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[.]

Advances in Mechanical Engineering

2016, Vol 8(8) 1–10

Ó The Author(s) 2016 DOI: 10.1177/1687814016664489 aime.sagepub.com

An augmented formulation of

distributed compliant mechanism

optimization using a level set method

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 effi-cient 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 liter-ature It is shown that the proposed augmented objective function can prevent the de facto hinges in the obtained com-pliant 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: 7 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.1,2Over the past decades, compliant

mechan-isms have been extensively studied Several approaches

have been developed for the design of compliant

mechanisms Generally, these approaches can be

cate-gorized 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.1,3Although the method has been

success-fully used in designing compliant mechanisms for

preci-sion 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,9and the top-down struc-tural 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,8,14and 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 Santosa21extended the method by introducing the

1

School of Automation Science and Engineering, South China University

of Technology, Guangzhou, P.R China

2

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

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shape sensitivity analysis into the framework In level

set–based topology optimization methods, the

struc-tural boundary is treated as the design parameter and is

implicitly embedded into a scalar function and updated

by solving the level set equation.22Therefore, topology

changes, for example, merging and splitting, can be

eas-ily handled Furthermore, several numerical instabilities

that usually occurred in the density-based topology

optimization approaches, such as checkerboard

pat-terns and gray scales, can be eliminated23–25as well

When using continuum topology optimization

meth-ods to design compliant mechanisms, one of the

signifi-cant 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

possi-ble to redesign the de facto hinge regions as continuous

material bridges.26However, the obtained mechanisms

will deviate from the original mechanism Alternatively,

one may use the procedures derived from the image

processing method,27such as the filtering method,28to

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 Swan4developed a new

spring-based method, to prevent de facto hinges Another

method can be regarded as a so-called hybrid

discreti-zation method.29,30In this method, during each

optimi-zation 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.32and Chen et al.33In 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 Zhang6developed 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

its validity by designing several two-dimensional (2D) numerical examples that are widely studied in the liter-ature of compliant mechanism optimization

Level set method

In level set method, the structural boundary ∂O is impli-citly embedded in a scalar function f18,19 as its zero level set Therefore

f(x, t).0 if x2 O f(x, t) = 0 if x2 ∂O f(x, t)\0 if x2 DnO

8

<

where D is the reference domain to contain all permissi-ble 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 follow-ing Hamilton–Jacobi equation

∂f

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 formu-lated 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 compli-ance 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 illu-strated in Figure 1 where Gd indicates the Dirichlet boundary An input force finis applied at the input port

i uinand uoutindicate the displacements occurred at the input port i and the output port o due to fin, respec-tively A spring with stiffness kout is attached to the out-put port to imitate the reaction force from the workpiece by fout= koutuout

In this study, in selecting the stiffness of kout, the bounding spring value method4is employed The value

of koutis set to be 104kbwhere kbis computed by apply-ing a unit load at the input port of the reference domain

D when D is fully occupied by the structural material

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

for-mulated as follows37

min : GA(u, f) = uout

u in

s:t: : uin umax

in Vol = Ð D

H (f)dO Volmax a(u, v, f) = l(v, f)8v 2 U

ð3Þ

where uinis constrained by an upper limit umax

in for indir-ectly controlling the maximum stress level.2Voldenotes

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

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

a(u, v, f) =

ð

D

Eijkleij(u)ekl(u)H(f)dO ð4Þ

l(v, f) =

ð

D

where Eijkl and eijare 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

u1, i koutu1, iu2, o+ koutu1, ou2, i

ð6Þ

where u1, i, u1, o, u2, i and u2, o are the displacements included in the displacement field u1 and u2, respec-tively 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 f2at the output port of the design domain u1, iand u1, o are the displacements at the input and the output ports, respectively, caused by f1 u2, iand

u2, o are the displacements at the input and the output ports, respectively, caused by f2 For more details, please refer to Sigmund2and Chen.37

H (f) is the Heaviside function defined as follows

H (f) = 1

0

if if

f 0 f\0



ð7Þ

and d(f) is the one-dimensional delta function defined

as follows

d(f) =dH(f)

De facto hinge problem

A hinged compliant inverter which is obtained using equation (3) is shown in Figure 2 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

Figure 1 The design domain of topology optimization of

compliant mechanisms.

Figure 2 A compliant displacement inverter mechanism

suffers from the de facto hinges, which are marked with dashed

line circle.

