Digital Object Identifier (DOI) 10.1007/s00158-002-0257-4

Struct Multidisc Optim 24, 441–448 Springer-Verlag 2003

Improving efficiency of evolutionary structural optimizationby implementing fixed grid mesh�

H. Kim, O.M. Querin, G.P. Steven and Y.M. Xie

Abstract Evolutionary structural optimization (ESO)has been shown through much published research to bea simple and yet effective method for structural shapeand topology optimization. However, attention has beendrawn to shortcomings in the method related to the com-putational efficiency of the algorithm as well as the jaggededge representation of the Finite Element optimal solu-tions. In this paper a fixed grid (FG) mesh is implementedand an improved ESO methodology is introduced in orderto address these shortcomings. The examples show a sig-nificant improvement in the solution time as well as elim-inating the jagged edges and checkerboard patterns thatmay appear in current solution topologies. In addition,FG is applied to both stress based and stiffness optimiza-tion. This paper demonstrates a simple implementationof FG and the consequent improvement in the efficiencyand practicality of the FG ESO formulation.

Key words topology optimization, ESO, fixed grid,checkerboard pattern

1Introduction

The competitive and rapidly developing environment ofthe engineering design industry demands faster and more

Received November 14, 2000

H. Kim1, O.M. Querin2, G.P. Steven3 and Y.M. Xie4

1Department of Mechanical Engineering, University of Bath,Bath BA2 7AY, UKe-mail: h.a.kim@bath.ac.uk2 School of Mechanical Engineering, University of Leeds, LeedsLS2 9JT, UK3 School of Engineering, University of Durham, Durham DH13LE UK4 School of the Built Environment, Victoria University, POBox 14428, MCMC VIC 8001, Australia

� Extended version of a paper presented at ASMO-2, held inSwansea, UK, 2000

practical topology optimization methods. Evolutionarystructural optimization (ESO) has been introduced asa simple and effective method for size, shape and topologyoptimization. The wide range of ESO applications havebecome proof of its versatility and its potential as a designtool (Xie and Steven 1997).

However, the disadvantages of ESO have become ap-parent in high solution time and the appearance of theoptimum topology output. Due to the iterative and slownature of the ESO process a typical solution time is oftenexcessive for practical problems.

The other unfavourable feature is the finite elementrepresentation of the final optimum topology, identifiedby the profile of the remaining elements’ edges. Whilstthis format can be viewed satisfactorily on a computerscreen, it takes a significant time and effort to extractthe boundary in a format that most CAD/CAM systemsrecognize. Furthermore, the boundaries are representedby the jagged edges of the finite elements, which requiresome smoothing or image filtering processing in order tomanufacture a smooth topology.

This paper addresses these problems by incorporat-ing Fixed Grid (FG) into ESO. FG has been used tomodel problems where the geometry or the physical prop-erties of the domain change with time (Voller et al. 1990).The advantage of FG is that simple modifications en-able the existing numerical formulation and solution to beadapted to a changing environment. Garcıa and Steven(2000) applied FG to elasticity problems. The studyshowed that the reduction in solution time outweighs theloss of accuracy in the analysis. The accuracy of a solu-tion was found to be adequate for the initial stage of thedesign process, where a design is subjected to frequentchanges. Also, despite the accuracy reduction, the formof the stress field is correct thereby making the ESO logicwork correctly.

Kim et al. (2000a) have made the implementation ofa FG into ESO, where a FG mesh represented the initialdesign domain and ESO was applied to remove elementswith low stress values. This study clearly demonstratedthe benefit to be gained in time reduction. It also pro-posed an algorithm, which converts a FG representationto a boundary representation of a topology. However, op-timization is still based on the presence or absence of

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elements, and a conversion algorithm must be applied toobtain a boundary representation of the topology.

This paper presents an improved FG ESO method,which optimises the given design domain by modify-ing and adapting the boundaries. This new method notonly results in a reduction of solution time, but also de-termines an optimum topology with a more favourableboundary representation form at every iteration, withoutthe typical finite elements’ “jagged-edges”. The followingsections briefly review the ESO methodology and the FGformulation for elasticity problems; some examples arepresented followed by some concluding remarks.

