The geometry of the 120member truss dome
1. Introduction
The weight minimization of the shallow truss structures is a challenging but sometimes frustrating engineering optimization problem. Theoretically, the optimal design searching process can be formulated as an implicit nonlinear mixed integer optimization problem with a huge number of variables. The flexibility of the shallow truss structures might cause different types of structural instability. According to the nonlinear behavior of the resulted lightweight truss structures, a special treatment is required in order to tackle the “hidden” global stability problems during the optimization process. Therefore, we have to replace the traditional “design variables → response variables” like approach with a more timeconsuming "design variables → response functions" like approach, where the response functions describe the structural response history of the loading process up to the maximal load intensity without constraint violation.
In this study, a higher order pathfollowing method [1] is embedded into a hybrid heuristic optimization method in order to tackle the structural stability constraints within the truss optimization. The proposed pathfollowing method is based on the perturbation technique of the stability theory and a nonlinear modification of the classical linear homotopy method.
The nonlinear function of the total potential energy for conservative systems can be expressed in terms of nodal displacements and the load parameter. The equilibrium equations are given from the principle of stationary value of total potential energy. The stability investigation is based on the eigenvalue computation of the Hessian matrix. In each step of the pathfollowing process, we get information about the displacement, stresses, local, and global stability of the structure.
With the help of the higherorder predictorcorrector algorithm, we are able to follow the loadresponse path and detect the hidden bifurcation points along the path in time. During the optimization process, the optimal design is characterized by the maximal load intensity factor along the equilibrium path. Consequently, all the structural constraints are controlled by a fitness function in terms of the maximal feasible load intensity factor. Because the function evaluation is very expensive (for example, we have to call a professional system like ANSYS to carry out an "eigenvalue buckling analysis") we have to select the appropriate populationbased metaheuristic frame very carefully. In everyday language, a populationbased metaheuristic means a good tale usually inspired by the nature, a set of operators, which describes the daily life of the population, and a set of rules which controls the life or death of individuals. In the heuristic frame developing process we applied a "minimal art" like approach to reach the "good quality solution within reasonable time" goal. According to our approach, we decreased the number of operators and tunableparameters, and simplified the significant operators and rules coming from different tales as much as possible.
In this chapter we present the result, which is a simple but very efficient hybrid metaheuristic for truss weight minimization with continuous and discrete design variables, and global and local stability constraints.
The presented "supernatural" ANGEL method [26] combines ant colony optimization (AN), genetic algorithm (GE) and gradientbased local search (L) strategy. In the algorithm, AN and GE search alternately and cooperatively in the design space. The powerful L algorithm, which is based on the local linearization of the constraint set, is applied to yield a more feasible or less unfeasible solution, when AN or GE obtains a solution.
The highly nonlinear and nonconvex largespan and largescale shallow truss examples with continuous and discrete design variables and response curves show that ANGEL may be more efficient and robust than the conventional gradient based deterministic or the traditional population based heuristic (metaheuristic) methods in solving explicit (implicit) optimization problems. ANGEL produces highly competitive results [1618] in significantly shorter runtimes than the previously described pure approaches.
The benefit of synergy can be demonstrated by standard statistical tests. To the best of our knowledge, no such work has been done in the literature for truss weight minimization with response curves. The reason is simple: the question of the global stability loss (the collapse of the structure as a whole) was not investigated very carefully in the truss optimization literature so far, according to a popular but totally misleading "assumption" of the truss optimization community that the local stability loss (local buckling) always precedes the global stability loss (the collapse), therefore the timeconsuming investigation of the global stability is meaningless (see in Hanahara and Tada [20]).
2. Structural optimization
Generally, the traditional implicit “design variables → response variables” weight minimization problem with continuous and discrete design variables can be written as follows:
where
The investigated new "design variables → response functions" weight minimization approach can be described as follows:
where
In the pathfollowing algorithm (details of the nonlinear structural investigation see in [1]), a design is represented by the set of
where
Our feasibilityoriented fitness function is based on the following set of criteria:
Any feasible solution is preferred to any infeasible solution,
Between two feasible solutions, the one having a smaller weight is preferred,
Between two infeasible solutions, the one having a larger load intensity factor is preferred.
The minimal weight design problem can be formulated in terms of member crosssections (a member crosssection may be a continuous variable or discrete value taken from a given catalogue) and nodal point shifts (to modify the shape), and may be constrained by the allowable nodalpoint displacements, element stresses and the global stability requirement which simple means a nonsingular Hessian on the loadresponse path.
We have to mention, that in our study we investigated only truss structures therefore the applied structural model was a large deflection truss model without simplifications. To avoid any type of stability loss even a structural collapse, a pathfollowing approach was used to compute the structural response.
The applied measure of design infeasibility was defined as the maximal load intensity factor subject to all of the structural constraints. Naturally, ANGEL which is presented in the next section can be used in the traditional “design variables → response variables” approach and may be easily adopted for other types of optimization problems including the traditional explicit function minimization problems.
3. ANGEL
