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Engineering Computations,Vol. 17 No. 4, 2000, pp. 427-439.

# MCB University Press, 0264-4401

Received November 1999Revised March 2000

Accepted March 2000

Introduction of fixed grid inevolutionary structural

optimisationH. Kim, M.J. Garcia, O.M. Querin and G.P. Steven

Department of Aeronautical Engineering, University of Sydney,Australia, and

Y.M.XieSchool of the Built Environment, Victoria University of Technology,

Melbourne, Australia

Keywords Topology, Structural optimization, Grids, Finite element analysis

Abstract Introduces a faster and improved structural optimisation method which combinesfixed grid finite element analysis (FG FEA) and evolutionary structural optimisation (ESO). ESOoptimises a structure by removing a few elements at every iteration. FG methods allow fast meshgeneration, fast solution and fast re-evaluation of the modified meshes. The implementation ofFG into the ESO process eliminates the need for regenerating the mesh and a few arithmeticcalculations replace the full regeneration of the stiffness matrix every time the structure ismodified. This greatly reduces the solution time, and the examples presented in this paperdemonstrate and validate the method.

IntroductionGeneralIn the busy and competitive environment of the engineering/design industry,there are demands for structural optimisation methods to be faster and morepractical. Over the past few decades, much research in the field of structuraloptimisation together with the advancement of computer technology has led tothe more practical and applicable methods of structural optimisation.Structural optimisation methods, like any design process, are iterative in natureand require FEA at every iteration. Thus optimisation on a common PC tendsto be a time-consuming process, typically taking hours and sometimes evendays for large problems.

In this study, the evolutionary structural optimisation (ESO) method isconsidered. An ESO process is started by generating a stiffness matrix of thegiven design. Once the matrix is defined, it is solved for displacements and thestress values of each element. ESO then removes a small percentage ofelements with low stress values. This completes one cycle of the ESO process.Repeating this process leads the design towards an optimum. It has been foundthat, during each iterative cycle, it is the generation of stiffness matrix whichtakes the most time. Thus the optimisation time is largely dependent upon theformulation of FEA as well as its efficiency and solution time.

Implementation of the fixed grid (FG) methodology not only simplifies themesh generation process, but also allows a very significant reduction in the

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arithmetic calculations to update the stiffness matrix for the modified topology,instead of a full regeneration of the matrix. This paper discusses theapplications of FG to ESO in an attempt to reduce the solution time.

Development of evolutionary structural optimisationESO in its original form optimises a structure by slowly removing elementswith low stress, approaching towards a fully stressed design (Xie and Steven,1992; 1993).

The primary goal of the research and development of ESO is to provide theengineering industry with a practical and `̀ user-friendly'' optimisation methodto assist in the design process. Hence ESO has been extended to accommodatevarious optimisation criteria and is becoming a more practical method. Some ofthese developments include the implementation of stiffness and displacementsas optimisation criteria (Xie and Steven, 1997) and the applications in multipleload (Xie and Steven, 1994), non-linear (Querin et al., 1996), dynamic (Zhao et al.,1996) and buckling problems (Manickarajah et al., 1995). Querin et al. (1998)extended the ESO method to add as well as remove elements, namely bi-directional ESO (BESO). This meant that the initial design no longer had to bethe maximum design domain. Thus the solution time may be reduced especiallyif the user specifies a near optimal topology to be the initial design. However,this knowledge is not always available and the typical long solution time of ESOhas been an obstacle to its practical applicability as a design tool.

Previous work on fixed gridFixed grid (FG) has commonly been employed in problems where the geometryor the physical properties of the domain changes with time, such as the fillingsof reservoirs and phase change problems (Voller et al., 1990). The advantage ofFG is that simple modifications enable the existing numerical formulation andsolution of a problem to be adapted in a changing environment. This has beendisplayed in modelling various complex phase change systems. Some of theexamples include solute transport (Bennon and Incropera, 1987) and convectivetransport (Dantzig, 1989).

GarcõÂa and Steven (1996) introduced an FG methodology to solve elasticityproblems. The advantages of using FG are simplicity and speed at apermissible level of accuracy. The stress error was seen to increase near theregion of stress concentration, with a maximum stress error beingapproximately 10 per cent for a reasonable sized mesh. However, the averagestress error was found to be about 5 per cent or below and stiffness errors wereeven lower, around 1 per cent (GarcõÂa, 1999). Thus the FG method was deemedappropriate for interactive design and structural optimisation.

