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Structural topology optimization using ant colony optimization algorithm
Applied Soft Computing 9 (2009) 1343–1353
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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
Structural topology optimization using ant colony optimization algorithm Guan-Chun Luh *, Chun-Yi Lin Department of Mechanical Engineering, Tatung University, Taipei 104, Taiwan, ROC
A R T I C L E I N F O
A B S T R A C T
Article history: Received 10 February 2008 Received in revised form 14 February 2009 Accepted 1 June 2009 Available online 10 June 2009
The ant colony optimization (ACO) algorithm, a relatively recent bio-inspired approach to solve combinatorial optimization problems mimicking the behavior of real ant colonies, is applied to problems of continuum structural topology design. An overview of the ACO algorithm is first described. A discretized topology design representation and the method for mapping ant’s trail into this representation are then detailed. Subsequently, a modified ACO algorithm with elitist ants, niche strategy and memory of multiple colonies is illustrated. Several well-studied examples from structural topology optimization problems of minimum weight and minimum compliance are used to demonstrate its efficiency and versatility. The results indicate the effectiveness of the proposed algorithm and its ability to find families of multi-modal optimal design. ß 2009 Elsevier B.V. All rights reserved.
Keywords: Ant colony optimization algorithm Continuum structural topology optimization Elitist ants Niche strategy Multiple colonies
1. Introduction The structure optimum design is a very interesting and important topic in the field of engineering optimization [33]. Structural optimization problems focus on minimizing the amount of material required in a defined domain for a specified loading subject to some constraints. The optimal design of structures including sizing, shape and topology forms the basic issues for a structural design process [18,25] as illustrated in Fig. 1(a)–(c). In sizing optimization (Fig. 1(a)), the parameterized shape and topology are assumed to be fixed while an optimal set of sizing parameters such as length, thickness, cross-sectional area are found. Nevertheless, the optimal design of a sizing optimization is only the best design from the predetermined structural geometric definition. Extending the defining ability on curve lines and surfaces, shape optimization (Fig. 1(b)) seeks the geometric definitions of the boundaries of outer circumference and inner holes of the structure. In general, the locations of key points defining a curve or parameters defining a specified predetermined geometric shape are the design variables. Shape optimization adds great flexibility to sizing optimization; however, the configuration of the structure remains unchanged. The optimum structure derived from sizing and shape optimization is only a result of the initial topology design. In other words, the last structure from sizing and shape optimization may not be the true optimum structure if the initial topology design is not an optimal one. The
* Corresponding author. Tel.: +886 2 25925252x3410/806; fax: +886 2 25997142. E-mail address: [email protected] (G.-C. Luh). 1568-4946/$ – see front matter ß 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2009.06.001
fact that finding a good structural configuration before the shape and sizing optimization is an important but difficult task stresses the requirement for topology optimization. In comparison to shape optimization, topology optimization (Fig. 1(c)) is an order of magnitude more complex since it involves the optimization of both the external boundary and distribution of the internal material within a structure. In other words, holes in the interior of structure can be created. Topology optimization is utilized to find a preliminary structural configuration that meets a predefined criterion. Occasionally it gives a design that can be completely new and innovative. Continuum structural topology optimization is one of the most challenging research topics in the field of structural optimization which aims to find the best possible structure that meets different multidisciplinary requirements [2]. It has received extensive attention and experienced considerable progress recently due to its great potential of application in many industrial areas. In the past decades numerous innovative approaches to topology optimization have been developed [6,15]. The domain variation, also termed sensitivity analysis, is the first approach proposed by Kibsgaard [24] for topological optimization. Based on the computation of the gradient of the objective function with respect to the domain, it consists of successive small variations of the initial design domain. However, this approach has two major defects: first, it requires a good initial guess, as it demonstrated to be unstable for large variations of the domain; second, it does not allow modification of the initial domain topology (e.g. add or remove holes). An alternative popular method, the homogenization method [28] first proposed by Bendsøe and Kikuchi [5] consists in dealing with a continuous density of material. In the end
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Fig. 1. Sizing, shape, and topology optimization for continuum structural design problems [17].
of this scheme, the final density is forced toward value 1 or 0 (material present or absent). Nevertheless, this approach requires the design of the homogenized operator, as thoroughly described by Allaire and Kohn [3], and is insofar limited to the linear elasticity case. In addition, it cannot address loadings that apply on the actual boundary of the shape to be determined, and hardly handles optimization for multiple loadings [21]. Another recognized family of structural optimization approach named evolutionary structural optimization (ESO) method has been developed by Xie and Steven [42]. The primary concept of ESO approach is to gradually remove lowly stressed elements from the structure after each finite element (FE) analysis. In addition, the element removal criterion is established by sensitivity analysis. Hence, the topology of the resulting design is gradually improved to achieve the optimal design. A fundamental potential drawback of this method pointed out by Liu et al. [29] is the strong dependence of the solution on the mesh of finite element from which it is evolved and on the sequence of the element removal. In addition, it may also easily lead to a non-optimal design [43] and produce truss-like topologies [37]. Although the capability to add or reinstate elements has recently been added to the ESO through the bidirectional evolutionary structural optimization (BESO) method [32], it is still restricted to previous element positions or to the area/volume predefined by the mesh of finite element. A possible approach to overcome those difficulties of topological optimization mentioned above is to adopt bio-inspired computation methods imitating natural phenomena and physical processes. Among these include simulated annealing, genetic algorithm (GA), and immune algorithm. In this study, ant colony optimization (ACO) algorithm [11–13], another recognized family of the biologically inspired computation methods, is implemented to solve the topology optimization of structure. The rest of the paper is organized as follows. Section 2 reviews the literature on the bio-inspired computation-based structural topology optimization. The detail of the ant colony optimization algorithm is introduced in Section 3. Section 4 describes the application of the modified ACO algorithm to the structural topology problem. The results of several well-known topology optimization problems from literature to evaluate the performance of the proposed algorithm are given in Section 5. Finally, the paper concludes in Section 6. 2. Literature on bio-inspired computation-based structural topology optimization Several bio-inspired computation methods have been utilized for structural topology optimization. Shim and Manoocheer [36] developed a combinatorial optimization procedure based on the simulated annealing approach for structure optimal configuration design. The configuration of the finite element structural was altered by removing or restoring elements to minimize the volume subject to maximum allowable stress constraints. The simulated
