In engineering, uncertainties are a result of defects in materials properties (Young modulus, density, ...), and manufacturing processes (thickness, other geometrical variables, ...). In the preliminary design phase, these uncertainties are introduced to take into account the much of knowledge inherent in certain design variables.

The deterministic approaches of optimization neglect the effects of uncertainty in the design space; solutions may violate the physical reality. To mitigate this difficulty, we proposed in this paper a stochastic optimization method. The goal of proposed method is to find the robust solutions by introducing additional cost functions (known as the robustness functions) for each original cost functions. This robustness function is defined to have the dispersion of the original cost function. The robustness functions and original functions are evaluated simultaneously. A Monte Carlo sampling is used to take into account uncertainties on the design parameters. This is performed by using an evolutionary multi-objective optimization in order to find all Pareto optimal solutions.

To reduce considerably the computing time, we used a neural network in order to approximate the cost functions and to evaluate the robustness functions. In this case, A multi-layer perceptron is used and a back propagation algorithm is employed to train the network [3]. Neural networks can be used to simulate multi-input, multi-output systems. They have been used as fast structural analysis and design tools. An advantage of neural networks is that they can simulate very complex, highly non-linear systems problems.

The development of a neural network consists of the following steps:

• select the network architecture : number of hidden layers, number of neurones in each layer, the transfer functions of the neurones and the training algorithm,
• train the network, and
• test the network.

The optimal solutions resulting from the stochastic optimization must be analyzed, only a few solutions were examined in detail. This is a typical case that computer produces/accumulates too much data, datamining techniques are needed. One of popular datamining techniques is the Self Organizing Map (SOM) by Kohonen [4]. The SOM is one of the neural network models. The SOM algorithm is based on unsupervised, competitive learning. It provides a topology preserving mapping from the high dimensional space to map, usually from a two-dimensional lattice and thus the mapping is a mapping from high dimensional space onto a plane. The property of topology preserving means that the mapping preserves the relative distance between the points, and each points are near other in the input space are mapped to nearby map units in the SOM. The SOM can thus serve as a cluster analyzing tool of high-dimensional data.

In this paper, the SOM is applied to map Pareto solutions obtained by the stochastic optimization. This will reveal the global tradeoffs between objective functions. Furthermore, from clusters obtained in the SOM, the relations between design variables are mapped onto another SOM. This will indicate the relative importance of design variables and their interactions.

The proposed examples illustrate the interest of suggested methodology to solve stochastic multi objective optimisation problems in structural dynamics

1

D.E. Goldberg, "Genetic Algorithms in Search Optimization, and Machine learning", Addison- Wesley, 1989.

2

J. Andersson, "Multiobjective Optimization in Engineering Design, Institute of Technology", Linköpings Universitet, Sweden, PhD thesis, 2001.

3

C. Zhang, S.H. Huang, "Applications of neural networks in manufacturing: a state-of-the-art survey", International Journal of Production Research Vol. 33, No. 3, 1995, pp. 705-728. doi:10.1080/00207549508930175

4

Y. Kohonen, "Self Organizing Maps", Springer, New York, 3rd edition, 2000.

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