Ministerio del medio ambiente de Chile

It would be interesting to find the most complex and the simplest structures. Or, in a more general problem, to target the value of the complexity of the structure. Different strategies are possible:

1. keeping fixed the position of the elevation nodes and the topology of the structure and changing the stiffnesses of the single elements;

2. keeping fixed the geometrical and material properties of the elements and changing the

position of the nodes (nodal coordinates);

3. keeping fixed the position of the elevation nodes and changing the topology of the struc- ture by adding/removing elements.

Obviously, a combination of the previous approaches can be done. Anyway, practically, the position of the loads is fixed and thus solution 2. is not applicable. Analogously, it is not possible to increase the number of the elements of the structure, since the cost increases. The more feasible solution seems to be the one that consider stiffness variation across the elements. In this sense, the distribution of stiffnesses in the structural scheme for which the com- plexity is minimised or maximised is worked out in the following.

The strategy for the solution of the problem takes into account the fact that the values of NSCI can be found only after the distribution of stiffnesses is given. In other words, as detailed in the examples of Chapter 4, the evaluation of NSCI is made by:

1. giving geometry and material properties of the elements of the structure; 2. computing the work of deformation of the whole structure;

3. evaluating the work of deformation of the set of fundamental structures and, conse- quently, the ψ-ratios;

4. determining SCI and NSCI values.

Because of the impossibility of inverting the problem and getting a possible distribution of sizes for a given value of complexity, since the problem is highly nonlinear and has a large number of variables, an evolutionary optimisation algorithm is used. This is an implementa- tion of a metaheuristic4_{modelled on the behaviour of biological evolution.}

The idea at the base of the evolution strategies sprouted in the Sixties at the Technical University of Berlin (TUB) in Germany (Rechenberg, 1965, Schwefel, 1965). The solution algorithm is based on the collective learning process within a population of individuals, each of which represents a search point in the space of potential solutions to a given problem. The population is arbitrarily initialised and it evolves toward better and better regions of the search space by means of randomised processes of selection, mutation, and recombination. The envi- ronment delivers quality information (fitness value) about the search points, and the selection process favours those individuals of higher fitness to reproduce more often than those of lower fitness. The recombination mechanism allows the mixing of parental information while pass- ing it to their descendants, and mutation introduces innovation into the population (Bäck and Schwefel, 1993). Evolution techniques are used in many fields, even in structural engineering for topological cost optimisation, see Hajela and Lee (1995) and further publications.

4_{Wikipedia defines a metaheuristic as an higher-level procedure designed to find, generate, or select a lower-level} procedure or a partial search algorithm that may provide a sufficiently good solution to an optimisation problem

Chapter 6 - Digression on structural complexity - 121

In the optimisation problem herein discussed, the dimensions of the element cross-sections are the unknown parameters, the length of the beams and the coordinates of the connections are fixed. In order to simplify the computations, a square cross-section is supposed. As demonstrated in Section 6.2, since the load set represented in Eqn. (6.1) is consistent, the value of the complexity is geometric and loading scale free. In this sense, the size of the elements identifies a class of structures. Because of that, there are ∞1_{possible solutions to the problem}

of complexity maximisation (or minimisation). To overcome this fact, the cross-section size of beam joining nodes A and B is set equal to a reference length (say one). Therefore, the sizes of the other elements are related to the one of the reference beam.

The evolution strategy implemented is based on a simple mutation following a random vector generator. The following procedure is applied: (i) a population of random solutions is generated (as said, element AB is set equal to one throughout the optimisation process). For each solution, (ii) the corresponding structure is defined and its complexity is computed. To achieve complexity maximisation, (iii) the structures are ordered in decreasing order based on their complexity. A subset of cardinality q of the population is chosen (the best q structures) and, randomly, one element Φ of the subset is selected as “father” of the future generation of parameters. (iv) The parameters of structure Φ are varied using a normally distributed random number generator (σ = 0.005) and a novel population (generation) of solutions, i.e. struc- tures, is found. The evolution process, consisting of steps (ii) to (iv) is repeated until the cost function, i.e. the NSCI, does not change between subsequent generations and optimisation is achieved. The initial population is about 24 structures; the size of the bests is about 12 structures (q = 12). The process is stopped at the 750th generation. The following structures are obtained. The structure with minimum complexity has NSCI = 1.6 × 10−6_{, while the}

structure with maximum complexity has a NSCI = 0.9936. The schemes are presented in Figures 6.6 and 6.7.

As shown in the histogram of Figure 6.8, the grey bars indicate that an extremely large number of performance factors is equal to zero (logarithmic scale does not represent properly that the performance factors are 1 × 10−8_{) except one that has performance factor equal to}

one. It is possible to identify the load path in Figure 6.6 with the largest beams. The structure with maximum complexity has a NSCI equal to 0.9936. As reported in the histogram of Figure 6.8, white bars shows that the performance factors are, more or less, around the order of magnitude 10−2_{. Few structures have ψ-values larger or smaller.}

Throughout the years, quicker and more precise evolution techniques have been developed by mathematicians and physicians. For complexity optimisation, the procedure implemented is based on the Covariance Matrix Adaptation Evolution Strategy (CMA-ES), which has been promoted by Hansen and Ostermeier (1996). Briefly and without claiming to be complete, in order to minimise a nonlinear objective function that is a mapping from search space S ⊆ Rn

to R, the search steps are are taken by stochastic variation by means of a normally distributed random vector that is adjusted following the covariance matrix of the population of parents. The effectiveness of the procedure has been tested and the results are positive in terms of computation time (Hansen and Ostermeier, 2001, Hansen and Kern, 2004). Implementing

Figure 6.6: Cross-section dimensions for the reference structure of Figure 6.1 for which the NSCIis minimised (NSCI = 1.6 × 10−6). The size of the beams (which is reported on each element) is proportional to the value of the multiplier of the reference length, i.e. the size of beam AB which is equal to one (top right of both schemes). The scheme is loaded similarly to the one of Figure 6.7.

Chapter 6 - Digression on structural complexity - 123

Figure 6.7: Cross-section dimensions for the reference structure of Figure 6.1 for which the NSCI is minimised (NSCI = 0.9936). The size of the beams (which is reported on each element) is proportional to the value of the multiplier of the reference length, i.e. the size of beam AB which is equal to one (top right of both schemes). The scheme is loaded similarly to the one of Figure 6.6.

10ï3 10ï2 10ï1 100 0 200 400 600 800 1000 1200 Performance factor si Number of occurencies Reference frame Minimum complexity Maximum complexity

Figure 6.8: Histogram nesting the performance factors of the reference structure (NSCI = 0.9389), the maximum complex structure (NSCI = 0.9936) and the minimum complex struc- ture (NSCI = 1.6 × 10−6_).

the algorithm for solving the problem of complexity maximisation and minimisation, better computational times are recorded. This strategy is suited for large structures requiring large computational effort.