annealing method searches for the best configuration based on a statistical analysis of the cost distribution. A comprehensive review of the applications of evolutionary computation (EC) in structural design is given and chronologically classified by Kicinger et al. [25]. They introduced the field of evolutionary design and its relevance to structure design. Further, they discussed the issue of creativity/novelty and suggested possible ways of achieving it during a structural design process. The EC approach to the continuum topology optimization design problem based on genetic algorithm has been developed by Sandgren et al. [34] and Jensen [20]. In their work, the design domain was discretized into small elements containing materials or voids in a cantilever plate so that the structure’s weight was minimized subject to displacement and/or stress constraints. Subsequently, a lot of researchers have extensively employed genetic algorithm based methods for structural optimization in the optimal design of topology. Chapman and the associated researchers [8,9,14,18] present summary examples of the GA-based approach to topological optimization. A variety of different structural design fitness functions including stiffness, area, perimeter, and hole are employed to find optimal cantilevered plate topologies. Additionally, fitness sharing, restricted mating, and cluster analysis techniques are implemented for obtaining families of highly fit topologies. Based on graph theory [38], a valid topology is represented by a connected simple graph consisting of vertices and simple undirected cubic Be´zier curves with varying thickness. The derived results show that the graph representation GA can generate clearly defined and distinct geometries and perform a global search with more computational cost. Afterward, a bit-array representation GA [39] was implemented for topology optimization. The design connectivity and constraint handling are further developed to improve the efficiency of the GA. In addition, a violation penalty method is proposed to drive the GA search towards the topologies with higher structural performance, less unusable material and fewer separate objects in the design domain. Further, a multi-GA system [41] and variable chromosome length genetic algorithm [26] were proposed for continuum structures topological optimization. Recently, a two-stage adaptive genetic algorithm (TSAGA) [4] was developed in bit-array represented topology optimization. Compared with other approaches, the authors demonstrate the efficiency and effectiveness of TSAGA in reaching the global optimal solutions on several case problems. Also, Hamda et al. [16] and Aguilar [1] considered a continuum topology optimization problem as a multi-objective problem employing genetic algorithm. Finally, a biological immunity-based optimization approach [30] has been implemented to overcome the particular drawbacks, lack of local search ability and premature convergence, implicit in genetic algorithms in recent times. The authors apply multi-modal immune algorithm (MMIA) for structure topology optimal design. Two well-studied benchmark examples in structural topology optimization problems are used to evaluate the proposed approach. The results indicate the effectiveness of MMIA. The ACO is a relatively recent bio-inspired approach to solve combinatorial optimization problems mimicking the behavior of real ant colonies. It has been successfully applied to solve numerous optimization problems [10,19]. Ant colony optimization algorithms are good candidates for structural topology optimization due to their discrete feature similar to genetic algorithm and immune algorithm mentioned previously. Nevertheless, only a few of researchers have employed ant algorithms for truss structure optimization [7,22,31,35]. Rarely utilizes ant algorithm to optimum design of continuum structural topology. Until recently, Kaveh et al. [23] applied ACO and finite element analysis in topology optimization of two- and three-dimensional structure models. Based on the element’s contribution to the strain energy,
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the authors find the stiffest structure with a certain amount of material (defined solid material fraction). In addition, a noise cleaning technique was used to prevent creation of undesirable tiny members in the resulted optimum layout. Four examples were presented in their study to show the capability and efficiency of the proposed method. 3. Ant colony optimization algorithm Ant colony optimization algorithms are a recently developed, population-based methodology applied to numerous NP-hard combinatorial optimization problems. They have been inspired by the behavior of real ant colonies especially by their foraging behavior. One of the main features of ant algorithms is the indirect communication of a colony of (artificial) ants, based on pheromone trails that are a kind of distributed numeric information used to reflect their experience while solving a particular optimization problem. Ant system (AS), the first ACO algorithm was proposed by Droigo [11,12] to solve the traveling salesman problem (TSP). Each ant in AS builds up a solution step-by-step employing transition rule until a solution is found. Ants that found a good solution mark their paths by laying some amount of pheromone on the edge of the path. The following ants are then attracted by the pheromone so that they will be able to search in the solution space near good solutions. The traveling salesman problem is expressed as a graph GhN,Ei where N denotes the set of cities and E represents the set of edges between cities. The objective is to find the minimal length closed tour that visits each city once. Each ant is a simple agent to fulfill the task and obeys the following rules: It lives in a discrete-time environment. It chooses the next city associated with a probability which is a function of pheromone laid on the connecting edge and of the amount of the visible distance. It cannot choose the cities which has been visited before a tour is completed. It lays pheromone on each edge visited when a tour is completed. The steps for ant system simulation imitating real ant’s behavior are described as follows. 3.1. Initialization The initialization of the AS algorithm consists of the graph problem representation and the ants’ initial distribution. The optimization problem should first be represented in terms of a graph form GhN,Ei, where N is the set of nodes/cities and E is the set of connections/edges between nodes. Subsequently, a number of ants are randomly placed on the nodes and each of them will perform a tour according to the node transition rule described below. In general, the initial pheromones laid on the connections are set to small constant values. 3.2. Node transition rule The ants move from node to node following a node transition rule, which is problem dependent. The transition probability pkij ðtÞ from node i to node j for the kth ant is defined as
pkij ðtÞ
¼
8 > < > :
P 0
½t i j ðtÞa ½hi j b
t
a
h 2 tabuk ½ ih ðtÞ
½hih
b
if j 2 tabuk
(1)
otherwise
where tabuk is used to define the set of cities that the kth ant located on city i still has to visit. When the kth ant finishes a cycle, the tabu list tabuk is then emptied and the ant is free again to start
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another tour. By exploiting the memory therefore an ant k can build feasible solutions by an implicit state-space graph generation, which corresponds to visiting a city exactly once. tij and hij represent the concentration of pheromone laid on connection (i, j) and the value of visibility of connection (i, j), respectively. For the TSP hij = 1/dij, and dij denotes the distance between the ith and the jth cities. Parameters a and b control the relative importance between the intensity of pheromone versus visibility. Dorigo [11] have investigated the behavior of the ant-cycle algorithm for different combination of a and b. Three different classes were resulted: bad solutions and stagnation, bad solutions and no stagnation, and good solutions. The intensity of pheromone reflects the previous experience of the ants and provides the indirect communication between ants. Pheromone information provides a global view about the goodness of the solution found while the value of visibility offers the local information on edge (i, j) determined by the greedy heuristic for the original problem. 3.3. Pheromone updating rule Let na and n be the number of ants and cities, respectively. An iteration of the AS algorithm means all the na ants carry their next movements during discrete time interval (t, t + 1) and thus n iterations constitute a cycle. Therefore, each ant will complete a tour in one cycle. The trail intensity is updated according to the formula:
t i j ðt þ nÞ ¼ ð1 rÞt i j ðtÞ þ Dt i j
(2)
where tij(t) denotes the intensity of pheromone on edge (i, j) at time t and r represents the evaporation of trail in time interval [t, t + n]. The main purpose of pheromone evaporation is to avoid stagnation, that is, the situation in which all ants end up doing the same tour. Dt kij denotes the quantity of trail substance laid on edge (i, j) by the kth ant between time t and t + n.
