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Size Optimization of Truss Structures By Cellular Automata M.H.Afshar and A.Faramarzi Abstract— This article presents an application of a cellular automata method to size optimization of truss structures. Each bar of truss is defined as a cell, and cross-sectional area of each bar denotes the corresponding state of the cell. The local updating rule is derived by setting the first derivative of the total penalized cost of the problem with respect to the state of the cell (crosssectional area) equal to zero. The penalized cost is defined as a weighted sum of the objective function (truss volume) and constraints violations of each design. The state of each cell is updated at each iteration by implementing the local updating rule to the corresponding cell. The updating process is repeated for all of the cells until convergence is achieved. The efficiency of the proposed method is examined by its application to some truss structures and the results are presented and compared to those of an alternative method, namely fully stressed design (FSD) approach. The results show that the proposed CA based method is more efficient than the FSD method requiring less computational effort to yield the same solution as the FSD approach. Index Terms— Size Optimization, Cellular Automata, Truss Structure.

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1 INTRODUCTION

T

HE history of structural optimization goes back to 1904 with presentation of a paper by Michell [1] on the optimization of truss structures. The paper of Michell presents theoretical optimum shapes for statically determinate trusses. Since then, many researchers have worked on different approaches for structural optimization, but no unique method has proved to be efficient and robust for the all range of structural engineering optimization problems [2]. Recent methods of optimization involved continuous design variables for size and shape optimization problems. However these methods would have to contemplate discrete design variables to satisfy standard profiles available on the market. Continuous methods have, therefore, been modified to be used for discrete optimization. If a structure is optimized using a continuous method, design variables would have to be rounded to the nearest discrete values. This approach has been used by Grieson and Lee [3]. Rounding design variables, however, could cause these problems; (1) the result may be sub optimal (2) overly conservative (3) or even infeasible design, indicating that rounding continuous design variables cannot be effective [4]. A later approach, namely branch and bound that has been used extensively, treated the design variables as discrete [5], [6], [7], [8], [9] and [10]. The problem with the branch and bound method was broken into continuous optimizations subproblems and solved which can take a large amount of time. In the early 1960s, some researchers modeled genetic

information on computers [11], [12] and [13]. Benefits of modeling DNA on the computer have led to developing Genetic algorithms to optimize systems including structures [14]. In the year 1991, ant colony optimization (ACO) was introduced by Dorigo et al [15]. Many application of GA has been reported in the literature for the optimization of structures [16] and [17]. Some other heuristic search methods such as Simulated Annealing [18] and Genetic programming [19] have been also employed for the optimum design of truss structures with respect to size, shape and topology. Fourie and Groenwold [20] used PSO for topology optimization and later extended the method for size and shape optimization [21]. Reactive Taboo Search was employed by Hamza et al [22] for topology, size and shape optimization of N-shaped roof trusses. Shih et al [23] used modified double-cut approach in fuzzy truss size optimization. Lamberti [24] presented an efficient simulated annealing for size and shape optimization. More recently GA and force method was used by Rahami et al [25] for size, shape and topology optimization of 2-D and 3-D truss structures.

2 CELLULAR AUTOMATA

Behavior of most physical systems in engineering and physics are often described by their governed generally non-linear partial differential equation. These systems can be very complex and many different methods have been suggested to simulate them. Cellular Automata (CA) supply an alternative method to describe, figure out and ———————————————— • M.H. Afshar is with the department of Civil Engineering, Iran University simulate the behavior of complex systems [26]. Cellular of Science and Technology, Narmak, Tehran, Iran automata are generally attributed to ulam [27] and von Neumann [28] that introduced the CA concept in the late • A. Faramarzi is with the Department of Civil Engineering, Iran University forties, and were used as early as 1946 by Weiner and of Science and Technology, Narmak, Tehran, Iran

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2 JOURNAL OF COMPUTER SCIENCE AND ENGINEERING, VOLUME 3, ISSUE 1, SEPTEMBER 2010

Rosenblunth to describe the operation of the heart muscle [29]. Initially CA techniques were introduced under the name of automata network, which were used to model discrete dynamical system in time and space [30]. They are used as a finite or infinite graph and there is a finite set of discrete values that each vertex of graph can take them on. The state of each vertex can be altered through transition rules that are based on the vertex’s current state as well as that of its neighbors in the graph. The whole graph is then updated repetitively either in synchronous or sequential manner. Synchronous mode is also called parallel mode. In this mode all sites are updated in the discrete time step simultaneously. The sequential update is practicable to only finite network, in which the sites are updated one by one. A particular case of automata networks is cellular automata in which the graph is a regular lattice of cells and the updating mode is synchronous. Each cell may be in one of the finite number of states. A cell or automata changes its state at the fixed discrete time step according to the local rule. This rule depends on the present state of the cell and its neighbors within a certain proximity. It’s proved that by CA model the overall global behavior of a large system can be computed by local rules over cells that only know local conditions. Since the local rule is defined so as to govern only local relationships among the neighboring cells, the use of governing equation for the whole domain is not necessary. Therefore the cellular automaton is considered to be very effective for simulating physical phenomena whose governing equations are unknown. In physics the CA has been used to simulate dynamic phenomena such as traffic flow [31], population dynamics [32],crack propagation, crystal structures, microstructures evolution, among many other applications [33] and [34].

