Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
Engineering Optimization Vol. 39, No. 6, September 2007, 737–756
Optimization of truss bridges within a specified design domain using evolution strategies ˙ O. HASANÇEBI* Middle East Technical University, Department of Civil Engineering, 06531 Ankara, Turkey (Received 31 July 2006; revised 18 December 2006; in final form 1 March 2007) This article reports and investigates the application of evolution strategies (ESs) to optimize the design of truss bridges. This is a challenging optimization problem associated with mixed design variables, since it involves identification of the bridge’s shape and topology configurations in addition to the sizing of the structural members for minimum weight. A solution algorithm to this problem is developed by combining different variable-wise versions of adaptive ESs under a common optimization routine. In this regard, size and shape optimizations are implemented using discrete and continuous ESs, respectively, while topology optimization is achieved through a discrete version coupled with a particular methodology for generating topological variations. In the study, a design domain approach is employed in conjunction with ESs to seek the optimal shape and topology configuration of a bridge in a large and flexible design space. It is shown that the resulting algorithm performs very well and produces improved results for the problems of interest. Keywords: Structural optimization; Evolutionary algorithms (EAs); Evolution strategies (ESs); Truss bridges; Optimum bridge design
1.
Introduction
Successful applications of genetic algorithms (GAs) in a large spectrum of disciplines have accelerated research on other evolutionary algorithm (EA) techniques. Evolution strategies (ESs) are one of these techniques and are robust and encouraging methods to deal with complex optimization problems effectively. Early studies in ESs were credited to Rechenberg (1965, 1973) and Schwefel (1965). They were first developed in a rather simple form referred to as (1 + 1) − ES in the literature, but soon after were modernized by Schwefel (1977, 1981) towards their universally accepted, state-of-the-art variants known as (μ + λ) − ES and (μ, λ) − ES. Both of these variants were intended to operate in continuous design spaces. The development of discrete versions of (μ + λ) − ES and (μ, λ) − ES to deal with combinatorial optimization problems was accomplished by Bäck and Schütz (1995). It is essential to emphasize that both continuous and discrete *Email:
[email protected]
Engineering Optimization ISSN 0305-215X print/ISSN 1029-0273 online © 2007 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/03052150701335071
Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
738
O. Hasançebi
versions of ESs in these studies employ a set of strategy parameters to produce a self-adaptive search mechanism. Adaptive ESs have been applied to structural optimization problems in several studies in the literature, and successful results have been reported (Rechenberg 2000, Papadrakakis et al. 2003, Franco et al. 2004, Garcia and Gonzalez 2004, Ebenau et al. 2005). Yet, neither the number nor the extents of these applications are comparable to those of GAs. It is believed that further research in the field is required to explore their potential in different problem areas, as well as to provide an insight into their robust search mechanism. This study is focussed on investigating the potential use and computational effectiveness of adaptive ESs in the optimum design of truss bridges. This is a difficult task since the bridge’s shape and topology configurations need to be determined in addition to the sizing of the structural members for the minimum weight. Accordingly, design variables in such a problem consist of cross-sectional areas of the members (size); rectangular coordinates of the nodes (shape) and existence/non-existence of the members and nodes (topology); referring to discrete, continuous and 0–1 optimization types. An effective and straightforward treatment of this problem with ESs entails implementation of different variable-wise versions of the technique under a common optimization routine. In this regard, size and shape optimizations are implemented using discrete and continuous ESs, respectively. In addition, noting that 0–1 optimization actually represents a particular instance of discrete optimization, topology optimization is also performed here with a discrete ES, however it is coupled with a particular method for generating topological configurations in the designs evolved. It is important to note that all the design variables are considered simultaneously during the search to enhance the robustness and reliability of the optimization procedure. A design domain approach is implemented in the solution algorithm to minimize restrictions and presumptions on the selection of shape and topology configurations. In this approach, a bridge is facilitated to attain any shape and topology within a specified physical design area. The efficiency of the resulting algorithm is demonstrated using a numerical example that has been previously studied in the literature using various global optimization techniques.
2.
Optimum design of truss bridges
The optimum design of a truss bridge is a typical size, shape and topology optimization problem in which the weight of the bridge is minimized under a set of behavioural and geometric constraints imposed on members’ stresses, stabilities and lengths, in addition to nodal points’ displacements. In this regard, size optimization is applied to select cross-sectional areas of the bridge members. A discrete treatment of size variables is essential in agreement with real-world applications. For this, a predefined profile list consisting of a certain number of standard sections is prepared in order to size the members. In shape optimization, the real-valued x and y coordinates of the bridge nodes (joints) are introduced as design variables, and their most favourable positions are sought in the bridge design. The ranges of the possible movements of the nodes should be defined prior to the optimization process. A continuous treatment of the shape variables is then considered within these bounds. Finally, topology optimization interrogates the presence/absence of the members and nodes in the optimum design model. It is noted that all the size, shape and topology design variables defined in this process should be considered simultaneously for the most effective optimization procedure. A complete mathematical formulation of this problem is stated in Hasançebi and Erbatur (2002).
Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
Optimization of truss bridges using evolution strategies
3.
