An Investigation of Nonlinearity of Reliability-Based Design Optimization Approaches

Because deterministic optimum designs obtained without taking uncertainty into account could lead to unreliable designs, a reliability-based approach to design optimization is preferable using a Reliability-Based Design Optimization (RBDO) method. A typical RBDO process iteratively carries out a design optimization in an original random space $(X$-space) and a reliability analysis in an independent and standard normal random space $(U$-space). This process requires numerous nonlinear mappings between $X$- and $U$-spaces for various probability distributions. Therefore, the nonlinearity of the RBDO problem will depend on the type of distribution of random parameters, since a transformation between $X$- and $U$-spaces introduces additional nonlinearity into the reliability-based performance measures evaluated during the RBDO process. The evaluation of probabilistic constraints in RBDO can be carried out in two ways: using either the Reliability Index Approach (RIA), or the Performance Measure Approach (PMA). Different reliability analysis approaches employed in RIA and PMA result in different behaviors of nonlinearity for RIA and PMA in the RBDO process. In this paper, it is shown that RIA becomes much more difficult to solve for non-normally distributed random parameters because of the highly nonlinear transformations that are involved. However, PMA is rather independent of probability distributions because it only has a small involvement with a nonlinear transformation.