TOSCA: a Tool for Optimisation in Structural and Civil engineering Analyses
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Abstract
Many structural engineering problems, e.g. parameter identification, optimal design and topology optimisation, involve the use of optimisation algorithms. Genetic algorithms (GA), in particular, have proved to be an effective framework for blackbox problems and general enough to be applied to the most disparate problems of engineering practice. In this paper, the code TOSCA, which employs genetic algorithms in the search for the optimum, is described. It has been developed by the authors with the aim of providing a flexible tool for the solution of several optimisation problems arising in structural engineering. The interface has been developed to couple the programme to general solvers using text input/output files and in particular widely used finite element codes. The problem of GA parameter tuning is systematically dealt with by proposing some guidelines based on the role and behaviour of each operator. Two numerical applications are proposed to show how to assess the results and modify GA parameters accordingly, and to demonstrate the flexibility of the integrated approach proposed on a realistic case of seismic retrofitting optimal design.
Keywords
Genetic algorithms Parameter tuning Multiobjective optimisation Optimal designIntroduction
According to the number and type of the functions involved, i.e. objectives and constraints, and the input variables, different methods may be used, from closedform Lagrange multiplier (Vapnyarskii 2002) to simplex (Dantzig and Thapa 1997) and gradientbased methods (Byrd et al. 1987; Box et al. 1969). For problems where multimodality, multiple objectives or noncontinuous variables/functions are present, or when functions involved are not known explicitly (blackbox problems), iterative methods are the only feasible approach. Among them, growing interest towards metaheuristics has been manifested by the scientific community. In the definition by Sörensen and Glover (2013) “a metaheuristic is a highlevel problemindependent algorithmic framework that provides a set of guidelines or strategies to develop heuristic optimization algorithms”. The most important metaheuristics, mainly developed in the 1970s and 1980s, but still in use in the field of numerical optimisation, are evolutionary strategies (Beyer and Schwefel 2002), genetic algorithms [GA (Holland 1975; Goldberg 1989)] and simulated annealing (Kirkpatrick et al. 1983). A comprehensive review of works approaching structural optimisation by means of metaheuristic algorithms may be found in Zavala et al. (2014).
Genetic algorithms have been extensively used by the authors in a variety of problems (Amadio et al. 2008; Chisari and Bedon 2016; Chisari et al. 2015, 2017,2016; Poh’sie et al. 2016a, b). This research effort has led to the creation of the software application TOSCA, acronym for Tool for Optimisation in Structural and Civil engineering Analyses. Written in C#, TOSCA code builds on preliminary work developed by Lucia (2008) and Zamparo (2009) for the optimal design of steel–concrete composite bridges, and has been extended to general optimisation problems in Chisari (2015). The aim of this paper is to describe the general structure of the code (“Main principles and interface”), giving an insight on the operators implemented and some recommendations for their selection and postanalysis assessment (“Tuning GA parameters”) and providing some examples to show the effectiveness of the approach proposed (“Examples”). Some conclusions are finally drawn in “Conclusions”.
Main principles and interface
Motivations and general concepts
Many tasks in structural engineering may be transformed into optimisation problems and solved accordingly. However, such an approach is not widespread in the professional community and sometimes not even in academia. In the authors’ opinion, the reason is threefold. Firstly, the most efficient way of solving an optimisation problem depends on the particular formulation of the objective function: see for example Zou et al. (2007) as an optimal design problem, or Bedon and Morassi (2014) as a parameter calibration problem. It implies that a considerable, problemdependent research effort should be directed to understanding the problem structure and selecting the best approach to its solution. Secondly, interfacing an optimisation software application with a finite element (FE) solver, which is a typical requirement in structural optimisation problems, demands programming skills which are outside the expertise of a designer. Finally, a structural engineer is often familiar with the issues of modelling a structure and performing an FE analysis, but has no specific competence on optimisation methods, parameters and appraisal of the results.

Rather than developing a specific formulation for each problem, it may be more convenient to provide a tool that, although possibly less efficient, is general enough to be applied routinely. In this context, GA has the capability to be applied to continuous/discrete, differentiable/nondifferentiable, analytical/blackbox, mono/multiobjective optimisation problems. The previously mentioned examples (Zou et al. 2007) and (Bedon and Morassi 2014) were solved as blackbox problems by means of GA in Chisari and Bedon (2016) and Chisari et al. (2015), respectively.

