Structural Topology Optimization using Multi-objective

Genetic Algorithm with Constructive Solid Geometry

Representation

Faez Ahmed, Kalyanmoy Deb, Bishakh Bhattacharya

Department of Mechanical Engineering

Indian Institute of Technology, Kanpur

Kanpur 208016, India

KanGAL Report Number 2012018

Abstract

This paper proposes a constructive solid geometry based approach (namedCG-TOM) for structural topology optimization. The use of evolutionary al-gorithms for topology optimisation is considered ineffective due to a consider-ably large number of topological design variables. To alleviate this problem,a novel representation scheme is proposed, which encodes the topology us-ing position of few joints and width of segments connecting them. Unionof overlapping rectangular primitives is calculated using constructive solidgeometry technique to obtain the structural topology and fine triangularmesh is used to improve the accuracy of finite element compliance approx-imation. A valid topology in the design domain is ensured by representingthe topology as a connected simple graph of nodes. A graph repair operatoris applied to ensure a physically meaningful connected structure. The al-gorithm is integrated with single and multi-objective genetic algorithm andits performance is compared with those of other methods like SIMP. Thetrade-off between compliance and material availability is explored in themulti-objective analysis and common design principles for optimized solu-tions are discussed. The proposed method is generic and can be extendedto any two or three-dimensional topology optimization problem by utilizingdifferent shape primitives.

Preprint submitted to Computers and Structures November 7, 2012

Keywords: Topology Optimization, Minimum compliance, Geneticalgorithm, Multi-objective optimization

1. Introduction

Structural optimization deals with the determination of the topology,shape and size of the structures and mechanism, starting with a domain ofmaterial to which the external loads and supports are applied [16]. Struc-tural topology optimization can be considered as determination of materialconnectivity among different ports such as input, output and support ports(boundary conditions). These special ports and other material intersectionports can be termed as nodes and the topology defines the connection be-tween such nodes. The objective function is often the compliance, that is,the flexibility of the structure under the given loads, subject to a volumeconstraint. The optimum distribution of material is measured in terms ofthe overall stiffness of the structure such that the higher the stiffness themore optimal the distribution of the allotted material in the domain.

Solid isotropic material with penalisation (SIMP) [2, 18] and homogeni-sation method [1] are the two well-established approaches for topology opti-misation found in literature. Known problems with these methods are pointflexure and mesh dependency, which are often dealt with using different fil-tering techniques. Gradient-based methods such as the method of movingasymptotes (MMA) and sequential linear programming (SLP) are mostlypreferred for optimization.

There have been some attempts to use evolutionary algorithms (EAs)for this type of optimisation problem. Genetic algorithms (GAs) were usedby Jakiela et al. [11] for the optimal topology search of continuum struc-tures. The design space was discretized into small elements with all of thefinite elements forming a binary-coded bit-string chromosome, 0 and 1 forabsence and presence of an element in the structure, respectively. Tai et al.[20] utilized spline based arrangements of skeleton and flesh surrounding thebones to represent structural geometry. Other recent works in non-gradientmethods include simulated biological growth (SBG) [13], bidirectional ESO(BESO) [14] and cellular automata [3]. It was found that their search per-formances are incomparable to the gradient-based methods. This is becausemost, if not all of topology optimisation problems have a great many de-sign variables. Since the EAs somewhat base their searching strategies on

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direct search, they are not as powerful when solving such a large scale designproblem [10, 19]. Coarse mesh gives inaccurate finite element results whilemesh refinement may further increase the number of design variables. Hencea need to improve the non-gradient based EA is understood. With an im-proved searching performance, using EAs for topology optimisation would beadvantageous because of the global search and the possibility of easily deal-ing with unconventional topological design problems, which may be difficultor even impossible to be solved by using the gradient-based optimisers.

Most of the work in topology optimization is based on ISE topologies i.e.Isotropic Solid or Empty ground elements of fixed boundaries [15]. In theseany ground element is either filled completely by a given isotropic materialor contains no material. Each ground element may consist of one or severalfinite elements. Such grids not only causes large number of design variablesbut also limits formation of thin sections in the topology. In the currentwork, a different variable encoding scheme using Constructive Solid Geome-try (CSG) is proposed. CSG is a technique widely used in solid modelling. Ituses Boolean operators to combine simple objects called solids or primitives,constructed according to geometric rules, and form complex two or three di-mensional geometries. Simple shapes like rectangle, circle, ellipse or a genericpolygon can be used as a CSG primitives in 2-D. The boolean operations canbe summarized as Union, Intersection and Difference as shown in Fig. 2(a). The operations are shown on a rectangular and circular primitive in it.The idea of utilizing CSG primitives to reduce the design variables providedthe motivation for the proposed technique. In the proposed technique, CSGunion of many overlapping rectangular shaped primitives is taken to formcomplex shape segments. The material where primitives do not appear isleft out as holes.