Dt i j ¼
na X
Dt kij
(3)
k¼1
Three AS algorithms have been defined according to the way pheromone updated: ant-density, ant-quantity, and ant-cycle. In the ant-cycle model: 8 Q > > if thekth ant uses edgeði; jÞ < Lk k (4) Dt i j ¼ in its tour between timet andt þ n > > : 0 otherwise where Q is a constant and Lk is the tour length of the kth ant. In general, the intensity of trail at time 0, i.e. tij(0), is set to a small positive constant. As to the ant-density model: ( Q if thekth ant goes fromi to j between timet andt þ 1 k Dt i j ¼ 0 otherwise (5) Finally, in 8 Q > > < di j k Dt i j ¼ > > : 0
the ant quantity model: if thekth ant goes fromi to j in its tour between timet andt þ 1
(6)
otherwise
where dij is the distance between node i and node j. In ant-density and ant-quantity, ants deposit pheromone while building a solution. On the contrary, ants deposit pheromone after they have
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Fig. 2. Mapping from design domain to topology and element position.
built a complete tour in ant-cycle case. Accordingly, research on AS was directed towards a better understanding of the characteristics of ant-cycle, which is now known as ant system, while the other two algorithms were abandoned. In this study, ACO algorithm was adopted and modified to solve structural topology optimization described in the following section.
4.2. Element transition rule Instead of node transition rule employed in ant system, the ants move from element to element following an ‘‘element transition rule’’. The transition probability from element E(x, y) to element E(i, j) (i.e. one of the eight neighboring elements E(x 1, y 1), E(x 1, y), E(x 1, y + 1), E(x, y 1), E(x, y + 1), E(x + 1, y 1), E(x + 1, y), E(x + 1, y + 1) shown in Fig. 3) for the kth ant is defined as
4. Continuum structural topology optimization using ACO algorithm pkij ðtÞ Corresponding to the 2D topological optimization problems, a two-dimensional design is discretized into small, square elements (E(x, y), x = 1, . . . ,X; y = 1, . . . ,Y) where each element represents either material (with code value of 1) or void (with code value of 0) as Fig. 2 illustrated. This binary, material-void design domain results in a discrete non-convex [36] search space and allows for a precise discretized topology boundary. A population of ants is then placed randomly in elements of the design domain. The elements where each ant has visited will be marked (material). The states of the individual elements define the distribution of material and void within the design domain. After an ant finished its tour, all the connecting elements (material) between loading regions and supporting regions of the structure establish a valid topology, physically meaningful connected structure. The resulting topology’s trail intensity is determined according pheromone updating rule and thus a finite element analysis is performed on the topology. The modified ACO algorithm for structure topology optimization proposed in this study follows the following steps.
¼
8 > < > :
P 0
½t i j ðtÞa ½s i j ðtÞb
t
h2 = tabuk ½ ih ðtÞ
a
½s ih ðtÞb
if j 2 = tabuk
(7)
otherwise
where tabuk is used to define the element has been visited for the kth ant in the preceding stage at time instant t 1. In the foraging process, each ant is not allowed to go back to the element most recent time it has visited. The purpose is to avoid to moving and fro between two elements. Therefore, the set of elements that have been visited by ants in the preceding status is stored in the tabu list. tij(t) and sij(t) represent the current concentration of pheromone laid on elements E(i, j) and the normalized stress value of elements E(i, j), respectively. In this study, a special scheme is proposed to choose the visited element E(i, j) for the kth ant according to the transition probability pkij ðtÞ described as following. The seven transition probability values of E(i, j) (except the one stored in tabu list) are normalized and sorted descendent from maximum to minimum. Afterward, a uniform random number ranged between [0, 1] is generated and compared with the maximal transition probability. If the random number is smaller, the kth ant will visit the element E(i, j) with the maximal
4.1. Initialization A population of na ants is placed randomly in elements of the discretized design domain. Then each ant located at E(x, y) will move to one of the eight neighboring elements shown in Fig. 3 according to ‘‘element transition rule’’ instead of ‘‘node transition rule’’ employed in ACO algorithm. Different from the fourneighborhood connectivity used in [40], this study adopts eightneighborhood connectivity scheme due to its diverse property. The elements where the ant has visited will be marked (material). Moreover, the initial pheromone intensity at each element E(x, y) is assigned according to its normalized stress value assumed all elements are with materials rather than a small constant value adopted in ACO. Fig. 4 demonstrates a short cantilever plate subjected to a downward concentrated loading applied at a finite element node on its right hand center side and the associated normalized stress graph. The value of the normalized stress on each element will be assigned as initial pheromone intensity.
Fig. 3. Eight-neighborhood connectivity scheme and moving directions for ant.
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structure, and the corresponding element set to a binary value of 0 (void). 4.4. Pheromone updating rule The trail intensity at each element E(i, j) is updated according to the following formula:
t i j ðt þ nÞ ¼ ð1 rÞt i j ðtÞ þ Dt i j þ
eQ Ob j
(8)
where e indicates the number of elitist ants, Q is a positive constant value, and Obj*is the best objective value of solution/topology found from the beginning of the trail. In an effort to improve performance, ‘‘elitist ants’’ (similar to the elitist strategy used in genetic algorithm) introduced by Dorigo et al. [12] is included in the pheromone updating rule.
Dt i j ¼
na X
Dt kij
(9)
k¼1
and
Fig. 4. Cantilever plate and its associated stress/initial pheromone intensity distribution.