If the optimisation programme had an interface with an FE solver, it would release the analyst from the programming need, helping focus on the structural problem under study.

Although the problem of optimally tuning GA parameters is a longlasting issue in the scientific community and an optimisation problem per se, it is in the authors’ opinion and experience that some simple rules based on experience can be formulated.
The main aim of this paper is to provide a comprehensive description of the TOSCA software system and show how all these points can be accomplished. Academic or commercial licences for the program can be released upon request to the authors.
Overview of genetic algorithms
Genetic algorithms are a metaheuristic approach to optimisation developed in the 1970s and 1980s by Holland (1975) and Goldberg (1989), and based on the concepts of adaptation and evolution taken from natural biology. Resembling what happens in nature, a set (population) of potential solutions (individuals) evolves (i.e. increases its average fitness) through the application of specific operators.
The first step of the procedure consists of the chromosome definition for the problem under study and its correct representation. The chromosome collects the parameters which are varied during the process. While the genotypic representation of the genes is called a chromosome, each phenotypic instance is an individual, and a population is a collection of different individuals. Each parameter (called a gene) may be a binary variable (original formulation), a discrete variable (integer representation) or a continuous variable (realcoded GA). The initial population can be generated randomly or with quasirandom techniques. The correct generation of the initial population is of the greatest importance for the GA analysis. It should be well distributed over the parameter space to let the algorithm explore all possibilities.
Processing a population consists of evaluating the objective function values for each individual; after that ranking and selection are applied. During the former phase, the results of the evaluation are inspected and the population is ranked according to fitness. In the simplest monoobjective problem, the fitness is equal to the objective function value, sometimes suitably scaled. If the optimisation formulation contains any constraints, the fitness value must account for constraint satisfaction too (MezuraMontes and Coello Coello 2011). If the problem is multiobjective, ranking may be based on nondomination and crowding distance, as in NSGAII (Deb et al. 2002). Selection is the operator responsible of creating a “mating pool”, i.e. a set of individuals that will be coupled to apply the crossover operator. With selection, no different individuals are created, but the previous population is rearranged in such a way that the most promising individuals are cloned and the worst deleted. Afterwards, a new population is generated: given two parents, two offspring are generated through application of the crossover (or recombination) operator, with a probability p_{c}. For the sake of completeness, it must be pointed out that, while this approach is the most widespread, some crossover operators handling more than two parents and creating more than two children have been proposed in the literature (Sánchez et al. 2009).
To improve convergence, an elitist approach can be used, in which the best N individuals are always placed (without undergoing the crossover operator) in the subsequent population. Once the new population has been created, mutation is applied to some individuals. Basically, mutation consists of randomly changing some genes of an individual according to a probability p_{m}; it is useful to prevent the loss of diversity in the population, but it is highly disruptive with respect to convergence. For this reason, special care must be taken in the choice of both the type and probability of mutation.
Operators
Operators implemented in TOSCA
GA phase  Operator  Parameters  References 