Section 2 describes the details of the algorithm with a sample exampleof cantilever beam and section 3 proposes a graph repair operator to ensurevalidity of topology. Single and multi-objective problem formulation is donein section 4 and compliance minimization test problems are solved next insection 5. Five single objective test cases are solved to give minimum com-pliance optimized structure, while its multi-objective counterpart in section6 throws light on trade-off existing between material availability and compli-ance. Finally the concluding remarks in section 7 summarize the work anddiscuss scope of future studies.

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(a) (b)

Figure 1: (a) CSG Boolean operations on circular and rectangular 2-D primitives (b)Problem domain for cantilever with end loading.

2. Proposed Methodology

The synthesis of compliant mechanisms has traditionally been viewed asa domain with presence or absence of holes. On the contrary, we propose touse a building-block model where different segments (primitives) overlap togive shape and volume to the final topology. The key idea is to view a generictopology to be comprising of joints and rectangular segments connecting thejoints. The optimization aims to find the connectivity and optimum positionof joints along with the dimensions of segments. For a topology optimizationproblem, domain, boundary conditions (supports) and loads are specifiedinitially. The scheme is explained by an example of cantilever domain ofdimensions 120 mm × 40 mm fixed at one end and with a point load appliedat the lower right corner, as shown in Fig. 2(b). The decoding scheme canbe summarized in following steps

• Define joint positions

• Define connectivity between joints

• Ensure topology validity

• Define the shape of segments connecting joints

• Obtain topology by union of overlapping primitives

• Meshing and finite element analysis

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Initialize Population Size=N

Objective Function Calculation

Non Dominated Ranking of Population

If Generation>Maximum Generation

Selection

Crossover

Mutation

Calculation of objective function

Non Dominated Ranking of combined

Population size=2N

Select first N members on basis of rank and crowding

Replace old population by selected members

STOP

a\) Node Generation

b\) Triangulation

c\) Connectivity and Repair (if needed\)

d\) Bar allotment

e\) CSG Operations

f\) Meshing

g\) FE AnalysisSample gene decoding and evaluation

Figure 2: Flowchart of proposed GA

Figure 2 diagrammatically denotes the multi-objective optimization flowchartalong with variable decoding scheme. The scheme us explained for Fig. 2(b)problem domain.

Defining nodes. The representation scheme completely defines the structuraltopology by 2n +

(n+k2

)real variables in computational space, which are

decoded to form a topology in geometrical space. Here n is number of variablenodes chosen by user and k is number of fixed nodes. After considering theboundary condition and loads, user defines k fixed nodes, representing spatial

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locations where material must necessarily be present. The representationscheme ensures that segments connecting these fixed nodes to the topologyare always present. Hence they are generally chosen at point of applicationof loads or boundary conditions. As shown in Fig. 2(a), fixed nodes at (0,0),(120,0) and (0,40) are taken for this example. The utility of fixed nodes willbe explained in Section 3.

The n variable nodes (N1, N2, ..., Nn) represent joints in geometrical space.Hence the 2n real variables represent the nodal co-ordinates (xi, yi,∀i ∈[1, ...n]). Fig. 2 (a) shows the spatial location of three fixed and nine variablenodes for the example problem. For a rectangular domain of length L andwidth W, the variable bounds are 0 ≤ xi ≤ L, 0 ≤ yi ≤ W, and the nodeposition P y

i for node Ni is same as yi and P xi are allotted after sorting xi

This positioning of nodes ensures that in any configuration, node Ni−1 willalways be situated to the left of node Ni. A little thought will reveal thatnodes are arranged in ascending order of their tag values, variable-wise re-combination becomes meaningful in the sense that the leftmost node in oneconfiguration will mate with another leftmost node, and a node lying to theright extreme with another such node. This is essential to have meaningfulcrossover between different genes and has been found to improve convergence.