2ESO methodology

The concept of ESO states that by slowly removing in-efficient materials from a structure, a structure evolvestowards an optimum (Xie and Steven 1997). The ineffi-ciency of any portion of the material (i.e. a finite element)is determined by low sensitivity number which is meas-ured against the optimality objective function. For stressbased optimization, an element’s von Mises stress is com-monly used as the sensitivity number (Xie and Steven1997). This is the original form of ESO where it is saidthat a reliable sign of inefficient material use is low stress,and an optimum design is where, for a single load case,every part of a structure is near a constant stress level, i.e.a fully stressed design.

Chu (1997) derived a sensitivity number for compli-ance or stiffness design. This compliance sensitivity num-ber, si indicates the change in the compliance as a resultof removing element i, as defined in (1). si is also referredto as an element contribution to the structure’s total com-pliance, and the sum of si over all elements equates tothe compliance of the structure. Removing elements withlow compliance sensitivity number minimizes the increasein compliance as the volume is reduced, leading towardsa minimum compliance design. Note that although com-pliance will always increase when an element is removedthe endeavour is to minimize the specific compliance,equivalent to maximizing the specific stiffness,

si =1

2

{ui}T [Ki] {ui}, (1)

where si = sensitivity number of element i; u= nodal dis-placement vector;Ki = stiffness matrix of element i. ESOremoves the material slowly. The slowness of the removalis ensured by the rejection ratio. The material is removedwhen the stress satisfies the ESO inequality, (2).

si ≤RR · smax , (2)

where smax = maximum elemental sensitivity number

RR = rejection ratio = a0+a1 ·SS ,

(Xie and Steven 1997) . (3)

The rejection ratio increases as the structure is evolvedand its function is to delay the element removal process sothat the design does not change significantly after each it-eration. The steady state number, SS is a positive integercounter which increases as the structure evolves. Figure 1summarizes the ESO methodology.

Specify design problem;

SS = 0;

Do while (optimum is not reached),

Perform FEA;

Remove elements according to (2);

If (number of removed elements = 0),

SS = SS + 1;

Evaluate RR using (3);

End if ;

End do;

Fig. 1 Summary of ESO process

3Fixed grid for elasticity problems

A FG is generated by superimposing a rectangular gridof equal sized elements on the given structure instead ofgenerating a mesh to fit the structure. Some of these elem-ents are inside the structure (I), some are outside (O)and some are on the boundary, namely Neither-In-nor-Out (NIO) elements as illustrated in Fig. 2. An O elem-ent is given a material property significantly less than anI element, resulting in a bi-material problem.

Fig. 2 FG mesh

A NIO element is partially inside the structure andits material property value is not constant nor continu-ous over the element. Such an element is approximatedinto a homogeneous isotropic element, with its materialproperty defined by (4),

[D(NIO)i

]= α[D(I)i

], (4)

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where[D(NIO)i

]= elemental material property of a NIO

element;[D(I)i

]= elemental material property of elem-

ent i,

α=AI/Ae , (5)

where α = area ratio; AI = area inside of the structurewithin NIO element i; Ae = total area of element i.

As the sizes of all elements are identical, the stiffnessmatrix entries are linearly proportional to their elemen-tal area ratio. This greatly reduces the time taken in thestiffness matrix generation and regeneration.

For FEA the preconditioned conjugate gradient solveris implemented as it is known to be fast and efficient(Garcia 1999; Garcıa and Steven 2000). In addition, thesolution vector from the previous optimization iterationsoffers an excellent initial solution vector for reanalysis, re-ducing the solution time even further.

4Methodology

4.1Boundary modification

The following explanation of the method uses stress asthe optimization criterion as ESO in its original format isstress based. However, it should be noted that replacingstress with compliance sensitivity number of (1) achievescompliance based optimization.

