Research Article Efficient Local Level Set Method without ...MathematicalProblems in Engineering Edge of narrow band Zero level set D\ 2 c0 c 0 0 x D\ F : Narrow-band model and local

$Page 1: Research Article Efficient Local Level Set Method without ...MathematicalProblems in Engineering Edge of narrow band Zero level set D\ 2 c0 c 0 0 x D\ F : Narrow-band model and local$
Research ArticleEfficient Local Level Set Method without Reinitialization andIts Appliance to Topology Optimization

Wenhui Zhang and Yaoting Zhang

School of Civil Engineering andMechanics Huazhong University of Science and Technology 1037 Luoyu Road Wuhan 430074 China

Correspondence should be addressed to Yaoting Zhang zyt1965mailhusteducn

Received 5 July 2015 Accepted 21 December 2015

Academic Editor Manuel Pastor

Copyright copy 2016 W Zhang and Y Zhang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The local level set method (LLSM) is higher than the LSMs with global models in computational efficiency because of theuse of narrow-band model The computational efficiency of the LLSM can be further increased by avoiding the reinitializationprocedure by introducing a distance regularized equation (DRE)The numerical stability of the DRE can be ensured by a proposedconditionally stable difference scheme under reverse diffusion constraints Nevertheless the proposed method possesses nomechanism to nucleate new holes in the material domain for two-dimensional structures so that a bidirectional evolutionaryalgorithm based on discrete level set functions is combined with the LLSM to replace the numerical process of hole nucleationNumerical examples are given to show high computational efficiency and numerical stability of this algorithm for topologyoptimization

1 Introduction

Topology optimization is a numerical iterative procedurefor making an optimal layout of a structure or the bestdistribution ofmaterial in the conceptual design stage [1]Thelevel set method (LSM) is a recently developed approach totopology optimization that uses a flexible implicit descriptionof the material domain [2] The central idea of the LSM is toemploy an implicit boundary describing model to parame-terize the geometric model and the boundary of a structureis embedded in a high-dimensional level set function thatis called its zero level set [3] The level set-based methodis able to not only fundamentally avoid checkerboards andmesh-dependence but alsomaintain smooth boundaries anddistinct material interfaces during the topological designprocess [4] Hence many level set-based methods [5] havebeen developed for topology optimization since the LSMwasfirst introduced into structure optimization

With an implicit local level set model the computationalefficiency of the local level set method (LLSM) [6] is muchhigher than that of the global level set methods especiallyfor shape optimization However the main shortcoming ofthe conventional LSM is that it possesses no mechanism

to nucleate new holes in the material domain for two-dimensional structures resulting in the final design heavilydependent on the initial guess [4] A mechanism named thebubble-method [7] was first proposed to create new holesinside the structures in topology and shape optimizationThis idea has been further developed into the mathemat-ical concept of topological derivatives [8] In the shape-sensitivity-based level set approaches topological derivativesare incorporated to indicate the best place for introducing anew hole in a separate step of the optimization process [9] oras an additional term in the Hamilton-Jacobi equation [10]The globally supported radial basis function (RBF) [11] andcompactly supported RBF (CSRBF) [12] are typically used todiscretize the original time-dependent initial value probleminto an interpolation problem The CSRBF brings about thestrictly positive definiteness and sparseness properties ofmatrices under certain conditions Hence the CSRBF hasgeneralized the practical applications of RBFs to a larger setof scattered data [12]

In the conventional LSM [3] a reinitialization procedureusually needs to reshape the level set function (LSF) to asigned distance function (SDF) periodically However thezero level set may drift away from its initial position by

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 6392901 14 pageshttpdxdoiorg10115520166392901

2 Mathematical Problems in Engineering

iteratively solving a classical reinitialization equation [13] Tosuppress this drift an interface preserving level set redis-tancing algorithm is proposed by Sussman and Fatemi [14]Nevertheless it has been proved that the SDF is not a feasiblesolution to theH-J equation [15] In practice it not only raisesserious problems as when and how it should be performedbut also affects numerical accuracy in an undesirable way andthus should be avoided as much as possible [16] The needfor reinitialization was originally eliminated by introducinga penalty term [17] into a variational level set approach [18]The undesirable boundary effect of the penalty term can beeliminated by taking a distance regularized equation (DRE)instead of this term Hence a so-called distance regularizedlevel set evolution (DRLSE) [16] is realized based on thevariational approach As an unnecessary diffusion effect ofthe DRLSE was found in some locations where the surfaceis too flat the DRE was recently modified with a new andbalanced formulation to eliminate this effect [19] Althoughparts of the diffusion rates in the DRE are negative thenumerical stability can still be maintained by incorporatingreverse diffusion constraints in the difference schemes of theDRE as can the reverse diffusion equations with all negativediffusion rates [20]

The aim of this work is to solve the aforementionednumerical issues that still exist in the LLSM for topologyoptimization of two-dimensional structures A bidirectionalevolutionary algorithm based on the discrete level set func-tions (DLSFs) is proposed to find a stable topological solutionfirst and then combined with the LLSM to further evolvethe local details of the topology and shape of the structureTransforming the DLSFs into the local level set function ofthe LLSM is achieved by iteratively solving the DRE Afterthat the DRE is incorporated into the LLSM to avoid thereinitialization procedure A difference scheme under reversediffusion constraints is formulated for the DRE to improve itsnumerical stability Typical examples are given to show theeffectiveness of the proposed algorithm in terms of conver-gence computational efficiency and numerical stability

2 Optimization Algorithm

21 Local Level Set Method Using Narrow-BandModel In thelocal level set method (LLSM) [6] the local level set equationis defined as

120597120601

120597119905+ 119862 (120601)119881

119899

1003816100381610038161003816nabla1206011003816100381610038161003816 = 0

(1)

where 120601(119909 119905) is defined as the local level set function (LLSF)and 119881

119899is the normal velocity in normal direction 119899 =

nabla120601|nabla120601| the truncation function 119862(120601) is

119862 (120601) =

1 if 10038161003816100381610038161206011003816100381610038161003816 le Δ

(10038161003816100381610038161206011003816100381610038161003816 minus 120574)2

(210038161003816100381610038161206011003816100381610038161003816 + 120574 minus 3Δ)

(120574 minus Δ)3

if Δ lt 10038161003816100381610038161206011003816100381610038161003816 le 120574

0 if 10038161003816100381610038161206011003816100381610038161003816 gt 120574

(2)

with 1198790= 119909

1003816100381610038161003816120601(119909)1003816100381610038161003816 lt 120574 being a narrow band with

the half-band width 120574 The narrow-band model and thecorresponding LLSF are described as shown in Figure 1

It can be seen from Figure 1 that only the LLSF within thenarrow-band 119879

0needs to be updated during each iteration

Hence the LLSM is higher in computational efficiency thanthe LSMs based on global level set models

22 Bidirectional Evolutionary Algorithm with Discrete LevelSet Functions A two-dimensional structural model is builtin the work region 119863 sub 119877

2 And the set 119878 = 1198781cup 1198782cup 1198783

represents the finite elements in the domain 119863 It can bedivided into three parts 119878

1that consists of the solid elements

with a full-material density 1198782that covers the elements

with intermediate material densities 1198783that involves the

void elements with a weak-material density Accordingly thenodal sets corresponding to the elemental sets 119878

1 1198782 and 119878

3

are defined as 1198781198991 1198781198992 and 119878119899

3 respectively If it is assumed that

119878119899

2sub 119878119899

3 1198781198991cup 119878119899

3consist of all the nodes within the region 119863

then a discrete level set function (DLSF) for node 119895 can bedefined as

120601119895=

minus1198880119895 isin 119878119899

1

1198880

119895 isin 119878119899

3

(3)

where 1198880is a predefined constant set as 1 in this study

The values of elemental densities can be derived from theDLSFs If the 119894th element belongs to 119878

1 that is 119894 isin 119878

1 then

the element density 120588119894= 1 if 119894 isin 119878

3 then 120588

119894= 120588min where

120588min is a small value 0001 if 119894 isin 1198782 then 120588

119894isin (120588min 1)

where 120588119894is calculated in terms of the interpolation criterion

given in the code manual [21] In the structural model therectangular element is split into four triangles first and thevalue of theDLSF at the commonpoint of the triangles is thengiven by the average of the values of the four points Afterthat each triangle is examined separately in the same logicFinally the elemental density is found to be the average of thecontributions of the triangles

The structural stiffness design has been widely inves-tigated in numerous literatures for topological sensitivityanalysis The standard notion [1] of minimum compliancedesign problems under a global volume constraint can bemathematically defined as follows

minimize120601

119869 (Ω) =1

2119906119879

119870119906

Subject to119873

sum

119894=1

119881119894120588119894(120601) = 119881

lowast

120588119894isin [120588min 1]

(4)

where 119869 is known as the mean compliance the open set Ωrepresents all admissible shapes in the design region 119863 119906(120601)is the nodal displacement vector and 119870 denotes the globalstiffness matrix 119881

119894is the volume of an individual element

119881lowast is the prescribed total volume and 119873 is the number of

elements

Mathematical Problems in Engineering 3

Edge of narrow band

Zero level set

Ω

DΩ

2120574

120601

c0

minusc0

x0

120574

Ω

Ω

D

Figure 1 Narrow-band model and local level set function in LLSM

Similar to the bubble-method [7] and the level set-basedoptimization methods [4 10] topological derivatives aretaken as topological sensitivities in this studyThe topologicalderivative for node 119895 is given by [9]

120572119899

119895= 119863119879119869 (120601119895) =

120587 (120582 + 2120583)

4120583 (120582 + 120583)4120583119864119895120576 (119906) 120576 (119906)

+ (120582 minus 120583) tr (119864119895120576 (119906)) tr (120576 (119906))

(5)

where 119864119895denotes the material elasticity tensor for node 119895 120576

is the strain tensor and the lame constants 120582 = 1198640](1 minus ]2)

120583 = 11986402(1+])with the Poisson ratio ] and Youngrsquos modulus

of solid materials 1198640 tr(119860) denotes the trace of a matrix 119860

Based on an interpolation function proposed by Shepard[22] a filter scheme of mesh independence is proposed toavoid the checkerboard patterns and mesh dependencies Acircular domain Ω

119908is first defined as the influence region

centered round point x with cut-off radius 119903119908 and 119873

Ω

denotes the number of points located inside the influencedomain The sensitivity filtering using the Shepard methodwith scattered points is then defined by

