,

12 = 1 + d1 exp[−�(p + d2)], (16)

which for any � yields a solution d2 = −p and d1 = − 12 . Eq.

(15) thus becomes

f (xi) = 1 − 1

2exp[−�(xi − p)], p < xi �1. (17)

The resulting material interpolation is finally expressed as

f (xi) =⎧⎨⎩

12 exp[−�(p − xi)], ∈ �xi �p,

1 − 12 exp[−�(xi − p)], p < xi �1,

� > 0,

(18)

which is a SIGMOID function with two parameters � and p tomanipulate its shape. Note that f (xi) in Eq. (18) is positive for∈ �xi �1, and differentiable overall and particularly at xi =p.The roles of two parameters � and p can be comprehended asbelow. For p = 1

2 , the behavior of the SIGMOID material in-terpolation is shown in Fig. 13(a). For � = 1, the material val-ues are within the range 0.3–0.7 suggesting that some mate-rial is assigned to every cell. As � increases, the material rangegets closer to [0,1]. For very large values of � (say, � = 100 inFig. 9a), the SIGMOID function approximates the binary (0,1)assignment about p= 1

2 . That is, for xi < p= 12 , a material value

close to 0 is obtained while for xi > p, a material value closeto 1 results. Thus, � has a role similar to the penalization expo-nent in the SIMP model. The parameter p acts as a thresholdsuch that only those material values for variables greater thanp are allowed to approach the “solid” state. For fixed and large�(=100), Fig. 13(b) shows the variation of the SIGMOID in-terpolation with respect to p. As p decreases from 1

2 to a smallchosen value, all variables xi slightly larger than their lowerbound ∈ are encouraged to exhibit the “solid” material state forthe cell. In other words, if the probability that there should bematerial within a cell is slightly larger than 0 (for small p), thematerial is readily assigned to that cell. Note, however, that forp approaching 1, for large �, the SIGMOID function is similarto the SIMP and PEAK functions.

During optimization, parameters � and p can be chosen verylarge and small, respectively, at the outset and held constantthrough the iterations. Alternatively, they can be varied withthe increase in the number of iterations. If � is increased bya factor f1 and p decreased by f2 after every iteration, in the(k + 1)th iteration, �k+1 = f1�k and pk+1 = f1pk . For initialand final values of � as �0 and �N , respectively, and that for pas p0 and pN , respectively, after N iterations, one gets

�N = (f1)N�0,

pN = (f2)Np0, (19)


Fig. 14. The example in Fig. 5 solved using SIGMOID material interpolation. (a)–(b) Optimal topology and material states for N =50, (c)–(d) optimal topologyand material states for N = 150, (e) solution with V ∗ =10% and (f) solution with V ∗ =5%.

from where the factors f1 and f2 can be computed as

f1 = exp

(1

Nlog

�N

�0

),

f2 = exp

(1

Nlog

pN

p0

). (20)

In addition, if one desires to specify the minimum materialvalue f∈ to be attained by the SIGMOID function at xi= ∈,p = pN and � = �N , from Eq. (18), one gets

f∈ = 12 exp[−�N(pN− ∈)]. (21)

Thus, by choosing f∈ and pN in Eq. (21), one may get anestimate of �N as

�N = log 2f∈∈ −pN

≈ − log 2f∈pN

, (22)

which when used with specified N iterations would yield themultiplying factors. Otherwise, Eq. (20) can be used directly byspecifying the initial and final values of the penalizing exponent� and threshold p, respectively, to be attained in N iterations,without controlling the minimum material value f∈.

The example in Fig. 5 is solved with the SIGMOID materialfunction with starting values �0 and p0 as 1 and 1

2 , respectively.For f∈ = 0.01 and pN = 0.05, �N is obtained using Eq. (22)as 78, and for 50 iterations, Eq. (20) provides the respectivemultiplying factors as f1 = 1.091 and f2 = 0.955. The opti-mal result with the flexibility-stiffness objective is shown inFig. 14(a) and material status for design variables is providedin Fig. 14(b). As can be observed, not all cells occupy the voidor solid states completely, and some cells remain “gray” anddisconnected. This may be attributed to the smaller number ofiterations (N = 50) used in optimization. It is recommendedthat the factors in Eq. (20) be as close to 1 as possible, that is,


Fig. 15. Synthesis of compliant pliers using the SIGMOID material interpolation (a)–(b) Optimal result and material distribution for V ∗ =20%, N = 160;(c)–(d) optimal result and material distribution for V ∗ =10%, N = 300.

if varying, the change in � and p over the iterations should begradual enough to avoid any abrupt change in the material stateof a cell after an iteration. A way to do so can be to choose alarge value of N a priori. Thus, with increased N as 150, theexample is solved again with the optimal result and materialdistribution is shown in Fig. 14(c) and (d), respectively. Here,the factors f1 and f2 are 1.029 and 0.985, respectively. In thiscase, a “black and white” solution is obtained with no mate-rial discontinuities, and the flexibility stiffness objective mini-mized to −0.03 for a volume constraint of 20%. The solutionin Fig. 14(e) is for a resource constraint of 10% while that inFig. 14(f) is for 5% and in both cases, the material states inhexagonal cells are very close to “0” or “1”.