transition probability. Otherwise, another random number will be generated and compared with the second rank transition probability. The kth ant will visit this element E(i, j) if its transition probability value is bigger than the random number. Otherwise, similar procedure will be continuous for the next rank transition probability until all the remaining elements are checked. If the kth ant does not visit any element in the final stage, Roulette Wheel method will be adopted to select the element the ant should approach. 4.3. Connectivity analysis The resulting topology (mapping the elements visited by the kth ant into the design domain) may form a disconnected checkerboard due to the eight-neighborhood connectivity scheme utilized in this study. For any two elements in a topology to be considered as connected they must share at least one edge while element sharing only one corner are considered as disconnected. Elements connected merely at a corner cannot withstand applied torques and thus could lead to an unstable structure which cannot support various loads. A topology contained disconnected elements requires to undergo a structure modification procedure. In this procedure, the removal of disconnected elements or the adding of elements to neighboring disconnected element will be done randomly until the discontinuous structure is compensated. The continuous topology will be further analyzed via the finite element computation to obtain the required displacements and stresses. To reduce computation time, elements with a stress value lower than the user-defined level of average stress (which do not break continuity requirements) will be removed from the
Dt kij ¼
Q Ob jkNiche
where Ob jkNiche denotes the niche objective value of the kth ant/ topology solution, and Q is a user-defined constant value. In topology optimization, usually there exist many solutions such as one global and many local minima to a given problem [43]. Moreover Chapman and the associated researchers [8,9,14] have demonstrated the similar concept of ‘‘design families’’. In other words, structural topology optimization may provide the designer with a family of possibly-optimal designs and the designer can then evaluate the designs to determine which best satisfies several performance criteria, much like a pareto optimization study. Consequently niche strategy is utilized in this study to find the multiple solutions since topology optimization of structure seems to a kind of multi-modal problem [30]. Sharing scheme is thus employed to calculate the similarity for each ant/topology and described below: Ob jkNiche ¼ and
Ob jk ð1 þ penaltyk Þ SC k
Ob jk ¼
(10)
1 dmax Areak k
where Objk denotes the original objective value of the kth ant/ topology solution. Referring to [8,9,14,18], Objk indicates the kth ant’s/topology’s stiffness-to-weight ratio and stiffness is presented by inverse of topology’s maximum displacement ð1=dmax Þ at the k point of loading application. To determine dmax , a finite element k analysis is performed on the topology. dmax was set equal to the k magnitude of the displacement of the node where the point load was applied for the kth ant/topology. It should be noted that the number of connected material element of topology is used as a qualitative measure of topology’s weight (Areak). Moreover, SCk represents the similarity count of the kth ant/topology with all other ants/topologies and is expressed as SC k ¼
na X countk j ;
k ¼ 1; 2; . . . ; na ; j ¼ 1; 2; . . . ; na
(11)
j¼1
with
1; if ant k j < Th count k j ¼ 0; else qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ant k j ¼ ðastress astress Þ ðstdstress stdstress Þ j j k k
where Th is an user-defined threshold value illustrating the allowable difference between ants/topologies, antkj representing
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distance in average-standard deviation stress space indicates a relationship between the kth and the jth ant/topology. Note that astress and stdstress are normalized average and normalized standard k k deviation of stress of the kth ant/topology. Furthermore, penaltyk of the kth topology is treated as the penalty term for the stress violation and expressed as follows: NC NC X X penaltyk ¼ amount kj countkj j¼1
with
amount kj ¼ countkj ¼
(12)
j¼1
8 <
js kj j
was predefined and the solutions saved in memories will be replaced by the best solution found currently if it satisfies the following conditions: k Ob jk > Ob jmemory \ SCmemory d
where Objk and Objmemory denote the objective value of the kth ant/ topology and that of the ants/topologies saved in memories, k respectively. The variable SCmemory represents their associated similarities and d is a predefined similarity threshold. 4.6. Stopping criteria
1
if
js kj j > s allowable
: s allowable 0 else 1 if js kj j > s allowable 0 else
where penaltyk represents the penalty value for the kth ant/ topology; Nc is the number of material elements for the kth ant/ topology; amount kj and count kj correspond to the normalized values of the summation and total number of the jth element stress violation, respectively; s kj denotes the jth element stress value of the kth ant/topology whereas sallowable indicates the maximal allowable stress. 4.5. Memories of multiple colonies In this study, multiple-colony memories were employed to save the multi-modal topologies/ants found. The number of memories
Whenever the kth ant finishes visiting all the ‘‘seed’’ elements, it completes a tour and stops. A ‘‘seed’’ element [8,18] is an element that is required to contain material so that it may serve as a support boundary condition or point of load application. Consequently, the trail of the ant constitutes a structural topology. 5. Simulation results and discussions Four topological optimization examples were utilized to evaluate the effectiveness and performance of the proposed ACO algorithm. All the mechanical model and material properties are tabulated in Table 1. These examples describe the optimization of a cantilever plate (with aspect ratio 1.6) subject a downward concentrated load 3 kN applied at an FE node on its right hand center [17,27,29], top [30], bottom [17,30] and 2/5 of the distance from the bottom [8,9,18], respectively. In addition, nodes on the
Fig. 5. Simulation window for case 1.
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Table 1 Illustration of topological optimization examples. Case 1 [17,27,29]
Case 2 [30]
Case 3 [18]
Case 4 [9,18,30]
Loading position
FE node on the mid-point of the right side
FE node on the top-point of the right side
FE node on the bottom-point of the right side
FE node on the right hand surface 2/5 of the distance from the bottom
Fixed position Material properties
Left-top and left-bottom Young’s modulus E = 200 Gpa, Poisson’s ratio v ¼ 0:33, allowable stress sallowable = 200 Mpa and thickness t = 0.001 m
Design domain
Fig. 6. Selected topologies form 10 ACO runs with cluster scheme of case 1.
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left-hand surface corresponding to the support points were defined to have zero displacement in the FE analysis. Moreover, the design domain is discretized according to a 20 32 plane stress element FE model. The goal of optimization was to minimize the stiffness-to-weight ratio described in the previous section. To carry out these computations, a computer program was developed with C++ programming tools and a graphical user interface. Fig. 5
illustrates the running process of simulation window for case 1, its setting parameters, and part of the resulting topologies saved in memory pool. One execution of the computer model requires around 36,000 functional evaluations (60 ants by 60 iterations and 10 different runs), taking approximately 120 min with a Pentium 4 processor running at 1.5 GHz. Table 2 lists the associated parameters utilized in the proposed ACO algorithm. These
Fig. 7. Selected topologies form 10 ACO runs with cluster scheme of case 2.
G.-C. Luh, C.-Y. Lin / Applied Soft Computing 9 (2009) 1343–1353 Table 2 Parameters employed in ACO algorithm. Number of ants na Number of iterations N Run time Number of elitist ants e Pheromone decay factor r Similarity threshold d Threshold value Th Q
a b
60 60 10 4 0.05 0.75 0.1 4.0 1 1
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parameters were determined through numerical experiments after multiple simulation runs. Figs. 6–9 demonstrate a family of topologies and their corresponding weights (Area) for each case. These typical topologies are part of the results selected from the combination of 10 ACO runs with a maximum displacement less than 1.0 mm and a cluster threshold value of 0.75. All the topologies have welldefined, solid-material outer boundaries. Moreover, the interior regions have a ‘composite-like’ internal structure comprised of equally distributed material and void. As can be seen from the diverse range of resultant topologies shown in these figures, the
Fig. 8. Selected topologies form 10 ACO runs with cluster scheme of case 3.
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Fig. 9. Selected topologies form 10 ACO runs with cluster scheme of case 4.
structures therefore provide the designer with a set of possible solutions. The designer can then evaluate the designs to determine which topology is best for a particular application. Part of the structures show very well-defined truss-like members of constant cross-sectional area with large voids between members. The designer may choose these truss-like topologies (i.e. large voids between members) if manufacturability is the prime consideration. Compared with the results derived in the other studies [8,9,17,18,27,29,30], the proposed ACO algorithm generates much more diverse and interesting structures. These topologies could then possibly serve as the initial seed for a hierarchical subdivision-based optimization at finer discretizations or a preliminary structural configuration for shape and/or sizing optimization.
6. Conclusions In this study, a novel concept for handling multi-modal topological optimization has been presented by using a modified ACO algorithm mimicking the behavior of real ant colonies. Elitist ants were employed to increase the performance while memories of multiple colonies were implemented to save the family of designs. In addition, visibility of connection is not included in the calculation of transition probability. The potential of the proposed algorithm as a tool for investigating optimal topologies and for automatically creating innovative solutions to structural design problems has been illustrated in the examples presented. The results show that diverse topology structures could be derived using ACO algorithm and part of the structures show very well defined truss-like members.