119899

(x) =119873Ω

sum

119894=1

119882119894(x) sdot 120572119899 (x)

119882119894(x) = 119863

119894(x)

sum119873

119895=1119863119895(x)

(119894 = 1 119873)

(6)

where 119882119894(x) is the Shepard interpolation with the basis

function 119863119894(x) = 1radic(119889

119894(x))2 + 1198882 in which 119889

119894(x) = x minus 119909

119894

denotes the radial distance from point 119909 to 119909119894 and if 119889

119894(x) ge

119903119908 then119863

119894(x) is set to zero 119888 is a positive constant and chosen

as a onefold mesh size in terms of numerical experiencesOver the last two decades many topology description

models have been developed for topology optimization ofstructures which can roughly be classified into two cate-gories the material distribution model and the boundarydescription model [23] Based on the material distribu-tion model the ESO (Evolutionary Structural Optimization)methodhaswon a great deal of popularity in recent years [24]The bidirectional ESO (BESO) method [25] as an extensionof the ESO method allows efficient material to be added tothe structure while the inefficient one is removed simultane-ously So a bidirectional evolutionary algorithm is developedby integrating both the DLSFs and topological derivativesinto the optimization criteria of the BESOmethod [25] Notethat the design variables and topological sensitivities in theBESO method are based on the elemental pseudo densitieswhile those in the proposed algorithm are based on thediscrete level set functions

It is assumed that the volume in the 119896th iteration 119881119896is

known and 119896 ge 0The target volume119881119896+1

in the next iterationis then updated by

119881119896+1=

min (119881119896(1 + ER) 119881lowast) if 119881

119896le 119881lowast

max (119881119896(1 minus ER) 119881lowast) if 119881

119896gt 119881lowast

(7)

with the evolutionary volume ratio ER and the volume limit119881lowast defined in (4)


A parameter AR119899 is defined as the adding numberof nodes in the set 119878119899

1divided by the total numbers of

nodes and AR119899 le AR119899max where AR119899max is a predefinedpositive constant The definition of AR119899 is different fromthat of AR in the original BESO method since the formerparameter corresponds to nodal sensitivity while the latterone corresponds to elemental sensitivity

It is assumed that the DLSF 120601119896119895of node 119895 is known in

the 119896th iteration Then the DLSF 120601119896+1119895

in the next iterationis updated by

120601119896+1

119895=

minus1198880120572119899

119895ge 120572

addth 120601

119896

119895= 1198880

1198880

120572119899

119895ge 120572

delth 120601119896

119895= minus1198880

(8)

where the threshold sensitivity numbers 120572delth and 120572addth aredetermined as the number of nodes decreased from theset 1198781198991and that increased from the set 119878119899

3 respectively

These thresholds are similar to those based on elementalsensitivities given in the original BESO method Full detailsof determining these thresholds are described in [25]

Finally a stable topological solution is obtained when thefollowing convergence criterion is satisfied

10038161003816100381610038161003816sum119896minus5

119896minus9119869 (Ω) minus sum

119896

119896minus4119869 (Ω)

10038161003816100381610038161003816

sum119896

119896minus4119869 (Ω)

le 120591 (9)

where 120591 is an allowable convergence error with typical valuesranging from 0001 to 001

The majority of logical steps of the bidirectional evolu-tionary algorithm are presented in Figure 2

23 Distance Regularized Equation (DRE) and Its Improve-ment In the distance regularized level set evolution (DRLSE)[16] the DRE can retain the signed distance feature |nabla120601| =1 at least within the narrow-band region near boundarieswithout reinitialization whose formula is expressed in thestandard form of the diffusion equation as

120597120601

120597119905= 120583 div (120572

1(120601) nabla120601) (10)

with the diffusion rate 1205721(120601) = 120583119889

1199011(|nabla120601|) where the

diffusion function is set to 1198891199011(119904) = 119901

1015840

1(119904)119904 with 119904 = |nabla120601|

In the original DRLSE the energy density 1199011(119904) was

defined as

1199011(119904) =

(1 minus cos (2120587119904))(2120587)2

if 119904 le 1

(119904 minus 1)2

2if 119904 gt 1

(11)

which is a double-well potential function because there aretwo minimum points of 119901

1(119904) at 119904 = 1 and 119904 = 0 So the

diffusion function 1198891199011(119904) is given by

1198891199011(119904) =

sin (2120587119904)2120587119904

if 119904 le 1

1 minus1

119904if 119904 gt 1

(12)

It is easy to verify the boundedness of the diffusion rate1205721(120601) = 120583119889

1199011(|nabla120601|) and |120572

1(120601)| le 120583 It can be seen from (12)

that for |nabla120601| gt 1 1198891199011(|nabla120601|) is positive and |nabla120601|will decrease

and approach 1 for 05 lt |nabla120601| le 1 1198891199011(|nabla120601|) is negative and

|nabla120601| will increase and approach 1 for |nabla120601| le 05 1198891199011(|nabla120601|) is

positive and |nabla120601| will decrease and approach 0If |nabla120601

0| le 05 is satisfied for all the initial values 120601

0 the

diffusion effect of (10) will make |nabla120601| approach 0 So it losesthe ability to regularize |nabla120601| to 1 An improved diffusion rate1205722(120601)with a diffusion function like the following is proposed

in [19]

1198891199012(1003816100381610038161003816nabla1206011003816100381610038161003816) =

2

120587 arctan ((1003816100381610038161003816nabla1206011003816100381610038161003816 minus 1) 120590)

10038161003816100381610038161206011003816100381610038161003816 le 120576

100381610038161003816100381610038161003816100381610038161003816

2


100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161206011003816100381610038161003816 gt 120576

(13)

where 120590 is a positive constant and is chosen as fourfold meshsizes in terms of numerical experiences 119879

1= 119909

1003816100381610038161003816120601(119909)1003816100381610038161003816 lt 120576

is a narrow band with a half-band width 120576The diffusion effect can be divided into two parts the

forward diffusion for |nabla120601| ge 1 and the backward diffusionfor |nabla120601| lt 1 It will make |nabla120601| approach one within 119879

1but

zero outside 1198791 However the two parts are balanced within

1198791but unbalanced outside 119879

1so that multiple iterations are

required to retain a flatter level set surface outside 1198791 In this

paper the diffusion function 1198891199012(|nabla120601|) is further localized by

introducing the half-band width 120574 of the narrow-band 1198790in

LLSM thereby resulting in an improved diffusion rate 1205723(120601)

using the diffusion function

1198891199013(1003816100381610038161003816nabla1206011003816100381610038161003816) =

2


10038161003816100381610038161206011003816100381610038161003816 lt 120574

010038161003816100381610038161206011003816100381610038161003816 ge 120574

(14)

With the diffusion rate 1205723(120601) the two parts are balanced

within 1198790without influencing the level set surface outside 119879

0

24 A Conditionally Stable Difference Scheme for DRE Itis noted that the common difference schemes for the DREwith parts of the negative diffusion rates are incapable ofremaining stable during an iterative process according to thestability definition of the difference equation In our numer-ical experiments |120601| is apt to gradually become divergentalongwith the process of iterations To enhance the numerical


Start

Determine the target volume

Update the topological derivatives by sensitivity filtering using Shepard

interpolation function

Define the maximum design domaininitial design loads and supports

No

a stable topological solution

Yes

optimization criterion of BESO method

Is the convergence criterion satisfied

Increase the iteration

Carry out the finite element analysiscalculate the topological derivative

of each node

the initial DLSFs 1206010jDefine the elemental set S and

for the next iteration Vk+1

Update the DLSFs 120601k+1j according to the

Stop iteration and obtain

number k = k + 1

Figure 2 Flow chart depicting logical steps of the bidirectional evolutionary algorithm

stability of the DRE a difference scheme similar to that of themean curvature given in [18] is developed and described as

120601119896+1

119894119895minus 120601119896

119894119895

Δ119905

=

120572 (120601119896

)119894+12119895

120575119909

+120601119894119895minus 120572 (120601

119896

)119894minus12119895

120575119909

minus120601119894119895

Δ119909

+

120572 (120601119896

)119894119895+12

120575119910

+120601119894119895minus 120572 (120601

119896

)119894119895minus12

120575119910

minus120601119894119895

Δ119910

(15)


where

120575119909

+120601119894119895=(120601119896

119894+1119895minus 120601119896

119894119895)

Δ119909

120575119909

119888120601119894119895=(120601119896

119894+1119895minus 120601119896

119894minus1119895)

(2Δ119909)

120575119909

minus120601119894119895=(120601119896

119894119895minus 120601119896

119894minus1119895)

Δ119909

120575119910

+120601119894119895=(120601119896

119894119895+1minus 120601119896

119894119895)

Δ119910

120575119910

119888120601119894119895=(120601119896

119894119895+1minus 120601119896

119894119895minus1)

(2Δ119910)

120575119910

minus120601119894119895=(120601119896

119894119895minus 120601119896

119894119895minus1)

Δ119910

(16)

and 120572(120601) fl 1205723(120601) in which the difference schemes of

|nabla120601|119894plusmn12119895

|nabla120601|119894119895plusmn12

are given by

1003816100381610038161003816nabla1206011003816100381610038161003816119894plusmn12119895 =

radic(120575119909plusmn120601119894119895)2

+ [(120575119910

119888120601119894119895+ 120575119910

119888120601119894plusmn1119895)

2]

2

1003816100381610038161003816nabla1206011003816100381610038161003816119894119895plusmn12 =

radic(120575119910

plusmn120601119894119895)2

+ [(120575119909

119888120601119894119895+ 120575119909

119888120601119894plusmn1119895)

2]

2

(17)

It has been verified by our numerical experiments thatthe evolution of level set can remain bounded stability evenafter a large number of iterations for solving (15) Howeverthe maximum of |120601| often exceeds the initial value 119888

0defined

by (1) which leads to the level set surface unsmoothednear the edges of the narrow-band 119879

0 thereby reducing

the computational accuracy of (15) Furthermore multipleiterations are required to find a suitable Courant-Friedrichs-Lewy (CFL) condition to ensure the stability of (15) Theissues related to the numerical instability of the DRE canbe resolved by imposing reverse diffusion constraints on thedifference scheme (see (15)) since it has been proved that theconstraints can ensure the numerical stability of the diffusionequations with all negative diffusion rates [20]