Connectivity determination. The next(n+k2

)variables represent the width

wi,j of rectangular primitives between each pair of node Ni, Nj. Here wi,j =wj,i,∀i ∈ [1, ...n], j ∈ [1, ...n], i 6= j. The connectivity between the n+k jointsis determined in two steps. First the base skeleton of possible geometry isformed and later the presence or absence of segments of the skeleton aredetermined. To find the possible connections between joints, we use trian-gulation of the nodes as shown in Fig. 2(b). In our simulation we have usedDelaunay triangulation [4] algorithm. The triangulation returns the sym-metric triangulation matrix Ci,j where 0 and 1 imply absence and presenceof connection between node Ni and node Nj respectively.

It should be noted that the variable range of widths wi,j is initially chosensuch that they are allowed to take negative values also. After determinationof the skeleton, the width values are updated as Wi,j = Ci,j × wi,j. Finallyconnections are allotted between nodes Ni and Nj, only if Wi,j > 0 to obtainthe decoded connectivity. In the example taken, suppose some of the neg-ative width connections are removed and resultant connectivity of nodes isshown in Fig. 2(c). After obtaining the decoded connectivity, the validity oftopology is checked and if necessary, repair operator is invoked. In Fig. 2(c),

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the point of application of load is not connected to main strucuture, hencethe topology repair is required in this case. The graph based repair operatorexplained in section 3 repairs this invalid topology by adding a segment toconnect it to point of load application.

Shape determination and Analysis. Having obtained the final connectivitybetween nodes, we replace the edges present, say between node Ni and Nj,by rectangular shaped bars of width Wi,j (Fig. 2(d)). The bars overlap witheach other and to give the resultant topology, union operation of all rectan-gular primitives is executed, as shown in Fig. 2(e). For a given topology,triangular meshing (using delaunay triangulation algorithm) is done usingmesh generator software (Fig. 2(f)).

Finally, the obtained mesh of topology, boundary conditions and loaddefinitions are translated as input to ABAQUS FE solver for analysis. Therequired deflection or stress output is read to calculate the objective andconstraint values. Fig. 2(g) shows the deflected geometry of example taken.The values are returned to a Genetic Algorithm for each function evaluation.

Discussion. The method defines a mapping between different topologies andreal numbers denoting positions and widths. The number of variables re-quired are small, as compared to grid based methods as only few nodes aresufficient to represent even complex topologies. The position variables areresponsible for determining the outline of structure, as the triangulation de-pends only on the position of nodes. On the other hand, the width variablesplay a dual role, determining the absence or presence of connections in thetriangulated skeleton as well as the width of the primitives connecting them.It is observed that CSG is capable of producing a model that appears visuallycomplex, but is actually little more than cleverly combined simple rectangles.CSG geometries can be used for any finite element analysis since the meshgenerator can interpret the geometry. Due to extensive usage of CSG tech-niques in game development, many mesh generators support the capabilityto mesh 2-D and 3-D geometries [17, 12].

After the union of bars, it is possible that a small portion of the ma-terial may protrude outside the box domain. Hence the CSG operation ofsubtraction is used to trim any portion outside the domain (Fig. 2(e)). Dur-ing the meshing operation, the maximum edge length of triangle element ispre-specified, hence a fine mesh with more elements in thin sections is ob-tained. This is in contrast to fixed mesh approaches which are found to give

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inaccurate results in regions where thin elements should exist. Meshing eachtopology is a computationally expensive step, which has been employed inthis work to ensure calculation accuracy. The proposed method can also beused with fixed grid meshes which is shown later in section 5.

3. Graph based Geometry Repair

The optimization algorithm initializes the variables randomly. It is pos-sible that many width variable values are zero or negative (corresponding toabsent connections), such that the resultant topology formed is inconsistent.An inconsistent topology can be formed if the point of application of load isnot connected to the topology or the topology is not connected to the point(or area) of application of boundary conditions or existence of two or moredisconnected parts.

In any such scenario, the fitness evaluation function must detect incon-sistent topology and take corrective actions before FE analysis. Otherwise,the FE solver will throw error and optimization will prematurely terminate.Rejecting or penalization of such inconsistent topologies is also not desir-able, as it will reduce the feasible population members and some genes willbe wasted. Hence we have employed a graph theory approach for anomalydetection and to take corrective action by repairing the geometry.