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Contents lists available at ScienceDirect
Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
Structural topology optimization using ant colony optimization algorithm Guan-Chun Luh *, Chun-Yi Lin Department of Mechanical Engineering, Tatung University, Taipei 104, Taiwan, ROC
A R T I C L E I N F O
A B S T R A C T
Article history: Received 10 February 2008 Received in revised form 14 February 2009 Accepted 1 June 2009 Available online 10 June 2009
The ant colony optimization (ACO) algorithm, a relatively recent bio-inspired approach to solve combinatorial optimization problems mimicking the behavior of real ant colonies, is applied to problems of continuum structural topology design. An overview of the ACO algorithm is first described. A discretized topology design representation and the method for mapping ant’s trail into this representation are then detailed. Subsequently, a modified ACO algorithm with elitist ants, niche strategy and memory of multiple colonies is illustrated. Several well-studied examples from structural topology optimization problems of minimum weight and minimum compliance are used to demonstrate its efficiency and versatility. The results indicate the effectiveness of the proposed algorithm and its ability to find families of multi-modal optimal design. ß 2009 Elsevier B.V. All rights reserved.
Keywords: Ant colony optimization algorithm Continuum structural topology optimization Elitist ants Niche strategy Multiple colonies
1. Introduction The structure optimum design is a very interesting and important topic in the field of engineering optimization [33]. Structural optimization problems focus on minimizing the amount of material required in a defined domain for a specified loading subject to some constraints. The optimal design of structures including sizing, shape and topology forms the basic issues for a structural design process [18,25] as illustrated in Fig. 1(a)–(c). In sizing optimization (Fig. 1(a)), the parameterized shape and topology are assumed to be fixed while an optimal set of sizing parameters such as length, thickness, cross-sectional area are found. Nevertheless, the optimal design of a sizing optimization is only the best design from the predetermined structural geometric definition. Extending the defining ability on curve lines and surfaces, shape optimization (Fig. 1(b)) seeks the geometric definitions of the boundaries of outer circumference and inner holes of the structure. In general, the locations of key points defining a curve or parameters defining a specified predetermined geometric shape are the design variables. Shape optimization adds great flexibility to sizing optimization; however, the configuration of the structure remains unchanged. The optimum structure derived from sizing and shape optimization is only a result of the initial topology design. In other words, the last structure from sizing and shape optimization may not be the true optimum structure if the initial topology design is not an optimal one. The
* Corresponding author. Tel.: +886 2 25925252x3410/806; fax: +886 2 25997142. E-mail address: [email protected] (G.-C. Luh). 1568-4946/$ – see front matter ß 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2009.06.001
fact that finding a good structural configuration before the shape and sizing optimization is an important but difficult task stresses the requirement for topology optimization. In comparison to shape optimization, topology optimization (Fig. 1(c)) is an order of magnitude more complex since it involves the optimization of both the external boundary and distribution of the internal material within a structure. In other words, holes in the interior of structure can be created. Topology optimization is utilized to find a preliminary structural configuration that meets a predefined criterion. Occasionally it gives a design that can be completely new and innovative. Continuum structural topology optimization is one of the most challenging research topics in the field of structural optimization which aims to find the best possible structure that meets different multidisciplinary requirements [2]. It has received extensive attention and experienced considerable progress recently due to its great potential of application in many industrial areas. In the past decades numerous innovative approaches to topology optimization have been developed [6,15]. The domain variation, also termed sensitivity analysis, is the first approach proposed by Kibsgaard [24] for topological optimization. Based on the computation of the gradient of the objective function with respect to the domain, it consists of successive small variations of the initial design domain. However, this approach has two major defects: first, it requires a good initial guess, as it demonstrated to be unstable for large variations of the domain; second, it does not allow modification of the initial domain topology (e.g. add or remove holes). An alternative popular method, the homogenization method [28] first proposed by Bendsøe and Kikuchi [5] consists in dealing with a continuous density of material. In the end
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Fig. 1. Sizing, shape, and topology optimization for continuum structural design problems [17].
of this scheme, the final density is forced toward value 1 or 0 (material present or absent). Nevertheless, this approach requires the design of the homogenized operator, as thoroughly described by Allaire and Kohn [3], and is insofar limited to the linear elasticity case. In addition, it cannot address loadings that apply on the actual boundary of the shape to be determined, and hardly handles optimization for multiple loadings [21]. Another recognized family of structural optimization approach named evolutionary structural optimization (ESO) method has been developed by Xie and Steven [42]. The primary concept of ESO approach is to gradually remove lowly stressed elements from the structure after each finite element (FE) analysis. In addition, the element removal criterion is established by sensitivity analysis. Hence, the topology of the resulting design is gradually improved to achieve the optimal design. A fundamental potential drawback of this method pointed out by Liu et al. [29] is the strong dependence of the solution on the mesh of finite element from which it is evolved and on the sequence of the element removal. In addition, it may also easily lead to a non-optimal design [43] and produce truss-like topologies [37]. Although the capability to add or reinstate elements has recently been added to the ESO through the bidirectional evolutionary structural optimization (BESO) method [32], it is still restricted to previous element positions or to the area/volume predefined by the mesh of finite element. A possible approach to overcome those difficulties of topological optimization mentioned above is to adopt bio-inspired computation methods imitating natural phenomena and physical processes. Among these include simulated annealing, genetic algorithm (GA), and immune algorithm. In this study, ant colony optimization (ACO) algorithm [11–13], another recognized family of the biologically inspired computation methods, is implemented to solve the topology optimization of structure. The rest of the paper is organized as follows. Section 2 reviews the literature on the bio-inspired computation-based structural topology optimization. The detail of the ant colony optimization algorithm is introduced in Section 3. Section 4 describes the application of the modified ACO algorithm to the structural topology problem. The results of several well-known topology optimization problems from literature to evaluate the performance of the proposed algorithm are given in Section 5. Finally, the paper concludes in Section 6. 2. Literature on bio-inspired computation-based structural topology optimization Several bio-inspired computation methods have been utilized for structural topology optimization. Shim and Manoocheer [36] developed a combinatorial optimization procedure based on the simulated annealing approach for structure optimal configuration design. The configuration of the finite element structural was altered by removing or restoring elements to minimize the volume subject to maximum allowable stress constraints. The simulated