First, we have to note, that ANGEL as a name of a combined populationbased metaheuristic for the resourceconstrained project scheduling problem was introduced by Tseng and Chen [15]. We use this name in a different context with a different content. Our ANGEL algorithm, according to the systematic simplification, is based only three operators: random selection (RS), random perturbation (RP), and random combination (RC). In ANGEL the traditional mutation operator was replaced by the local search procedure (L) as a "locally best" form of mutation. That is, rather than introducing small random perturbations into the offspring solution, a gradientbased local search is applied to improve the solution until a local optimum is reached. The first result of our systematic simplification work is trivial: hard to imagine a populationbased heuristic without an RS operator. The RS operator is in a special position in the heuristic community therefore the populationbased heuristic literature is full with many general and problemspecific selection mechanisms (a good overview can be found in the work of Sivaraj and Ravichandran [13]). When we imagine the population as a matrix in which the rows are individuals and the columns are variables and the fitness function values of the individuals form a corresponding column vector, then very easy to identify the two basic selection possibilities: the columnwise (AN like) and the rowwise (GE like) selection mechanisms (see Figure 1). In Figure 1 we used a greyscale to show the fitness of individuals (the lighter the grey color the better the individual) and we assumed that the individuals are ordered according to their fitness values. To demonstrate the possibilities we presented two similar cases (two parents (P2) → one child (C1)).
The AN mechanism selects at least one "more or less good parent" from each column step by step and after that applies the RP or RC operator procedure for each selected variable or variable set independently to get a child, from which L try to make a "better child".
The GE mechanism selects at least one "more or less good parent" in exactly one step for each case. In other words, GE selects at least one complete row. After that the algorithm repeats the previous steps to generate the child by applying the RP or RC operator for each variable or variable set of the selected parent or parents and after that L try to improve the quality of the child to get a "locally best" child.
In AN approach, by definition, the RS means a set of randomly selected "more or less good" element or elements according to the taledependent fitness function. This approach always imitates a "route" independently from its reality. When we imagine a bee flying from flower to flower or a salesperson travelling from city to city, the reality of the abstraction is trivial. But when we have to solve an optimal truss design problem minimizing its total weight on the set of element crosssections as design variables, subject to the displacement and stress constraints, the local and global stability requirements and load conditions and imagine the construction as a whole, then the "from crosssection to crosssection" route may be totally meaningless and misleading abstraction. We may become the slave of the tale, which may yield a "brutalforcesearch" like efficiency, because in our case the function evaluation is very expensive and timeconsuming according to the implicit dependency between the design variables (element crosssections) and the response variables (for example: global stability loss).
According to the optimal structural design problem, it is very easy to imagine the GE selection strategy, in which we select randomly at least one "more or less good design" and after that, according to the other operators of the tale, we try to make a better one (less unfeasible or lighter feasible) by RP or RC.
Easy to imagine, that the combination of the two selection mechanisms may increase the variability of the searching process as a synergism. The two selection mechanisms are very general: from singleparent to multiparents they are able to manage every case using only the RP and RC operators. In this study, "tradition is a tradition" we used the generally accepted operator types. Namely we used the ANP1C1 and GEP2C1 operators alternately and cooperatively using only the RP, RC, and L operators, which are invariant to the selection direction.
In the ANGEL developing process, we tried to simplify the three operators (RS, RP, and RC) and decrease the number of tunableparameters, namely the size of the problemspecific "goldennumber" set, as much as possible, to minimize the time requirement of the socalled "preliminary investigation". In our case, the preliminary investigation may be an "experimental design and analysis" like problem in the problem with terrible large computational cost which yields only 'good" problemspecific goldennumberset after several "tryanderror" iterations.
The flowchart of the proposed simplified heuristic ANGEL method is presented in Figure 2.The main procedure of the proposed hybrid metaheuristic follows the repetition of these two steps:
AN with LS and
GE with LS.
According to the systematic simplification, the hybrid algorithm is based only three operators:
random selection (AN+GE),
random perturbation (AN), and
random combination (GE).
In the presented form, the populationbased ANGEL has only three "tunable" parameters
The parameter pair
which means, that ANGEL is practically a “tuningfree” algorithm.
The monotonically decreasing standard deviation function for each continuous design variable can be defined in the following way:
In our approach, the case of the discrete design variables can be managed in a similar way. The only difference is that we replace the value set with the equivalent index set and carry out all the operations on the index set.
The main procedure of the proposed metaheuristic method follows the repetition of these two steps:
AN with L and
GE with L.
In other words, metaheuristic ANGEL firstly generates an initial population, after that, in an iterative process AN and GE search alternately and cooperatively on the current design set. The initial population is a totally random set. The random perturbation and random combination procedures which are based on the normal distribution, call therandom selection function which uses the discrete inverse method, to select a “more or less good” design (GE) or a set of "more or less good" design variable values from the current population. The higher the fitness values of a design (a design variable value) the higher the chance is that it will be selected by the function (see Figure 3).
The random perturbation procedure uses the continuous inverse method to generate a new solution from the old one (see Figure 4). The random combination procedure generates an offspring solution from the selected mother and father solutions (see Figure 5). The offspring solution is generated from the combined distribution, where the combined distribution is the weighted sum of the parent’s distributions. The two procedures are controlled by the standard deviation, which is decreasing exponentially from generation to generation.