3.1. Repair algorithm

A brief overview of the algorithm is given below. Here the topologiesare viewed as graph of connected nodes. A group of interconnected nodesis termed as a set. As discussed before, the fixed nodes are chosen on theboundary region and point of application of load hence if the graph of topol-ogy is connected to all fixed nodes, then the resultant topology is alwaysvalid. Hence to ensure connections between load and boundary conditionsas well as no disconnected section, the algorithm aims to form a single setcontaining all the fixed nodes. In case of absence of such a set, geometry isrepaired by adding minimum possible segments to form the connected set.Other disconnected sets are ignored, hence removing hanging sections. Theworking is explained by an example below.

3.2. Repair example

To illustrate the algorithm, an example of the broken geometry of a struc-ture is derived from the example topology in Fig. 2 by randomly removing

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some connections. The inconsistent topology is shown in Fig. 3(a). Fixednode 3 at point of application of load and fixed node 1 on fixed boundaryconnection are not connected to the topology, and topology comprises ofthree disconnected parts. The graph of topology is shown in Fig. 3 (c) whileFig. 3(b) shows the node numbering on an imaginary topology, formed bytaking the absolute value of all width variables.

Initially all Sets of interconnected nodes are found as shown in Fig. 3(c).Then the sets containing the fixed nodes are found, which are Set 1, 2 and3 in the example problem. These are termed Fixed Sets. If single such setexists, then the topology is consistent and repair is not required. Usingthe initial triangulation of nodes, the connectivity between each pair of setsis found. A set ’A’ is connected to set ’B’ if any of its nodes is allowedto be connected to any of the nodes of set ’B’ after triangulation. Obtainthe graph of connectivity between sets as shown in Fig. 3(d). Weight ofconnection between two sets is the length of the shortest segment whichcan connect them. Using BFS (breadth-first search) from each fixed set inthe graph, the minimum connections required to join all the Fixed Sets isfound. Such connections between fixed sets are denoted in Fig. 3(d) byred lines. Having obtained the connectivity between all fixed sets, segmentsbetween the corresponding nodes of the sets are added. Fig. 3(e) shows thecorresponding connection between nodes to form a single connected set. Thewidth of the new connections are taken to be the absolute value of originalnegative widths between the nodes which are shown in Fig. 3(b). All othersets are ignored and the final topology is formed using the connected set.The repaired topology after CSG union and trimming operations is shown inFig. 3(f). The complexity of repair algorithm is found to be O(n+e) where nis the total number of nodes and e is the number of edges after triangulation.Hence using the repair operator, it is ensured that all genes decode to giveconsistent topologies.

4. Single and Multi-objective Optimization Aspects

We have modified the elitist non-dominated sorting genetic algorithmor NSGA-II algorithm for this purpose[8] to solve the resulting optimiza-tion problem. The GA code was integrated with MATLAB CSG tool andABAQUS FEA solver to decode real variables to geometry, mesh the geom-etry and finally after carrying the FEA analysis with prescribed conditions,reading the gene fitness from FEA output file for every function evaluation.

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(a) (b)

(c)

(d) (f)

(e)

Figure 3: Repair operation for inconsistent topologies

For details of NSGA-II, readers are encouraged to refer to the original NSGA-II study [12]. The single objective GA is derived from NSGA-II with onlyone objective, retaining other characteristics such as elitism, SBX crossover[6]and polynomial mutation[5]. The algorithm flowchart is shown in Fig. 2.

NSGA-II. First, a population is initialized randomly. Thereafter, the popu-lation members are sorted based on their non-domination level in the pop-ulation. In the so-called non-dominated sorting, the first front members arenon-dominated members of the entire population. The second front membersare those that are non-dominated population members for which first frontmembers are excluded, and so on. Each individual in a front is assigned acrowding distance based on the distance of the neighboring solutions from

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it. A solution has a large crowding distance if its nearest neighbor lies faraway from it in the objective space. In a tournament of comparing twosolutions, a solution is considered better if it lies on a better front or hasa larger crowding distance value (thereby indicating an isolated solution).After the mating pool is created, they are used to create a new populationusing crossover and mutation operator exactly the same way as they are donein the case of single-objective GA. In the NSGA-II approach, both parentand newly created populations are merged and the combined population issorted for their non-domination level again. Population members lying onthe better fronts are chosen one at a time till the new population cannotaccommodate any new front. To maintain the population size, all membersof the last front, which could not be accommodated as a whole, are comparedfor their crowding distance values – a measure of emptiness in a solution’svicinity in the objective space [12] and the ones having the largest crowdingdistance values are selected. This process is continued till the terminationcondition is met.