739
Design domain approach
In the present study, the design domain approach is implemented in cooperation with ESs to explore the optimal shape and topology configuration of a bridge in a large and flexible design space. In this approach, only the lower chord of a bridge is geometrically defined. To make up the superstructure (i.e. upper chord, diagonals and verticals, etc.), a design domain is introduced in the form of a rectangular physical design area (PDA) with an arbitrary height (h), figure 1. The bridge is permitted to evolve any shape and topology configuration provided that it is fully enclosed by this area. First, a reasonable assumption is made regarding the maximum number of nodes nPDA that will be considered for the superstructure design. It is recommended that nPDA is chosen to be high enough to account for all promising and practical design configurations of the bridge. The nPDA number of nodes with varying x and y coordinates are then generated within the PDA. Design intervals for the resulting shape variables are assigned to cause all the nodes to remain and occupy any position within the PDA. Next, a ground structure is generated for the topology model of the superstructure. Here, every connectivity between two nodes in the PDA or between a node in the PDA and a node on the lower chord is treated as a potential member (figure 1). Accordingly, a total of nPDA (nPDA /2 + nLC − 1/2) member connectivities (potential members) are considered for the superstructure, where nLC refers to the number of nodes on the lower chord. The question of which member connectivities and nodes really exist in the optimum design model of the bridge is sought by employing topology variables associated with them all. Among the entire set of topology variables defined in this way, however, only those representing member connectivities are independent, while others (representing nodes) are dependent. The reason for this is that existences of nodes are implicitly governed by member connectivities, i.e. a node will exist provided that there is at least one connectivity to it in the design model and will not exist otherwise (if there is not any). The dependent variables are introduced here as a part of the application of mutation, which is discussed later in section 8.3.
Figure 1.
Design domain approach.
Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
740
4.
O. Hasançebi
Optimization routine
As in all EA techniques, the underlying idea of ESs rests on simulation of natural evolution in an effort to evolve a population of individuals (designs) towards the optimum. In this framework, both continuous and discrete adaptive ESs make use of a common optimization routine featuring a generation-based iteration of the technique. This routine is outlined in the flowchart shown in figure 2, where P (t) and P (t) denote the parent and offspring populations at a generation (iteration) t, respectively. Concerning this flowchart, the first two steps consist of setting the generation counter t to 0 and creating an initial population P (0). The initial population consists of μ parent individuals, which are customarily created through a random initialization. Hence, it is highly
Figure 2.
Flowchart of the optimization routine with ESs.
Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
Optimization of truss bridges using evolution strategies
741
likely that the initial population consists of a high number of unfit individuals that violate the constraints or highly overestimate the optimum. The next step is to evaluate the individuals’ performances, where each individual is assigned a fitness score according to how well it satisfies the objective function and constraints of the problem at hand. In the following step, an offspring population P (t) is created through a sequential application of recombination and mutation operators to the parent population. The offspring population consists of λ individuals, which also undergo an evaluation process (step 5 in figure 2) to attain a fitness scores. Next, in step 6, the survivors of parent and offspring populations are determined via the selection operator, which identifies the only difference between (μ + λ) and (μ, λ) variants of ESs. In the (μ + λ) − ES, the operator is implemented by choosing deterministically the best μ individuals from a sum of μ + λ parent and offspring individuals. On the other hand, the (μ, λ) − ES excludes the parent population from the selection mechanism – instead the best μ individuals are chosen only from the λ offspring individuals. This completes one generation in the optimization procedure, accompanied by an increase of the generation counter by one (step 7). The surviving individuals in generation t make up the parent population P (t + 1) of the next generation. The loop between steps 4 and 8 is iterated in the same way for each new value of the generation counter until a termination criterion is satisfied. The optimization routine discussed above forms the basic framework of the solution algorithm developed in the present work for a simultaneous optimum design of steel truss bridges using adaptive ESs. In the following sections, computational implementations and further details of this algorithm are explained, as applied to the problems under consideration.
5.
Representation of an individual
In order to better distinguish various components of an individual (design) for the problem of interest, representation of an individual will first be discussed separately for the size, shape and topology optimization of each. Next, a combined formulation will be given to form a complete representation of the individual with all the size, shape and topology design variables and strategy parameters. 5.1
Shape variables and strategy parameters
As far as shape optimization is concerned, an individual d consists of two components, defined as: d = (c, σ ). (1) In this formulation, the first component c = [c1 , . . . , cnc ] represents the continuous shape design variable vector with nc number of independent shape variables. The second component σ is referred to as the vector of standard deviations, defining the set of strategy parameters employed in continuous optimization with ESs. Standard deviations are operated by the technique so as to achieve a self-adaptive search during the optimization process. That is, as the search continues, these parameters automatically adjust themselves to appropriate values according to topological features of the design space for a successfully implemented optimization process. Standard deviations σ act very similarly to step-size parameters in a traditional optimization algorithm, and control the variation ranges of shape variables during implementation of the mutation operator. In general, each shape variable ci must be associated with a standard deviation σi , and the number of independent standard deviations nσ used may vary between 1 and nc . For a general case of 1 < nσ < nc , the first nσ − 1 standard deviations (σ1 , . . . , σnσ −1 )
Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
742
O. Hasançebi
are coupled with the first nσ − 1 shape variables (c1 , . . . , cnσ −1 ) one by one, and the last one (σnσ ) is used to control mutation of the remaining shape variables cnσ , . . . , cnc jointly (Bäck 1996). Apart from σ , the original (μ + λ) and (μ, λ) variants of ESs developed by Schwefel (1977) incorporate a second set of strategy parameters referred to as rotation angles (or correlation coefficients) α. These parameters serve to relate the standard deviations of separate variables and thus to perform their mutations in a correlated manner. However, it has been found that calculation of correlated mutations as enabled by α leads to a prohibitively heavy computational burden, which is not suited to large-scale optimization problems; they are thus omitted from the solution algorithm. 5.2 Size variables and strategy parameters For a size optimum design of truss bridges, it is required that each individual d consists of two components, which can be defined as follows: d = (a, pa ),
(2)
where the first component a = [a1 , . . . , ana ] represents the discrete size design variable vector with na number of independent size variables. The second component pa ∈ (0, 1), referred to as mutation probabilities for size variables, represents the set of strategy parameters used in discrete optimization. They substitute the original strategy parameters σ and α in continuous optimization, and were introduced into the algorithm by Bäck and Schütz (1995) as an extension of the technique to handle discrete variables. Mutation probabilities are manipulated as self-adaptive parameters throughout the optimization process, adjusting the probability of size variables to undergo mutation for an efficient search. Hence, each size variable aj is coupled with a mutation probability pa,j , and the number of independent mutation probabilities npa employed may vary between 1 and na . For a general case of 1 < npa < na , the first npa − 1 mutation probabilities (pa,1 , . . . , pa,npa −1 ) are coupled with the first npa − 1 size variables (a1 , . . . , anpa −1 ) one by one, and the last one (pa,npa ) is used for remaining size variables anpa , . . . , ana jointly. 5.3 Topology variables and strategy parameters Topology optimization represents a particular instance of discrete optimization such that any topology variable is chosen from a discrete set of two characters {0, 1}, representing the existence of a member connectivity or a node in the case of {1} and its non-existence otherwise. Hence, topology optimization is also performed here using discrete ESs and each individual is formulated to have two components: d = (t, pt ),
(3)
where the first component t = [t1 , . . . , tnt ] ∈ {0, 1} represents the topology design vector with ntm and ntn member and node-related topology variables, respectively, resulting in a total of (nt = ntm + ntn ) topology design variables. As stated previously, only member-related topology variables are independent, i.e. node-related topology variables are dependent. The second component pt ∈ (0, 1) is referred to as the mutation probability for these variables, and has the same description and functionality with its counterpart in size optimization. The coupling between mutation probabilities and topology variables is processed in precisely the same way as in size optimization.
Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
Optimization of truss bridges using evolution strategies
743
5.4 A combined representation Given the above different components of an individual for size, shape and topology optimization alone, its complete formulation for a simultaneous optimum design process is obtained by a direct combination of these components, as follows: d = (c, a, t, σ , pa , pt ).
6.
(4)
Initial population
The creation of an initial population requires initialization of μ parent individuals with their entire set of components (design variables and strategy parameters). As a usual procedure in any EA technique, a random initialization of design variable vectors (c, a, t) is implemented for this purpose. That is, for each individual, the size and topology design variables are assigned arbitrary values from their associated discrete sets at random. Likewise, shape design variables are selected randomly between their lower and upper bounds. The initialization of strategy parameters is conducted based on extensive numerical experimentations performed here. Accordingly, for shape variables, the initial values of the standard deviations σi are set to 0.25δi , where δi denotes the real-valued difference between the upper and lower bounds of a shape variable. In addition, mutation probabilities are initially set to pa,j = 0.25 for size variables, while they are set to pt,k = 0.20 and 0.05 for member and node-related topology variables, respectively.
7.
Recombination
Recombination is applied to create an offspring population, such that μ parent individuals undergo an exchange of design characteristics to produce λ offspring individuals. A variety of distinct recombination operators exists and, in principle, recombination of different components of an individual can be implemented using different operators. Assuming that s represents an arbitrary component of an individual, i.e. s ∈ (c, a, t, σ , pa , pt ), a formulation of these operators is given in equation (5) as applied to produce the recombined s : ⎧ ⎪ (1)-no recombination sa,i ⎪ ⎪ ⎪ ⎪ ⎪ s or s (2)-discrete a,i b,i ⎨ si = sa,i or sbj,i (5) (3)-global discrete ⎪ ⎪ ⎪sa,i + (sb,i − sa,i )/2 (4)-intermediate ⎪ ⎪ ⎪ ⎩s + (s − s )/2 (5)-global intermediate. a,i bj,i a,i In equation (5), sa and sb represent the s component of any two parent individuals that are chosen from the parent population at random. Accordingly, in type (1) no recombination takes place; rather s is simply formed by duplicating sa . Type (2) refers to discrete recombination in which each element of s is selected from one of the two parents (sa and sb ) under equal probability. Type (3) denotes the global version of discrete recombination such that the first parent is selected and held unchanged, while a second parent is randomly determined anew for each element of s, and then si is chosen from one of these two parents (sa , sbj ) under equal probability. Intermediate forms of types (2) and (3) are given in types (4) and (5), respectively, which are identical to the former except that arithmetic means of the elements are calculated. In the
Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
744
O. Hasançebi
present study, the choice of recombination operators for particular components is performed in line with the recommendations of Bäck and Schwefel (1993), Bäck and Schütz (1995) and Bäck (1996) as follows: (c, a, t, σ , pa , pt ) → recombination types (5, 3, 3, 5, 4, 4).
8.
Mutation
Every offspring individual of the form d = (c, a, t, σ , pa , pt ) is subject to mutation, which results in a new (and expectantly improved) set of design variables (c , a , t ) and strategy parameters (σ , pa , pt ) for the individual. The application of mutation to size, shape and topology design variables is performed in a different manner, as explained in the following. 8.1
Mutation of shape variables and strategy parameters
Equations (6) and (7) describe a complete formulation of the mutation as applied to a shape variable ci and its associated standard deviation σi . σi = σi · exp[τ · N (0, 1) + τ · Ni (0, 1)]
(6)
ci
(7)
= ci +
σi
· Ni (0, 1).