In our algorithm an offspring will not necessarily be the member of the current population, and a parent will not necessarily die after mating. The reason is straightforward, because our algorithm uses very simple rule without explicit pheromone evaporation handling: If the current design is better than the worst solution of the current population than the worst one will be replaced by the better one.
In this work, without loss of generality, we only deal with the two fundamental cases when the design variables are only element (elementgroup) crosssection areas. In the continuous case a crosssection area may be any value from a given interval and in the discrete case a crosssection area has to be taken from a discrete catalogue. Additionally, also without loss of generality, it is assumed that we are interested only in the local and global stability investigation without displacement constraints. We assume that the allowed maximal positive (stretching) stress defined by a constant, and the allowed minimal negative (compressive) stress is constrained by a local buckling function, which is a function of the material properties, the element length, and the element crosssectional area. The global stability investigation is based on the loadeigenvalue curves. From the global stability point of view a truss design is feasible, when during the loading process each loadeigenvalue curve remains in the positive segments up to the end of the process.
In the pure continuous case (when only the crosssection range is fixed) the iterative local search procedure (L) alternates two approaches according to the current feasibility indicator value.
If the current design is feasible, namely:
If the current design is infeasible, namely:
In the pure discrete case (when the crosssections are taken from a catalogue) we have two possibilities to develop a local search procedure.
We can define a simple "thumb rule" used to improve the quality of the generated discrete solutions. The starting base of the thumb rule is a discrete solution given by applying the usual "rounding to the next catalogue value" rule. When the discrete solution is feasible (infeasible) then, in a cyclically repairable process, we try to decrease the crosssectional areas step by step selecting always the "best" element (element group), where "best" means an element (element group) for which the element stress is minimal (maximal) in absolute value.
An improvement, namely a crosssectional area decreasing (increasing) is accepted, when the starting design feasibility level is not decreased by the current modification. The process terminates, when no such an element exists. We have to emphasize that in the presented path following approach the design feasibility is measured by the maximal load intensity factor, and therefore, the designs satisfy the stress constraints up to the maximal load intensity factor computed by the applied path following method.
The other possibility would be a “locally exact” binary formulation.The proposed binary linear (or quadratic) programming (BLP or BQP) approach exploits the fact, that using a "stateoftheart" solver the solution time of a local BLP (or BQP) problem is competitive with the solution time of the "thumb rule" heuristic.
Naturally, a local BLP (BQP) formulation can give better results, as a pure heuristic approach. Using a "dense" catalogue the problem can be managed as linear programming problem, when the catalogue is "sparse", we have to use a quadratic formulation to describe the possible stress changes accurately, in the function of the "local" catalogue values. The immediate predecessor (successor) of the current catalogue value defines the “local catalogue”, for each element (element group), if such a value exists. According to the "local environment", an element (element group) can be described by at least three binary variables.
Naturally, using the standard trick (special ordered set (SOS) constraint management) of the operations research (OR), the formulation which has at least three binary variables, can be replaced by an equivalent formulation which has only at least two binary variables. Let
Let
In the local model exact analytical derivatives were used. To generate the symbolic derivatives, optimized to speed, Wolfram Mathematica 8.0 was used. Naturally, a linearized model can be replaced by a quadratic one, and the simplified assumption that the stress change of membergroup
The local search algorithm, in an iterative process, minimizes the weight increment (maximizes the weight decrement) needed to get a better (a lighter feasible or less unfeasible) solution. The OR formulation follows the conception of the "thumb rule", the "at least as good" quality requirement is managed by nonsmoothed formulation, namely in the formulation the maximal constraint violation is constrained.
Naturally, the nonsmooth
4. Numerical example
4.1. Sizing optimization with buckling constraints  120bar truss dome
In this paper, in order to demonstrate the proposed solution method a wellknown space dome structure is presented as a simple sizing problem, where two basic sub problems, continuous and discrete optimization problems are distinguished.
Saka and Ülker [12], as a continuous optimization problem, have introduced first time the 120bar example. The minimal weight design subjected to structural constraints imposed on the member stress and nodal displacements based on linear and nonlinear analysis. Subsequently, Soh and Yang [14] have been analyzed the same structure to obtain the optimal design related to sizing and configuration variables Kaveh and Talatahari [7] presented a heuristic method where the particle swarm optimizer, ant colony strategy and harmony search are hybridized.Therefore, several techniques have been incorporated to handle the constraints. Similar to Lee and Geem [10], Kelesoglu and Ülker [9], only sizing variables are considered to minimize the structural weight. According to the complexity of the concerned problems, another method has been proposed by Kaveh and Talatahari [8], namely a hybrid big bang–big brunch (HBB–BC) algorithm.The comparisons of numerical results using the HBB–BC method with the results obtained by other heuristic approaches are performed to demonstrate the robustness of the present algorithm. With respect to the big bang–big brunch (BB–BC) approach, HBB–BC has better solutions and standard deviations. In addition, HBB–BC has low computational time and high convergence speed compared to BB–BC. However, when the number of design variables increases the hybrid BB–BC shows better performance. The effects of nonlinear behavior to the optimal results have been investigated by Hadi and Alvani [19] and Lemonge and Barbosa [21].
The geometry and nodal coordinates are presented in Figure 8 and in Table 1. According to the structural symmetry, truss members are grouped into seven membergroups (see in Table 2). The truss is subjected to the given applied external loads in Table 3. The truss members as design variables are grouped into seven group variables (Table 4).
1  0.  0.  7.000 
4  6.01108  3.4705  5.850 
5  6.94100  0.  5.850 
18  10.82532  6.2500  3.000 
19  11.66266  3.1250  3.000 
20  12.50000  0.  3.000 
40  13.76114  7.9450  0. 
41  15.89000  0.  0. 
Load [ 
60  30  10 