Unlike the standard ESO formation the stress alongthe boundaries is considered separately from that ofa non-boundary region. The optimization algorithmfirstly examines stress on a boundary, which is deter-mined by a linear interpolation of the two adjacent nodalstresses. If this boundary stress is lower than the deletioncriterion, i.e. the ESO inequality of (6) is satisfied, theboundary is modified to be the contour line of the deletioncriterion. Therefore stress along the modified boundary iseither equal to or higher than the deletion criterion. Thisremoves material with low stress of any shape and size.Hence a boundary is no longer restricted to right angles,but can be defined at any angle,

σb < σdel , (6)

where σb = stress of a boundary point; σdel = deletioncriterion.

After modifying all existing boundaries, stresses of thenon-boundary nodes are examined. If a non-boundarynodal stress satisfies the ESO inequality of (6), the stressvalues of the adjacent nodes are examined for a newboundary, i.e. a cavity. If stress of an adjacent node doesnot satisfy the ESO inequality, there exists a point be-tween the two nodes, where its stress equates the deletioncriterion. This point is again obtained by a linear inter-polation of the two nodal stresses, and becomes a bound-ary point. The optimization algorithm then searches for

a series of new boundary points which define an enclosedboundary of a low stress region. This region of material isremoved to create a cavity or a new boundary.

One iteration of optimization is completed when allboundary and nodal stresses are examined. The area ra-tios of the modified elements are computed and the elem-ental stiffness values are updated according to the arearatios for FEA of the modified topology.

4.2Calculation of deletion criterion

The ESO’s definition of deletion criterion is a percentageof the maximum value of the criterion. When the deletioncriterion is too small, it is increased by a pre-specified stepuntil the deletion criterion value is large enough to removeelements. This sometimes requires a series of calculationsof the deletion criterion and checking all elemental crite-ria before the deletion criterion is finally large enough.

However, the minimum value of the criterion can bedetermined and the deletion criterion can thus be com-puted to be always greater than the minimum criterion.This ensures that a modification is made at every iter-ation and increases the efficiency of ESO. Here, a newdefinition of deletion criterion is proposed:

σdel = σmin+aσmean , (7)

where σmin = minimum value of the criteria; σmean =mean value of the criteria; a= evolutionary rate constant,a positive real value which ensures that deletion criterionis slightly greater than the minimum criterion value, typ-ically 0.01 deletion criterion.

The value of the evolutionary rate constant, a defineshow much greater σdel is relative to σmin. The larger thevalue of a, the more material ESO removes during an iter-ation and hence a faster optimization. However increasingthe rate of optimization leads to a less refined optimumsolution.

4.3Definition of steady state

In standard ESO, a steady state is reached when no elem-ents have criteria less than the deletion criterion. Whena topology reaches a steady state, the deletion criterion isincreased to further optimize the structure if desired.

For FG ESO, a steady state would be equivalent toa state where no more modifications are made. However,unlike standard ESO, FG ESO allows almost infinitelysmall amount of material removal. A small boundarymodification still adjusts the nodal stress values, whichleads to another small modification. Therefore FG ESOoften continues to remove or add a small percentage ofmaterial, e.g. 0.0001% of the total volume. In such cases,it is said that a steady state is reached and a new deletioncriterion is calculated to obtain a significant modification

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of the topology. Thus a topology is said to have reacheda steady state if the total volume change during an itera-tion is less than the elemental volume, (8),

∆V < Velem , (8)

where∆V = volume of removed material during an itera-tion; Velem = volume of an element.

4.4Merging two boundaries

As a topology is optimized and material is removed, twoseparate boundaries may merge into one. FG ESO identi-fies such a case by examining each element that containssome boundary. When two boundaries lie on one elementas shown in Fig. 3a, the two boundaries, A and B be-come one boundary. The assumption here is that an areasmaller than the element width is negligible, as it wouldbe impractical to manufacture.