3 CA LATTICE A basic CA composed of: (1) Regular lattice of cells in a d-dimensional space: (2) A set of k discrete variables, i.e., states, S (r , t ) = [ S1 ,..., S k ] associated with each cell located in r, at the discrete time t=0,1,2,…; and (3) A set of k rules R = [ R1 ,...Rk ] which specifies the time evolution of the states as: (1) S (r , t + 1) = Rk ( S (r , t ), S (r + δ1 , t ),..., S (r + δ j , t ) Where r + δ j specifies the cells belonging to a given neighborhood of the cell r. The state of each cell consists of a single bit S1 (r , t ) of information. The set of rules R is identical for whole domain, it means that the rules does not depend on the position r. There are different forms of lattice, three samples of which are shown in Fig. 1. The form of the cellular space strongly depends on the domain of the problem and the nature of physical phenomenon. The most common lattice that is often used in different problems is rectangular lattice, but other shapes of lattice like triangular and hexagonal are also employed. Wolfram used the hexagonal lattice in his problem [35], but Frisch, Hasslacher and Pomeau used triangular one for a cellular automaton fluid model [36].

(a) Rectangular lattice tice

(b) Triangular lattice

(c) Hexagonal lat-

5 THE NEIGHBORHOOD One of the important characteristics of a CA model is its neighborhood structure. In updating the state of a cell, it is necessary to consider the state of local cells in its defined neighborhood. Some common examples of neighborhood structures are shown in Fig. 2. The cell to be updated is labeled as C, and neighbors in each example are labeled according to its direction. These are not the only neighborhood structures possible. There are other kinds of neighborhood structure too. For example "MvonN neighborhood" is another structure that combines the nine sites of the Moor neighborhood shown in the Fig. 2. With four more cells lying two sites away in the north, south, east and west direction.

6 STRUCTURAL OPTIMIZATION There are three distinct forms of structural optimization each with a different solution strategy. Topology Optimization also known as Topological Optimum Design (TOD) is a problem looking for an optimal material layout of an engineering system. In other word, topology optimization is faced where the actual form of the structure is unknown. What is known are the environments of the structure that has to live in and the optimality criteria and design constraints need to be applied. Initial investigations in the late 1970’s and early 1980’s were conducted using formal methods. Generally TOD problems can be divided into two major category: (1) Continuum TOD (2) discrete TOD. In continuous environments the domain is discretized into small elements or grids where each element contains material or void. Some of the most practical methods that are effective in solving topology optimizations problems are as follows: (1) homogenization [37] (2) evolutionary structural optimization [38] (3) genetic algorithm [39], (4) cellular automata [40]. Discrete TOD problems consist of determining the optimal element connectivity from a finite, albeit large, number of possible connections. Two major discrete structures are truss and frame structure. Initial applications of GAs to optimize topology of discrete-member trusses were conducted by Shankar and Hajela [41] and Hajela et al. [42] Soh and Yang [43] introduced GP-based approach to TOD of truss

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structures. Fig. 3 schematically shows topology optimiza-

tion of a truss structrures.

Fig. 2. Cellular Neighborhoods

Fig. 3. Topology optimization of truss structure

Shape optimization maintains a fixed topology of structural designs but changes their shape or node locations. The object of shape optimization is to find the best shape that will have the best stress outcome. Similar to the TOD case, shape optimization problems can also be divided into two major groups: Continuum SO and discrete SO. Traditionally, in continuum SO, a shape is defined by the oriented boundary curves [2D structures] or boundary surfaces [3D structures] of the body, and the optimal form of these boundaries is computed. Kita and Tanie [44], Annicchiarico and Cerrolaza [45] used GAs to optimize the shape of continuum 2D structures through B-spline functions. Discrete SO methods conduct shape

optimization through variations in geometry of discrete truss and frame structures introduced through changes in locations of nodes. In the case of shape optimization of truss structures, discrete TOD methods using the ground structure have been extended to include optimization of the nodal point locations for a given number and connectivity of nodal points [46]. Initial applications of EC methods to discrete SO problems have been conducted by Grierson and Pak [47] in the context of truss structures. Soh and Yang [48] applied fuzzy controlled GAs to optimize the shape of planar and spatial truss structures. Fig. 4 shows shape optimization of truss structure.