Volume Correction Operator. To improve the computational efficiency, a vol-ume correction operator is used In compliance minimization problems, theamount of material available to form the optimum topology is limited. HenceV/V0 ≤ η acts as a constraint (where V is topology volume, η is volume frac-tion and V0 is domain volume). Any topology with volume greater than η%is deemed as an infeasible population member. Since the optima is found onthe constraint boundary, it was observed that near convergence, most of theparent population members had volume nearly η × V0. New child membercreated with slight variation of bar width or position of node may slightlyexceed the volume limit and would become infeasible. Hence such membersare repaired to satisfy volume constraint.

After the CSG operations to form the final topology, its volume is checked.If the volume is above the constraint value (say 50%) its deviation ε fromconstraint boundary is calculated. If ε ≤ δ, each variable width is reducedby a ratio, such that the final topology satisfies the volume constraint. Inour simulations, we have taken δ = 20. The volume operator was found tobe very effective in improving convergence time as most of the topologiesformed were made to satisfy volume constraints. FEA analysis is carried outafter the volume correction step, hence saving computational effort.

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(a) (b)

Figure 4: (a) Optimized topology for cantilever problem in Example 1 (b) Rectangularbars overlapping to form the optimized solution

5. Compliance Minimization Examples

In the current section we have undertaken compliance minimization teststudies to demonstrate the performance of the proposed method. The mate-rial data for problems are E = 10 GPa and ν = 0.3, unless specified other-wise. For all the minimum compliance design problems, standard NSGA-IIevolution is used with problem dependent population size and number ofgenerations. A crossover rate of 0.9, crossover distribution index of 10; amutation rate of 0.05 and a mutation distribution index of 20 are taken, runis terminated after the prescribed maximum number of generations or if noimprovement has been obtained over large number of generations (20% oftotal generations).

5.1. Example 1

The test case considered is benchmark problem taken from [21] to compareour methods performance with other existing methods in literature. Thedesign domain and boundaries for the compliance minimization problem wasshown in Fig. 1(b) for a cantilever system. Here the domain is 120 mm ×40 mm and a point load of 100 N is applied at the end. The volume fractioninequality constraint is V/V0 ≤ 0.5 and fitness is reported in N-mm.

Three fixed nodes have been taken and nine variable nodes. Hence thetotal number of real variables are 84. A small population size of 40 members isinitialized randomly and GA is run for 200 generations. The width variablesare allowed to vary between -10 to 20 units while the range of x and y positionvariables varies from 0 to 120 and 0 to 40, respectively. Fixed nodes are takenat (0,0), (120,0) and (0,40), using Cartesian co-ordinate system with origin atlower left corner of domain. The optimized solution for benchmark test using

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Gen 1 Gen 80 Gen 160

Gen 200Gen 120Gen 40

Figure 5: Variation of fitness and volume with generations for Example 1

(a) (b)

Figure 6: (a) Solution obtained using SIMP algorithm with fitness 185.5568 (b) Optimizedsolution approximated by grid discretization with fitness 186.9

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CG-TOM has been shown in Fig. 4(a). Fig. 4(b) shows the primitive barsthat combine by CSG operations to form this optimized solution. Note thatany material outside the box domain is trimmed during function evaluationand not all nodes are utilized to obtain the solution topology.

In Fig. 5 the variation of minimum fitness of feasible population member(in every generation) is shown with generations. Mean volume of entirepopulation is also calculated for each generation and plotted. It can be seenthat although the problem has inequality constraint, the volume of entirepopulation quickly converges towards the constraint boundary. The initialsharp dip in volume is explained by the usage of volume repair operatorwhich helps the population members to remain feasible. To get a furtherinsight into the progress of evolution of structure with generations, we lookat the best population members after each interval of 40 generations in Fig.5 during the optimization process. It can be observed that GA recognizes theoptimum shape quickly (at around 40 generations) and thereafter small widthand node position variations lead it to the optimized solution. An importantquestion to ask is that although the volume upper bound was restriction,did the final solution utilize all the material available. The final solutionhas a volume fraction of 0.499 while it can be seen from the figure that theentire population average volume also converges to the constraint boundary.Further generations would result in slight improvement in fitness value andthe termination criteria is chosen depending on the problem complexity andhistory of convergence. Moreover, the mean volume of population has alsobeen more or less fixed by 200 generations and only slight improvements infitness are observed, hence the optimization is terminated.