For the pliers example in Fig. 10(a), continuing with the mul-tiplying factors f1 and f2 are 1.029 and 0.985, respectively,after 160 iterations, pN is 0.04 and �N is 105. The optimal re-sult for formulation (P1) is shown in Fig. 15(a) and the materialdistribution is shown in Fig. 15(b) as varying with the designvariables. As conjectured, the optimal design is black and whitewith all cells either close to “0” or “1” states. The minimummaterial value f∈ can be determined from Eq. (21) as 2×10−3.Note the appearance of flexural pivots in this result, which isexpected when using the flexibility-stiffness objective [21]. Infact, each flexural pivot is a “solid” hexagonal cell with finite(non-zero) torsional stiffness and can be manufactured with-out requiring user interpretation. Fig. 15(c) and (d) show the


Fig. 16. Results with the flexibility-strength objective using the SIGMOID material model (a)–(b) Optimal solution and material states of the optimal continuumfor the example in Fig. 5. (c)–(e) Solutions and states for the pliers.

optimal solution and material distribution for the pliers withthe resource constraint set to 10%. Here, the final penalizationexponent and threshold values are set to 78.26 and 0.05, re-spectively over 300 iterations for which the multiplying factorsare f1 = 1.015 and f2 = 0.9924. Fig. 15(d) suggests that, byand large, the material states are close to “0” or “1”, however,for some cells, they are in the “gray” region. The observationhighlights the tendency of the flexibility-stiffness multi-criteriaobjective to yield flexural pivots, in this case of lesser torsionalstiffness than that in Fig. 15(a). Moreover, it suggests that theSIGMOID material approximation may not always result in bi-nary material assignment and that a few cells may appear grayin optimal solutions, particularly for problems wherein a strictrecourse constraint is set.

Synthesis examples with the flexibility-strength formulation(P2) are solved next using the SIGMOID material function.Fig. 16(a) shows the optimal solution for the example withspecifications of Fig. 5(a) and the material states are given inFig. 16(b). With starting values of � as 1 and p as 1

2 , the fi-nal values of the same are obtained as 100 and 0.07, respec-tively, after 150 iterations. Allowable stress limit is 5 N mm−2

and the stress relaxation parameter is decreased from 0.1 to0.001. The resource constraint used is 10%. As can be conjec-tured from Fig. 16(b), the material states mainly correspondto either “void” or “solid” with a few cells still in the grayregion. Fig. 16(a) shows the “black and white” solution with

gray cells removed. Likewise, synthesis of compliant pliers(design specifications for Fig. 10(a) is accomplished using theflexibility-strength formulation. Figs. 16(c) and (d) show theoptimal topology and material states. The solution has some dis-connected regions and also, quite a few cells are in the gray re-gion in the primary layout that includes the input–output ports.Fig. 16(e) shows the deformed profile after the dangling cellsare removed.

7. Discussion and closure

Implementation with three material models, namely, theSIMP, SIGMOID and PEAK functions, is discussed in thispaper with the newly proposed honeycomb representation. Theaim is to closely approximate the “binary” material assign-ment to the cells for singularity-free analysis and optimizationand interpretation-free manufacture of optimal solutions. TheSIMP and PEAK models approximate the 0–1 assignment forlarge values of penalization coefficient and small values ofstandard deviation respectively. The SIGMOID function be-haves similarly for large values of the penalization coefficient.In addition, further flexibility is rendered to the SIGMOID ma-terial interpolation by choosing a parameter p as a threshold.That is, design variables less than p are assigned the “void”states while those greater than p are rendered the “solid”states. Note that the three interpolation functions are similar


in behavior for large penalization coefficients, small standarddeviations and for values of the threshold p (for SIGMOID)near 1. For small values of the threshold, however, the PEAKand SIGMOID functions show different behavior. For smallerstandard deviations, the former is very restrictive and encour-ages material assignment only for design variables close to1. For smaller values of the threshold and large penalizationcoefficient, the SIGMOID function, on the other hand, assignsno material to the cell only if it is absolutely not required, i.e.,if the design variable is very close to zero. In other words, ifthere is a slight probability that the material is required, thecell is totally filled with material for xi �p.

