To cope with the uncertainties existing in an engineering product design, the failure probability methodology is used based on random variables. In this methodology, uncertainties are assumed as random variables and the failure criterion is expressed by the performance function. The failure region is a region where the performance function value is smaller than zero. The performance function includes random variables. The failure probability is obtained by integrating the joint probability density function of the random variables over the failure region.
Most of this integration usually be approximated by numerical computation. Among various numerical methods, the Monte Carlo simulation gives a relatively accurate numerical solution for nonlinear problems and it has an advantage of evaluating the error of the numerical solution. However, it consumes a lot of numerical costs.
The importance sampling technique and the Kriging metamodel application technique have been used to reduce the numerical cost of the Monte Carlo simulation. Several researchers have proposed a numerical calculation method combining the importance sampling and the Kriging metamodel. In this study, the improvement of the numerical method combining the importance sampling method and the Kriging metamodel is proposed.
The improvement of the importance sampling technique is related to the importance sampling function construction. In order to approximate an ideal importance sampling function, a kernel density function based on the points extracted from the failure region is implemented. Points are extracted in the failure region by the Metropolis-Hastings algorithm. With each point as a mean, kernels are assumed. In order to reduce numerical cost, kernels that are far from MPFPs(most probable failure points) are excluded. The MPFP has a large impact on the failure probability. A kernel density function is constructed based on kernels. The importance sampling density is assumed as a kernel density function.
The first improvement of the Kriging metamodel is about a scheme of adding experimental points. The performance function is approximated by the Kriging metamodel. Initially, the Kriging metamodel is constructed based on points extracted from the Metropolis-Hastings algorithm. The added experimental points of the Kriging metamodel are selected through a learning function. But, only points within a certain range are added as the experimental points of the Kriging metamodel.
The second improvement of the Kriging metamodel is about a stop criterion of adding experimental points. The more experimental points are added, the more accurate the Kriging metamodel, but the higher the numerical cost. Therefore, adding the experimental points to the Kriging metamodel should be terminated by the appropriate criterion. The parameter for evaluating the accuracy of the Kriging metamodel is presented focusing the importance region. The proposed methodology is verified by numerical examples.