Fig. 4. Shape optimization of truss structure

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Size optimization problems involve finding optimal cross-sections, or dimensions, of the elements of a structural system whose topology and shape is fixed. Research on formal methods of solving these kinds of problems has a long history and widespread literature is available on this topic [49]. First applications of EC to structural optimization problems involved sizing optimization. Lawo and Thierauf [50] used ES to optimize members of a pla-

nar six-story frame subjected to earthquake loading. Goldberg and Samtani [51] applied a GA to optimize cross-sections of the members of a 10-bar plane truss. Hajela [52] investigated cross-section optimization of discrete member trusses using GAs. Rajeev and Krishnamoorthy [53] applied GAs to optimize cross-sections of generalized trusses. Fig. 5 shows size optimization of truss structure.

Fig. 5. Size optimization of truss structure variation of the state of the cell ( A + ∆A) we have:

wi = ( Ai + ∆Ai ) 2 + α1,i (csv1,i +

6 STRUCTURAL SIZE OPTIMIZATION USING CELLULAR AUTOMATA We will consider the problem of minimization of volume V of a plane truss, N

V =

∑ Al

(2)

i i

i =1

Where, Ai and li , are the cross-sectional area and length of each element, respectively, and N is the number of truss elements. Minimization is to be carried out under the following constraints:

σ i − σ imax < 0 σ imin − σ i < 0

i=(1,…,N) i=(1,…,N)

(3) (4)

Where, σ imax , σ imin are allowable tensile and compressive stress for each bar, respectively, and σ i is the stress in each element. In this paper, each bar of the truss is defined as a cell, and cross-section of each bar is denoted the corresponding state of the cell. The process of derivation of local rule can be stated as follows: The penalized objective function for each cell is defined by adding the corresponding constraint condition to the original objective function. 2

wi = Ai + α1,i (

σi σi − 1) 2 + α 2,i ( min − 1) 2 σ imax σi

(5)

+ α 2,i (csv2,i +

∆σ i

σ imin

)

∆σ i 2 ) σ imax

(6)

2

Where:

σi −1 σ imax σi csv2,i = min −1 σi

(7)

csv1,i =

(8)

The stress variation is defined as: ∆σ i =

∂σ i ∆Ai ∂Ai

(9)

Or

∆σ i = β i ∆Ai

(10)

Assuming that

βi = (

∂σ )i ∂A

(11)

The derivative of the stress with respect to the crosssectional area of each bar can be calculated using a finite difference method. Substituting “(10)” in “(6)” will result in the final form of the penalized local objective function defined as:

Where α 1,i , α 2 ,i are penalty coefficients.

wi = ( Ai + ∆Ai ) 2 + α 1,i (csv1,i +

By expanding the objective function (w) around the

βi + α 2,i (csv2,i + min ∆Ai ) 2 σi

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βi ∆A ) 2 σ imax i

(12)

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The updating rule can now be obtained by setting the first derivative of the penalized objective function with respect to ∆Ai equal to zero: ∂Wi = 2( Ai + ∆Ai ) ∂ (∆Ai ) (13) βi βi + 2α 1,i max (csv1,i + max ∆Ai ) σi σi

vergence curves for the FSD and proposed CA algorithms, respectively.

βi βi (csv2,i + min ∆Ai ) = 0 σ imin σi Leading to the local updating rule as follows: α csv α csv Ai + βi ( 1,i 1,i max + 2,i 2,i min ) σi σi (14) ∆Ai = − β β 2 2 i i 1 + α1,i ( ) + α2,i ( ) σ imin σ imax The updated cross sectional area can be calculated as: new old (15) Ai = Ai + ∆Ai + 2α 2,i

Fig. 6. Scheme of truss (example 1) Table 1. CA solution for example 1 Member

7 THE FULLY STRESSED DESIGN (FSD) Fully stressed design (FSD) or stress-ratio method is a direct method to optimize a structure. This method is used when only stress constraints are taken into account. In this method the new design is obtained by multiplying two factors:

Area (cm2)

Member

Area (cm2)

1

8.3271

6

0.01

2

0.01

7

5.9013

3

8.3395

8

5.8838

4 5

4.1605 0.01

9 10

5.8838 0.01

(1).design variables (2).ratio of current of constraint to the constraint limit [54].

Ain+1 = Ain

σ in σ all

(16)

n +1

Where, Ai denotes the improved value of cross-section of ith bar (i=1,2,…,N) at iteration (n+1), Ain is the old valn ue, and σ i is the current stress. And σ all is allowable stress. According to the Morris [55] in the case of statically determinate structures subject to a single load, a FSD design is an optimum design but for multiple loading conditions, it can’t be guaranteed that the solution is in fact optimal. It is also noted that FSD method is basically independent to the initial value of set of design variables.