An example of Fig. 3 is used here in order to displaythe boundary merge mechanism more clearly. Elemente with two boundaries A and B of Fig. 3a is enlargedin Fig. 3b. The shaded areas represent the structure andthe clear areas represent the outside or voids. As bound-aries are modified, FG ESO checks all edges of the bound-ary elements. When modifying element e of boundary A,FG ESO checks for another boundary on the elementand would find that boundary B also lies on element e.Boundary A is then modified such that boundary B be-comes a part of A, as shown in Fig. 3c. This thereforereduces the total number of boundaries or cavities by one.

Fig. 3 Merging two boundaries

4.5FG ESO formulation

The standard formulation of ESO removes material byelements, where the existence of an element becomes thedesign variable. However, FG ESO removes a region ofmaterial with low stress values, and the design variablebecomes the area ratio, α of each element. Therefore, themathematical representation of FG ESO is modified toreflect this change, (9),

minimize f(x) =

∑ni=1 σiαiνi

FLsubject to V ≤ V ∗ , σ− (σmin+aσmean)≥ 0 . (9)

The objective function and the finite element formulationinclude αi to incorporate the use of FG elements. Thesecond constraint represents the ESO inequality whichensures stress in the domain is always greater than thedeletion criterion, (6).

The following step-by-step procedure gives an over-view of the boundary based FG ESO algorithm and hasbeen summarized in Fig. 4.

1. The user is required to define an optimization problemby defining the maximum domain, the design environ-ment and optimization parameters. Von Mises stressis usually specified as the optimization criterion fora fully stressed design, hence it will be used as the op-timization criterion here. However, it should be notedthat other criteria such as compliance and frequencysensitivity numbers could be used instead.

2. FG mesh is generated.3. FEA is conducted to determine displacement and

stress at all nodes.4. The minimum stress value is determined and the dele-

tion criterion is calculated, (7).5. Using the nodal stress values, the stress along the

boundary is examined. If the stress on a boundaryis less than the deletion criterion, a contour line ofthe deletion criterion becomes the new boundary. Thisboundary is obtained by a linear interpolation of twonodal stresses.

6. The stress values of the non-boundary nodes are ex-amined, and a new boundary, ie. a cavity along thedeletion criterion contour is initiated if it exists.

7. While modifying the boundaries, if two boundariespass through a single element, they merge and becomeone boundary.

8. If an optimum is reached, the optimization process isterminated.

9. Otherwise area ratios of the elements are obtained andthe new stiffness matrix is generated. Another FEA iscarried out and the nodal stress values are determined.

10. If a steady state is reached according to (8), the pro-cess is repeated from step 5 to compute a new deletioncriterion. If a steady state is not reached, the processis repeated from step 6 and continues to modify thestructure.

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Specify design problem;

Generate FG mesh;

Generate stiffness matrix;

Conduct FEA;

Evaluate the initial deletion criterion, (7);

Do while (optimum is not reached),

Modify all existing boundaries;

Initiate new boundaries;

Update stiffness matrix;

Conduct FEA;

If (SS is reached),

Evaluate deletion criterion, (7);

End if ;

End do;

Fig. 4 Summary of FG ESO process

5Examples

5.1MBB beam

To demonstrate the method, the well-known MBB beam(Olhoff et al. 1991; Zhou and Rozvany 1991) is optimizedusing standard ESO with the traditional finite elementformulation and FG ESO for stress and compliance. Thedesign environment and the maximum domain are asshown in Fig. 5. As only the qualitative result is of in-terest, the nondimensional physical parameters are cho-sen. Due to symmetry, only the left half of the model isoptimized with the mesh density of 75 × 25. For stressbased standard ESO, the evolutionary rate constants area1 = 0.0001 and a0 = 0.0, and for FG ESO, a= 0.006 forstress and a = 0.003 for compliance. The computer usedfor all problems presented in this paper is a Pentium133 MHz with 32 MB RAM.

The optimum solutions from standard ESO; stressbased FG ESO; compliance based FG ESO are selectedat an equivalent volume level for comparison and they aredisplayed in Fig. 6, together with the known optimal gril-lage layout.