The reason for de facto hinges lies behind the

objec-tive formulations.36 In fact, for designing compliant

mechanism, many developed formulations specify two

main purposes, that is, maximizing the elastic

deforma-tion at the output port, meanwhile minimizing the

over-all 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

char-acteristic stiffness formulation.37A 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.5Here,

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.32proposed 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

topol-ogy 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,

Eoutcan be illustrated as follows

Eout= ð

∂D

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

displa-cement 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

intui-tive as shown in Figure 2 However, when the

mechan-ism is completely free of de facto hinges as is shown in

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

This means a hinged compliant mechanism corre-sponds to a large displacement while a hinge-free com-pliant mechanism corresponds to a small one Conversely, by minimizing the displacement at the out-put 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 mathe-matical formulation of the optimization problem can

be rewritten as follows

Figure 3 Schematic for determining Eout.

Figure 4 Schematic of a (a) hinged and (b) hinge-free compliant mechanism.

min : GA(u, f) + aEout(u, f)

s:t: : uin(f) umax

in Ð

D

H (f)dO Volmax a(u, v, f) = l(v, f)8u 2 U

ð11Þ

where a is the weighting factor of Eout The new

objec-tive function derived from the original function

(equa-tion (3)) is augmented with the energy func(equa-tion 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

func-tion with respect to the boundary perturbafunc-tions, the

shape derivative method39,40 is employed The

optimi-zation problem represented in equation (11) is

reformu-lated using Lagrange’s method of undetermined

multipliers as follows

J =GA + aEout+ lu in(uin umax

in + u2uin) + lVol(Vol Volmax+ u2Vol) ð12Þ

where uuin, uVol are slack variables to convert the

inequality constraint into the equality one and lu in, lVol

are Lagrange multipliers

Applying the Kuhn–Tucker conditions of J leads to

∂J

∂f, u

= ∂GA

∂f , u

+ a ∂Eout

∂f , u

+ lu in

∂uin

∂f , u

+ lVol

∂Vol

∂f , u

= 0 ð13Þ

∂J

∂luin = uin umax

in + u2

u in= 0 ð14Þ

∂J

∂lVol = Vol Volmax+ u2Vol= 0 ð15Þ

∂J

∂uuin = 2luinuuin= 0 ð16Þ

∂J

∂uVol= 2lVoluVol= 0 ð17Þ

lu in  0, lVol 0 ð18Þ where ∂J =∂f, uh i indicates the Fre´chet derivative of J

with respect to f in the direction of u

Equation (13) can be expressed as

∂J

∂f, u

=

ð

D

SGA+ aSEout + zuinSu in+ zVolSVol

d(f)udO ð19Þ

where SGA, SEout, Suin, and SVol can be derived from

∂GA=∂f, u

h i, h∂Eout=∂f, ui, h∂uin=∂f, ui, and

∂Vol=∂f, u

h i, respectively And zuin, zVol can be deter-mined from the mentioned Kuhn–Tucker optimally condition as follows

zuin= luin if uin.umax

in

0 if uin umax

in



ð20Þ

zVol= lVol if Vol.Vol

max

0 if Vol Volmax



ð21Þ

In order to calculate the shape sensitivity of J, we need to calculate the shape sensitivities of the individ-ual functions, that is, h∂GA=∂f, ui, h∂Eout=∂f, ui,

∂uin=∂f, u

h i, and ∂Vol=∂f, uh i

Taking Fre´chet derivative of the GA with respect to

fin the direction of u leads to

∂GA

∂f , u

=∂GA

∂u1, i

∂u1, i

∂f , u

+ ∂GA

∂u1, o

∂u1, o

∂f , u

+ ∂GA

∂u2, i

∂u2, i

∂f , u

+ ∂GA

∂u2, o

∂u2, o

∂f , u

ð22Þ Furthermore, taking Fre´chet derivative of the uin with respect to f in the direction of u leads to

∂uin

∂f , u

=∂uin

∂u1, i

∂u1, i

∂f , u

+ ∂uin

∂u1, o

∂u1, o

∂f , u

+ ∂uin

∂u2, i

∂u2, i

∂f , u

+ ∂uin

∂u2, o

∂u2, o

∂f , u

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 ∂uh 1, i=∂f, ui,

∂u1, o=∂f, u

h i, ∂uh 2, i=∂f, ui, and ∂uh 2, o=∂f, ui need

to be solved

The Fre´chet derivative of u1, iwith respect to f in the direction of u can be written as