Fig. 7. The history of optimization using FSD approach (example 1)

8 NUMERICAL EXAMPLES 8.1 Example 1: 10-bar truss The first example considers optimization of the truss structure shown in Fig. 6. As stated earlier, the cell state is considered as the cross-sectional area of each bar with the range of values 0.01cm 2 < A < 20cm 2 . The truss is made of steel (st 37), with the modulus of elasticity equal to E = 2.1 × 10 6 (kg/cm2). The allowable stresses are limited to ± 24000 (kg/cm2), and vertical loads on nodes 2 and 4 have the same magnitude of 1MN . The CA results are shown in Table 1 which is the same as those obtained using FSD approach. These results, however, are obtained within 25 iteration compared to 33 iteration required by FSD approach. This is clearly evident from Fig. 7 and Fig. 8 which show the con-

Fig. 8. The history of optimization using proposed algorithm (example 1)

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8.2 Example 2: 18-bar truss The second example is the 18-bar truss shown in Fig. 9 along with its loading condition. The material properties and all other data including geometric constraint are the same as example one. Table 2 show the results obtained using the proposed CA approach which are again the same as the FSD

approach. Fig. 10 and 11 show the convergence history of both methods. It is again seen that the proposed CA method is more efficient than the FSD approach due to faster convergence characteristics.

Fig. 9. Scheme of truss (example 2)

Table 2. Optimized cross-section of example 2 Member 1 2 3 4 5 6

Area(cm2) 3.9063 0.01 0.01 3.9063 7.3703 4.1658

Member 7 8 9 10 11 12

Fig. 10. The history of optimization using FSD approach (example 2)

Area(cm2) 6.6908 6.9793 0.511 5.5207 8.3333 0.511

Member 13 14 15 16 17 18

Area(cm2) 5.5207 6.9793 4.1658 7.3703 6.6908 0.01

Fig. 11. The history of optimization using proposed algorithm (example 2)

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8.3 Example 3: 60-bar truss The next example considers the size optimization of a sixty-bar truss structure. Geometry and loading conditions of this example is shown in Fig. 12. The problem characteristics are the same as previous examples except for the crosssectional constraints which is assumed to be as 0.01cm 2 < A < 30cm 2 . Table 3 shows optimized cross-sections obtained using the proposed CA method. The history of convergence for FSD solution and proposed CA algorithm

are also shown in Fig. 13, 14. Once again it is seen the CA approach is faster than FSD approach while producing the same results.

Fig. 12. Scheme of truss (example 3)

Table 3. Optimized cross-section of example 3 Member 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Area(cm2) 5.8969 0.01 0.0102 8.319 5.8846 0.01 4.1577 5.9039 4.1663 5.8831 4.1577 0.01 4.1663 0.01 11.7959

Member 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Area(cm2) 8.3154 0.01 0.01 0.01 8.3154 8.313 23.5555 0.011 0.01 23.5711 0.011 0.01 0.01 16.6668 0.01

Member 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

Area(cm2) 0.01 0.01 16.6715 0.01 0.01 0.01 16.6762 8.323 0.01 11.7705 24.9992 8.334 8.332 11.7861 30

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Member 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Area(cm2) 8.3333 30 11.7851 24.9998 8.3333 24.9998 11.7851 16.6666 8.3333 16.6666 11.7851 8.3333 8.3333 8.3333 11.7851

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Fig. 13. The history of optimization using FSD approach (Example 3)

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9 CONCLUSION A Cellular Automata approach was introduced in this paper for the optimal design of truss structures. Each member of the truss structure was taken as a CA cell with the corresponding cross sectional area as the cell state. The local updating rule was derived by setting the first derivative of the total penalized cost of the problem with respect to the state of the cell equal to zero. The penalized cost was defined as a weighted sum of the objective function and constraints violations of each design. The efficiency of the proposed method was examined by its application to some truss structures and the results were presented and compared to those of an alternative method, namely fully stressed design (FSD) approach. The results show that the proposed CA based method is more efficient than the FSD method requiring less computational effort to yield the same solution as the FSD approach.

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M.H. Afshar obtained his BSc in Civil Engineering at Tehran University, Faculty of Engineering in 1984. He completed his MSc and PhD at University College of Swansea, Swansea, and U.K in 1993. He is now an academic staff of the Iran University of Science and Technology, Civil Eng. Faculty. He has published more than 80 journal paper and 60 conference paper. He has also supervised more than 36 Msc students and graduated 3 Ph.D. students. A. Faramarzi obtained his BSc in Mechanical Engineering at University of Guilan in 2006. He received his MSc in Civil Engineering at Iran University of Science and Technology in 2009. He has also submitted 3 journal papers.

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