First (15) can be subdivided along the direction of 119909 and119910 into

120601119896+12

119894119895minus 120601119896

119894119895= (

Δ119905

Δ119909)

sdot [(120601119896

119894+1119895minus 120601119896

119894119895) 120572119894+12119895

+ (120601119896

119894minus1119895minus 120601119896

119894119895) 120572119894minus12119895

]

120601119896+1

119894119895minus 120601119896+12

119894119895= (

Δ119905

Δ119910)

sdot [(120601119896

119894119895+1minus 120601119896

119894119895) 120572119894119895+12

+ (120601119896

119894119895minus1minus 120601119896

119894119895) 120572119894119895minus12

]

(18)

with 120572119894plusmn12119895

= (1Δ119909)120572(120601119896

)119894+12119895

and 120572119894119895plusmn12

=

(1Δ119910)120572(120601119896

)119894119895plusmn12

Then the four flow functions are defined as

119865119909

119894119895= (120601119896

119894+1119895minus 120601119896

119894119895) 120572119894+12119895

119865119909

119894minus1119895= (120601119896

119894minus1119895minus 120601119896

119894119895) 120572119894minus12119895

119865119910

119894119895= (120601119896

119894119895+1minus 120601119896

119894119895) 120572119894119895+12

119865119910

119894minus1119895= (120601119896

119894119895minus1minus 120601119896

119894119895) 120572119894119895minus12

(19)

where 119865119909119894119895denotes the change from 120601

119896

119894119895to 120601119896119894+1119895

in one timestep Δ119905 and in the 119909 direction and the definitions of 119865119909

119894119895 119865119910119894119895

and 119865119910119894119895minus1

are similar to that of 119865119909119894119895 The lowest and highest

limit values of these flow functions are defined as

119865low = min119901119902=minus101

120601119896

119894+119901119895+119902minus 120601119896

119894119895

119865high = max119901119902=minus101

120601119896

119894+119901119895+119902minus 120601119896

119894119895

(20)

It can be seen that the four diffusion rates in (15) satisfythe boundedness |120572(120601119896)| le 120583 If 120583 le Δ119909 the reverse diffusionconstraints can be defined by

119865low le 119865119909

119894119895 119865119909

119894minus1119895 119865119910

119894119895 119865119910

119894119895minus1le 119865high (21)

If Δ119905Δ119909 le 14 first substituting (20) into inequalities(21) and then substituting the result into (18) one can obtaina solution using the inequalities

min119901119902=minus101

120601119896

119894+119901119895+119902le 120601119896+1

119894119895le max119901119902=minus101

120601119896

119894+119901119895+119902 (22)

It can be seen that the absolute values of 120601119896119894+119901119895+119902

for 119901 119902 =minus1 0 1 are lower than their initial value 119888

0Thatmeans that all

the absolute values |120601| le 1198880if the CFL conditions are satisfied

120583Δ119905

Δ1199092le1

4

120583Δ119905

Δ1199102le1

4

(23)

25 Flow Chart for Difference Schemes to LLSM with DREThe procedure for the LLSM with the DRE consists of twomain parts transforming the models of discrete level setfunctions into the local level set function and solving thedifference schemes of the LLSE associated with the DREThefinal DLSFs corresponding to the stable topological solutioncan be transformed into the LLSF within the initial narrow-band119879

0by iteratively solving theDREunder reverse diffusion

constraints The LLSE can be solved by difference schemesusing the third-order Runge-Kutta (R-K) scheme for tem-poral discretization and the fifth-order weighted essentiallynonoscillatory (WENO) scheme for spatial discretizationThe reader is referred to [26] for more numerical details Thelogical steps of the two parts can be described by a flow chartgiven in Figure 3


End

No

Yes

Is the convergencecritertion satisfied

Transforming DLSFs into LLSF


The DLSF 120601nj of stable topological solution

Define an initial diffusion rate 1205720(120601)

Iteratively solve the DRE

gain the half-band width 120574

with 1205720(120601) 100 times and

Restart from the DLSFs accordingto the stable topological solution 120601nj

under reverse diffusion constraintsby iteratively solving the DRE with 1205723(120601)

Transform the DLSFs into the LLSF 120601

Solve the LLSE while iteratively solving the DRE

until the LLSF 120601 is sufficiently smooth within T0

with 1205723(120601) under reverse diffusion constraints

number k = k + 1

with dp0(|nabla120601|) = 2 ((|nabla120601| minus 1)120590)120587 arctan

Figure 3 Flow chart depicting logical steps of the LLSM


26 Shape Derivatives and Normal Extension Velocities Withthe classical level set model [4] the minimum compliancedesign problem given in (4) can be converted to an uncon-strained problem with the Lagrangian method

Minimize120601

119871 (119906 120601)

= 119869 (119906 120601) + 120582+(int119863

119867(120601) 119889Ω minus 119881lowast

)

(24)

where the Lagrangian function 119871(119906 120601) is the objective func-tional 120582

+is the Lagrangian multiplier of the volume con-

straint119867(120601) is theHeaviside functionThemean compliance119869(119906 120601) is reformulated as

119869 (119906 120601) =1

2int119863

119864120576 (119906) 120576 (119906)119867 (120601) 119889Ω (25)

For a number of level set-based approaches [4 9 11 12]the steepest descent method is used to ensure the decrease ofthe objective function by directly setting the normal velocityfield 119881

119899as the negative shape derivative of 119871(119906 120601) For the

particular case of a 2-D model of a linear elastic structurethe boundary traction is fixed and remains unchanged thedisplacement constraint is fixed and the body force is set tozero thus 119881

119899can be given by

119881119899= 05119864120576 (119906) 120576 (119906) minus 120582

+ (26)

The reader is referred to the article [4] for more detailedtheoretical proofs In addition a bisectioning algorithm isused to find the Lagrangian multiplier 120582

+to guarantee

that the volume constraint be exactly satisfied during eachiteration

The normal velocity field can be naturally extended to thewhole domain using the so-called ldquoersatz materialrdquo approachwhich fills the void areas with a weak material and then thematerial density is assumed to be piecewise constant in eachelement and is adequately interpolated in those elements cutby the zero level set (the shape boundary) [4] In the LLSMthe extension velocity field is localized within the narrow-band 119879

0 By iteratively solving the difference scheme for the

DRE (see (15)) one can obtain a smooth velocity field in theregion near the edges of the narrow-band 119879

0to improve the

computational accuracy of the extension velocity

3 Numerical Examples

In this section two widely researched examples the can-tilever beam and the arch bridge are presented in the contextof structuralminimumcompliance design to demonstrate thecharacteristics of the proposed method Some of the systemparameters using the same values are defined as follows

Youngrsquos elasticity modulus for the solid material is 1198640=

200GPa and for weak material is 119864min = 10minus3 Pa and the

Poisson ratio is 03 The volume constraint 119881lowast = 05119881allwhere 119881all is the total volume in the design region 119863 Theconvergence tolerance 120591 is set as 001 in the bidirectionalevolutionary algorithm and 0001 in the LLSM

31 First Cantilever-Beam Model Shown in Figure 4 is thedesign domain of a cantilever beam with a size 40mm times

25mm The left side of the domain is fixed as the Dirichletboundary and a concentrated force 119875 = 100N is verticallyapplied at the central point of the right side as a nonhomoge-neous Neumann boundary In the bidirectional evolutionaryalgorithm ER = 2 AR119899max = 5 and 119903

119908= 2mm In the

difference schemes for the LLSE Δ119905 = 01 Δ = 03 and120574 = 0999 In the difference schemes for the DRE 120583 = 05 and119889119905 = 0001 The design domain is discretized with a mesh of80 times 50 quadrilateral elements

In the design domain as shown in Figure 4 the initialvolume 119881

0of the solid region Ω is set to 119881

0= 119881all The

structural topologies corresponding to the zero level set andrelated level set surfaces are shown in Figures 5 and 6 respec-tively The convergence histories of the mean complianceand volume fraction are depicted in Figure 7 The result inFigure 5(e) stands for a stable topological solution obtainedfrom the bidirectional evolutionary algorithm Topologicalresults given by this algorithm are characterized by a smoothboundary attributed to the structural model described by theDLSFs By comparing Figures 6(e) and 6(f) the sharp levelset surface corresponding to the DLSFs has been successfullyconverted into a smooth one related to a local level setfunction by iteratively solving the DRE at the initial stageof the LLSM Then the shape of the boundary is furtherimproved by iteratively solving the LLSE In addition allthe absolute values of the level set function are less thanthe initial value 119888

0 Therefore it verifies the effectiveness of

reverse diffusion constraints on the numerical stability of thedifference scheme for the DRE

32 Second Cantilever-Beam Model This study has alsoinvestigated the influence of different initial models of struc-ture on the final design Figure 8 depicts the design domainwith a size 40mm times 25mm The left side of the domain isfixed and a concentrated force 119875 = 1N is vertically appliedat the bottom of the right side All the parameters but 119903

119908=

02mm and AR119899max = 1 remain unchanged as those ofthe first cantilever-beam model Figure 9 shows two cases ofthe initial configurations with full materials and the least butessential materials and their topologies during the processof optimization The two final designs are made with thesame topology and almost similar shape of the structurewhich shows the complexity of the final topology is notchanging appreciably with different initial structures There-fore the numerical process of the bidirectional evolutionaryalgorithm can be used to replace the numerical process ofhole nucleation in the LLSM to avoid the final design heavilydependent on the initial guess

Figure 10 shows the topological topologies for almost119903119908= 02mm using several mesh sizes It can be seen that

the optimal topology does not depend on the discretizationin terms of layout and number of bars

33 Arch-BridgeModel Thedesign domain of an arch-bridgemodel with a size 20mm times 12mm is shown in Figure 11Both the bottom corners of the domain are the fixed supportA uniform static pressure is vertically applied on the upper


Design domain

P

40mm

25m

m

Figure 4 Design domain of the first cantilever beam and its boundary conditions

(a) (b) (c)

(d) (e) (f)

Figure 5 Topologies of zero level set at different steps (a) Step 1 (b) Step 10 (c) Step 20 (d) Step 30 (e) Step 43 (f) Step 57

side and the sum of the pressure is 5N In accordance withthe same definitions of the cantilever-beam parameters theparameters are set to ER = 3 AR119899max = 5 119903