Representation  Integer  –  – 
Initial population  Random  Population size  – 
Sobol  Population size  Sobol (1967)  
Diagonal  Population size  –  
From file  Initial population file, population size  –  
Replacement  Children  –  – 
Elitism  Number of elitist individuals  Srinivas and Patnaik (1994)  
Competence  –  –  
Ranking  Normalizing  \(\gamma \in \left] {0, 1} \right[\)  Gen and Cheng (1997) 
Linear  \(\alpha_{\text{r}} \in \left[ {1,2} \right]\)  Hancock (1994)  
Exponential  \(\alpha_{\text{r}} \in \left] {0,1} \right]\)  Hancock (1994)  
Selection  Tournament  Tournament size T_{s}  Goldberg and Deb (1991) 
Roulette wheel  –  Goldberg (1989)  
Stochastic universal sampling (SUS)  –  Baker (1987)  
Crossover  Multipoint  Number of crossover points, crossover probability p_{c}  Goldberg (1989) 
Directional  Crossover probability p_{c}  Michalewicz et al. (1994)  
Fixed arithmetical  Crossover probability p_{c}  –  
Probabilistic arithmetical  Interval modifier α, crossover probability p_{c}  Michalewicz (1996)  
Discrete  Number of offsprings, crossover probability p_{c}  Goldberg (1989)  
Blendα  Interval modifier α, crossover probability p_{c}  Eshelman and Schaffer (1992)  
Mutation  Aleatory  Mutation probability p_{m}  – 
Directional  Mutation probability p_{m}  –  
Local  Nondimensional mutation range, mutation probability p_{m}  –  
Constraint penalty function  Statical  Constraint weight  – 
Dynamic  Constraint weight  Gen and Cheng (1996) 
Interface
Once the GA framework has been completely defined by selecting the values for parameters listed in Table 1, the optimisation process consists of evaluating each individual in the population for a number of generations. Therefore, the user’s task entails instructing the programme on how to perform this basic operation. In other words, the user must specify: (a) the variables to write on the input file for the single evaluation; (b) the format of the input file; c) the actions to carry out to perform a single evaluation; (d) the variables to be extracted from the output file of the single evaluation and how to combine them into objectives and constraints; (e) the format of the output file.

GA_Parameters.txt. This file contains the information regarding operators and parameters described in Table 1. The typical syntax of an instruction for the program is: operatorName: value.

InputVariables.txt. This file contains all information regarding the input parameters in the optimisation problem (variables x in Eq. (1)). They can be defined as constant (variableName: value), variable within an interval (variableName: lowerBound, upperBound, increment), variable with predefined values extracted from a file (variableName: filename_columnName), or depending on other variables (variableName: expression).

OutputVariables.txt. In this file, the output parameters, objectives and constraints are defined. The variables may be extracted from the output file (variableName) or evaluated from other variables (variableName: expression); the constraints are defined as constraintName: expression, weight, lowerAdmissibleValue, upperAdmissibleValue; the objectives are defined as objectiveName: expression, minimizemaximize, [tol]. tol is an optional tolerance for the objective value.

InputTemplate. This file (whose actual name is defined in GA_Parameters.txt) is copied in the directory where the single evaluation is performed. When the special string < !variableName! > is encountered in the template, this is replaced by the actual value of the input variable variableName defined in InputVariables.txt.

OutputTemplate. This file (whose actual name is defined in GA_Parameters.txt) should have the same structure as the output file written by the script at each evaluation. Each variable variableName is identified by its position with respect to a reference referenceName. This is the multiline text block included between < !referenceName!b! > and < !referenceName!e! > in the template. The position of the special string < !variableName!referenceName! > with respect to this block determines the location of the variable value in the actual output file.