∂u 1, i

∂f , u

D

2½kf1u1+∂(f1 u 1 )

∂n 

Eijkleij(u1)ekl(u1)

d(f)udO + 2Ð

G d(f) jrfj

∂f

∂nf1u1udG

ð24Þ

where k is the mean curvature of the structural bound-ary 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

∂u1, i

∂f , u

= ð

D

Eijkleij(u1)ekl(u1)d(f)udO ð25Þ

Similarly, the Fre´chet derivatives of u1, o, u2, i, Eout,

and u2, ocan be written as

∂u1, o

∂f , u

= ∂u2, i

∂f , u

= ð

D

Eijkleij(u1)ekl(u2)d(f)udO

ð26Þ

∂Eout

∂f , u

= ∂u2, o

∂f , u

=  ð

D

Eijkleij(u2)ekl(u2)d(f)udO

ð27Þ

The derivative of allowable material usage Vol with

respect to f in the direction u can be written as follows

∂Vol

∂f , u

= ð

D

dH (f)

df udO =

ð

D d(f)udO ð28Þ

For more details, please refer to previous

stud-ies.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)18is employed for solving equation (2)

In order to ensure stability, the following

re-initialization equation42is used

∂f

∂t + S(f0)(jrfj  1) = 0 ð29Þ

where S(f0) is a sign function taken as 1 in O, 1 in

DnO, and 0 on the interface In numerical

implementa-tion, S(f) is approximated by

S(f0) = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif0

f20+ (Dx)2

where Dx is set to the mesh size

In order to avoid the numerical difficulties, we

adopted the method proposed in Wang et al.15in which

the Heaviside function is approximated using equation

H (f) =

3

43e

4

D f3 3D3

+1 + e

2 if D  f  D

8

<

:

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

43e 4

Df2

D 3

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 = 1 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

Displacement inverter

The designing of the displacement inverter is first con-sidered The design domain is shown in Figure 5 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 = 1 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 3 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 6 For all studied five cases, the created compli-ant mechanisms do not suffer de facto hinges When large a is used, the elastic hinges can also be eliminated

Figure 5 The design domain of the displacement inverter topology optimization problem.

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 7 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 diffi-culty of selection, one may rephrase the objection func-tion in (equafunc-tion (11)) using the normalizafunc-tion method

so that the value of a lies between 0 and 1

In fact, equation (11) can prevent de facto hinges not only in the final topology but also during the optimiza-tion process This can be seen from some intermediate designs of the displacement inverter with a = 0:002 in Figure 8 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 cer-tain number of holes) is needed for obcer-taining a reason-able solution The convergence histories of using the proposed method with the case of a = 0:002 are shown

in Figure 9 It needs more than 190 iterations before convergence Future works will be focused on reducing

Figure 6 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 7 The effect of a on the geometric advantages and the

iterations of the obtained mechanisms.

Figure 8 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 9 The convergence histories of the displacement inverter problem with a = 0:002.

the computational effort without affecting the

optimi-zation outcomes

Push gripper

The design of the push gripper has been widely studied

previously, and the design domain and boundary

con-ditions 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 3 80 finite elements for the elastic

analysis The void area (gap size) is set to be 30 3 30

finite elements Due to symmetry, only half of the

design domain is taken into consideration The

weight-ing factor a is set to 0.004 The maximum material

usage Volmaxis set to 30% and the umaxis set to 60

The final designs obtained using the proposed for-mulation 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 for-mulation (equation (11)) for designing hinge-free com-pliant mechanisms

Conclusion

A simple yet efficient formulation for topology optimiza-tion of hinge-free compliant mechanisms is presented by taking into consideration an output stiffness The level set method is used for modeling the optimization prob-lem The proposed method is examined by topology opti-mization of two benchmark compliant mechanisms, that

is, the displacement inverter and the push gripper It is shown that the augmented objective formulation can pre-vent de facto hinges Although only the level set method

is considered, the implementation of the proposed formu-lation 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) large-displacement 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 sup-port for the research, authorship, and/or publication of this

Figure 10 The design domain of the push gripper topology

optimization problem.

Figure 11 The final designs of the push gripper mechanism.

Figure 12 Level set surfaces of the optimal topologies of the push gripper mechanism (only half of the design domain is plotted).

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.

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