119908= 01mm

Δ119905 = 002 Δ = 03 120574 = 099 119889119905 = 0001 and 120583 = 00167The design domain is discretized with a mesh of 120 times 72quadrilateral elements

This example focuses on the new characteristic of theproposed algorithm for improving the convergence of thebidirectional evolutionary algorithm using the LLSM Thestructural topologies corresponding to the zero level setare depicted in Figure 12 Note that it starts from theinitial model with the volume 119881 = 05119881all and remainsunchanged so as to maintain the stability of the evolutionprocess of the object function The evolutionary historiesof the objective and the volume constraint starting from

the initial models are plotted in Figure 13 A design of thestructure shown in Figure 12(b) corresponding to a stabletopological solution is also the final design of the topology(not shape)mademerely using the bidirectional evolutionaryalgorithm in our numerical experiments The subsequenttopologies given in Figures 12(c)ndash12(f) show that the LLSMcan further optimize the topology of the structure to improveconvergenceMoreover the LLSMcan also improve the shapeof the boundary to obtain a smoother shape design till itreaches the convergence tolerance in the 38th iteration Itis worth noticing that the final topology obtained by theLLSM is just a local optimal design because of the use ofthe steepest descent method Despite the optimal solutionof this arch-bridge model obtained by an element-wise ESOmethod [27] in this case the optimized topology obtained


(a) (b) (c)

(d) (e) (f)

Figure 6 The corresponding level set surface at different steps (a) Step 1 (b) Step 10 (c) Step 20 (d) Step 30 (e) Step 43 (f) Step 57

0 10 20 30 40 50 6001

02

03

04

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

04

06

08

1

Volu

me f

ract

ion

Mean complianceVolume fraction

Figure 7 Convergent histories of the objective function and theconstraint

by the proposed bidirectional evolutionary algorithm is notreasonable compared with the final optimal solution shownin Figure 12(f)

Starting from different initial models the final modelsobtained by the bidirectional evolutionary algorithm and theLLSM respectively are shown in Figure 14

It can be seen from Figures 12 and 14 that the finaltopologies obtained by the bidirectional evolutionary algo-rithm are inconsistent In contrast the final optimized resultsfound using the LLSM subsequently are of the same topologyand similar shape Although the local optimal solution ofthe arch-bridge model can also be obtained by using theESOBESO methods with elemental variables numerical

Design domain

4mm25

mm

P

Figure 8 Design domain of the second cantilever beam and itsboundary conditions

instabilities and zigzag boundaries can result in these ele-mental variables-based methods Hence nodal variables areneeded to take the place of the elemental variables in thesemethods The combined algorithm with the bidirectionalevolutionary algorithm and the LLSM can also resolve thisproblem and achieve at least the consistent local optimalsolution for the different cases of initial models

4 Conclusions

The LLSM is intended to remarkably increase the compu-tational efficiency of the conventional LSMs using globalmodels To overcome the issue of hole nucleation of the


Initial Step 20 Step 40 Step 43

(a) The case starts from the full-material initial configurations


(b) The case starts from the least-material initial configurations

Figure 9 Topologies of zero level set in the two cases of the second cantilever beam

(a) (b) (c) (d)

Figure 10 Mesh-independent solutions of the second cantilever beam (a) 40 times 25 (b) 96 times 60 (c) 128 times 80 (d) 160 times 100

P

Design domain

20mm

12

mm

Figure 11 Design domain of the arch bridge


(a) (b) (c)

(d) (e) (f)

Figure 12 Topologies of zero level set at different steps starting from the case of 119881 = 05119881all (a) Initial (b) Step 19 (c) Step 20 (d) Step 21(e) Step 22 (f) Step 38

0 5 10 15 20 25 30 35 400

010203040506070809

1

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

0

05

1

Volu

me f

ract

ion


times10minus3

Figure 13 Convergent histories of the objective function and the constraint

LLSM a bidirectional evolutionary algorithm is combinedwith the LLSM This proposed algorithm has been usedsuccessfully in topology optimization of two-dimensional(2-D) structures and it is easy to be extended to 3-Dstructures The main features of this algorithm unknown tothe conventional LSMs and the LLSM can be summarized asfollows

(a) The discrete level set functions can be efficientlytransformed into the local level set function byiteratively solving the distance regularized equation(DRE)

(b) The DRE can be used instead of the reinitializationequation to further increase the computational effi-ciency of the LLSM

(c) A conditionally stable difference scheme underreverse diffusion constraints is formulated to ensurethe numerical stability of the DRE

(d) If the stable topological solutions of the bidirectionalevolutionary algorithm are inconsistent the LLSMcan achieve at least the consistent local optimalsolution for the different cases of initial models


(a)

(b)

(c)

Figure 14 Topologies of zero level set for the initial model and the final models obtained by the bidirectional evolutionary algorithm andthe LLSM respectively (a) case 1 119881 = 04119881all (b) case 2 119881 = 06119881all (c) case 3 119881 = 08119881all

High computational efficiency and numerical stability ofthe proposed algorithm have been verified by three typicalnumerical examples

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgment

The financial support from National Natural Science Foun-dation of China (no 51278218) is gratefully acknowledged

References

[1] M P Bendsoslashe and O Sigmund Topology Optimization TheoryMethods and Applications Springer Berlin Germany 2003

[2] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural and Multidisciplinary Optimization vol 49no 1 pp 1ndash38 2014

[3] S Osher and R Fedkiw Level SetMethods andDynamic ImplicitSurfaces Springer New York NY USA 2003

[4] G Allaire F Jouve and A-M Toader ldquoStructural optimizationusing sensitivity analysis and a level-set methodrdquo Journal ofComputational Physics vol 194 no 1 pp 363ndash393 2004

[5] N P van Dijk K Maute M Langelaar and F van KeulenldquoLevel-set methods for structural topology optimization areviewrdquo Structural and Multidisciplinary Optimization vol 48no 3 pp 437ndash472 2013

[6] D Peng B Merriman S Osher H K Zhao and M Kang ldquoAPDE-based fast local level set methodrdquo Journal of Computa-tional Physics vol 155 no 2 pp 410ndash438 1999

[7] H A Eschenauer V V Kobelev and A Schumacher ldquoBubblemethod for topology and shape optimization of structuresrdquoStructural Optimization vol 8 no 1 pp 42ndash51 1994


[8] J Sokolowski and A Zochowski ldquoOn the topological derivativein shape optimizationrdquo SIAM Journal on Control and Optimiza-tion vol 37 no 4 pp 1251ndash1272 1999

[9] GAllaire F deGournay F Jouve andA-MToader ldquoStructuraloptimization using topological and shape sensitivity via a levelset methodrdquo Control and Cybernetics vol 34 no 1 pp 59ndash812005

[10] M Burger and S J Osher ldquoA survey on level set methodsfor inverse problems and optimal designrdquo European Journal ofApplied Mathematics vol 16 no 2 pp 263ndash301 2005

[11] S Wang and M Y Wang ldquoRadial basis functions and levelset method for structural topology optimizationrdquo InternationalJournal for NumericalMethods in Engineering vol 65 no 12 pp2060ndash2090 2006

[12] Z Luo M Y Wang S Y Wang and P Wei ldquoA level set-basedparameterization method for structural shape and topologyoptimizationrdquo International Journal for Numerical Methods inEngineering vol 76 no 1 pp 1ndash26 2008

[13] M Sussman P Smereka and S Osher ldquoA level set approach forcomputing solutions to incompressible two-phase flowrdquo Journalof Computational Physics vol 114 no 1 pp 146ndash159 1994

[14] M Sussman and E Fatemi ldquoAn efficient interface-preservinglevel set redistancing algorithm and its application to interfacialincompressible fluid flowrdquo SIAM Journal on Scientific Comput-ing vol 20 no 4 pp 1165ndash1191 1999

[15] G Barles HM Soner and P E Souganidis ldquoFront propagationand phase field theoryrdquo SIAM Journal on Control and Optimiza-tion vol 31 no 2 pp 439ndash469 1993

[16] C Li C Xu C Gui and M D Fox ldquoDistance regularized levelset evolution and its application to image segmentationrdquo IEEETransactions on Image Processing vol 19 no 12 pp 3243ndash32542010

[17] C Li C Xu C Gui andM D Fox ldquoLevel set evolution withoutre-initialization a new variational formulationrdquo in Proceedingsof the IEEE Computer Society Conference on Computer Visionand Pattern Recognition (CVPR rsquo05) pp 430ndash436 IEEE Wash-ington DC USA June 2005

[18] H-K Zhao T Chan B Merriman and S Osher ldquoA variationallevel set approach to multiphase motionrdquo Journal of Computa-tional Physics vol 127 no 1 pp 179ndash195 1996

[19] W F Wu Y Wu and Q Huang ldquoAn improved distanceregularized level set evolution without re-initializationrdquo in Pro-ceedings of the IEEE 5th International Conference on AdvancedComputational Intelligence (ICACI rsquo12) pp 631ndash636 NanjingChina October 2012

[20] O Salvado C M Hillenbrand and D L Wilson ldquoPartialvolume reduction by interpolation with reverse diffusionrdquoInternational Journal of Biomedical Imaging vol 2006 ArticleID 92092 13 pages 2006

[21] G Allaire A Karrman and G Michailidis ldquoScilab CodeManualrdquo 2012 httpwwwcmappolytechniquefrsimallairelevelsetmanualpdf

[22] D Shepard ldquoA two-dimensional interpolation function forirregularly-spaced datardquo in Proceedings of the 23rd ACMNational Conference ACM New York NY USA August 1968

[23] Z Kang and Y Q Wang ldquoStructural topology optimizationbased on non-local Shepard interpolation of density fieldrdquoComputer Methods in Applied Mechanics and Engineering vol200 no 49 pp 3515ndash3525 2011

[24] X Huang and M Y Xie Evolutionary Topology Optimization ofContinuum Structures Methods and Applications John Wiley ampSons 2010

[25] X Huang and Y M Xie ldquoConvergent and mesh-independentsolutions for the bi-directional evolutionary structural opti-mization methodrdquo Finite Elements in Analysis and Design vol43 no 14 pp 1039ndash1049 2007

[26] C W Shu Essentially Non-Oscillatory and Weighted EssentiallyNon-Oscillatory Schemes for Hyperbolic Conservation LawsICASE-NASA Langley Reasearch Center Hampton Va USA1997