The mean compliance value obtained using CG-TOM is 183.4 N-mm withvolume of 50%. Triangular mesh with 2416 elements is used in the final so-lution for finite element calculations. Author solved the same problem usingSigmund’s code [18] for SIMP method with penalization power 3 and filtersize 1.5. The optimized solution using SIMP is shown in 6(a) with compli-ance value 185.56, obtained after 117 iterations. In [21], Xu et. al comparesdifferent methods and filtering schemes for compliance minimization on thisdomain and reports 179.1 mean compliance as the minimum value. Thesolutions reported in it are visually similar to the solution obtained by CG-TOM. Hence, the authors show that their approach allows, for the presentedexample being a cantilever beam, to obtain the topology comparable withresults of other authors. CG-TOM used only 84 real variables in contrastto 4800 design variables used in methods like SIMP. To explore the effect

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of triangular meshing in FE calculation, the domain was discretized to 4800cells (120 × 40) and obtained solution was converted to the grid based 0-1representation as shown in 6(b). Although the volume ratio was ensured tobe exactly 0.5, such conversion can give only approximately similar structuredue to conversion of continuous design to discrete elements. Fitness value of185.1 was obtained after FE analysis of discretized design using the rectan-gular mesh, which is within margin of error to the originally computed valueusing triangular mesh.

5.2. Example 2

(a) (b)

Figure 7: Problem domain for example 2 with symmetric loading (b) Optimized solutionusing CG-TOM

The domain of second benchmark problem shown in Fig. 7(a) is takenfrom [3]. For this compliance minimization problem with volume constraintV/V0 ≤ 0.5, a symmetric loading of 100 N force is applied on a domain fixedat two corners. The domain dimensions are 80 mm × 40 mm.

The problem has symmetrical loading and boundary conditions, hence theresultant optimized geometry will also be symmetric. We utilize the symme-try information by using a symmetric representation scheme. CG-TOM ismodified to impose symmetry within variable representation by taking mir-ror image of nodes and widths about the line of symmetry. Three fixed nodeshave been taken on load and boundary positions and only four variable nodesin one half of the topology. Hence although total 11 nodes will represent the

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(a)

10 20 30 40 50 60 70 80

5

10

15

20

25

30

35

40

(b)

Figure 8: Deflection obtained using ABAQUS (b) Solution obtained using SIMP withfitness 9.8356

Gen 1 Gen 40 Gen 80

Gen 100Gen 60 Gen 20

Figure 9: Variation of fitness and volume with generations for Example 2

geometry, the position of only 4 nodes needs to be optimized. Cross connec-tions between nodes across the symmetry line are allowed, hence the totalnumber of design variables are 48 (= 8 +

(112

)−

(62

)). A population size of

40 is initialized and the GA is run for 100 generations only. We terminatethe GA run at 100 generations, as the rate of the change in objective valueis small at this generation.

The optimized solution for benchmark test using symmetrical CG-TOMhas been shown in Fig.7(b) and its deflection obtained in ABAQUS is shownin Fig.8 (a). The fitness obtained by CG-TOM is 8.26 N-mm after 100 gen-

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erations. It is notable that the optimized solution does not utilize the entireheight of domain available to form a wheel-like structure as in [3]. A mini-mum compliance of 9.83 N-mm is reported in [3] for the same problem, givinga half-wheel shaped structure. To compare the result with SIMP method,Sigmund’s 99-line code was used and minimum compliance of 9.84 was ob-tained by it after 98 iterations. The solution is shown in Fig.8(b). Thus,our solution is slightly better compared to it. Fig. 9 shows the evolutionprocess with variation of fitness and average volume of population. Due tosmall number of design variables, the basic shape of optimized structure isquickly evolved by GA within just 20 generations (800 function evaluations)and further generations are required for minor variations.

5.3. Other examples

(a) (b)

Figure 10: (a) L-shaped domain for test example 3 (b)Solution obtained using CG-TOM

Figure 10(a) shows the L-shaped domain. For this compliance minimiza-tion problem with volume constraint V/V0 ≤ 0.3, a loading of 100 N force isapplied at the end. Nine variable nodes are used and two fixed nodes havebeen taken at fixed boundary extremes along with one at point of applicationof load. Hence the total number of real variables are 84. The optimizationrequires the nodes to always lie within the L-shaped domain. For this, they-axis position variables vary between 0 to 1 and are defined relative to x-position variables. If x-co-ordinate of a node is in first half of domain, the

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corresponding nodes y-co-ordinate is obtained by scaling the y variable by 80units while if it lies in second half, the y variable is scaled by 40 units. Thisensures that, by definition all the nodes will always lie within the domainboundary. As before, CSG subtraction is used to trim any portion protrudingoutside the domain. The optimized solution, shown in Fig. 10(b) is obtainedwith a fitness of 136.9 N-mm after 200 generations.