Similar truss-like topologies are obtained for all cases:The locations and angles of the members as well as thegeneral topology of the solutions compare closely. Theyalso agree favourably with the exact analytical (Fig. 6d,after Lewinski et al. 1994) optimal layout and with dis-

Fig. 5 MBB beam design problem

Fig. 6 Optimum solutions of MBB beam problem. (a) Stan-dard ESO solution, (b) stress based FG ESO solution, (c)compliance based FG ESO solution, and (d) exact analyticaloptimal truss layout (Lewinski et al. 1994)

cretized solutions by other researchers (Zhou and Roz-vany 1991; Olhoff et al. 1991; Hassani and Hinton 1998;Sigmund 1994).

Table 1 presents the details of the solutions. The ad-vantage of using FG can easily be viewed by comparingthe solutions time, where the FG ESO solution times aresignificantly lower than that of standard ESO: For stressoptimization, using FG reduced the solution time by 87%.The mean to maximum stress ratio is an indication ofthe even stress distribution, and as a topology is opti-mized, the stress ratio is expected to rise. The relativemean to maximum stress in Table 1 is measured relativeto its initial value. Both standard ESO and FG ESO so-lutions display approximately 20% increase in the relative

Table 1 Comparison of MBB beam optimization

Standard Stress ComplianceESO FG ESO FG ESO

volume (%) 65 65 61solution time 13:06:24 1:41:49 1:25:04relative mean/max 1.22 1.23 –stress ratiomaximum – 41.6 41.8displacement

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mean to maximum stress ratio values, again indicatingthe equivalence of the two solutions.

As mentioned earlier, stress based and compliancebased FG ESO both produced similar solutions. Somedifferences in their internal structural member arrange-ments in Fig. 6b and c are primarily due to stress ap-proximation. However the equivalence of these solutionsis demonstrated clearly in their maximum displacementvalues in Table 1. This is in agreement with Li et al.(1999) and Rozvany and Karolyi (1999)’s finding, wherean optimal stress design is equivalent to a compliance de-sign for single load case problems.

5.2Michell’s beam design

A typical Michell type problem (Michell 1904) has beenoptimized. The design environment and the maximumdomain are as shown in Fig. 7. Due to symmetry, only theleft half of the model is optimized and the mesh size of50 × 50 is used. For standard ESO, the evolutionary rateconstants are a1 = 0.0001 and a0 = 0.0; a= 0.04 for stressFG ESO; and a= 0.008 for compliance FG ESO.

The optimal solutions for all three optimization areobtained at around 40% volume level as shown in Fig. 8.Again, the expected topology of a truss-like structure(Michell 1904) is observed for all cases. The standardstress based ESO solution of Fig. 8a is obtained after22 hours 55 minutes and 30 seconds whilst stress basedFG ESO obtained its solution of Fig. 8b in 2 hours and22 minutes and 52 seconds, which is almost 90% reduc-tion in time. The compliance FG ESO solution of Fig. 8crequired 50 minutes and 40 seconds.

However, three cavities are created in the topology bystandard ESO while 5 cavities are created in FG ESO’stopology. This demonstrates the benefits of the bound-ary based optimization where a part of an element may beremoved according to the stress patterns. This not onlygains a boundary representation of the solution, it leadsto a more refined and detailed topology, indicated by theincreased number of cavities.

A greater number of cavities can be obtained by in-creasing the mesh density and/or reducing the evolution-

Fig. 7 Michell’s design problem of beam AR2

Fig. 8 Optimum solutions of Michell’s beam problem

ary rate (Kim et al. 2000b). It can therefore be under-stood that applying standard ESO with a reduced evolu-tionary rate may gain a more comparable topology withthe same number of cavities. ESO is applied again tothe same problem with the same sized mesh but witha slower rate of a1 = 1×10−5. After 982 iterations and24 hours 6 minutes and 24 seconds, the same topologywith three cavities of Fig. 8a is obtained again. There-fore, it is induced that an even slower evolutionary rateand a finer mesh density must be applied in order to ob-tain a topology with 5 cavities by standard ESO. Thiswill increase the solution time even further. Thus, thebenefits of using FG ESO can be appreciated from thisillustration.