[27] X Y Yang Y M Xie and G P Steven ldquoEvolutionary methodsfor topology optimisation of continuous structures with designdependent loadsrdquo Computers amp Structures vol 83 no 12-13 pp956ndash963 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


iteratively solving a classical reinitialization equation [13] Tosuppress this drift an interface preserving level set redis-tancing algorithm is proposed by Sussman and Fatemi [14]Nevertheless it has been proved that the SDF is not a feasiblesolution to theH-J equation [15] In practice it not only raisesserious problems as when and how it should be performedbut also affects numerical accuracy in an undesirable way andthus should be avoided as much as possible [16] The needfor reinitialization was originally eliminated by introducinga penalty term [17] into a variational level set approach [18]The undesirable boundary effect of the penalty term can beeliminated by taking a distance regularized equation (DRE)instead of this term Hence a so-called distance regularizedlevel set evolution (DRLSE) [16] is realized based on thevariational approach As an unnecessary diffusion effect ofthe DRLSE was found in some locations where the surfaceis too flat the DRE was recently modified with a new andbalanced formulation to eliminate this effect [19] Althoughparts of the diffusion rates in the DRE are negative thenumerical stability can still be maintained by incorporatingreverse diffusion constraints in the difference schemes of theDRE as can the reverse diffusion equations with all negativediffusion rates [20]

The aim of this work is to solve the aforementionednumerical issues that still exist in the LLSM for topologyoptimization of two-dimensional structures A bidirectionalevolutionary algorithm based on the discrete level set func-tions (DLSFs) is proposed to find a stable topological solutionfirst and then combined with the LLSM to further evolvethe local details of the topology and shape of the structureTransforming the DLSFs into the local level set function ofthe LLSM is achieved by iteratively solving the DRE Afterthat the DRE is incorporated into the LLSM to avoid thereinitialization procedure A difference scheme under reversediffusion constraints is formulated for the DRE to improve itsnumerical stability Typical examples are given to show theeffectiveness of the proposed algorithm in terms of conver-gence computational efficiency and numerical stability

2 Optimization Algorithm

21 Local Level Set Method Using Narrow-BandModel In thelocal level set method (LLSM) [6] the local level set equationis defined as

120597120601

120597119905+ 119862 (120601)119881

119899

1003816100381610038161003816nabla1206011003816100381610038161003816 = 0

(1)

where 120601(119909 119905) is defined as the local level set function (LLSF)and 119881

119899is the normal velocity in normal direction 119899 =

nabla120601|nabla120601| the truncation function 119862(120601) is

119862 (120601) =

1 if 10038161003816100381610038161206011003816100381610038161003816 le Δ

(10038161003816100381610038161206011003816100381610038161003816 minus 120574)2

(210038161003816100381610038161206011003816100381610038161003816 + 120574 minus 3Δ)

(120574 minus Δ)3

if Δ lt 10038161003816100381610038161206011003816100381610038161003816 le 120574

0 if 10038161003816100381610038161206011003816100381610038161003816 gt 120574

(2)

with 1198790= 119909

1003816100381610038161003816120601(119909)1003816100381610038161003816 lt 120574 being a narrow band with

the half-band width 120574 The narrow-band model and thecorresponding LLSF are described as shown in Figure 1

It can be seen from Figure 1 that only the LLSF within thenarrow-band 119879

0needs to be updated during each iteration

Hence the LLSM is higher in computational efficiency thanthe LSMs based on global level set models

22 Bidirectional Evolutionary Algorithm with Discrete LevelSet Functions A two-dimensional structural model is builtin the work region 119863 sub 119877

2 And the set 119878 = 1198781cup 1198782cup 1198783

represents the finite elements in the domain 119863 It can bedivided into three parts 119878

1that consists of the solid elements

with a full-material density 1198782that covers the elements

with intermediate material densities 1198783that involves the

void elements with a weak-material density Accordingly thenodal sets corresponding to the elemental sets 119878

1 1198782 and 119878

3

are defined as 1198781198991 1198781198992 and 119878119899

3 respectively If it is assumed that

119878119899

2sub 119878119899

3 1198781198991cup 119878119899

3consist of all the nodes within the region 119863

then a discrete level set function (DLSF) for node 119895 can bedefined as

120601119895=

minus1198880119895 isin 119878119899

1

1198880

119895 isin 119878119899

3

(3)

where 1198880is a predefined constant set as 1 in this study

The values of elemental densities can be derived from theDLSFs If the 119894th element belongs to 119878

1 that is 119894 isin 119878

1 then

the element density 120588119894= 1 if 119894 isin 119878

3 then 120588

119894= 120588min where

120588min is a small value 0001 if 119894 isin 1198782 then 120588

119894isin (120588min 1)

where 120588119894is calculated in terms of the interpolation criterion

given in the code manual [21] In the structural model therectangular element is split into four triangles first and thevalue of theDLSF at the commonpoint of the triangles is thengiven by the average of the values of the four points Afterthat each triangle is examined separately in the same logicFinally the elemental density is found to be the average of thecontributions of the triangles

The structural stiffness design has been widely inves-tigated in numerous literatures for topological sensitivityanalysis The standard notion [1] of minimum compliancedesign problems under a global volume constraint can bemathematically defined as follows

minimize120601

119869 (Ω) =1

2119906119879

119870119906

Subject to119873

sum

119894=1

119881119894120588119894(120601) = 119881

lowast

120588119894isin [120588min 1]

(4)

where 119869 is known as the mean compliance the open set Ωrepresents all admissible shapes in the design region 119863 119906(120601)is the nodal displacement vector and 119870 denotes the globalstiffness matrix 119881

119894is the volume of an individual element

119881lowast is the prescribed total volume and 119873 is the number of

elements


Edge of narrow band

Zero level set

Ω

DΩ

2120574

120601

c0

minusc0

x0

120574

Ω

Ω

D



120572119899

119895= 119863119879119869 (120601119895) =

120587 (120582 + 2120583)

4120583 (120582 + 120583)4120583119864119895120576 (119906) 120576 (119906)

+ (120582 minus 120583) tr (119864119895120576 (119906)) tr (120576 (119906))

(5)








Ω


119899

(x) =119873Ω

sum

119894=1

119882119894(x) sdot 120572119899 (x)

119882119894(x) = 119863

119894(x)

sum119873

119895=1119863119895(x)

(119894 = 1 119873)

(6)


function 119863119894(x) = 1radic(119889

119894(x))2 + 1198882 in which 119889

119894(x) = x minus 119909

119894


119894(x) ge

119903119908 then119863







119881119896+1=

min (119881119896(1 + ER) 119881lowast) if 119881

119896le 119881lowast


119896gt 119881lowast

(7)









120601119896+1

119895=

minus1198880120572119899

119895ge 120572

addth 120601

119896

119895= 1198880

1198880

120572119899

119895ge 120572

delth 120601119896

119895= minus1198880

(8)


3 respectively



10038161003816100381610038161003816sum119896minus5


119896

119896minus4119869 (Ω)

10038161003816100381610038161003816

sum119896

119896minus4119869 (Ω)

le 120591 (9)




120597120601

120597119905= 120583 div (120572

1(120601) nabla120601) (10)




1015840

1(119904)119904 with 119904 = |nabla120601|


defined as

1199011(119904) =

(1 minus cos (2120587119904))(2120587)2

if 119904 le 1

(119904 minus 1)2

2if 119904 gt 1

(11)


1(119904) at 119904 = 1 and 119904 = 0 So the


1198891199011(119904) =

sin (2120587119904)2120587119904

if 119904 le 1

1 minus1

119904if 119904 gt 1

(12)


1199011(|nabla120601|) and |120572







0 the


in [19]

1198891199012(1003816100381610038161003816nabla1206011003816100381610038161003816) =

2


10038161003816100381610038161206011003816100381610038161003816 le 120576

100381610038161003816100381610038161003816100381610038161003816

2


100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161206011003816100381610038161003816 gt 120576

(13)


1= 119909

1003816100381610038161003816120601(119909)1003816100381610038161003816 lt 120576



1but









1198891199013(1003816100381610038161003816nabla1206011003816100381610038161003816) =

2


10038161003816100381610038161206011003816100381610038161003816 lt 120574

010038161003816100381610038161206011003816100381610038161003816 ge 120574

(14)



0



Start





No


Yes





of each node





number k = k + 1



120601119896+1

119894119895minus 120601119896

119894119895

Δ119905

=

120572 (120601119896

)119894+12119895

120575119909

+120601119894119895minus 120572 (120601

119896

)119894minus12119895

120575119909

minus120601119894119895

Δ119909

+

120572 (120601119896

)119894119895+12

120575119910

+120601119894119895minus 120572 (120601

119896

)119894119895minus12

120575119910

minus120601119894119895

Δ119910

(15)


where

120575119909

+120601119894119895=(120601119896

119894+1119895minus 120601119896

119894119895)

Δ119909

120575119909

119888120601119894119895=(120601119896

119894+1119895minus 120601119896

119894minus1119895)

(2Δ119909)

120575119909

minus120601119894119895=(120601119896

119894119895minus 120601119896

119894minus1119895)

Δ119909

120575119910

+120601119894119895=(120601119896

119894119895+1minus 120601119896

119894119895)

Δ119910

120575119910

119888120601119894119895=(120601119896

119894119895+1minus 120601119896

119894119895minus1)

(2Δ119910)

120575119910

minus120601119894119895=(120601119896

119894119895minus 120601119896

119894119895minus1)

Δ119910

(16)


|nabla120601|119894plusmn12119895

|nabla120601|119894119895plusmn12

are given by

1003816100381610038161003816nabla1206011003816100381610038161003816119894plusmn12119895 =

radic(120575119909plusmn120601119894119895)2

+ [(120575119910

119888120601119894119895+ 120575119910

119888120601119894plusmn1119895)

2]

2

1003816100381610038161003816nabla1206011003816100381610038161003816119894119895plusmn12 =

radic(120575119910

plusmn120601119894119895)2

+ [(120575119909

119888120601119894119895+ 120575119909

119888120601119894plusmn1119895)

2]

2

(17)


0defined


0 thereby reducing



120601119896+12

119894119895minus 120601119896

119894119895= (

Δ119905

Δ119909)

sdot [(120601119896

119894+1119895minus 120601119896

119894119895) 120572119894+12119895

+ (120601119896

119894minus1119895minus 120601119896

119894119895) 120572119894minus12119895

]