Scope of present studyThis study proposes an optimisation method which implements two-dimensional FG in ESO. The primary advantage is a decrease in time to reach

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an optimal topology due to the reduction in FEA solution time. In addition, theintroduction of a fixed grid allows a finer and less mesh dependent solutionthan that obtained when the traditional finite elements were used.

The second and third sections briefly discuss the concepts and formulationsof ESO and FG FEA respectively. Owing to the different definition of elementsin FG, the mesh generation process was addressed and the element removalprocess of ESO was modified as outlined in the fourth section. The fourthsection also explains the conversion of FG representation of a topology to aboundary representation. Some examples are presented to demonstrate FG ESOin the fifth section, followed by a conclusion.

Classical evolutionary structural optimisationThe basic concept of the stress based ESO is that a structure evolves towards afully stressed design, i.e. an optimum, by slowly removing the lightly stressedelements. The elements that satisfy the ESO inequality of equation (1) are notefficiently utilised in carrying the applied load and hence can be removed witha minimal effect on the structural integrity. The number of elements removedat each iteration is limited to ensure the slow modification of the design:

�VM ; e � RR � �VM ;max �1�

RR � a0 � a1 � SS � a2 � SS2 � a3 � SS3 �2�where

�VM, e = Von Mises stress or selected criterion of element, e;

�VM, max = maximum Von Mises stress or selected criterion of thestructure;

RR = rejection ratio calculated using equation (2);

a0, a1, a2, a3 = user specified evolutionary rate constants based onexperiences (Xie and Steven, 1997);

SS = steady state number (Xie and Steven, 1997).

The ESO method has been summarised in Figure 1. The authors refer to thebook by Xie and Steven (1997) for more details of the algorithm.

Figure 1.ESO method

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Implementation of fixed grid (FG) in elasticity problemsA fixed grid is generated by superimposing a rectangular grid of equal sizedelements on the given structure instead of generating a mesh to fit thestructure. Some of these elements are inside the structure (I), some are outside(O) and some are on the boundary, namely neither-in-nor-out (NIO) elements asillustrated in Figure 2 (GarcõÂa, 1999). An O element is given a material propertysignificantly less than an I element and the problem becomes a bimaterial one.

A NIO element is partially inside the structure and its material propertyvalue is not constant nor continuous over the element. Such an element isapproximated by transforming the bimaterial element into a homogeneousisotropic element. The material property matrix of a NIO element is computedusing equation (3):

D�NIO�e� � � � D�I�e� � �3�

� � AI=Ae �4�where

[D(NIO)e] = elemental material property of a NIO element;

[D(I) e] = elemental material property of inside;

� = area ratio;

AI = area inside the structure within the NIO element, e; and

Ae = total area of an element, e.

Using these values of [D], the stiffness matrix can be computed and a standardFEA can be applied to determine the displacements and hence stress values ofelements. For this study, the pre-conditioned conjugate gradient solver wasimplemented, as it is known to be fast and efficient (GarcõÂa, 1999).

Figure 2.FG mesh

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Fixed grid and ESOGeneration of meshAs mentioned in the previous section, the FG mesh is generated bysuperimposing a rectangular mesh on the given design domain. When themaximum design domain is specified, the minimum and maximum coordinatesare determined for the coordinates of the rectangular FG domain. Using thespecified mesh density, the FG mesh is generated and each element is identifiedas I/O/NIO elements by their property values.

During an ESO process, the removed element becomes an O element and thenew stiffness matrix is obtained by a simple subtraction instead of having toregenerate an entire stiffness matrix. This significantly reduces the FEAsolution time.

Modification of ESOThe original ESO method is a binary operation where an element either does ordoes not exist. However, a NIO element in FG represents a partially insideelement where its material property value is proportional to its area ratio,equation (4). Thus incorporating NIO elements in ESO allows a partial removalof elements by reducing the area ratio. This allows a finer optimisation withoutincreasing the mesh size.