Next minimum compliance design problem is a short cantilever plate withlength to breadth ratio of 1.5:1. The left boundary is fixed with support anda unit point force is applied vertically downward at half-height of the rightboundary and a volume fraction of 0.5 is prescribed as the volume constraint.Figure 11(a) shows this short cantilever beam domain with symmetric load-ing. Hence the resultant optimized geometry will also be symmetric aboutcentral axis parallel to x-axis. Similar to example 2 in section 5.2, symme-try is imposed by taking mirror image of nodes and widths about the lineof symmetry. Three fixed nodes have been taken at and only four variablenodes (at (0,0), (60,0) and (0,40)) with total 48 variables. Hence total 11nodes will represent the geometry, 8 position variables (for 4 variable nodes)and 40 width variables are required. A population size of 40 is initialized andGA is terminated after 100 generations. The optimized solution is shown inFig. 11(b) with fitness 32.5 N-mm. Using similar optimization conditions,the long cantilever beam (120 mm × 40 mm) with symmetric loading shownis Fig. 12(a) is also solved and the resultant solution with fitness 176.7 N-mmis shown in Fig. 12(b).

The current method provides a tool to solve topology optimization prob-lems using evolutionary methods with usage of small number of design vari-ables compared to conventional methods in literature. Despite the smallnumber of design variables used and low computational cost, performanceon standard compliance minimization problems has been shown to be com-parable to literature. Usage of larger number of design variables (more nodes)can provide more detailed optimized design but may require higher compu-tational effort to converge. Secondly, sharp edges may arise due to usageof rectangular primitives, which can be corrected in post processing. In thenext section we discuss the extension of the method to multi-objective opti-mization cases and its advantages as a design tool.

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(a) (b)

Figure 11: (a) Problem domain for short cantilever with symmetric loading (b) Solutionfor example 4 obtained using CG-TOM

6. Bi-objective Examples

In this section, we discuss the bi-objective optimization for complianceminimization problem. In practical scenario, the material availability is nota hard constraint and its usage is linked to the manufacturing cost. Henceit becomes imperative for a design engineer to know the trade-off betweenmaterial usage and objective function-in this case compliance, to make deci-sions. In single objective analysis discussed in previous chapter, fraction ofmaterial used is taken as an inequality constraint for compliance minimiza-tion problem. It has been observed that the optima lies on the constraintboundary due to conflicting nature of material used and stiffness in topology.With more amount of material available to construct the topology, the opti-mal topology is observed to possess more stiffness while less material makesthe structure flimsy and hence compliant. Hence a trade-off exists betweenvolume used and compliance value. To obtain a design for different volumefractions, currently existing topology optimization methods like SIMP, ESOmake multiple runs with different volume constraints each time. This proce-dure is time consuming and suffers with multiple disadvantages like inabilityto detect solution in non-convex region of trade-off front.

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(a) (b)

Figure 12: (a) Problem domain for long cantilever with symmetric loading (b) Solutionfor example 5 obtained using CG-TOM

Bi-objective Example 1. In this problem we undertake simultaneous mini-mization of compliance and fraction of material used for cantilever problemin benchmark test 1. A point load of 100 N is applied at corner of the beamas shown in Fig. 1(b). For simplicity, five variable nodes are taken along withthree fixed nodes. Hence total 38 design variables are used. The result is ob-tained by employing a small population size of 40 members optimized for 300generations. Using multi-objective analysis using evolutionary algorithm, thetrade-off front is obtained in a single run with solutions for different volumefractions. Figure 13 shows the obtained trade-off front. Each point on thefront denotes a design for the corresponding volume fraction on y axis. Theright extreme of the trade-off front denotes flimsy solutions while the leftextreme shows solutions with low compliance and more material usage.