Based on the observations from the examples solved in thispaper, it may not be always possible for the aforementionedmaterial functions to guarantee a perfect “black and white”optimal solution for a generic topology optimization problemat hand. The SIMP models are known to yield gray results[12–15]. The PEAK function also yields gray cells and discon-nected sub-regions for very small values of the standard devi-ation. A common observation with the four examples solvedusing PEAK material model is that small values of standard de-viation hinders many design variables close to their upper bound(the ‘�’ plots in Figs. 9(b), 10(c), 11(d) and 12(b)) from attain-ing the “solid” state. With design variables at convergence, ifthe standard deviation is relaxed or increased to a larger value,resultant material states obtained are close to the 0–1 assign-ment (the ‘*’ plots in Figs. 9(c), 10(d), 11(e) and 12(c), andthe topologies obtained are well connected. Investigation alsoshows that the change in the objective function value for cor-responding increase in standard deviation is negative, imply-ing that the flexibility-stiffness objective is decreased further.This suggests that the PEAK model is more restrictive in thatfor smaller standard deviation values, it does not allow mate-rial assignment for cells for which the corresponding designvariables suggest a higher possibility for the same. The SIG-MOID material assignment rectifies this anomaly by assigningmaterial even if the design variable indicates a slight possibil-ity (even close to zero) of material deposition. For strict re-source and stress constraints, the SIGMOID material functionmay fail to fully render the topological landscape “black andwhite”, as observed in Figs. 15(c) and 16(c). Upon close obser-vation (Figs. 15(d) and 16(d)), the gray states correspond to afew design variables very close to the threshold value p. Suchstates might get eliminated if the threshold is further reducedand the penalization coefficient is further increased. In a lim-iting case, there will exist a “jump” at the threshold at whichit would be equally likely to and not to assign the material. Insuch cases, material may be assigned with some trade-off withthe objective.Note that performance of the SIGMOID materialassignment will depend significantly on the initial choice ofparameters, the initial and final values of the penalization co-efficient and the threshold, and/or the (respectively, increasingand decreasing) multiplying factors. A very low value of theinitial penalization coefficient assigns the material value of 0.5throughout the domain. The penalization coefficient can thenbe gradually increased by a multiplying factor close to 1. Theinitial value of the threshold may be chosen anywhere between

Fig. 17. A composite Hermite curve approximating the boundary contour.

0 and 1. A lower initial value will mean that the material as-signment will be more probable at the start. Finally, the pe-nalization coefficient and threshold may be assigned very highand low values, respectively.

The primary intent in this paper is to study the performanceof honeycomb parameterization in conjunction with the existingand newly proposed SIGMOID material assignment functions.Though linear deformation analysis is employed to synthesizecompliant mechanisms in this work, geometrically/materiallynonlinear models can readily be incorporated to capture dis-placements accurately for synthesis of, say, path generatingcompliant mechanisms. Another issue when using the honey-comb based representation is the presence of notches in thecontinuum. While this is also a concern with traditional rect-angular cells, the hexagonal cells, at the boundary of the con-tinuum, can be smoothened using a composite Hermite curveas shown in Fig. 17. Though this will help eliminate notches, acost to pay would be the slight compromise on the accuracy ofthe behavior of the mechanism. Alternatively, outer and innercontours may be formed by the procedure suggested by Fig.17 and a solid model may be generated and meshed each timethe objectives and constraints evaluations are to be performedwith additional computational cost.

It is preferred to have distributed compliance over lumpedcompliance in optimal compliant mechanisms for reasons ofprevention of static/fatigue failure. Currently, with existing de-sign representation (beam and continuum) and material models,lumped compliance is observed in solutions. A few suggestionshave been made in this direction [21,36] to explicitly addressand seek distributivity. Yin and Ananthasuresh [21] propose topenalize the local relative rotation. They mention that strain en-ergy minimization is responsible for the appearance of lumpedcompliance. Rahmatalla and Swan [36] solve the optimizationproblem in two stages; they maximize the mutual potential en-ergy alone in the continuum in presence of artificial springsof high stiffness at both the input and output ports. In otherwords, they do not additionally minimize the strain energy butindirectly achieve a similar goal by constraining the input portusing a high-stiffness artificial spring at the input. Thus, sincestrain energy minimization within the continuum is not a directrequirement, hinges do not appear. In the second stage, after de-facto hinges are eliminated, the artificial springs are removedand an optimal continuum is obtained by optimizing the desireddeflection characteristics only in presence of the output spring.


Hull and Canfield [37] use a variable grid based subdivisionmethodology to avoid the formation of single point flexure intheir designs. In context of the work in this paper, the honey-comb parameterization does not directly address the issue ofdistributivity in compliance. Optimal solutions obtained in thiswork do contain hinges though with larger torsional stiffnessesimplying that local rotations are constrained. While the abovetwo methods by Yin and Ananthasuresh [21] and Rahmatallaand Swan [36] are indirect in that they seek distributivity toaddress static/fatigue failure, optimal methods will be soughtin future to address such issues directly.

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