As seen from equation (6), mutation of the standard deviation σi is performed by application of a multiplicative lognormal distribution based variation, which ensures that the mutated value (σi ) of the standard deviation always remains positive. Here, the notations N (0, 1) and Ni (0, 1) symbolize two random variables sampled according to a normal distribution with expectation 0 and standard deviation 1. In this context, the first random variable N (0, 1) controls an overall change of the mutability for all standard deviations of an individual, and is sampled once per individual (Bäck 1996). A change of the mutability on parameter level is achieved by the second random variable Ni (0, 1), which is sampled anew for each standard deviation. The factors τ and τ here refer to learning rates of the standard deviations, which are set to the following recommended values (Schwefel 1981): τ =
1 (2nc )1/2
and
τ=
1 . [2(nc )1/2 ]1/2
(8)
Following equation (6), the mutated value (σi ) of the strategy parameter is used to mutate the shape variable ci through an additive normal distribution based variation, as formulated in equation (7). This order of mutation has experimentally proved to be efficient and essential in terms of the effectiveness of self-adaptation, and is rationalized by the argument that it permits the design information to be modified in the light of improved search information. 8.2 Mutation of size variables and strategy parameters When applied, mutation creates a new set of values for size variables a and strategy parameters pa of an individual. Again here, mutation of the strategy parameters pa is performed first according to a logistic normal distribution (equation (9)), which ensures that the values of mutation probabilities pa are restricted within a range (0, 1). −1 1 − pa,j = 1+ · exp[−γ · Nj (0, 1)] . (9) pa,j pa,j stands for the mutated value of pa,j , and indicates the probability of a In equation (9), pa,j size variable aj to go through mutation. The factor γ here refers to the learning rate of this
Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
Optimization of truss bridges using evolution strategies
745
parameter, and is set to the following recommended value (Bäck and Schütz 1995): γ =
1 . [2(na )1/2 ]1/2
(10)
Once pa,j is obtained from equation (9), a uniform random variable uj is generated in the range of (0, 1), and is compared with pa,j to decide whether the size variable aj is to be mutated or not, as follows: if uj ≤ pa,j ; aj −→ aj (mutation) if uj > pa,j ; aj −→ aj (no mutation)
(11)
where aj denotes the mutated value of the size variable aj . In Bäck and Schütz (1995), (whenever applied) mutation of the variable is performed in accordance with a uniform distribution based variation in which the variable is assigned a new value from the discrete set at random. 8.3
Mutation of topology variables and strategy parameters
The application of mutation for topology optimization is conceptually identical to that for size optimization, except in the way topology variables are mutated. Accordingly, strategy parameters pt are mutated by means of a logistic normal distribution (equation (12)) and mutated values of the strategy parameters are then utilized to determine whether the topology variables t will be mutated or not (equation (13)). −1 1 − pt,k pt,k = 1 + · exp[−γ · Nk (0, 1)] (12) pt,k if uk ≤ pt,k ; tk −→ tk (mutation) ; tk −→ tk (no mutation). if uk > pt,k
(13)
However, unlike size variables, the mutation of topology variables calls for an exclusive method specialized for sampling new topology configurations with ESs fruitfully. A member and node restoring/removing method is proposed here to accomplish this task. 8.3.1 Member and node restoring/removing method. The method consists of two separate, yet complementary approaches, referred to as (i) member restoring/removing (MRR) and (ii) node restoring/removing (NRR). The objective of MRR is to generate primarily member-level variations in the topology model of an individual; it is applied to memberrelated variables only. On the contrary, NRR intends to create mainly node-level variations and thus only node-related variables go through this approach. It is important to emphasize that only one of these two approaches is applied to an individual at a time under equal probability. Hence, each time a decision is made to determine the approach, that approach will be implemented. At times when MRR is selected, those member-related topology variables that pass through the mutation test in equation (13) are processed with this approach. Otherwise, the same task is performed on node-related topology variables only, using NRR. When applied to a member-related topology variable tk , MRR leads to removal of the associated member connectivity if it already exists in the model, or its restoration if it does not. This is performed by switching the value of the topology variable with the other character in the discrete set, i.e. if tk = {1} → tk = {0} or tk = {0} → tk = {1}. Implementing this approach on a number of variables, each time a different group of structural members in the physical
Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
746
O. Hasançebi
design area will be restored or removed from the topology model of an individual during mutation. A point to be noted here is that, occasionally, the approach can come up with removal and/or restoring of the nodes as well. This happens when all the connectivities of an existing node are removed, or some connectivities of a non-existing node are restored to produce a stable configuration for the structural system. The same possibility may also occur during recombination when generating offspring individuals. However, these events take place occasionally and coincidentally, suffering from a lack of systematic basis. Thus, noticing the abovementioned shortcoming of MRR alone, NRR is incorporated into the solution algorithm as a complementary approach dealing specifically with nodes. NRR is implemented in a similar manner. When applied to a node-related topology variable, the approach removes the node if it exists in the topology model and, if not, restores it. The removal of an existing node is carried out by removing all of its member connectivities with other nodes. Therefore, it is essential to make certain that not only the value of the node-related variable, but also all member-related variables representing a connectivity of this node, are set to {0} in this process. In contrast, restoring a non-existing node does not necessarily require restoring all of its member connectivities. In the present study, to restore a node using a variety of alternatives, the number and the set of connectivities restored are randomized using the following procedure. First, all the connectivities of the node are identified. Next, a uniform random variable (u1 ) is sampled in the range [0, 1]. Finally, for each connectivity of the node, a new uniform random variable (u2 ) is sampled in the same range [0, 1] such that if u2 ≤ u1 and the other node of the connectivity exists in the model, it is restored by setting its corresponding topology variable to {1}, otherwise it is not restored. In this procedure, the first random variable u1 controls the number of connectivities in restoring a node and u2 chooses the set of connectivities used for this purpose. This way, each time a different set and number of connectivities are used to restore a node during the course of the optimization process. Since, in general, the restoring or removal of nodes gives rise to more significant variations in the topology configuration of a bridge, it is recommended that the initial values of mutation probabilities for node-related topology variables should be kept lower than those for memberrelated ones. In fact, experiments performed using various test problems have indicated that the choice of initial values of pt,k as 0.20 and 0.05 for member and node-related topology variables, respectively, will be useful settings for these parameters.
9.
Evaluation
Not only the objective function (weight), but also the problem constraints have to be taken into account during the evaluation process of parent and offspring individuals. A variety of different approaches and/or specialized operators have been proposed in the literature to handle constraints in conjunction with EAs; a complete overview of these methods can be found in Michalewicz (1995). In the present study, the constraints are dealt with using an external penalty function approach, such that a new (or constrained) objective function is defined: ⎡ ⎛ ⎞⎤ ng Wc = W [1 + Penalty (d)] = W ⎣1 + r ⎝ (gj )⎠⎦. (14) j =1
In equation (14), W and Wc are unconstrained and constrained objective functions, respectively, and gj , j = 1, . . . ng denotes the entire set of normalized constraints. The factor r here refers to the penalty coefficient, and is used to adjust the intensity of penalization as a whole. A value of r = 1 will be a useful setting for this parameter.
Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
Optimization of truss bridges using evolution strategies
10.
747
Selection
Selection is the only deterministic operator of ESs, and is implemented on the basis of two different schemes characterized as (μ + λ) and (μ, λ) selections. Unlike the (μ, λ) variant, the (μ + λ) − ES always comes up with a promise of guaranteed evolution. Since the parents are also involved in this variant, the parent population at any generation consists of the best μ individuals sampled thus far throughout the process. Hence, at first sight it may seem to be more advantageous as compared to the (μ, λ) − ES. According to Bäck and Schwefel (1993, 1995), however, this advantage may turn into a more serious disadvantage when interpreted in view of adaptation of the strategy parameters. They argue that retreat from mis-adapted strategy parameters and local optima is more difficult in the (μ + λ) variant. In this work, both variants are studied, where the ratio of offspring to parent individuals (λ/μ) is set to a value around 5–7 for a satisfactory performance of the algorithm. 11. Additional components In the work of Hasançebi (2006), the performance of ESs in continuous and discrete parameter optimizations is greatly enhanced through a number of additional parameters. Furthermore, an adaptive penalty function implementation is experimented, where the penalty coefficient r in equation (14) is facilitated to modify itself automatically in order that the algorithm is enforced to adopt a search direction along the constraint boundaries. Along with these topics, further components of the solution algorithm are discussed in that work. A comparison of ESs with other techniques was presented by Hasançebi and Ulusoy (2005) for some benchmark problems of structural optimization. It has been demonstrated that ESs are very robust and reliable search tools. 12.
Numerical example
Figure 3 shows a numerical example that has been previously studied in the literature using various global optimization techniques (Shrestha and Ghaboussi 1998, Hasançebi and Erbatur
Figure 3.
Numerical example (single-span truss bridge).
Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
748
O. Hasançebi
2002,Yang and Soh 2002). In the problem, the goal is to achieve the minimum weight design of a single-span truss bridge that crosses an opening of 70 m. Only the lower chord of the bridge is geometrically defined and it consists of eight panel points (including the end supports) equally spaced at 10 m. The design loading consists of equivalent panel point loads of P = 500 kN on the lower chord. It is required that the bridge is designed according to the provisions of the American Institute of Steel Construction specification (AISC 1989). Hence, the problem constraints are imposed as follows: (i) the stress and slenderness limitations of the structural members are calculated as per AISC (1989); (ii) the allowable displacements of all nodes in any direction are assigned to be 7.0 cm, which is equal to 1/1000 of the span length; and (iii) the lengths of the structural members are restricted to a minimum of 5.0 m and a maximum of 35.0 m. Finally, the structural members will be selected from a total of 30 standard steel sections between W14 × 22 and W14 × 426 in a W-shape profile list. The material properties of the standard sections are taken as follows: modulus of elasticity (E) = 2 038 936 kg/cm2 (29 000 ksi) and yield stress (fy ) = 2531 kg/cm2 (36 ksi). In engineering practice, an economic design of a truss bridge is achieved by keeping its height-to-span ratio around one-fifth to one-eighth, depending on a number of factors such as truss type, loading, constraints, span length, etc. Therefore, for the application of design domain approach to the problem under consideration, normally it would suffice to choose a PDA with a height (h) equal to one-fourth of the span length. Yet, as depicted in figure 3, a relatively larger area with 35 m height and 70 m width is used in the current application to comply with original treatment of the problem in Shrestha and Ghaboussi (1998). Reasonably assuming that the number of nodes (joints) used in the overall design of such a bridge system will never exceed 40, including those nLC = 8 nodes on the lower chord, nPDA = 32 nodes are generated in the PDA. The connectivities between these 40 nodes are defined as explained before, resulting in 752 possible structural members in the PDA. Hence, the maximum number of members used is limited to 759, including those seven members on the lower chord. Independent shape variables and their corresponding design intervals are assigned in consideration of a desired symmetry of the bridge about the y-axis. In principle, the design intervals for shape variables might be chosen such that the nodes can occupy any position within the PDA. However, it may be expected that this leads to extremely large design sets for the shape variables. To avoid this problem, a more efficient use of PDA is achieved in the current application by taking advantage of the symmetry of the bridge. For this, 32 nodes in the PDA are grouped in three sets with different design intervals (figure 3). The first group consists of 14 independent nodes that are located in the left half of the PDA, excluding the y-axis. Hence the design intervals of the first group nodes are defined as −35.0 m ≤ xj < 0.0 and 0.0 < yj ≤ 35.0 m. The second group incorporates those 14 dependent nodes that are linked to first group nodes one by one to yield a symmetrical shape of the bridge about the y-axis. Accordingly, the second group nodes are situated in the right half of the PDA (excluding the y-axis again) with design intervals 0.0 < xj ≤ 35.0 and 0.0 < yj ≤ 35.0 m. Both group nodes are prohibited to lie at x = 0 (y-axis) to prevent overlapping of the symmetric nodes on this axis, in which case the connectivities between the nodes lead to development of zero-length members. The third group members are introduced to put into use this part of the PDA, which are not covered by the first two group nodes. Four independent nodes are defined in the third group having design sets along the y-axis only, i.e. xj = 0.0 and 0.0 < yj ≤ 35.0 m. In this way, a total of 32 independent shape variables are used in the current application, considering the x- and y-coordinates of the first group nodes plus the y coordinates of the third group nodes. Independent size and topology design variables are also appointed considering the symmetry of the bridge. For this reason, structural members that are connected to symmetric nodes in the design model are linked together to form a single design variable. In this way, a total of 390 independent size variables are introduced–four on the lower chord and 386 in the PDA. Except
Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
Optimization of truss bridges using evolution strategies
749
for those four on the lower chord, the presence or absence of all other pairs in the optimum topology model is interrogated by associating a separate topology variable for each of them, i.e. ntm = 386 independent member-related topology variables are used in all. Furthermore, ntn = 20 dependent node-related topology variables addressing the first and third group nodes are specified for the application of topological mutation. In summary, the optimum design of the bridge is sought by employing 808 independent (390 size + 32 shape + 386 topology) plus 20 dependent design variables. To perform numerical experiments, first the ES self-adaptation parameters were set to appropriate values in line with the recommendations of former studies (Bäck and Schütz 1995, Bäck 1996). In this regard, shape optimization was performed employing the general case of adaptation (nσ = nc = 32), where each shape variable is associated with a different strategy parameter. It has been observed that the self-adaptation capability of the technique for continuous parameter optimization is best exploited when used with the general case. On the contrary, for size and topology optimizations the number of mutation probability parameters employed by each individual was set to one, i.e. npa = 1 and npt = 1. It has been verified that in discrete optimization the learning process works very well with one mutation probability employed, and the general cases npa = na and npt = nt suffer from a poor convergence behaviour. Next, useful settings of the ES population parameters (μ and λ) were determined. As a general rule, it can be stated that as the population size parameters increase, the algorithm tends to locate the optimum more reliably and accurately at the expense of increased computational cost. Taking into account the complexity of the current application, three pairs of relatively large population parameters were first identified, as follows: (15:100), (25:150) and (50:250). Extensive numerical experimentations performed with these three parameter sets indicated that the sets (15:100) and (25:150) mostly led to non-optimal topology configurations of the bridge, whereas the performance of (50:250) was generally satisfactory. Hence, the population parameters were set to μ = 50 and λ = 250 in the current application. Table 1. The optimum design reached in this study using (50, 250)-ES. Size and topology design Lower chord members
PDA members
Shape design Joints of the first group
Joints of the third group Weight (kg)
Design variables
Ready section
Area (cm2 )
1–2 2–3 3–4 4–9 1–5 5–6 6–7 7–8 2–5 2–6 3–6 3–7 4–7 4–8
W14 × 61 W14 × 68 W14 × 68 W14 × 74 W14 × 145 W14 × 145 W14 × 132 W14 × 109 W14 × 22 W14 × 43 W14 × 48 W14 × 61 W14 × 68 W14 × 61
115.48 129.03 129.03 140.64 275.48 275.48 250.32 206.45 41.87 81.29 90.97 115.48 129.03 115.48
Design variables
Position (cm)
x5 y5 x6 y6 x7 y7 y8
−2822.2 638.6 −1833.4 1184.7 −833.4 1474.5 1544.3 35 573 kg
Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
750
O. Hasançebi
Considering the stochastic performance of ESs, both (50, 250) and (50 + 250) variants of the solution algorithm were run six times independently over 2000 generations. It follows that a total of 250 × 2000 = 500 000 offspring individuals were generated (evolved) in each run of the algorithm, resulting in a total computing time of 488 min on a PC with a Pentium IV 2.4 GHz processor. In general, a better performance of the (50, 250) − ES was observed
Figure 4. The reported optimum designs of the bridge in the literature: (a) this work (W = 35 573 kg); (b) Shrestha and Ghaboussi (1998), when h = 35 m (W = 73 937 kg); (c) Shrestha and Ghaboussi (1998), when h = 10 m (W = 60 329 kg); (d) Hasançebi and Erbatur (2002) (W = 52661 kg); (e) Yang and Soh (2002) (W = 45404 kg).
Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
Optimization of truss bridges using evolution strategies
751
compared with the other. The six runs performed with the (50, 250) and (50 + 250) variants produced designs with different weights ranging between 35 573–47 242 kg in the former and 40 503–49 813 kg in the latter. The difference in the experimental performances of the variants may be attributed to the argument that escaping from local optima is more difficult in the (50 + 250) − ES, as hypothesized by Bäck and Schwefel (1993, 1995). In fact, in all the runs with the (50 + 250) variant, it was observed that a promising, yet non-optimal topology configuration was located relatively early by the algorithm, and the subsequent designs were evolved based on this configuration to reach better solutions. On the contrary, the search for new and better topology configurations was prolonged for much longer in the (50, 250) − ES, reducing the probability of entrapment in a non-optimal configuration. The best design obtained by the (50, 250) − ES is reported to be the optimum design of the problem reached in the present study. This design is tabulated in table 1 and is also shown geometrically in figure 4(a). As can be seen from figure 4(a), the optimum design consists of a total of 15 nodes and 27 members, of which seven nodes and 20 members are defined in the PDA. It is interesting to note that the topological model of the optimum design actually corresponds to a well-known truss form named the Parker truss, which is frequently used in practice for the design of single-span truss bridges.
Table 2.