120601119896+1

119894119895minus 120601119896+12

119894119895= (

Δ119905

Δ119910)

sdot [(120601119896

119894119895+1minus 120601119896

119894119895) 120572119894119895+12

+ (120601119896

119894119895minus1minus 120601119896

119894119895) 120572119894119895minus12

]

(18)

with 120572119894plusmn12119895

= (1Δ119909)120572(120601119896

)119894+12119895

and 120572119894119895plusmn12

=

(1Δ119910)120572(120601119896

)119894119895plusmn12


119865119909

119894119895= (120601119896

119894+1119895minus 120601119896

119894119895) 120572119894+12119895

119865119909

119894minus1119895= (120601119896

119894minus1119895minus 120601119896

119894119895) 120572119894minus12119895

119865119910

119894119895= (120601119896

119894119895+1minus 120601119896

119894119895) 120572119894119895+12

119865119910

119894minus1119895= (120601119896

119894119895minus1minus 120601119896

119894119895) 120572119894119895minus12

(19)


119896

119894119895to 120601119896119894+1119895


119894119895 119865119910119894119895

and 119865119910119894119895minus1



119865low = min119901119902=minus101

120601119896

119894+119901119895+119902minus 120601119896

119894119895

119865high = max119901119902=minus101

120601119896

119894+119901119895+119902minus 120601119896

119894119895

(20)


119865low le 119865119909

119894119895 119865119909

119894minus1119895 119865119910

119894119895 119865119910

119894119895minus1le 119865high (21)


min119901119902=minus101

120601119896

119894+119901119895+119902le 120601119896+1

119894119895le max119901119902=minus101

120601119896

119894+119901119895+119902 (22)



0Thatmeans that all


120583Δ119905

Δ1199092le1

4

120583Δ119905

Δ1199102le1

4

(23)





End

No

Yes








with 1205720(120601) 100 times and







number k = k + 1





Minimize120601

119871 (119906 120601)

= 119869 (119906 120601) + 120582+(int119863

119867(120601) 119889Ω minus 119881lowast

)

(24)




119869 (119906 120601) =1

2int119863

119864120576 (119906) 120576 (119906)119867 (120601) 119889Ω (25)





119881119899= 05119864120576 (119906) 120576 (119906) minus 120582

+ (26)


+to guarantee





0to improve the









119908= 2mm In the




0= 119881all The





119908=






Design domain

P

40mm

25m

m


(a) (b) (c)

(d) (e) (f)



119908= 01mm





(a) (b) (c)

(d) (e) (f)


0 10 20 30 40 50 6001

02

03

04

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

04

06

08

1

Volu

me f

ract

ion






Design domain

4mm25

mm

P



4 Conclusions








(a) (b) (c) (d)


P

Design domain

20mm

12

mm



(a) (b) (c)

(d) (e) (f)


0 5 10 15 20 25 30 35 400

010203040506070809

1

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

0

05

1

Volu

me f

ract

ion


times10minus3








(a)

(b)

(c)





Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Edge of narrow band

Zero level set

Ω

DΩ

2120574

120601

c0

minusc0

x0

120574

Ω

Ω

D



120572119899

119895= 119863119879119869 (120601119895) =

120587 (120582 + 2120583)

4120583 (120582 + 120583)4120583119864119895120576 (119906) 120576 (119906)

+ (120582 minus 120583) tr (119864119895120576 (119906)) tr (120576 (119906))

(5)








Ω


119899

(x) =119873Ω

sum

119894=1

119882119894(x) sdot 120572119899 (x)

119882119894(x) = 119863

119894(x)

sum119873

119895=1119863119895(x)

(119894 = 1 119873)

(6)


function 119863119894(x) = 1radic(119889

119894(x))2 + 1198882 in which 119889

119894(x) = x minus 119909

119894


119894(x) ge

119903119908 then119863







119881119896+1=

min (119881119896(1 + ER) 119881lowast) if 119881

119896le 119881lowast


119896gt 119881lowast

(7)









120601119896+1

119895=

minus1198880120572119899

119895ge 120572

addth 120601

119896

119895= 1198880

1198880

120572119899

119895ge 120572

delth 120601119896

119895= minus1198880

(8)


3 respectively



10038161003816100381610038161003816sum119896minus5


119896

119896minus4119869 (Ω)

10038161003816100381610038161003816

sum119896

119896minus4119869 (Ω)

le 120591 (9)




120597120601

120597119905= 120583 div (120572

1(120601) nabla120601) (10)




1015840

1(119904)119904 with 119904 = |nabla120601|


defined as

1199011(119904) =

(1 minus cos (2120587119904))(2120587)2

if 119904 le 1

(119904 minus 1)2

2if 119904 gt 1

(11)


1(119904) at 119904 = 1 and 119904 = 0 So the


1198891199011(119904) =

sin (2120587119904)2120587119904

if 119904 le 1

1 minus1

119904if 119904 gt 1

(12)


1199011(|nabla120601|) and |120572







0 the


in [19]

1198891199012(1003816100381610038161003816nabla1206011003816100381610038161003816) =

2


10038161003816100381610038161206011003816100381610038161003816 le 120576

100381610038161003816100381610038161003816100381610038161003816

2


100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161206011003816100381610038161003816 gt 120576

(13)


1= 119909

1003816100381610038161003816120601(119909)1003816100381610038161003816 lt 120576



1but









1198891199013(1003816100381610038161003816nabla1206011003816100381610038161003816) =

2


10038161003816100381610038161206011003816100381610038161003816 lt 120574

010038161003816100381610038161206011003816100381610038161003816 ge 120574

(14)



0



Start





No


Yes





of each node





number k = k + 1



120601119896+1

119894119895minus 120601119896

119894119895

Δ119905

=

120572 (120601119896

)119894+12119895

120575119909

+120601119894119895minus 120572 (120601

119896

)119894minus12119895

120575119909

minus120601119894119895

Δ119909

+

120572 (120601119896

)119894119895+12

120575119910

+120601119894119895minus 120572 (120601

119896

)119894119895minus12

120575119910

minus120601119894119895

Δ119910

(15)


where

120575119909

+120601119894119895=(120601119896

119894+1119895minus 120601119896

119894119895)

Δ119909

120575119909

119888120601119894119895=(120601119896

119894+1119895minus 120601119896

119894minus1119895)

(2Δ119909)

120575119909

minus120601119894119895=(120601119896

119894119895minus 120601119896

119894minus1119895)

Δ119909

120575119910

+120601119894119895=(120601119896

119894119895+1minus 120601119896

119894119895)

Δ119910

120575119910

119888120601119894119895=(120601119896

119894119895+1minus 120601119896

119894119895minus1)

(2Δ119910)

120575119910

minus120601119894119895=(120601119896

119894119895minus 120601119896

119894119895minus1)

Δ119910

(16)


|nabla120601|119894plusmn12119895

|nabla120601|119894119895plusmn12

are given by

1003816100381610038161003816nabla1206011003816100381610038161003816119894plusmn12119895 =

radic(120575119909plusmn120601119894119895)2

+ [(120575119910

119888120601119894119895+ 120575119910

119888120601119894plusmn1119895)

2]

2

1003816100381610038161003816nabla1206011003816100381610038161003816119894119895plusmn12 =

radic(120575119910

plusmn120601119894119895)2

+ [(120575119909

119888120601119894119895+ 120575119909

119888120601119894plusmn1119895)

2]

2

(17)


0defined


0 thereby reducing



120601119896+12

119894119895minus 120601119896

119894119895= (

Δ119905

Δ119909)

sdot [(120601119896

119894+1119895minus 120601119896

119894119895) 120572119894+12119895

+ (120601119896

119894minus1119895minus 120601119896

119894119895) 120572119894minus12119895

]

120601119896+1

119894119895minus 120601119896+12

119894119895= (

Δ119905

Δ119910)

sdot [(120601119896

119894119895+1minus 120601119896

119894119895) 120572119894119895+12

+ (120601119896

119894119895minus1minus 120601119896

119894119895) 120572119894119895minus12

]

(18)

with 120572119894plusmn12119895

= (1Δ119909)120572(120601119896

)119894+12119895

and 120572119894119895plusmn12

=

(1Δ119910)120572(120601119896

)119894119895plusmn12


119865119909

119894119895= (120601119896

119894+1119895minus 120601119896

119894119895) 120572119894+12119895

119865119909

119894minus1119895= (120601119896

119894minus1119895minus 120601119896

119894119895) 120572119894minus12119895

119865119910

119894119895= (120601119896

119894119895+1minus 120601119896

119894119895) 120572119894119895+12

119865119910

119894minus1119895= (120601119896

119894119895minus1minus 120601119896

119894119895) 120572119894119895minus12

(19)


119896

119894119895to 120601119896119894+1119895


119894119895 119865119910119894119895

and 119865119910119894119895minus1



119865low = min119901119902=minus101

120601119896

119894+119901119895+119902minus 120601119896

119894119895

119865high = max119901119902=minus101

120601119896

119894+119901119895+119902minus 120601119896

119894119895

(20)


119865low le 119865119909

119894119895 119865119909

119894minus1119895 119865119910

119894119895 119865119910

119894119895minus1le 119865high (21)


min119901119902=minus101

120601119896

119894+119901119895+119902le 120601119896+1

119894119895le max119901119902=minus101

120601119896

119894+119901119895+119902 (22)



0Thatmeans that all


120583Δ119905

Δ1199092le1

4

120583Δ119905

Δ1199102le1

4

(23)





End

No

Yes








with 1205720(120601) 100 times and







number k = k + 1





Minimize120601

119871 (119906 120601)

= 119869 (119906 120601) + 120582+(int119863

119867(120601) 119889Ω minus 119881lowast

)

(24)




119869 (119906 120601) =1

2int119863

119864120576 (119906) 120576 (119906)119867 (120601) 119889Ω (25)





119881119899= 05119864120576 (119906) 120576 (119906) minus 120582

+ (26)


+to guarantee





0to improve the









119908= 2mm In the




0= 119881all The





119908=






Design domain

P

40mm

25m

m


(a) (b) (c)

(d) (e) (f)



119908= 01mm





(a) (b) (c)

(d) (e) (f)