Eight sample points for different volume fractions are marked on thetrade-off front and the corresponding topologies are shown in Fig. 13. So-lutions with V/V 0 ≤ 0.1 are not useful for practical designs, hence they areignored. Solutions at this extreme of trade off front are flimsy with veryhigh compliance value. Corresponding to volume fraction V/V0 = 0.5 solvedin single objective study, the solution using multi-objective run is shownin Fig. 13 by point 6 with compliance value of 195.2 N-mm. Few impor-tant inferences can be made from the solutions marked on trade-off frontusing the concepts of innovization [9]. It is seen that the skeleton of opti-mized structures are locally similar for different volume ratio. Solutions 1,2 and 3 corresponding to volume ratio from 0.1 to 0.2 have a similar shapebut variations in bar widths provide differences in volume and compliancevalues. These minimum volume solutions connect the load point from two

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1 2 3

65

7 8

4

112

3

4

5

6

7

8

Figure 13: Trade off front between compliance and volume for example 1

extreme support points and use just enough material to make the structuresafe against failure. To obtain a trade-off, the shape remains the same, butthe thickness of members are increased. Thereafter, different topologies areneeded and a better compliance is achieved with heavier structures. Even-tually, almost the complete rectangle becomes the most compliant structure.Similarity in solutions 4, 5 and 6 can also be observed while the high vol-ume solution 7 and 8 have similar skeletons. Global similarity is observed inpresence of a beak shape in all solutions on trade-off front. Such knowledgeabout generic shape of optimal structures can be further explored and usedto find optimal shape for solutions not found on the trade-off front by usingsimilar skeleton and varying the widths.

Bi-objective Example 2. In the previous example of cantilever beam optimi-sation, a small population of only 40 members was used. Hence the trade-offfront had inadequate points to denote all volume fractions. In the currentproblem we undertake simultaneous minimization of compliance and fractionof material used for Example 2 with 100 population members, keeping thetotal number of function evaluations same as before. A point load of 100 Nis applied at center of the beam as shown in Fig. 14. Symmetrical represen-tative scheme with four variable nodes of CG-TOM method has been usedsimilar to the single objective analysis.

Due to larger population used, the trade-off front obtained is dense. Com-

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1 2 3 4 5

9876

121110

Figure 14: Trade off front between compliance and volume for example 2

paring the solution with single objective result of Sec. 5.2, it is observed thatthe solution obtained by multi-objective analysis with fitness 8.57 Nmm for0.495% volume is similar to earlier single objective results of 8.26 Nmm using50 % volume obtained in section 5.2. Observations regarding generic skeletonof optimized solutions can be made in this example also. It is seen that lowvolume solutions have similar skeleton shape as seen from solutions 1, 2, 3, 4and 5 in Fig. 14. These solutions utilize only four variable nodes in skeletonformation. As shown the skeleton of higher volume fraction solutions are alsosimilar. Hence, the bi-objective example helps us to find multiple optimizedsolutions in a single run. It also provides cue about design principles amongthe optimum solutions on the trade-off front.

The obtained trade-off front comprises of a set of points in the objectivespace representing various structures. If the trade-off front has a knee-region,it is advisable to select a solution that lies in the knee region. Such a solu-tion is always preferred because deviating from the knee region means thata small change in the value of one of the objectives would come from alarge compromise in at least one other objective. A recent study suggesteda number of viable ways of identifying a knee point in a two-objective front[7]. Although a knee region may not exist in all trade-off frontiers, a visualinspection of trade-off fronts shown in Figs. 13 and 14 reveal that in gen-eral a compliance-material trade-off front exhibits a knee in most conditions.Applying the knee-finding methods on our trade-off front shown in Figure14, we observe that solution 8 lies near the knee region. It shows a rea-sonable compromise between volume and compliance as compared to other

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solutions. Hence such decision making approaches can be used to select thefinal structure from the obtained trade-off front.

7. Concluding Remarks

The current work proposed a new representation scheme for topology op-timization. Graph based repair operator and volume operator are proposedto improve computational performance by ensuring feasible topologies. Theproposed approach (CG-TOM) uses union of simple rectangular primitivesfor compliance minimization problem test. Single and multi-objective ge-netic algorithm is integrated with the approach for optimisation. Resultson standard test problems are found to be comparable to existing methodslike SIMP. Bi-objective optimization results for simultaneous minimization ofcompliance and material availability is shown to give a trade-off front, reveal-ing interesting design principles among solutions. The current work opensup many possibilities of usage of CSG techniques for topology optimizationand inverse finite element problems by integrating CSG modelling techniquewith optimization. Further research will focus on post-processing to obtainsmooth topologies and proposing more efficient GA algorithms to give opti-mized solutions within a low yet finite budget of function evaluations.

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