Details of the optimum design and constraints. Constraints
Members 1−2 2−3 3−4 4−9 1−5 5−6 6−7 7−8 2−5 2−6 3−6 3−7 4−7 4−8
Axial force (ton)
Stress (Si /Si,a ≤ 1.0)∗
Slenderness ratio (ηi /ηi,a ≤ 1.0)∗
Minimum length (li / lmin ≥ 1.0)∗
Maximum length (li / lmax ≤ 1.0)∗
162.35 186.49 195.94 198.09 −223.05 −213.99 −207.14 −204.33 55.45 1.70 46.03 7.32 34.79 17.92
0.93 0.95 1.00 0.93 0.78 0.88 0.91 0.93 0.87 0.01 0.33 0.04 0.18 0.10
0.54 0.53 0.53 0.53 0.46 0.56 0.55 0.44 0.90 0.94 0.85 0.87 0.81 0.87
2.00 2.00 2.00 2.00 1.86 2.26 2.08 1.67 1.43 2.72 2.46 3.24 3.02 3.25
0.29 0.29 0.29 0.29 0.27 0.32 0.30 0.24 0.20 0.39 0.35 0.46 0.43 0.46
Constraints
Joints 1 2 3 4 5 6 7 8 ∗
x-displacement (cm)
y-displacement (cm)
Displacement along x-axis (dj,x /dmax ≤ 1.0)∗
Displacement along y-axis (dj,y /dmax ≤ 1.0)∗
0.00 0.69 1.40 2.14 2.06 2.92 2.87 2.49
0.00 3.93 5.91 6.89 2.72 5.17 6.52 6.87
0.00 0.10 0.20 0.31 0.29 0.42 0.41 0.36
0.00 0.56 0.84 0.98 0.39 0.74 0.93 0.98
The symbols used are defined as follows: Si = actual stress in member i; Sa,i = allowable stress for member i; ηi = slenderness ratio of member i; ηa,i = allowable slenderness ratio for member i; li = length of member i; lmax , lmin = allowable maximum and minimum lengths for members; dj,x = displacement of node j along x-axis (horizontal); dj,y = displacement of node j along y-axis (vertical); dmax = maximum allowable displacement.
Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
752
O. Hasançebi Table 3.
Gen. no.
Evolution of the best designs until the first feasible design is obtained. Design
W (kg)
Wc
J
M
F/I
0
2 180 712
2.912E + 11
36
380
I
10
279 147
10 702 508
22
58
I
20
272 690
4 912 526
17
40
I
30
182 384
2 561 905
16
37
I
50
154 447
651 675
14
34
I
70
120 780
449 512
10
20
I
90
179 404
359 387
13
27
I
100
156 321
321 606
13
27
I
130
175 218
228 362
15
35
I
160
157 391
163 823
12
28
I
Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
Optimization of truss bridges using evolution strategies Table 4. Gen. no.
753
Evolution of the best designs after the first feasible design is obtained. Design
W (kg)
Wc
J
M
F/I
168
157 456
157 456
12
26
F
180
146 773
146 773
12
24
F
200
132 502
132 502
12
24
F
250
101 589
101 589
12
22
F
275
86 934
86 934
12
22
F
325
72 018
72 018
11
21
F
375
63 201
63 201
15
31
F
425
54 188
54 188
15
29
F
477
48 622
48 622
15
27
F
600
42 326
42 326
15
27
F
1000
38 277
38 277
15
27
F
1500
36 078
36 078
15
27
F
1876
35 573
35 573
15
27
F
Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
754
O. Hasançebi
As for the other solutions of the problem in the literature, Shrestha and Ghaboussi (1998) reported best design weights of 73 937 kg and 60 329 kg (figures 4(b) and 4(c)), when the height of the PDA was chosen as h = 35 and 10 m, respectively. These designs were both obtained using a GA based on the generation of 1 000 000 offspring individuals. The simulated annealing (SA) algorithm implemented in Hasançebi and Erbatur (2002) yielded an optimum design weight of 52 661 kg (figure 4(d)). This solution was reached by sampling only 76 914 designs, which took 72 min of computing time on a PC with a Pentium IV 2.4 GHz processor. Finally, the same problem was studied by Yang and Soh (2002) using a genetic programming (GP) technique, where an optimum design weight of 45 404 kg (figure 4(e)) was reported employing an algorithm based on the generation of 200 000 offspring individuals. The large differences among the solutions of various techniques are due to variations in the topology models of the optimum designs produced. It was found that non-optimal topology models of the bridge were located by Shrestha and Ghaboussi (1998) and Hasançebi and Erbatur (2002), resulting in a significant increase in the final design weight of the structure. Verification of the optimum design obtained here (table 1) was performed using commercial structural design software to demonstrate that the resulting structure is fully feasible. The results are presented in table 2, which shows the member forces and nodal point displacements in addition to constraint details of the optimum design. Table 2 shows that the discrete sizes of the lower and upper chord members are governed by the stress constraints, ranging between 0.78 and 1.00. On the other hand, the slenderness ratio appears to be a dominant constraint for the web members (i.e. diagonals), ranging between 0.81 and 0.94. It is also observed that the active displacement constraints occur for joints 4 and 7, where the vertical deflections are computed to be 6.89 cm and 6.87 cm, respectively. In the optimization process, the first feasible design of 157 456 kg was obtained at the 168th generation. The evolution of the best designs up to this point is shown in table 3 at some selected generations. Table 3 lists the generation number in the first column. Columns 2 to 4 show the geometric model, weight and constrained objective function value of the best design, respectively. The number of nodes and members in the current design model are denoted in the fifth and sixth columns, respectively. Finally, the feasibility (F) or infeasibility (I) of the best design is indicated in the last column. A similar table is also produced for the evolution of the best designs after the 168th generation (table 4). As can be seen from this table, the best design featuring a Parker truss model was first created by the algorithm at the 477th generation. Retaining this topology model, the best design was progressively improved until the 1876th generation to locate the optimum. The large variations in the geometry model of the best designs in tables 3 and 4 point out the effectiveness of the member and node restoring/removing method for generating topological variations.
13.