annealing method searches for the best configuration based on a statistical analysis of the cost distribution. A comprehensive review of the applications of evolutionary computation (EC) in structural design is given and chronologically classified by Kicinger et al. [25]. They introduced the field of evolutionary design and its relevance to structure design. Further, they discussed the issue of creativity/novelty and suggested possible ways of achieving it during a structural design process. The EC approach to the continuum topology optimization design problem based on genetic algorithm has been developed by Sandgren et al. [34] and Jensen [20]. In their work, the design domain was discretized into small elements containing materials or voids in a cantilever plate so that the structure’s weight was minimized subject to displacement and/or stress constraints. Subsequently, a lot of researchers have extensively employed genetic algorithm based methods for structural optimization in the optimal design of topology. Chapman and the associated researchers [8,9,14,18] present summary examples of the GA-based approach to topological optimization. A variety of different structural design fitness functions including stiffness, area, perimeter, and hole are employed to find optimal cantilevered plate topologies. Additionally, fitness sharing, restricted mating, and cluster analysis techniques are implemented for obtaining families of highly fit topologies. Based on graph theory [38], a valid topology is represented by a connected simple graph consisting of vertices and simple undirected cubic Be´zier curves with varying thickness. The derived results show that the graph representation GA can generate clearly defined and distinct geometries and perform a global search with more computational cost. Afterward, a bit-array representation GA [39] was implemented for topology optimization. The design connectivity and constraint handling are further developed to improve the efficiency of the GA. In addition, a violation penalty method is proposed to drive the GA search towards the topologies with higher structural performance, less unusable material and fewer separate objects in the design domain. Further, a multi-GA system [41] and variable chromosome length genetic algorithm [26] were proposed for continuum structures topological optimization. Recently, a two-stage adaptive genetic algorithm (TSAGA) [4] was developed in bit-array represented topology optimization. Compared with other approaches, the authors demonstrate the efficiency and effectiveness of TSAGA in reaching the global optimal solutions on several case problems. Also, Hamda et al. [16] and Aguilar [1] considered a continuum topology optimization problem as a multi-objective problem employing genetic algorithm. Finally, a biological immunity-based optimization approach [30] has been implemented to overcome the particular drawbacks, lack of local search ability and premature convergence, implicit in genetic algorithms in recent times. The authors apply multi-modal immune algorithm (MMIA) for structure topology optimal design. Two well-studied benchmark examples in structural topology optimization problems are used to evaluate the proposed approach. The results indicate the effectiveness of MMIA. The ACO is a relatively recent bio-inspired approach to solve combinatorial optimization problems mimicking the behavior of real ant colonies. It has been successfully applied to solve numerous optimization problems [10,19]. Ant colony optimization algorithms are good candidates for structural topology optimization due to their discrete feature similar to genetic algorithm and immune algorithm mentioned previously. Nevertheless, only a few of researchers have employed ant algorithms for truss structure optimization [7,22,31,35]. Rarely utilizes ant algorithm to optimum design of continuum structural topology. Until recently, Kaveh et al. [23] applied ACO and finite element analysis in topology optimization of two- and three-dimensional structure models. Based on the element’s contribution to the strain energy,
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the authors find the stiffest structure with a certain amount of material (defined solid material fraction). In addition, a noise cleaning technique was used to prevent creation of undesirable tiny members in the resulted optimum layout. Four examples were presented in their study to show the capability and efficiency of the proposed method. 3. Ant colony optimization algorithm Ant colony optimization algorithms are a recently developed, population-based methodology applied to numerous NP-hard combinatorial optimization problems. They have been inspired by the behavior of real ant colonies especially by their foraging behavior. One of the main features of ant algorithms is the indirect communication of a colony of (artificial) ants, based on pheromone trails that are a kind of distributed numeric information used to reflect their experience while solving a particular optimization problem. Ant system (AS), the first ACO algorithm was proposed by Droigo [11,12] to solve the traveling salesman problem (TSP). Each ant in AS builds up a solution step-by-step employing transition rule until a solution is found. Ants that found a good solution mark their paths by laying some amount of pheromone on the edge of the path. The following ants are then attracted by the pheromone so that they will be able to search in the solution space near good solutions. The traveling salesman problem is expressed as a graph GhN,Ei where N denotes the set of cities and E represents the set of edges between cities. The objective is to find the minimal length closed tour that visits each city once. Each ant is a simple agent to fulfill the task and obeys the following rules: It lives in a discrete-time environment. It chooses the next city associated with a probability which is a function of pheromone laid on the connecting edge and of the amount of the visible distance. It cannot choose the cities which has been visited before a tour is completed. It lays pheromone on each edge visited when a tour is completed. The steps for ant system simulation imitating real ant’s behavior are described as follows. 3.1. Initialization The initialization of the AS algorithm consists of the graph problem representation and the ants’ initial distribution. The optimization problem should first be represented in terms of a graph form GhN,Ei, where N is the set of nodes/cities and E is the set of connections/edges between nodes. Subsequently, a number of ants are randomly placed on the nodes and each of them will perform a tour according to the node transition rule described below. In general, the initial pheromones laid on the connections are set to small constant values. 3.2. Node transition rule The ants move from node to node following a node transition rule, which is problem dependent. The transition probability pkij ðtÞ from node i to node j for the kth ant is defined as
pkij ðtÞ
¼
8 > < > :
P 0
½t i j ðtÞa ½hi j b
t
a
h 2 tabuk ½ ih ðtÞ
½hih
b
if j 2 tabuk
(1)
otherwise
where tabuk is used to define the set of cities that the kth ant located on city i still has to visit. When the kth ant finishes a cycle, the tabu list tabuk is then emptied and the ant is free again to start
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another tour. By exploiting the memory therefore an ant k can build feasible solutions by an implicit state-space graph generation, which corresponds to visiting a city exactly once. tij and hij represent the concentration of pheromone laid on connection (i, j) and the value of visibility of connection (i, j), respectively. For the TSP hij = 1/dij, and dij denotes the distance between the ith and the jth cities. Parameters a and b control the relative importance between the intensity of pheromone versus visibility. Dorigo [11] have investigated the behavior of the ant-cycle algorithm for different combination of a and b. Three different classes were resulted: bad solutions and stagnation, bad solutions and no stagnation, and good solutions. The intensity of pheromone reflects the previous experience of the ants and provides the indirect communication between ants. Pheromone information provides a global view about the goodness of the solution found while the value of visibility offers the local information on edge (i, j) determined by the greedy heuristic for the original problem. 3.3. Pheromone updating rule Let na and n be the number of ants and cities, respectively. An iteration of the AS algorithm means all the na ants carry their next movements during discrete time interval (t, t + 1) and thus n iterations constitute a cycle. Therefore, each ant will complete a tour in one cycle. The trail intensity is updated according to the formula:
t i j ðt þ nÞ ¼ ð1 rÞt i j ðtÞ þ Dt i j
(2)
where tij(t) denotes the intensity of pheromone on edge (i, j) at time t and r represents the evaporation of trail in time interval [t, t + n]. The main purpose of pheromone evaporation is to avoid stagnation, that is, the situation in which all ants end up doing the same tour. Dt kij denotes the quantity of trail substance laid on edge (i, j) by the kth ant between time t and t + n.