0 10 20 30 40 50 6001

02

03

04

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

04

06

08

1

Volu

me f

ract

ion






Design domain

4mm25

mm

P



4 Conclusions








(a) (b) (c) (d)


P

Design domain

20mm

12

mm



(a) (b) (c)

(d) (e) (f)


0 5 10 15 20 25 30 35 400

010203040506070809

1

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

0

05

1

Volu

me f

ract

ion


times10minus3








(a)

(b)

(c)





Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















120601119896+1

119895=

minus1198880120572119899

119895ge 120572

addth 120601

119896

119895= 1198880

1198880

120572119899

119895ge 120572

delth 120601119896

119895= minus1198880

(8)


3 respectively



10038161003816100381610038161003816sum119896minus5


119896

119896minus4119869 (Ω)

10038161003816100381610038161003816

sum119896

119896minus4119869 (Ω)

le 120591 (9)




120597120601

120597119905= 120583 div (120572

1(120601) nabla120601) (10)




1015840

1(119904)119904 with 119904 = |nabla120601|


defined as

1199011(119904) =

(1 minus cos (2120587119904))(2120587)2

if 119904 le 1

(119904 minus 1)2

2if 119904 gt 1

(11)


1(119904) at 119904 = 1 and 119904 = 0 So the


1198891199011(119904) =

sin (2120587119904)2120587119904

if 119904 le 1

1 minus1

119904if 119904 gt 1

(12)


1199011(|nabla120601|) and |120572







0 the


in [19]

1198891199012(1003816100381610038161003816nabla1206011003816100381610038161003816) =

2


10038161003816100381610038161206011003816100381610038161003816 le 120576

100381610038161003816100381610038161003816100381610038161003816

2


100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161206011003816100381610038161003816 gt 120576

(13)


1= 119909

1003816100381610038161003816120601(119909)1003816100381610038161003816 lt 120576



1but









1198891199013(1003816100381610038161003816nabla1206011003816100381610038161003816) =

2


10038161003816100381610038161206011003816100381610038161003816 lt 120574

010038161003816100381610038161206011003816100381610038161003816 ge 120574

(14)



0



Start





No


Yes





of each node





number k = k + 1



120601119896+1

119894119895minus 120601119896

119894119895

Δ119905

=

120572 (120601119896

)119894+12119895

120575119909

+120601119894119895minus 120572 (120601

119896

)119894minus12119895

120575119909

minus120601119894119895

Δ119909

+

120572 (120601119896

)119894119895+12

120575119910

+120601119894119895minus 120572 (120601

119896

)119894119895minus12

120575119910

minus120601119894119895

Δ119910

(15)


where

120575119909

+120601119894119895=(120601119896

119894+1119895minus 120601119896

119894119895)

Δ119909

120575119909

119888120601119894119895=(120601119896

119894+1119895minus 120601119896

119894minus1119895)

(2Δ119909)

120575119909

minus120601119894119895=(120601119896

119894119895minus 120601119896

119894minus1119895)

Δ119909

120575119910

+120601119894119895=(120601119896

119894119895+1minus 120601119896

119894119895)

Δ119910

120575119910

119888120601119894119895=(120601119896

119894119895+1minus 120601119896

119894119895minus1)

(2Δ119910)

120575119910

minus120601119894119895=(120601119896

119894119895minus 120601119896

119894119895minus1)

Δ119910

(16)


|nabla120601|119894plusmn12119895

|nabla120601|119894119895plusmn12

are given by

1003816100381610038161003816nabla1206011003816100381610038161003816119894plusmn12119895 =

radic(120575119909plusmn120601119894119895)2

+ [(120575119910

119888120601119894119895+ 120575119910

119888120601119894plusmn1119895)

2]

2

1003816100381610038161003816nabla1206011003816100381610038161003816119894119895plusmn12 =

radic(120575119910

plusmn120601119894119895)2

+ [(120575119909

119888120601119894119895+ 120575119909

119888120601119894plusmn1119895)

2]

2

(17)


0defined


0 thereby reducing



120601119896+12

119894119895minus 120601119896

119894119895= (

Δ119905

Δ119909)

sdot [(120601119896

119894+1119895minus 120601119896

119894119895) 120572119894+12119895

+ (120601119896

119894minus1119895minus 120601119896

119894119895) 120572119894minus12119895

]

120601119896+1

119894119895minus 120601119896+12

119894119895= (

Δ119905

Δ119910)

sdot [(120601119896

119894119895+1minus 120601119896

119894119895) 120572119894119895+12

+ (120601119896

119894119895minus1minus 120601119896

119894119895) 120572119894119895minus12

]

(18)

with 120572119894plusmn12119895

= (1Δ119909)120572(120601119896

)119894+12119895

and 120572119894119895plusmn12

=

(1Δ119910)120572(120601119896

)119894119895plusmn12


119865119909

119894119895= (120601119896

119894+1119895minus 120601119896

119894119895) 120572119894+12119895

119865119909

119894minus1119895= (120601119896

119894minus1119895minus 120601119896

119894119895) 120572119894minus12119895

119865119910

119894119895= (120601119896

119894119895+1minus 120601119896

119894119895) 120572119894119895+12

119865119910

119894minus1119895= (120601119896

119894119895minus1minus 120601119896

119894119895) 120572119894119895minus12

(19)


119896

119894119895to 120601119896119894+1119895


119894119895 119865119910119894119895

and 119865119910119894119895minus1



119865low = min119901119902=minus101

120601119896

119894+119901119895+119902minus 120601119896

119894119895

119865high = max119901119902=minus101

120601119896

119894+119901119895+119902minus 120601119896

119894119895

(20)


119865low le 119865119909

119894119895 119865119909

119894minus1119895 119865119910

119894119895 119865119910

119894119895minus1le 119865high (21)


min119901119902=minus101

120601119896

119894+119901119895+119902le 120601119896+1

119894119895le max119901119902=minus101

120601119896

119894+119901119895+119902 (22)



0Thatmeans that all


120583Δ119905

Δ1199092le1

4

120583Δ119905

Δ1199102le1

4

(23)





End

No

Yes








with 1205720(120601) 100 times and







number k = k + 1





Minimize120601

119871 (119906 120601)

= 119869 (119906 120601) + 120582+(int119863

119867(120601) 119889Ω minus 119881lowast

)

(24)




119869 (119906 120601) =1

2int119863

119864120576 (119906) 120576 (119906)119867 (120601) 119889Ω (25)





119881119899= 05119864120576 (119906) 120576 (119906) minus 120582

+ (26)


+to guarantee





0to improve the









119908= 2mm In the




0= 119881all The





119908=






Design domain

P

40mm

25m

m


(a) (b) (c)

(d) (e) (f)



119908= 01mm





(a) (b) (c)

(d) (e) (f)


0 10 20 30 40 50 6001

02

03

04

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

04

06

08

1

Volu

me f

ract

ion






Design domain

4mm25

mm

P



4 Conclusions








(a) (b) (c) (d)


P

Design domain

20mm

12

mm



(a) (b) (c)

(d) (e) (f)


0 5 10 15 20 25 30 35 400

010203040506070809

1

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

0

05

1

Volu

me f

ract

ion


times10minus3








(a)

(b)

(c)





Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start





No


Yes





of each node





number k = k + 1



120601119896+1

119894119895minus 120601119896

119894119895

Δ119905

=

120572 (120601119896

)119894+12119895

120575119909

+120601119894119895minus 120572 (120601

119896

)119894minus12119895

120575119909

minus120601119894119895

Δ119909

+

120572 (120601119896

)119894119895+12

120575119910

+120601119894119895minus 120572 (120601

119896

)119894119895minus12

120575119910

minus120601119894119895

Δ119910

(15)


where

120575119909

+120601119894119895=(120601119896

119894+1119895minus 120601119896

119894119895)

Δ119909

120575119909

119888120601119894119895=(120601119896

119894+1119895minus 120601119896

119894minus1119895)

(2Δ119909)

120575119909

minus120601119894119895=(120601119896

119894119895minus 120601119896

119894minus1119895)

Δ119909

120575119910

+120601119894119895=(120601119896

119894119895+1minus 120601119896

119894119895)

Δ119910

120575119910

119888120601119894119895=(120601119896

119894119895+1minus 120601119896

119894119895minus1)

(2Δ119910)

120575119910

minus120601119894119895=(120601119896

119894119895minus 120601119896

119894119895minus1)

Δ119910

(16)


|nabla120601|119894plusmn12119895

|nabla120601|119894119895plusmn12

are given by

1003816100381610038161003816nabla1206011003816100381610038161003816119894plusmn12119895 =

radic(120575119909plusmn120601119894119895)2

+ [(120575119910

119888120601119894119895+ 120575119910

119888120601119894plusmn1119895)

2]

2

1003816100381610038161003816nabla1206011003816100381610038161003816119894119895plusmn12 =

radic(120575119910

plusmn120601119894119895)2

+ [(120575119909

119888120601119894119895+ 120575119909

119888120601119894plusmn1119895)

2]

2

(17)


0defined


0 thereby reducing



120601119896+12

119894119895minus 120601119896

119894119895= (

Δ119905

Δ119909)

sdot [(120601119896

119894+1119895minus 120601119896

119894119895) 120572119894+12119895

+ (120601119896

119894minus1119895minus 120601119896

119894119895) 120572119894minus12119895

]

120601119896+1

119894119895minus 120601119896+12

119894119895= (

Δ119905

Δ119910)

sdot [(120601119896

119894119895+1minus 120601119896

119894119895) 120572119894119895+12

+ (120601119896

119894119895minus1minus 120601119896

119894119895) 120572119894119895minus12

]

(18)

with 120572119894plusmn12119895

= (1Δ119909)120572(120601119896

)119894+12119895

and 120572119894119895plusmn12

=

(1Δ119910)120572(120601119896

)119894119895plusmn12


119865119909

119894119895= (120601119896

119894+1119895minus 120601119896

119894119895) 120572119894+12119895

119865119909

119894minus1119895= (120601119896

119894minus1119895minus 120601119896

119894119895) 120572119894minus12119895

119865119910

119894119895= (120601119896

119894119895+1minus 120601119896

119894119895) 120572119894119895+12

119865119910

119894minus1119895= (120601119896

119894119895minus1minus 120601119896

119894119895) 120572119894119895minus12

(19)