Script.bat. This file (whose actual name is defined in GA_Parameters.txt) is the script that will be run by the optimisation process at each evaluation. It is responsible of performing the analysis reading an input file with the structure defined in InputTemplate and writing an output file with the structure defined in OutputTemplate.
Tuning GA parameters
As stated in Sörensen and Glover (2013), GA should be considered as a framework more than a procedure, since many operators must be selected and calibrated for the problem at hand. Since the behaviour of the algorithm depends not only on the individual operators, but also on how they interact with each other, they are usually tuned by using a trialanderror procedure. Some recommendations are provided in this section.
Initial population generation, population size and number of generations
The basic rule for the generation of the initial population is that it should sample as much genetic material as possible. Considering this point, the problem of generating a good initial population resembles that of creating an optimal design of experiment (DOE) uniformly filling the sampling space. For this reason, it is well known (Sloan and Woźniakowski 1998) that lowdiscrepancy (or quasirandom) sequences as Sobol’s (1967) are to be preferred over pseudorandom generators.
The population size (\(P_{\text{s}}\)) is linked to the number of generations (\(N_{\text{g}}\)) by the need to limit the computational time of the analysis. The single run (individual evaluation) may be expensive in terms of the computing effort, i.e. it may be a nonlinear static or dynamic structural analysis, and so the number of total evaluations \(N_{\text{e}} = P_{\text{s}} \cdot N_{\text{g}}\) must be limited. The hypothesis here is that \(N_{\text{e}}\) is fixed (since, once the analysis time of a single run is known, the user can decide how long the optimisation process should last). To perform \(N_{\text{e}}\) runs, one can choose to have a large \(P_{\text{s}}\) and small \(N_{\text{g}}\) or vice versa.
Under this hypothesis, large population sizes reduce the power of GA. This is easily understandable, since, as a limit, with \(P_{\text{s}} = N_{\text{e}} ,\) the process reduces to a simple random search. On the contrary, some researchers (Krishnakumar 1990) have developed microGAs, i.e. a GA with small population size (typically only 5–10 individuals) that is repeatedly run for short durations and then restarted (while keeping a few optimal solutions from the previous runs) until convergence is achieved.
Apart from these extreme examples, the authors have usually found good results by applying the empirical rule \(P_{\text{s}} = N_{\text{g}}\), as a first attempt. Notwithstanding, a test in which the population size is increased stepbystep until convergence is always suggested.
Genetic drift control
In Fig. 3, the normalised standard deviation of each variable in the parameter space (a measure of how much it is distributed in the space) is plotted against the generation number. It is clear that, thanks to its deterministic nature, SUS is the only selection method able to maintain diversity in the population even in the case of small populations. The other selection procedures suffer from genetic drift and at about the generation 7 (tournament) and 30 (roulette wheel) the entire population consists of copies of just one individual.
Selection parameters
In general, the optimisation analysis should be a compromise between exploration and refinement, i.e. the need of covering the parameter space exhaustively and the need of accelerating the process by discarding the areas where poor solutions have been found so far, and focusing on the regions where particularly good individuals are present.
“In the GA, selection operation should be designed so as to gradually narrow the probability distribution function (p.d.f.) of the population, and the crossover operation should be designed so as to preserve the p.d.f. while keeping its ability of yielding novel solutions in finite population case.”
This functional specialisation hypothesis clearly divides the responsibilities of the operations. The selection operator, which mainly utilises the fitness values of solution candidates rather than their location information, should encourage the population in convergence towards an optimum, and crossover operation, which does vice versa, should explore the promising regions identified by the selection operation.
The selection pressure becomes almost equal for linear scaling with α = 1.8 and exponential scaling with α = 0.986 (Hancock 1994). The suggested values for an ordinary monoobjective optimisation problem are linear rank scaling with α = 1.5–1.8. In case of multiobjective problems and NSGAII (see “NSGAII”), selection is responsible of promoting more isolated points and thus a value α = 2.0 generally leads to good results.
Crossover parameters
The role of crossover is to explore the region of the space identified by the selection operator. Therefore, “the distribution of the offsprings generated by crossover operators should preserve the statistics such as the mean vector and the covariance matrix of the distribution of parents” (Kita and Yamamura 1999). A very good review of different crossover operators and optimal tuning of their parameters can be found in Someya (2008, 2012).

Multipoint and discrete crossovers have no parameters to tune, and they automatically satisfy the functional specialisation hypothesis.

Fixed arithmetical and directional crossovers have no parameters to tune, and they cannot satisfy the functional specialisation hypothesis.

