Keywords: structural optimization, multiple local minima, global optimization.

Optimization algorithms generally require many complete structural analyses during their iterative process. Moreover, most structural engineering design optimization problems are multimodal, with several local minima. A common approach to tackle this problem is to construct inexpensive global approximation models of the responses often called metamodels or surrogates. These are based on simulation results obtained for a limited number of designs using global data fitting. The optimization algorithm repetitive analysis needs are all performed using the inexpensive metamodels.

In this study a two-stage approach is employed based on the efficient global optimization algorithm (EGO). First an initial sample of designs is obtained using some design of experiments (DOE) technique. Parallel structural analyses for the initial sample are used to construct a kriging metamodel. A kriging model is a generalized linear regression model that accounts for the correlation in the residuals between the regression model and the observations. The wide use of Kriging is due to its ability to accommodate irregularly spaced data, to model general multimodal functions that have many peaks and valleys, and its exact interpolation of sample values. Another main advantage is that a best linear unbiased predictor as well as its mean squared error have been developed.

In the second stage the metamodel is used to guide the search for promising designs which are added to update the model until a suitable termination criterion is fulfilled. The selection of designs which are adaptively added to the sample is the infill sampling criterion (ISC). ISC should balance the need for improving the value of the objective function with that of improving the quality of the prediction so that one does not get trapped in a local minimum. In the EGO algorithm this balance is achieved through the use of the expected improvement merit function (EIF). The EIF measures the expectation that any point in the design space will produce a better solution than the current best.

In the original EGO algorithm, only one point is added to the sample per iteration. In this study the ISC of the original algorithm is modified to exploit parallelism. In order to include general nonlinear constraints an augmented Lagrangian merit function approach is used. Lagrange multipliers and penalty parameters are estimated and subsequently updated to drive a reduction in constraint violations of the iterates.

The modified EGO algorithm is applied to the optimization of a framed structure. The minimum weight design of the elastic frame is sought subject to stress constraints. It is shown that the problem is multimodal where optimum local solutions can be traced to the different possible load carrying mechanisms. The algorithm is able to identify the global optimum solution in all cases.

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