The layout of the stress and compliance solutionsof Fig. 8b and c, again compare closely, but with differ-ence number of cavities. This is due to the stress approx-imation procedure, where a nodal stress is approximatedby a volume weighted average of the von Mises stressesat the Gauss points of 4 surrounding elements. This in-troduces a smoothing effect on the stress distribution,similar to the patch smoothing technique (Li et al. 1999),thus reducing the number of cavities (Kim 2000). There-fore, a smaller number of cavities may be obtained in

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stress based solutions. However, the layout of both so-lutions display the features of the optimal arrangementdetermined by Michell (1904), albeit his work related topin-jointed frame and not continua.

5.3Bridge

A simple bridge model is optimized for compliance. Thedesign environment is as shown in Fig. 9. A deck of 4 unitthickness is placed on the top and is specified as the non-design domain. A uniformly distributed load is appliedon the top deck to simulate traffic. Applying a symmetrycondition, only half of the design domain is modelled witha mesh density of 50 × 30. An evolutionary rate constantof a= 0.01 is applied.

Acontinuous optimumof anarch-like support, Fig. 10ais reached at 55% volume and the stiffness sensitivity

Fig. 9 Bridge design problem

Fig. 10 Optimum topologies with uniformly distributed load

Fig. 11 Optimum topology with a point load

values of the boundary are uniform. Thus, applying FGESO creates the cavities in order to reduce the volumefurther and this leads to a discrete optimum at 27% vol-ume of Fig. 10b.

Stiffness FG ESO is also applied to the same designdomain subjected to a point load at the centre of thetop deck, instead of the uniformly distributed load, andits solution is displayed in Fig. 11. Due to the nature ofthe applied load, two diagonal members transmitting theload from its point of application to the fixed supports areobtained as the optimum configuration, at the volume of32% relative to its initial design domain.

6Conclusions

This paper has presented the FG ESO algorithm whichremoves material along the contour line of the deletioncriterion. In contrast to element based ESO with its elem-ent by element removal, FG ESO removes material in anyform. The definition of the deletion criterion is modifiedso that it always removes a small percentage of materials,which again increases the efficiency of the optimizationalgorithm.

In some structural designs, a large deflection is notfavourable and compliance becomes an important consid-eration. FG ESO was thus extended to compliance basedoptimization. When applied to single-load problems, thesolutions were comparable to the stress based optimiza-tion results. This confirms that the compliance design isequivalent to a fully stressed design (Li et al. 1999; Roz-vany and Karolyi 1999).

The “jagged-edges” which are a prominent feature ofstandard ESO topologies are not observed in the FG ESOtopologies, however their boundaries of the solutions pre-sented in the paper are not perfectly smooth. It shouldbe noted that these solutions are depicted by a series ofpoints, simply connected by straight lines. Another fac-tor contributing to the nonsmooth boundaries is the useof linear interpolation in the determination of boundarystresses. However, as the purpose of this study is to ap-preciate the solution time reduction and the feasibilityof the optimization method, the proposed algorithm isdeemed adequate to show the efficiency and effectivenessof boundary based FG ESO. Since the output format ofthe FG ESO results is the boundary points arranged inthe anticlockwise manner, a more sophisticated image fil-tering technique can be applied to the output in orderto obtain smooth boundaries. Equivalently a higher orderinterpolation of the nodal stresses may be applied to bet-ter approximate the boundary stress, leading to smoothboundaries.

The examples demonstrate that FG ESO is able toproduce an optimal topology in agreement with theknown solutions. The topologies are not represented bythe finite elements’ “jagged-edges” but by a series ofboundary points arranged in the anticlockwise direc-

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tion, and such an output of a topology is produced atevery iteration. This greatly simplifies further analysis orfurther design manipulation or the preparation for manu-facturing. However the most significant advantage is thereduction of solution time.

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