A convex hull approach for the reliability-based design optimization of nonlinear transient dynamic problems

Nonlinear transient dynamic problems exhibit structural responses that might be discontinuous due to numerous critical points. The discontinuous behavior hinders classical gradient-based or response surface-based optimization. However, these discontinuities help to classify the system's responses and identify regions of the design space corresponding to distinct dynamic behaviors. In this paper, data mining techniques are employed to group the responses into clusters. The regions of the design space corresponding to the clusters of unwanted behaviors are delimited with convex hulls. This allows an explicit definition of the boundaries of the failure region in terms of the design variables. In addition, the identification of response clusters, within which the responses might be considered continuous, enables the use of traditional response surface approximation for optimization. The proposed approach is applied to the reliability-based design of a tube impacting a rigid wall, which is optimized for a prescribed dynamic behavior.