Conclusions
This study reports the development of an ES-based solution algorithm for the optimum design of truss bridges. The major characteristics of the solution algorithm are identified as (i) ES optimization routine, (ii) design domain approach, (iii) simultaneous design optimization and (iv) member and node restoring/removing method. The design domain approach is implemented here as a supplementary method to ESs for seeking the optimal shape and topology configuration of a bridge in a large and flexible design space. The approach has been found to be practical, efficient and well-suited for applications of ESs provided that the initial number of nodes in the PDA (nPDA ) is chosen to be sufficiently large to cover all promising design configurations. A combined formulation of an individual
Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
Optimization of truss bridges using evolution strategies
755
is presented with all the size, shape and topology design variables and their associated strategy parameters. Different variable-wise versions of the technique are implemented together under a common optimization routine for the application of a simultaneous design optimization. Finally, the member and node restoring/removing method coupled with discrete ESs is proposed as an exclusive method for sampling new topology configurations with the technique. The success of the resulting algorithm was tested using a numerical example that has been previously studied in the literature using various global techniques such as GAs, SA and GP. Six independent runs were performed with both (50, 250) and (50 + 250) variants of the algorithm considering the complexity of the problem as well as stochastic performance of the technique. It was observed that the (50, 250)-ES generally performed better than (50 + 250) − ES due to a reduced probability of entrapment of the former in local optima. Nevertheless, both variants produced better results than those obtained with other techniques. The underlying reasons for the superior performance of ESs observed in this study can be attributed to the concept of self-adaptability plus an efficacious application of genetic principles in the technique. The inherent self-adaptation feature, which is missing in all the others (i.e. GAs, GP and SA), is probably the major reason for the exceptional search abilities of ESs. As a matter of fact, this feature facilitates the technique to refine its search strategies instantly according to varying characteristics of the design space. In this way, a dynamic and problem-specific search strategy is established for each problem automatically. Despite the fact that all EA techniques are somewhat similar in the sense that they all mimic the same conceptual mechanism of natural evolution, the manner in which the genetic principles are applied in these techniques may differ considerably. An additional success of ESs stems from an effective realization of the genetic principles for a successfully modelled evolutionary process. For instance, mutation, which is applied by GAs and GP as a secondary operator mainly to preserve genetic diversity within a design population, has been turned into a very robust search tool in ESs by means of normally distributed based variation of design variables. Nevertheless, the main disadvantage of ESs, in fact of all EAs (including GAs and GP), is that they are computationally intensive due to the necessity of maintaining a design population for genetic operations. The computational burden is comparatively less in the case of SA due to a point-by-point iteration of the search process. The integration of approximate analysis techniques can be considered as a remedy to eliminate this drawback of EAs.
References AISC, Manual of Steel Construction-Allowable Stress Design, 8th edn, 1989 (American Institute of Steel Construction: Chicago, IL). Bäck, T., Evolutionary Algorithms in Theory and Practice, 1996 (Oxford University Press: New York). Bäck, T. and Schütz, M., Evolutionary strategies for mixed-integer optimization of optical multilayer systems, in Proceedings of the 4th Annual Conference on Evolutionary Programming, 1995, pp. 33–51. Bäck, T. and Schwefel, H.-P., An overview of evolutionary algorithms for parameter optimization. Evolution. Comput., 1993, 1(1), 1–23. Bäck, T. and Schwefel, H.-P., Evolution strategies I: variants and their computational implementation. In Genetic Algorithms in Engineering and Computer Science, edited by G. Winter et al., pp. 111–126, 1995 (Wiley: Chichester). Ebenau, C., Rottschäfer, J. and Thierauf, G., An advanced evolutionary strategy with an adaptive penalty function for mixed-discrete structural optimization. Adv. Eng. Software, 2005, 36(1), 29–38. Franco, G., Betti, R. and Lu¸s, H., Identification of structural systems using an evolutionary strategy. J. Eng. Mech., 2004, 130(10), 1125–1139. Garcia, M.J. and Gonzalez, C.A., Shape optimisation of continuum structures via evolution strategies and fixed grid finite element analysis. Struct. Multidisciplin. Optimiz., 2004, 26(1-2), 92–98. Hasançebi, O., Adaptive evolution strategies in structural optimization: enhancing their computational performance with applications to large-scale structures. In press, 2006. Hasançebi, O. and Erbatur, F., Layout optimization of trusses using simulated annealing. Adv. Eng. Software, 2002, 33(7), 681–696.
Downloaded By: [Middle East Technical University] At: 10:57 24 September 2007
756
O. Hasançebi
Hasançebi, O. and Ulusoy, A.F., Discrete and continuous structural optimization using evolution strategies, in Proceedings of the 8th International Conference on the Application of Artificial Intelligence to Civil, Structural and Environmental Engineering, 2005, paper 23. Michalewicz, Z., A survey of constraint handling techniques in evolutionary computations methods, in Proceedings of the 4th Annual Conference on Evolutionary Programming, 1995, pp. 135–155. Papadrakakis, M., Lagaros, N.D. and Fragakis, Y., Parallel computational strategies for structural optimization. Int. J. Num. Methods Eng., 2003, 58(9), 1347–1380. Rechenberg, I., Cybernetic Solution Path of an Experimental Problem. Royal Aircraft Establishment, Farnborough, UK, Library translation No. 1122, 1965. Rechenberg, I., Evolutionsstrategie: Optimierung technischer Systeme nach Prinzipien der biologischen Evolution, 1973 (Frommann-Holzboog: Stuttgart). Rechenberg, I., Case studies in evolutionary experimentation and computation. Comput. Methods Appl. Mech. Eng., 2000, 186(2), 125–140. Schwefel, H.-P., Kybernetische evolution als strategie der experimentellen forschung in der strömungstechnik. Diplomarbeit, Technische Universität Berlin, 1965. Schwefel, H.-P., Numerische optimierung von computer-modellen mittels der evolutionsstrategie. In Interdisciplinary Systems Research, Vol. 26, 1977 (Birkhäuser: Basel). Schwefel, H.-P., Numerical Optimization of Computer Models, 1981 (Wiley: Chichester). Shrestha, S.M. and Ghaboussi, J., Evolution of optimum structural shapes using genetic algorithm. J. Struct. Eng., 1998, 124(11), 1331–1338. Yang, Y. and Soh, C.K., Automated optimum design of structures using genetic programming. Comput. Struct., 2002, 80(18-19), 1537–1546.