This partial removal of elements is analogous to morphing ESO, whichreduces the thickness of the elements by steps until the thickness becomes zero(Querin et al., 1998). However, morphing ESO reduces the thickness of an elementwith a constant elemental area, whilst FG ESO removes a part of an element witha constant thickness. Using the definition of FG with linear elements, a topologyoptimised by FG ESO can be represented by piecewise linear boundaries and aconstant thickness. This avoids the need for and difficulty of post-processing andmanufacturing of morphing ESO topologies with numerous discretisedelemental thicknesses and jagged edges.

Interpretation of optimal topologyIn traditional FEA, the geometry of the structure is represented by a set offinite elements, while most computer aided design (CAD) systems employboundary or constructive solid geometry representations. An ESO topology istypically represented by square finite elements; therefore the boundary hasjagged edges when an optimum is found. Thus the final topology necessitatesuser interactions and post-processing of the boundary approximations in orderto represent the topology in the CAD environment and/or to manufacture thestructure. This may result in a loss of some optimal features as well as the timeand cost involved in the post-processing.

When a topology is specified, the geometry is converted to an FG meshusing I, O and NIO elements. After carrying out optimisation, the optimaltopology can be converted back to the boundary representation. As a NIOelement is defined by the material property value proportional to its area ratio,the elemental material property value can be used again to determine the

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boundary within each element. Therefore the final topology can be obtained byreversing the conversion of the boundary representation to FG process usingthe values of the elemental material properties. This eliminates the jagged-edgerepresentation of the boundaries and the topology can then be used for furtherpost-processing or manufacturing.

It is assumed that there are three NIO element shapes: where the insidematerial is rectangular; lower triangular; upper triangular. The inside shape ofa NIO element is determined by the status of its surrounding elements. Figure 3shows an example of I, O and NIO elements.

The boundary points are defined to be the mid-point of an element'sboundary. One boundary point is determined per element, and these boundarypoints define the topology. Equations (5a-c) in Figure 4 compute thecoordinates of the boundary points for respective shapes of the NIO elements.

Figure 3.Surrounding elementstatus of NIO element

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ExamplesExample 1A typical Michell type problem was optimised using FG ESO (Michell, 1904).The beam of aspect ratio 2 was defined with a point load applied downwards inthe middle and both ends fixed (Figure 5). Von Mises stress was used as theoptimisation criterion and a slow rate of a1 = 0.0001 and a0 = a2 = a3 = 0.0 ofequation (2) was applied. Owing to symmetry, the problem was solved for onlythe left half. The optimisation was carried out twice: once using classical ESOand the other using FG ESO. The computer used for all problems was aPentium 133 with 32MB RAM.

Owing to the differences in FEA solvers, the optimisation path of FG ESOdiffers from that of classical ESO. Both methods optimised the structure to

Outside

Inside

Outside

Inside

Outside

Inside

(a) Rectangular

(b) Lower Triangular

(c) Upper Triangular

(5a)

(5b)

(5c)Figure 4.

Possible inside materialorientation of a NIO

element, where � and Ae

are known

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reduce its volume to 40 per cent of the initial volume and the evolutionaryhistory of the structure's volume and stress ratios is displayed in Figure 6 andFigure 7.

FG ESO reached 40 per cent of the initial volume in 1,298 iterations andclassical ESO required only 980 iterations. However, despite the greaternumber of FEA solutions required by FG ESO, it took only one hour nineminutes and 34 seconds, while classical ESO took 53 hours 34 minutes and 46seconds, which is a significant difference in time.

The stress ratio of Figure 7 is defined as the ratio of mean Von Mises stressto maximum Von Mises stress calculated at each iteration. The mean VonMises stress is evaluated by taking an average of all elemental Von Misesstresses of the design, and the maximum Von Mises stress is the maximum ofall elemental Von Mises stresses at the given iteration. Thus the stress ratio canbe seen as a measure of evenness in the stress distribution of the design, and in

Figure 5.Initial design domain ofMichell's optimisationproblem

Figure 6.Evolutionary history ofvolume

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Figure 7 it increased as the structure was modified, optimising towards a fullystressed design. It was also noted that the stress ratio of the final FG ESOsolution was greater than that of the classical ESO with the same volume.Figure 8 depicts the optimal topology obtained by FG ESO.

Example 2Classical ESO and FG ESO were applied to optimise a cantilevered beam ofaspect ratio 1.6. The optimisation problem and the maximum design domainhave been described in Figure 9. The optimisations were again carried out on aPentium 133 with 32MB RAM.