Dt i j ¼
na X
Dt kij
(3)
k¼1
Three AS algorithms have been defined according to the way pheromone updated: ant-density, ant-quantity, and ant-cycle. In the ant-cycle model: 8 Q > > if thekth ant uses edgeði; jÞ < Lk k (4) Dt i j ¼ in its tour between timet andt þ n > > : 0 otherwise where Q is a constant and Lk is the tour length of the kth ant. In general, the intensity of trail at time 0, i.e. tij(0), is set to a small positive constant. As to the ant-density model: ( Q if thekth ant goes fromi to j between timet andt þ 1 k Dt i j ¼ 0 otherwise (5) Finally, in 8 Q > > < di j k Dt i j ¼ > > : 0
the ant quantity model: if thekth ant goes fromi to j in its tour between timet andt þ 1
(6)
otherwise
where dij is the distance between node i and node j. In ant-density and ant-quantity, ants deposit pheromone while building a solution. On the contrary, ants deposit pheromone after they have
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Fig. 2. Mapping from design domain to topology and element position.
built a complete tour in ant-cycle case. Accordingly, research on AS was directed towards a better understanding of the characteristics of ant-cycle, which is now known as ant system, while the other two algorithms were abandoned. In this study, ACO algorithm was adopted and modified to solve structural topology optimization described in the following section.
4.2. Element transition rule Instead of node transition rule employed in ant system, the ants move from element to element following an ‘‘element transition rule’’. The transition probability from element E(x, y) to element E(i, j) (i.e. one of the eight neighboring elements E(x 1, y 1), E(x 1, y), E(x 1, y + 1), E(x, y 1), E(x, y + 1), E(x + 1, y 1), E(x + 1, y), E(x + 1, y + 1) shown in Fig. 3) for the kth ant is defined as
4. Continuum structural topology optimization using ACO algorithm pkij ðtÞ Corresponding to the 2D topological optimization problems, a two-dimensional design is discretized into small, square elements (E(x, y), x = 1, . . . ,X; y = 1, . . . ,Y) where each element represents either material (with code value of 1) or void (with code value of 0) as Fig. 2 illustrated. This binary, material-void design domain results in a discrete non-convex [36] search space and allows for a precise discretized topology boundary. A population of ants is then placed randomly in elements of the design domain. The elements where each ant has visited will be marked (material). The states of the individual elements define the distribution of material and void within the design domain. After an ant finished its tour, all the connecting elements (material) between loading regions and supporting regions of the structure establish a valid topology, physically meaningful connected structure. The resulting topology’s trail intensity is determined according pheromone updating rule and thus a finite element analysis is performed on the topology. The modified ACO algorithm for structure topology optimization proposed in this study follows the following steps.
¼
8 > < > :
P 0
½t i j ðtÞa ½s i j ðtÞb
t
h2 = tabuk ½ ih ðtÞ
a
½s ih ðtÞb
if j 2 = tabuk
(7)
otherwise
where tabuk is used to define the element has been visited for the kth ant in the preceding stage at time instant t 1. In the foraging process, each ant is not allowed to go back to the element most recent time it has visited. The purpose is to avoid to moving and fro between two elements. Therefore, the set of elements that have been visited by ants in the preceding status is stored in the tabu list. tij(t) and sij(t) represent the current concentration of pheromone laid on elements E(i, j) and the normalized stress value of elements E(i, j), respectively. In this study, a special scheme is proposed to choose the visited element E(i, j) for the kth ant according to the transition probability pkij ðtÞ described as following. The seven transition probability values of E(i, j) (except the one stored in tabu list) are normalized and sorted descendent from maximum to minimum. Afterward, a uniform random number ranged between [0, 1] is generated and compared with the maximal transition probability. If the random number is smaller, the kth ant will visit the element E(i, j) with the maximal
4.1. Initialization A population of na ants is placed randomly in elements of the discretized design domain. Then each ant located at E(x, y) will move to one of the eight neighboring elements shown in Fig. 3 according to ‘‘element transition rule’’ instead of ‘‘node transition rule’’ employed in ACO algorithm. Different from the fourneighborhood connectivity used in [40], this study adopts eightneighborhood connectivity scheme due to its diverse property. The elements where the ant has visited will be marked (material). Moreover, the initial pheromone intensity at each element E(x, y) is assigned according to its normalized stress value assumed all elements are with materials rather than a small constant value adopted in ACO. Fig. 4 demonstrates a short cantilever plate subjected to a downward concentrated loading applied at a finite element node on its right hand center side and the associated normalized stress graph. The value of the normalized stress on each element will be assigned as initial pheromone intensity.
Fig. 3. Eight-neighborhood connectivity scheme and moving directions for ant.
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structure, and the corresponding element set to a binary value of 0 (void). 4.4. Pheromone updating rule The trail intensity at each element E(i, j) is updated according to the following formula:
t i j ðt þ nÞ ¼ ð1 rÞt i j ðtÞ þ Dt i j þ
eQ Ob j
(8)
where e indicates the number of elitist ants, Q is a positive constant value, and Obj*is the best objective value of solution/topology found from the beginning of the trail. In an effort to improve performance, ‘‘elitist ants’’ (similar to the elitist strategy used in genetic algorithm) introduced by Dorigo et al. [12] is included in the pheromone updating rule.
Dt i j ¼
na X
Dt kij
(9)
k¼1
and
Fig. 4. Cantilever plate and its associated stress/initial pheromone intensity distribution.