Probabilistic arithmetical and blendα crossovers may satisfy the functional specialisation hypothesis if they are properly calibrated.
The probability of crossover should be set high (85–00%).
Elitism
Elitism can be useful to prevent the loss of good individuals. However, the user must be aware that it is another source of selection pressure together with scaling pressure (Someya 2011). So, it should be used when the nonconvexity and discontinuity of the function may cause the algorithm to lose good individuals. Another common situation is when the parameter space is too large and the population size limited, it is necessary to increase the disrupting power of mutation to explore the space as much as possible. In this case, good individuals may be easily lost. In any case, not more than one or very few elitist individuals should be considered.
Mutation
The basic objective of mutation is to increase the exploration of the parameter space. To limit its disrupting power, its probability should be kept under 1–2%. A special case is the mutation called local in subsection “Operators”. If associated with probabilistic arithmetical crossover, it can produce a hybrid operator which has an intermediate behaviour between probabilistic arithmetical crossover and BLXα. To obtain this, the mutation probability should be set at a high value (20–30%) and the additional parameter measuring the subset of the original range could be around 10%.
NSGAII
When the number of objectives is greater than one, the general solution of the optimisation problem is represented by the Pareto front (PF), composed of nondominated individuals (Miettinen 1999). Thanks to the implicit parallelism of population processing, genetic algorithms are naturally designed to converge to a set of solutions instead of a single one and thus are clearly superior to gradientbased methods. In this context, the stateofart approach to multiobjective optimisation is represented by the nondominated sorting genetic algorithm II (NSGAII) (Deb et al. 2002). It exploits the concepts of nondomination ranking and crowding distance to reach convergence to the PF while maintaining diversity in the population. At the end of each generation, the individuals are ranked based on nondomination fronts. The first front is composed of individuals which are not dominated by any other in the population; the second front by those dominated only by the first front and so on. Inside each front, the individuals are ranked according to a densityestimation metric, called crowding distance, which represents a measure of how close (in terms of objective values) an individual is to its neighbours, and more isolated points are favoured to increase diversity in the population. Even though in the original formulation the domination ranking is associated with tournament selection, this is not mandatory, and, as stated above, stochastic universal sampling is herein suggested as a selection operator. Constraint satisfaction is imposed without the use of any penalty function, but directly in the ranking stage. When comparing two individuals, if one satisfies the constraint and the other does not, the former is considered better than the latter regardless of the objective values; otherwise domination and then crowding distance govern the comparison between individuals.
 (i)
Refinement is achieved by a broader elitism operator (competence in Table 1), in which P_{s}, more fitting individuals among the 2 P_{s} belonging to the current and the previous generation, undergo the selection operator (while in ordinary GA without elitism, the P_{s} individuals belonging to the current generation do).
 (ii)
Exploration is achieved by the selection operator based on crowding distance. When comparing two individuals in tournament selection, or when ranking the population in SUS or roulette wheel, the objective value is not taken into account, whilst the fitness is based on the crowding distance value. This encourages exploration.
Suggested values for the GA implemented in TOSCA
Parameter  Suggested value  

Monoobjective  Multiobjective  
Population size  \(\sqrt {N_{\text{e}} }\)  \(\sqrt {N_{\text{e}} }\) 
Number of generations  \(\sqrt {N_{\text{e}} }\)  \(\sqrt {N_{\text{e}} }\) 
Initial population  Sobol  Sobol 
Selection  SUS  SUS 
Scaling  Linear α = 1.5–1.8  Linear α = 2.0 
Crossover  Blendα α = 2.0 p_{c}= 0.85–1.0  Blendα α = 2.0 p_{c}= 0.85–1.0 
Mutation  Aleatory p_{m}= 0.005–0.01  Aleatory p_{m}= 0.005–0.01 
Replacement  Elitism n = 1  Competence 
It is underlined herein that there cannot exist an algorithm which is the most efficient and effective for all optimisation problems (Wolpert and Macready 1997). The guidelines suggested in this paper regard optimisation problems where the number of variables does not exceed ten and the number of objectives is less then 4–5. Largescale problems (Mohamed 2017) and manyobjective problems (Farina and Amato 2004) require specialised algorithms which are not covered in this work.
Examples
Minimisation of a multimodal function

Initial population generation: Sobol sequence.

Population size: 100 individuals.

Number of generations: 100.

Crossover probability: 1.0.

Type of crossover: probabilistic arithmetical with parameter α = 2.1.

Mutation probability: 0.0.

Type of selection: SUS.

Linear scaling pressure: 1.4.

Replacement: elitism, 5 individuals.
Since GAs significantly rely on random procedures, to be sure that the solution is valid it is usually a good practice to perform more than one analysis with different random seeds. A random seed (or seed state, or just seed) is a number used to initialise a pseudorandom number generator, such as the one used by computers. This has been done, and the resulting solution has a value \(f_{\text{opt}} = 0.301,\) close to the solution previously found.
Considering that this paper is directed to users more than theoretical analysts, it seems useful to summarise some general guidelines to assess the results of an optimisation analysis. A good analysis should comprise two complementary stages. In the first part of the analysis, the average standard deviation of the fitness function should decrease considerably, until the population is concentrated around the best solution(s) found so far (“Convergence” in Fig. 11b). Afterwards, the analysis should explore the most promising area(s), in the search for the optimum (“Refinement” in Fig. 11b). At this stage, it is very important to maintain a minimum diversity in the population (nonzero standard deviation, unlike Fig. 8). As an empirical rule of thumb, good results have been generally observed when the two stages were approximately of the same duration (compare for instance Figs. 9 and 11). In the literature, hybrid methods have been proposed, in which the two stages are assigned to two different algorithms, i.e. simple GA for the “convergence” phase and a local gradient/nongradient method for the “refinement” one (Mahinthakumar and Sayeed 2005). This is not explored in this paper.
Optimal design of nonlinear viscous dampers
List of the selected seismic events at the damage limit state
Earthquake ID  Station ID  Earthquake name  Date  Mw  Epicentral distance (km)  PGA_{x} (m/s^{2})  EC8 site class 

