Keywords: uncertainty, multiobjective optimization, reliability, stochastic processes, approximation concepts, asymptotic methods.

When a structure is being designed the environmental loads that the built structure will experience in its life time are highly uncertain. The uncertain load time history needed in the dynamic analysis of a structure subjected to environmental loads such as earthquake, wind, water wave excitation, and aerodynamic turbulence is an uncertain value function, and it is best modeled by a stochastic process [1]. If the structural parameters are known precisely, the system response and system reliability can be calculated using well-known techniques from random vibration theory. In the more realistic case, the values of the structural parameters are uncertain and they can have a significant effect on the behavior of the structure. Therefore, it is necessary to consider their effect explicitly during the optimization process. In this paper, probabilistic methods are used for incorporating system uncertainties in the analysis by describing the uncertainties as random variables with a prescribed joint probability density function.

In practical optimization problems, usually more than one objective is required to be optimized. Generally, the approach in multiobjective optimization is to transform the original problem into a scalar problem which contains the influence of all objectives [2]. In the present work, the objective functions and constrained system responses are treated as design criteria characterized by a range of values and a possibility distribution that describes the preference of using a particular value within the range. The constrained system responses take the form of conventional structural parameters such as forces, stresses, and displacements, or other parameters such as costs and structural reliabilities. Once the possibility distributions for each design criterion have been defined, an overall design evaluation measure is obtained by a preference aggregation rule [3]. Such measure is then used as the objective function of the optimization process. The solution of the original optimization problem is replaced by the solution of a sequence of explicit approximate problems. These approximate problems are generated by constructing high quality approximations for system responses by using approximation concepts [4].

Structural reliabilities are evaluated and written in terms of the solution of a general linear structural system for a class of stochastic excitation. In this study, attention is directed toward problems in which the stochastic excitation is a stationary Gaussian white noise process with zero mean. A white noise process is a process whose power spectral density function is constant over the whole spectrum. Because of its mathematical simplicity, it is often used as an approximation to a great number of physical phenomena. As previously mentioned, response predictions made during the design process are usually based on system models with uncertain parameters since the properties which will be exhibited by the system when completed are not known precisely. System responses and system reliabilities that account for the uncertainties in the system parameters are given by the total probability theorem as particular integrals over all the uncertain parameters. In practice, these multidimensional integrals rarely, if ever, can be integrated analytically. Asymptotic methods are used here to provide accurate estimates of multidimensional probability integrals [5]. This technique is based on the Laplace's method for asymptotic approximation of multidimensional integrals.

The proposed methodology provides a general framework in which the optimal design of complex structural systems with uncertain properties subjected to stochastic excitation can be determined. The use of approximations proves to be efficient for the numerical implementation of the method. Numerical results have also shown that uncertainty in the model parameters may cause significant changes on the reliability of the system. In these situations, the errors or uncertainties in the specification of the system properties should be properly accounted for during the optimization process, since if they are not accounted for, the performance and reliability of the optimal design can be affected significantly. Therefore, under uncertain conditions, it is recommended to use the approach presented in this study instead of classical optimization approaches.

1

T.T. Soong, M. Grigoriu, "Random Vibration of Mechanical and Structural Systems", Prentice-Hall, Inc. Englewood Cliffs, N.J., 1993.

2

C.L. Hwang, S.M. Masud, "Multiple Objectives Decision-Making Methods and Applications", Notes in Economics and Mathematical Systems, Springer-Verlag, New York, 164-180, 1979.

3

K.N. Otto, "A Formal Representation Theory for Engineering Design", PhD Thesis, California Institute of Technology, Pasadena, California, May, 1992.

4

A.E. Sepulveda, H.L. Thomas, "New Approximation for Steady-State Response of General Damped Systems", AIAA Journal, Vol. 33, No 6, 1127-1133, 1995. doi:10.2514/3.12459

5

N. Bleistein, R. Handelsman, "Asymptotic Expansions for Integrals", Dover, New York, N.Y. 1986.

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