Material density 

Stress constraints for tension 

Stress constraints for compression 

T 


12  13  14  15  16  17 
18  19  110  111  112  113  

23  34  45  56  67  78 
89  910  1011  1112  1213  132  

214  316  418  520  622  724 
826  928  1030  1132  1234  1336  
G_{4}  215  317  419  521  623  725 
315  417  519  621  723  825  
827  929  1031  1133  1235  1337  
927  1029  1131  1233  1335  237  
G_{5}  1415  1617  1819  2021  2223  2425 
1516  1718  1920  2122  2324  2526  
2627  2829  3031  3233  3435  3637  
2728  2930  3132  3334  3536  3714  

1438  1639  1840  2041  2242  2443 
2644  2845  3046  3247  3448  3649  
G_{7}  1538  1739  1940  2141  2342  2543 
1539  1740  1941  2142  2343  2544  
2744  2945  3146  3347  3548  3749  
2745  2946  3147  3348  3549  3738 
Refer to the formerly presented papers (e.g. [1618]), in this study, stainless steel tubular crosssections are considered as design variables.According to the thinwall pipe structural behavior, the following local stability constraints are proposed. The stress constraint for against of Eulerbuckling or peripheral shelllike buckling is given in terms of the thickness ratio:
where
The obtained results for continuous problem using linear and nonlinear structural model are compared are presented in Table 5 and in Table 6. Comparing with the results of continuous optimizations shows that GA based approach [19] gives a better minimum weight than the optimality criteria approach [12]. It is observed that further reduction is possible in the weight of the space truss considering the geometrically nonlinear analysis as compared to linear one.
Worthy of note, that the optimal design obtained by the proposed hybrid ANGEL seems much better than the results of previously presented compared methods. Remarkable in this study  using the formula (9) that the related fitness value is
In this paper for discrete optimization problem two types of catalogue values are distinguished, a sparse (case 1) and a dense (case 2) with the following cross sections:
Case 1: {5.0; 10.0; 15.0; 20.0; 25.0;…; 50.0}
Case 2: {5.0; 7.5; 10.0; 12.5; 15.0; 17.5; 20.0; 22.5; 25.0;…; 50.0}
We have to note that the related fitness value is
Linear  Nonlinear  Nonlinear  Nonlinear  