2142  ST2557  South Iceland (aftershock)  21/06/2000  6.4  15  1.2481  A 
474  ST1258  Ano Liosia  07/09/1999  6  14  2.3842  B 
65  ST28  Friuli (aftershock)  15/09/1976  6  14  1.3841  B 
83  ST50  Volvi  20/06/1978  6.2  29  1.3649  C 
1635  ST2487  South Iceland  17/06/2000  6.5  13  1.2916  A 
83  ST50  Volvi  20/06/1978  6.2  29  1.3649  C 
2142  ST2488  South Iceland (aftershock)  21/06/2000  6.4  11  4.1226  B 
An FE model of the structure was created in ABAQUS 6.9 (Dassault Systemes 2009). The structural members were modelled as isotropic elastic beam elements B32, having the mechanical properties of the gross section without considering the steel reinforcement. Lumped masses accounting for selfweight of the members and additional vertical loads were applied at the nodes, with massproportional damping characterised by coefficient α_{d}= 0.45. The dampers to be designed were modelled as DASHPOTA element with nonlinear behaviour defined in the input file by a force–velocity piecewise linear law. This law is evaluated for each element type by means of Eq. (3) after defining the value assumed by the damping constant c and the exponent α. Fixed restraints were applied at the base of each column.
According to EC8, when nonlinear time history analysis with at least seven ground motions is performed used in the design, average quantities are to be used for the structural checks. In particular, at the DLS, the standard prescribes the interstorey drift to be less than 0.005 h for buildings having nonstructural elements of brittle materials attached to the structure, where h is the interstorey height. The bare frame without any dampers does not satisfy this prescription, as all interstorey drifts are outside the minimum limit. Thus, a retrofitting system is needed.
 (a)
The number of devices. This affects the costs associated with the intervention, as it would be preferable to locate as few dampers as possible.
 (b)
The damping constant. Stronger (i.e. with higher c) dampers are usually more expensive; for illustrative reasons the single damper cost will be herein assumed as proportional to its damping constant.
 (c)
The maximum forces transferred by the devices. If these forces are very high, expensive local reinforcement actions must be carried out on the existing structure.
Theoretically, knowing the relative weight of each of these factors it could be possible to express the total cost of the intervention and carry out the design by minimising it. However, this may be difficult in the preliminary stage, and it is preferable to carry out a multiobjective optimisation, postponing the choice of a single solution a posteriori.
In Eq. (4), c_{i} is the damping constant of the ith device type (with N number of damper types), which can assume values inside the interval \(\left[ {c_{\text{li}} ,c_{\text{ui}} } \right],\) while i_{i} is a binary variable which is equal to 1 if the ith damper is applied and 0 otherwise. Hence, f_{1} counts the number of dampers actually applied to the structure; f_{2}, according the hypothesis stated above, is proportional to the cost of the dampers; f_{3} is equal to the maximum force F_{max} transferred by the dampers, evaluated considering all dampers and all earthquakes. The constraint required by the code is enforced on \(\bar{d}_{j} ,\) which is the maximum interstorey drift of the jth storey averaged over the considered earthquakes and N_{S}= is the number of storeys. The lower and upper bounds for c_{i} were set as c_{li}= 0.0 kN (m/s)^{−α} and c_{ui} = 3000.0 kN (m/s)^{−α}. The damping coefficient was fixed as α = 0.15.

Initial population generation: Sobol sequence.

Population size: 20 individuals.

