Multimodal Reliability Assessment for Complex Engineering Applications using Efficient Global Optimization

As engineering applications become increasingly complex, they are often characterized by implicit response functions that are both expensive to evaluate and nonlinear in their behavior. Reliability assessment given this type of response is difficult with available methods. Current reliability methods focus on the discovery of a single most probable point of failure, and then build a low-order approximation to the limit state at this point. This creates inaccuracies when applied to engineering applications for which the limit state has a higher degree of nonlinearity or is multimodal. Sampling methods, on the other hand, do not rely on an approximation to the shape of the limit state and are therefore generally more accurate when applied to problems with nonlinear limit states. However, sampling methods typically require a large number of response function evaluations, which can make their application infeasible for computationally expensive problems. This paper describes the application of efficient global optimization to reliability assessment to provide a method that efficiently characterizes the limit state throughout the uncertain space. The method begins with a Gaussian process model built from a very small number of samples, and then intelligently chooses where to generate subsequent samples to ensure the model is accurate in the vicinity of the limit state. The resulting Gaussian process model is then sampled using multimodal adaptive importance sampling to calculate the probability of exceeding (or failing to exceed) the response level of interest. By locating multiple points on or near the limit state, more complex limit states can be modeled, leading to more accurate probability integration. By concentrating the samples in the area where accuracy is important (i.e. in the vicinity of the limit state), only a small number of true function evaluations are required to build a quality surrogate model. The resulting method is both accurate for any arbitrarily shaped limit state and computationally efficient even for expensive response functions. This new method is applied to a collection of example problems that currently available methods have difficulty solving either accurately or efficiently. The focus is on forward reliability analysis (calculating the probability of exceeding a specified response level), but some discussion of application to inverse reliability analysis (calculating the response level that corresponds to a specified probability) is included.