Structural Robust Optimization Based On Bayesian Method

As an advanced technology to improve product quality, robust optimization design is able to take into account the actual existence of various uncertainties. Performance fluctuation caused by various kinds of uncertainty in reality can be reduced without eliminating the uncertainty source by using robust optimization design. Therefore, this technology has been widely used in many areas, such as manufacturing, aeronautics and astronautics, mechanical design, electronics, finance, chemistry and pharmacy, et al. Such strong demand makes robust optimization design receive wide attention and remarkable achievements. Robust optimization design which is based on classical statistics is completed on a large number of sample data. But in many cases, it is hard to obtain large enough numbers of testing data for a practical engineering problem as limited by time consumption, financial support, technical level, etc. Therefore, a structural robust optimization based on Bayesian method is proposed to eliminate the limit of size of sample set by making full use of prior information in estimation of parameters on the basis of sample through Bayesian theorem in the present paper.At first, Bayesian method is used to estimate the probability distributions of parameters of ground motion models. Estimation of parameters is updated with increasing of sample information when the prior information and sample information are taken into consideration together.Then, because finding a reasonable prior distribution is an important part and a major problem in Bayesian theory, the posterior estimation including point estimation and interval estimation on the two parameters, mean and variance, of random variable with normal distribution is proposed under the same sample set. These results are discussed for different prior distributions and two different cases of sampling, one for a large number of sample and the other for a small number of sample, in order to make a qualitative explanation of how prior distributions influence posterior estimates. The results are also compared with those obtained from classical statistics. It is shown that prior distributions have significant effect on parameter estimation. So it is wise to choose the prior distribution properly when using Bayesian method.Finally, the posterior distribution of parameters is evaluated from samples of random variables and prior distribution by using Gibbs sampling in the framework of Bayesian theory. Optimization is completed by supposing point estimates equal to these posterior means which is looked as the input of uncertainty. In many cases, random variables can be assumed from normal population, so only normal distributions are discussed herein.It is validated in examples that results of robust optimization based on Bayesian method in which prior information is taken into consideration are more accurate than those of traditional robust optimization. It is also validated in the practical problems that robust optimization based on Bayesian method can use less samples to obtain similar results as the classical optimization based on large samples. Therefore, robust optimization based on Bayesian method provides a feasible idea for robust design, especially for cases with finite sample or small number of sample.