Number of Generations: 20.

Crossover probability: 1.0.

Type of crossover: Blend α with parameter α = 2.0.

Mutation probability: 0.005.

Type of selection: SUS.

Linear scaling pressure: 2.0.

Replacement: competence.
For multiobjective optimisation analyses, a plot as that in Fig. 11b allowing the assessment of the convergence, would be desirable. However, it cannot be created simply considering each objective separately: if the optimum of one objective implies low fitness in another objective (largespanning Pareto front), the average value of the former objective in the population does not approach the minimum at convergence like in the monoobjective case. For this reason, the domination front, evaluated at the end of the analysis considering all individuals, was used here as a metric. According to this rule, at the end of the analysis, domination front equal to 0 was assigned to the individuals belonging to the PF, 1 to those only dominated by the former, and so on. The domination front represents a sort of fitness value for each individual, and hence it is possible to describe the evolution of the population towards the solution, i.e. the Pareto front, by means of only one indicator, independently from the number of objectives.
Optimal solutions in the analysis with different population sizes
Population size  Minimum objective  f _{1}  f _{2}  f _{3}  C_{1} [kN (m/s)^{−α}]  C_{2} [kN (m/s)^{−α}] 

20  f _{2}  5  7880  1774  1794  1249 
f _{3}  5  8003  1714  1800  1300  
30  f _{1}  4  7702  2422  2477  1374 
f _{2}  6  7658  1312  1234  1362  
f _{3}  6  7684  1300  1239  1364 

Conversely, increasing the number of damper groups to six (Fig. 19b), it is possible to decrease the maximum force on the structure by 25% compared with the same optimum solution.

It is interesting to note that all solutions displayed in Table 4 are characterised by similar f_{2} values, i.e. cost. In this specific case, the designer has the possibility of selecting the final solution considering a posteriori if the primary cost of the device has a higher incidence than the reinforcement intervention on the existing structure.
Considering the results of this analysis, it appears clear that the population size selected as first attempt was excessively small and by increasing this parameter more fitting solutions can be found. It must be pointed out, however, that further enlargement of the population is not allowed by the need of limiting analysis time.
Conclusions
In this paper, an optimisation tool called TOSCA, specifically designed for structural and civil engineering problems, is described. It makes use of genetic algorithms and has a general interface which can easily be adapted to external solvers without the need of programming. Along with the description of the code, a thorough discussion on the role of each GA operator is presented, with the aim of giving some indications on parameter setting and analysis assessment. Basically, the initial population should be generated by algorithms able to explore efficiently the parameter space, a problem analogous to that of experimental design. Genetic drift, a phenomenon reducing the chromosome variability in the case of small populations, may be avoided by using stochastic universal sampling as the selection algorithm. To obtain a good balance between exploration of the space and refinement of the solution, the functional specialisation hypothesis prescribes that the crossover operator should not change the population variance of the mating pool in the parameter space, while the selection operator should gradually decrease it in the objective space.
Two examples are presented. The first example is the minimisation of an analytical function of ten variables (Rastrigin function). The large number of local optima makes it ideal to test the ability of GA to reach convergence without being trapped in the local minima. The importance of tuning scaling pressure according to the crossover type utilised and of applying the mutation operator is shown. Even though the algorithm is not able to reach the real optimum, very good nearoptimal solutions may be found by selecting the GA parameters according to the strategy presented above.
The second example concerns the optimal design of a seismic retrofitting system composed of nonlinear dampers for RC frames. It is shown how the design problem may be formulated as a multiobjective constrained optimisation problem. It is a mixedinteger optimisation problem, since some binary variables are used to model yes/no decisions, i.e. whether apply the damper or not. From the point of view of genetic algorithms, the presence of constraints, integer variables and multiple objectives can be easily handled. The results show the effectiveness of the approach, which is able to find solutions remarkably decreasing the interstorey drifts (which did not satisfy the standard prescription for the bare frame), but minimising different cost items. It is shown that by increasing the population size, a more fitting solution may be found, but this approach is necessarily limited by the computational effort needed by a single analysis.
Notes
Acknowledgements
The authors wish to thank Mr. Enrico Parcianello for his collaboration in the second numerical example reported in this paper.
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