119896

119894119895to 120601119896119894+1119895


119894119895 119865119910119894119895

and 119865119910119894119895minus1



119865low = min119901119902=minus101

120601119896

119894+119901119895+119902minus 120601119896

119894119895

119865high = max119901119902=minus101

120601119896

119894+119901119895+119902minus 120601119896

119894119895

(20)


119865low le 119865119909

119894119895 119865119909

119894minus1119895 119865119910

119894119895 119865119910

119894119895minus1le 119865high (21)


min119901119902=minus101

120601119896

119894+119901119895+119902le 120601119896+1

119894119895le max119901119902=minus101

120601119896

119894+119901119895+119902 (22)



0Thatmeans that all


120583Δ119905

Δ1199092le1

4

120583Δ119905

Δ1199102le1

4

(23)





End

No

Yes








with 1205720(120601) 100 times and







number k = k + 1





Minimize120601

119871 (119906 120601)

= 119869 (119906 120601) + 120582+(int119863

119867(120601) 119889Ω minus 119881lowast

)

(24)




119869 (119906 120601) =1

2int119863

119864120576 (119906) 120576 (119906)119867 (120601) 119889Ω (25)





119881119899= 05119864120576 (119906) 120576 (119906) minus 120582

+ (26)


+to guarantee





0to improve the









119908= 2mm In the




0= 119881all The





119908=






Design domain

P

40mm

25m

m


(a) (b) (c)

(d) (e) (f)



119908= 01mm





(a) (b) (c)

(d) (e) (f)


0 10 20 30 40 50 6001

02

03

04

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

04

06

08

1

Volu

me f

ract

ion






Design domain

4mm25

mm

P



4 Conclusions








(a) (b) (c) (d)


P

Design domain

20mm

12

mm



(a) (b) (c)

(d) (e) (f)


0 5 10 15 20 25 30 35 400

010203040506070809

1

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

0

05

1

Volu

me f

ract

ion


times10minus3








(a)

(b)

(c)





Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

120575119909

+120601119894119895=(120601119896

119894+1119895minus 120601119896

119894119895)

Δ119909

120575119909

119888120601119894119895=(120601119896

119894+1119895minus 120601119896

119894minus1119895)

(2Δ119909)

120575119909

minus120601119894119895=(120601119896

119894119895minus 120601119896

119894minus1119895)

Δ119909

120575119910

+120601119894119895=(120601119896

119894119895+1minus 120601119896

119894119895)

Δ119910

120575119910

119888120601119894119895=(120601119896

119894119895+1minus 120601119896

119894119895minus1)

(2Δ119910)

120575119910

minus120601119894119895=(120601119896

119894119895minus 120601119896

119894119895minus1)

Δ119910

(16)


|nabla120601|119894plusmn12119895

|nabla120601|119894119895plusmn12

are given by

1003816100381610038161003816nabla1206011003816100381610038161003816119894plusmn12119895 =

radic(120575119909plusmn120601119894119895)2

+ [(120575119910

119888120601119894119895+ 120575119910

119888120601119894plusmn1119895)

2]

2

1003816100381610038161003816nabla1206011003816100381610038161003816119894119895plusmn12 =

radic(120575119910

plusmn120601119894119895)2

+ [(120575119909

119888120601119894119895+ 120575119909

119888120601119894plusmn1119895)

2]

2

(17)


0defined


0 thereby reducing



120601119896+12

119894119895minus 120601119896

119894119895= (

Δ119905

Δ119909)

sdot [(120601119896

119894+1119895minus 120601119896

119894119895) 120572119894+12119895

+ (120601119896

119894minus1119895minus 120601119896

119894119895) 120572119894minus12119895

]

120601119896+1

119894119895minus 120601119896+12

119894119895= (

Δ119905

Δ119910)

sdot [(120601119896

119894119895+1minus 120601119896

119894119895) 120572119894119895+12

+ (120601119896

119894119895minus1minus 120601119896

119894119895) 120572119894119895minus12

]

(18)

with 120572119894plusmn12119895

= (1Δ119909)120572(120601119896

)119894+12119895

and 120572119894119895plusmn12

=

(1Δ119910)120572(120601119896

)119894119895plusmn12


119865119909

119894119895= (120601119896

119894+1119895minus 120601119896

119894119895) 120572119894+12119895

119865119909

119894minus1119895= (120601119896

119894minus1119895minus 120601119896

119894119895) 120572119894minus12119895

119865119910

119894119895= (120601119896

119894119895+1minus 120601119896

119894119895) 120572119894119895+12

119865119910

119894minus1119895= (120601119896

119894119895minus1minus 120601119896

119894119895) 120572119894119895minus12

(19)


119896

119894119895to 120601119896119894+1119895


119894119895 119865119910119894119895

and 119865119910119894119895minus1



119865low = min119901119902=minus101

120601119896

119894+119901119895+119902minus 120601119896

119894119895

119865high = max119901119902=minus101

120601119896

119894+119901119895+119902minus 120601119896

119894119895

(20)


119865low le 119865119909

119894119895 119865119909

119894minus1119895 119865119910

119894119895 119865119910

119894119895minus1le 119865high (21)


min119901119902=minus101

120601119896

119894+119901119895+119902le 120601119896+1

119894119895le max119901119902=minus101

120601119896

119894+119901119895+119902 (22)



0Thatmeans that all


120583Δ119905

Δ1199092le1

4

120583Δ119905

Δ1199102le1

4

(23)





End

No

Yes








with 1205720(120601) 100 times and







number k = k + 1





Minimize120601

119871 (119906 120601)

= 119869 (119906 120601) + 120582+(int119863

119867(120601) 119889Ω minus 119881lowast

)

(24)




119869 (119906 120601) =1

2int119863

119864120576 (119906) 120576 (119906)119867 (120601) 119889Ω (25)





119881119899= 05119864120576 (119906) 120576 (119906) minus 120582

+ (26)


+to guarantee





0to improve the









119908= 2mm In the




0= 119881all The





119908=






Design domain

P

40mm

25m

m


(a) (b) (c)

(d) (e) (f)



119908= 01mm





(a) (b) (c)

(d) (e) (f)


0 10 20 30 40 50 6001

02

03

04

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

04

06

08

1

Volu

me f

ract

ion






Design domain

4mm25

mm

P



4 Conclusions








(a) (b) (c) (d)


P

Design domain

20mm

12

mm



(a) (b) (c)

(d) (e) (f)


0 5 10 15 20 25 30 35 400

010203040506070809

1

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

0

05

1

Volu

me f

ract

ion


times10minus3








(a)

(b)

(c)





Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










End

No

Yes








with 1205720(120601) 100 times and







number k = k + 1





Minimize120601

119871 (119906 120601)

= 119869 (119906 120601) + 120582+(int119863

119867(120601) 119889Ω minus 119881lowast

)

(24)




119869 (119906 120601) =1

2int119863

119864120576 (119906) 120576 (119906)119867 (120601) 119889Ω (25)





119881119899= 05119864120576 (119906) 120576 (119906) minus 120582

+ (26)


+to guarantee





0to improve the









119908= 2mm In the




0= 119881all The





119908=






Design domain

P

40mm

25m

m


(a) (b) (c)

(d) (e) (f)



119908= 01mm





(a) (b) (c)

(d) (e) (f)


0 10 20 30 40 50 6001

02

03

04

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

04

06

08

1

Volu

me f

ract

ion






Design domain

4mm25

mm

P



4 Conclusions








(a) (b) (c) (d)


P

Design domain

20mm

12

mm



(a) (b) (c)

(d) (e) (f)


0 5 10 15 20 25 30 35 400

010203040506070809

1

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

0

05

1

Volu

me f

ract

ion


times10minus3








(a)

(b)

(c)





Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Minimize120601

119871 (119906 120601)

= 119869 (119906 120601) + 120582+(int119863

119867(120601) 119889Ω minus 119881lowast

)

(24)




119869 (119906 120601) =1

2int119863

119864120576 (119906) 120576 (119906)119867 (120601) 119889Ω (25)





119881119899= 05119864120576 (119906) 120576 (119906) minus 120582

+ (26)


+to guarantee





0to improve the









119908= 2mm In the




0= 119881all The





119908=






Design domain

P

40mm

25m

m


(a) (b) (c)

(d) (e) (f)



119908= 01mm





(a) (b) (c)

(d) (e) (f)


0 10 20 30 40 50 6001

02

03

04

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

04

06

08

1

Volu

me f

ract

ion






Design domain

4mm25

mm

P



4 Conclusions








(a) (b) (c) (d)


P

Design domain

20mm

12

mm



(a) (b) (c)

(d) (e) (f)


0 5 10 15 20 25 30 35 400

010203040506070809

1

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

0

05

1

Volu

me f

ract

ion


times10minus3








(a)

(b)

(c)





Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Design domain

P

40mm

25m

m


(a) (b) (c)

(d) (e) (f)



119908= 01mm





(a) (b) (c)

(d) (e) (f)


0 10 20 30 40 50 6001

02

03

04

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

04

06

08

1

Volu

me f

ract

ion






Design domain

4mm25

mm

P



4 Conclusions








(a) (b) (c) (d)


P

Design domain

20mm

12

mm



(a) (b) (c)

(d) (e) (f)


0 5 10 15 20 25 30 35 400

010203040506070809

1

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

0

05

1

Volu

me f

ract

ion


times10minus3








(a)

(b)

(c)





Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c)

(d) (e) (f)


0 10 20 30 40 50 6001

02

03

04

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

04

06

08

1

Volu

me f

ract

ion






Design domain

4mm25

mm

P



4 Conclusions








(a) (b) (c) (d)


P

Design domain

20mm

12

mm



(a) (b) (c)

(d) (e) (f)


0 5 10 15 20 25 30 35 400

010203040506070809

1

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

0

05

1

Volu

me f

ract

ion


times10minus3








(a)

(b)

(c)





Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















(a) (b) (c) (d)


P

Design domain

20mm

12

mm



(a) (b) (c)

(d) (e) (f)


0 5 10 15 20 25 30 35 400

010203040506070809

1

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

0

05

1

Volu

me f

ract

ion


times10minus3








(a)

(b)

(c)





Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c)

(d) (e) (f)


0 5 10 15 20 25 30 35 400

010203040506070809

1

Mea

n co

mpl

ianc

e (N

mm

)

Iteration

0

05

1

Volu

me f

ract

ion


times10minus3








(a)

(b)

(c)





Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a)

(b)

(c)





Acknowledgment


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

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