Nonlinear problems often 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 can be used in our advantage as they help classify the responses and identify regions of the design space corresponding to distinct behaviors. Support Vector Machines (SVM) is a powerful machine learning technique which enables the explicit identification of specific regions of the design space. As a result, explicit boundaries of failure (or infeasible) regions in terms of deterministic and random design variables are obtained. This allows an easy calculation of probabilities of failure. In addition, the explicit "decision function" can be used in a reliability-based design optimization (RBDO) formulation. However, the function evaluations by finite element analysis for obtaining the response at training points can be very costly. Presently, we are working on updating the SVM classification in order to obtain a more accurate design space decomposition with a minimum number of actual function evaluations (i.e., simulation calls). This approach is general and can be applied to various problems. Of particular interest: crashworthiness, fluid-structure interaction, and aeroelasticity.
Impact of a tube on a rigid wall and global buckling. Global buckling of the tube (Left).Example of explicit design space decomposition with SVM. Problem with 3 variables (Right).
Example of update of limit state function. The blue is the actual limit state function, the red is the predicted decision function.