transition probability. Otherwise, another random number will be generated and compared with the second rank transition probability. The kth ant will visit this element E(i, j) if its transition probability value is bigger than the random number. Otherwise, similar procedure will be continuous for the next rank transition probability until all the remaining elements are checked. If the kth ant does not visit any element in the final stage, Roulette Wheel method will be adopted to select the element the ant should approach. 4.3. Connectivity analysis The resulting topology (mapping the elements visited by the kth ant into the design domain) may form a disconnected checkerboard due to the eight-neighborhood connectivity scheme utilized in this study. For any two elements in a topology to be considered as connected they must share at least one edge while element sharing only one corner are considered as disconnected. Elements connected merely at a corner cannot withstand applied torques and thus could lead to an unstable structure which cannot support various loads. A topology contained disconnected elements requires to undergo a structure modification procedure. In this procedure, the removal of disconnected elements or the adding of elements to neighboring disconnected element will be done randomly until the discontinuous structure is compensated. The continuous topology will be further analyzed via the finite element computation to obtain the required displacements and stresses. To reduce computation time, elements with a stress value lower than the user-defined level of average stress (which do not break continuity requirements) will be removed from the
Dt kij ¼
Q Ob jkNiche
where Ob jkNiche denotes the niche objective value of the kth ant/ topology solution, and Q is a user-defined constant value. In topology optimization, usually there exist many solutions such as one global and many local minima to a given problem [43]. Moreover Chapman and the associated researchers [8,9,14] have demonstrated the similar concept of ‘‘design families’’. In other words, structural topology optimization may provide the designer with a family of possibly-optimal designs and the designer can then evaluate the designs to determine which best satisfies several performance criteria, much like a pareto optimization study. Consequently niche strategy is utilized in this study to find the multiple solutions since topology optimization of structure seems to a kind of multi-modal problem [30]. Sharing scheme is thus employed to calculate the similarity for each ant/topology and described below: Ob jkNiche ¼ and
Ob jk ð1 þ penaltyk Þ SC k
Ob jk ¼
(10)
1 dmax Areak k
where Objk denotes the original objective value of the kth ant/ topology solution. Referring to [8,9,14,18], Objk indicates the kth ant’s/topology’s stiffness-to-weight ratio and stiffness is presented by inverse of topology’s maximum displacement ð1=dmax Þ at the k point of loading application. To determine dmax , a finite element k analysis is performed on the topology. dmax was set equal to the k magnitude of the displacement of the node where the point load was applied for the kth ant/topology. It should be noted that the number of connected material element of topology is used as a qualitative measure of topology’s weight (Areak). Moreover, SCk represents the similarity count of the kth ant/topology with all other ants/topologies and is expressed as SC k ¼
na X countk j ;
k ¼ 1; 2; . . . ; na ; j ¼ 1; 2; . . . ; na
(11)
j¼1
with
1; if ant k j < Th count k j ¼ 0; else qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ant k j ¼ ðastress astress Þ ðstdstress stdstress Þ j j k k
where Th is an user-defined threshold value illustrating the allowable difference between ants/topologies, antkj representing
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distance in average-standard deviation stress space indicates a relationship between the kth and the jth ant/topology. Note that astress and stdstress are normalized average and normalized standard k k deviation of stress of the kth ant/topology. Furthermore, penaltyk of the kth topology is treated as the penalty term for the stress violation and expressed as follows: NC NC X X penaltyk ¼ amount kj countkj j¼1
with
amount kj ¼ countkj ¼
(12)
j¼1
8 <
js kj j
was predefined and the solutions saved in memories will be replaced by the best solution found currently if it satisfies the following conditions: k Ob jk > Ob jmemory \ SCmemory d
where Objk and Objmemory denote the objective value of the kth ant/ topology and that of the ants/topologies saved in memories, k respectively. The variable SCmemory represents their associated similarities and d is a predefined similarity threshold. 4.6. Stopping criteria
1
if
js kj j > s allowable
: s allowable 0 else 1 if js kj j > s allowable 0 else
where penaltyk represents the penalty value for the kth ant/ topology; Nc is the number of material elements for the kth ant/ topology; amount kj and count kj correspond to the normalized values of the summation and total number of the jth element stress violation, respectively; s kj denotes the jth element stress value of the kth ant/topology whereas sallowable indicates the maximal allowable stress. 4.5. Memories of multiple colonies In this study, multiple-colony memories were employed to save the multi-modal topologies/ants found. The number of memories
Whenever the kth ant finishes visiting all the ‘‘seed’’ elements, it completes a tour and stops. A ‘‘seed’’ element [8,18] is an element that is required to contain material so that it may serve as a support boundary condition or point of load application. Consequently, the trail of the ant constitutes a structural topology. 5. Simulation results and discussions Four topological optimization examples were utilized to evaluate the effectiveness and performance of the proposed ACO algorithm. All the mechanical model and material properties are tabulated in Table 1. These examples describe the optimization of a cantilever plate (with aspect ratio 1.6) subject a downward concentrated load 3 kN applied at an FE node on its right hand center [17,27,29], top [30], bottom [17,30] and 2/5 of the distance from the bottom [8,9,18], respectively. In addition, nodes on the
Fig. 5. Simulation window for case 1.
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Table 1 Illustration of topological optimization examples. Case 1 [17,27,29]
Case 2 [30]
Case 3 [18]
Case 4 [9,18,30]
Loading position
FE node on the mid-point of the right side
FE node on the top-point of the right side
FE node on the bottom-point of the right side
FE node on the right hand surface 2/5 of the distance from the bottom
Fixed position Material properties
Left-top and left-bottom Young’s modulus E = 200 Gpa, Poisson’s ratio v ¼ 0:33, allowable stress sallowable = 200 Mpa and thickness t = 0.001 m
Design domain
Fig. 6. Selected topologies form 10 ACO runs with cluster scheme of case 1.
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left-hand surface corresponding to the support points were defined to have zero displacement in the FE analysis. Moreover, the design domain is discretized according to a 20 32 plane stress element FE model. The goal of optimization was to minimize the stiffness-to-weight ratio described in the previous section. To carry out these computations, a computer program was developed with C++ programming tools and a graphical user interface. Fig. 5
illustrates the running process of simulation window for case 1, its setting parameters, and part of the resulting topologies saved in memory pool. One execution of the computer model requires around 36,000 functional evaluations (60 ants by 60 iterations and 10 different runs), taking approximately 120 min with a Pentium 4 processor running at 1.5 GHz. Table 2 lists the associated parameters utilized in the proposed ACO algorithm. These
Fig. 7. Selected topologies form 10 ACO runs with cluster scheme of case 2.
G.-C. Luh, C.-Y. Lin / Applied Soft Computing 9 (2009) 1343–1353 Table 2 Parameters employed in ACO algorithm. Number of ants na Number of iterations N Run time Number of elitist ants e Pheromone decay factor r Similarity threshold d Threshold value Th Q
a b
60 60 10 4 0.05 0.75 0.1 4.0 1 1
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parameters were determined through numerical experiments after multiple simulation runs. Figs. 6–9 demonstrate a family of topologies and their corresponding weights (Area) for each case. These typical topologies are part of the results selected from the combination of 10 ACO runs with a maximum displacement less than 1.0 mm and a cluster threshold value of 0.75. All the topologies have welldefined, solid-material outer boundaries. Moreover, the interior regions have a ‘composite-like’ internal structure comprised of equally distributed material and void. As can be seen from the diverse range of resultant topologies shown in these figures, the
Fig. 8. Selected topologies form 10 ACO runs with cluster scheme of case 3.
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Fig. 9. Selected topologies form 10 ACO runs with cluster scheme of case 4.
structures therefore provide the designer with a set of possible solutions. The designer can then evaluate the designs to determine which topology is best for a particular application. Part of the structures show very well-defined truss-like members of constant cross-sectional area with large voids between members. The designer may choose these truss-like topologies (i.e. large voids between members) if manufacturability is the prime consideration. Compared with the results derived in the other studies [8,9,17,18,27,29,30], the proposed ACO algorithm generates much more diverse and interesting structures. These topologies could then possibly serve as the initial seed for a hierarchical subdivision-based optimization at finer discretizations or a preliminary structural configuration for shape and/or sizing optimization.
6. Conclusions In this study, a novel concept for handling multi-modal topological optimization has been presented by using a modified ACO algorithm mimicking the behavior of real ant colonies. Elitist ants were employed to increase the performance while memories of multiple colonies were implemented to save the family of designs. In addition, visibility of connection is not included in the calculation of transition probability. The potential of the proposed algorithm as a tool for investigating optimal topologies and for automatically creating innovative solutions to structural design problems has been illustrated in the examples presented. The results show that diverse topology structures could be derived using ACO algorithm and part of the structures show very well defined truss-like members.
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