16.66  17.50  10.85  12.968 

44.89  45.56  38.70  8.282 

24.89  25.45  35.40  13.325 

9.66  8.44  5.23  7.964 

21.93  22.30  27.37  8.316 

16.59  15.96  15.30  7.776 

11.74  3.90  3.90  7.990 

8511  7587  7158.6  4650.659 
Using a stateoftheart callable BLP (BQP) solver, for example: CPLEX 12.0, the time requirement of the improved local search is compatible with the time requirement of the traditional "thumb rule" like approach. However, the improved approach is more efficient, because it is able to modify more than one crosssectional area in one iteration.
In the presented computational test, ANGEL was run with the following parameters:
the population size was 100,
the number of generations was 10, and
the maximal number of local search iterations was 10.
We note, that the maximal number of iterations does not necessarily mean that the number of iterations always 10.
4.2. Sizingshaping optimization with stability constraints 24bar truss dome
This academic example has been analyzed by the author previously [17] to demonstrate the difficulties of the stability investigation. The layout and the initial data are presented in Figure 9 and Table 78. At the central node, the load is 0.5, while at nodes 27 it is 1.0 unit.
Linear  Nonlinear  Nonlinear  Nonlinear  

15.00  12.30  10.0  10.0 

46.70  46.70  10.0  10.0 

27.00  27.00  15.0  12.5 

7.05  5.33  10.0  7.5 

27.60  24.70  5.0  5.0 

11.10  17.80  10.0  10.0 

1.82  1.53  10.0  7.5 

7264.6  7229.0  4979.681  4242.075 
1  0  0  8.216 
2  12.50  21.65063509  6.2.16 
3  25.00  0  6.216 
8  0  50.00  0 
9  43.330127019  25.00  0 
The equilibrium path that involves in this case four critical points has been determined inside of the optimization process. First is a single bifurcation (
In this paper, a weight optimization is considered subjected to global stability constraints. The crosssections as design variables are involved into three groups (Figure 9). The load intensity factor is changing from zero to one.
Using the proposed hybrid metaheuristic method, where the number of generations is 10 and the population size is 100, two optimization problems are considered.
In first case, a sizing optimization problem is solved for minimal volume optimization subjected to structural stability. The structure is loaded up to the maximal load intensity factor while the smallest eigenvalue becomes zero. The obtained best solution for the grouped design variables are the following:
In the second case, a sizingshaping optimization problem is presented. The three sizing variables are extended with three shift variables namely the vertical position of all free joints (


Load cases  Nodes  Z 
1  1 

2, 3, 4, 5, 6, 7 


Material properties  Modulus of elasticity 

5. Conclusion
The weight minimization of the shallow truss structures is a challenging but sometimes frustrating engineering optimization problem. Theoretically, the optimal design searching process can be formulated as an implicit nonlinear mixed integer optimization problem with a huge number of variables. The flexibility of the shallow truss structures might causes different type of structural instability. According to the nonlinear behavior of the resulted lightweight truss structures, a special treatment is required in order to tackle the “hidden” global stability problems during the optimization process. Therefore, we have to replace the traditional “design variables → response variables” like approach with a more timeconsuming "design variables → response functions" like approach, where the response functions describe the structural response history of the loading process up to the maximal load intensity without constraint violation.
In this study, a higher order pathfollowing method was embedded into a hybrid heuristic optimization frame in order to tackle the global structural stability constraints within the truss optimization. The proposed pathfollowing method is based on the perturbation technique of the stability theory and a nonlinear modification of the classical linear homotopy method.
In this chapter we presented a simple but very efficient hybrid metaheuristic for truss weight minimization with continuous and discrete design variables, and local and global stability constraints. The presented "supernatural" ANGEL method combines ant colony optimization (AN), genetic algorithm (GE) and gradientbased local search (L) strategy. In the algorithm, AN and GE search alternately and cooperatively in the design space. The powerful L algorithm, which is based on the local linearization of the constraint set, is applied to yield a more feasible or less unfeasible solution, when AN or GE obtains a solution.
The highly nonlinear and nonconvex largespan and largescale shallow truss examples with continuous and discrete design variables and nonlinear response curves show that ANGEL may be more efficient and robust than the conventional gradient based deterministic or the traditional population based heuristic (metaheuristic) methods in solving explicit (implicit) optimization problems. ANGEL produces highly competitive and from engineering point of view safe and accurate results in significantly shorter runtimes than the previously described pure approaches. The benefit of synergy was demonstrated by standard statistical tests. To the best of our knowledge, no such work has been done in the literature for truss weight minimization with nonlinear response curves so far.
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