Developments in Design of Experiments and Reliability-Based Design Optimization Using Saddlepoint Approximation

摘要

Computational optimization plays an important role in engineering design, leading to greatly improved performance. Deterministic optimization, however, can result in undesired choices because it neglects uncertainty. Reliability-Based Design Optimization (RBDO) can provide optimum designs in the presence of uncertainties. The traditional double-loop RBDO approach based on First-Order Reliability Method (FORM) may be inaccurate, since it requires transformation from non-normal random variable space to standard normal variable space which, in certain circumstances, increases the nonlinearity of the limit state function(s). FORM may also be inefficient, because an iterative search process for the Most Probable Point (MPP) is required, resulting in a costly double-loop optimization algorithm. An RBDO with Mean Value First Order Saddlepoint Approximation (MVFOSA) algorithm is proposed with enhanced accuracy and almost the same efficiency with deterministic optimization. MVFOSA estimates the Probability Density Function (PDF) and the Cumulative Distribution Function (CDF) of the response using an accurate Saddlepoint Approximation (SA). The limit state function is approximated using a first order Taylor series expansion at the mean values of the random input variables. MVFOSA is more accurate than FORM, because there is no transformation from non-normal to normal random variables and the iterative search process for the MPP is avoided. Examples demonstrate the proposed methodology. The extension to MVSOSA (Mean Value First Order Saddlepoint Approximation), based on a second order Taylor expansion is also proposed, in order to further increase the accuracy of the computed probability density functions, retaining the high efficiency by calculating the required Hessian matrix using quasi-second order saddlepoint approximation.;Real life problems usually exhibit a multidimensional and multimodal behavior requiring a global optimization approach which is computationally inefficient. We propose a Design of Experiments (DOE) algorithm, which constructs optimal space filling designs in many dimensions with good projective properties. The algorithm also creates DOE groups with space filling properties and unions of these groups also retain space filling properties. The DOEs are obtained without optimization, improving computational efficiency. The proposed DOE algorithm can be used to create accurate metamodels (model(s) of an original model) sequentially and efficiently. Examples illustrate the concepts and demonstrate the applicability of the proposed method. We combine the DOE algorithm and MVSOSA methods into an integrated RBDO algorithm. The objective is to use MVFOSA and MVSOSA for black-box optimization problems by replacing the computation of first and second-order derivatives with a quadratic response surface, trained on design sites defined by the proposed DOE.;The final step of this research was the development of a second-order Saddlepoint (SA) method for reliability analysis. The Advanced Mean-Value Second-Order Saddlepoint Approximation (AMVSOSA) is proposed as an extension to the Mean-Value Second-Order Saddlepoint Approximation (MVSOSA). The proposed method is based on a second-order Taylor expansion of the limit state function around an approximate Most Probable Point (MPP) computed using a Mean-Value First-Order Second-Moment (MVFOSM) approach rather than the mean value of the random parameters as in MVSOSA.