Figure 10 was obtained by the binary operation of FG ESO where an elementis completely removed rather than partially removed by reducing �. The mesh

Figure 8.Optimal topology by

FG ESO

Figure 7.Evolutionary history of

ratio of mean Von Misesstress to maximum

stress

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size was 48 � 30, and a1 = 0.1 and a0 = a2 = a3 = 0.0 for classical ESO and FGESO solutions. The volume of the final topologies using classical ESO and FGESO were 49.44 per cent and 49.31 per cent, and the solution time was reducedby 70 per cent by using FG ESO.

The same problem with a coarser mesh of 32� 20, was also optimised usingFG ESO allowing for the partial removal of elements. The comparable topologyof Figure 11 was obtained. The topologies in Figures 10 and 11 are alike despitethe differences in the mesh sizes; however, due to the decreased mesh density,the solution time reduced further by 42 per cent. Table I summarises thesolution times of these solutions. Therefore it can be seen that the use of NIOelements enables finer optimisation with a relatively coarse mesh with thebenefit of a reduced solution time.

Figure 10.Optimal topology usingbinary FG ESO

Figure 9.Design domain of shortcantilevered beam ofaspect ratio 1.6

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Example 3FG ESO was applied to a generic aircraft floor beam optimisation problem. Thedesign domain was a beam of aspect ratio 30 where both ends were fixed by thefuselage rings. Eight loads were applied symmetrically. The top of the beamwas to support the cabin floor; thus there were to be flat top and bottom flangesof 10mm. The remaining 80mm of the web was optimised using FG ESO with amesh size of 150 � 10. Figure 12 displays the initial design domain. Again dueto symmetry, only the left half was modelled.

A very slow rate of ESO with a1 = 0.00005 and a0 = a2 = a3 = 0.0 wasapplied to obtain an optimum of 50 per cent volume reduction at iteration 5,806.When compared to the initial design of a rectangular beam, the maximum VonMises stress was increased by 7.812 per cent, while the minimum increased byover 17 times the initial minimum Von Mises stress. The optimum topology isshown in Figure 13. It can be seen that FG ESO not only optimises towards afully stressed design, but the final topology is smooth and feasible.

Figure 11.Optimal topology using

FG ESO with NIOelements

Table I.Summary of solution

times

Classical ESO,48 � 30

Binary FG ESO,48 � 30

Partial FG ESO,32 � 20

Solution time 58min 5sec 17min 31sec 10min 8secReduction relative to classical ESO ± 70 per cent 83 per cent

Figure 12.Initial design domain of

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Such a topology may be manufactured by laying up unidirectional carbonfibres in the locations shown in Figure 13, while having only resin in the`̀ holes''. Subsequent resin infusion would produce a solid cross-section.

ConclusionThis manuscript has presented the implementation of FG into ESO anddemonstrated that the combination effectively optimises structures towardsfully stressed designs, as does the original ESO technique. The primaryadvantage of FG ESO is a significant reduction in solution time as shown in theexamples. The FG uses NIO elements that represent an area smaller than anelement. Applying these elements in ESO enabled a more refined optimisationthan the traditional ESO method for the same mesh density. This reduces thesolution time even further.

A method of interpreting the FG was also proposed here and was applied toconvert the FG to the smooth boundary representation of an optimal topology.As the method is based on the definition of the FG, it reduces the use ofnumerical approximations and maximises the optimal features in the finaltopology. It is also noted here that the optimal topologies do not containcheckerboard patterns, and therefore are more feasible and manufacturablerelative to typical optimised topologies. Thus it is concluded that FG ESOimproved the ESO method by making it more practical and applicable as adesign tool.

Further research is carried out to extend the FG methodology to the fullrange of ESO capabilities including multiple load cases, thermoelasticity andbi-directional ESO.

References

Bennon, W.D. and Incropera, F.P. (1987), `̀ Evolution of macro-segregation in statistically castbindary ingots'', Metallurgical Transactions B-Process Metallurgy, Vol. 18 No. 3, pp. 611-12.

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GarcõÂa, M.J. (1999), `̀ Fixed grid finite element analysis in structural design and optimisation'',PhD thesis, University of Sydney, Australia.

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Figure 13.Left half of optimaltopology using FG ESOwith NIO elements

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