Title: "The eDR method for Derivative-free Probability Analysis" and "Bayesian introduction, Bayesian reliability analysis and design" Speaker: Byeng D. Youn, Mechanical Engineering and Engineering Mechanics (MEEM), Michigan Technological University Date/Time: Thursday, June 15, 2006, 9:00 – 10:30 am Location: Building 980, Room 95 (Sandia NM) Brief Abstract: In this talk, a new method for uncertainty quantification is presented to estimate statistical moments of system responses. The proposed method is based on the Dimension-Reduction (DR) method. It has been acknowledged that the DR method is accurate and efficient for assessing statistical moments of nonlinear system responses. However, the recent investigation on the DR method has found two problems: stability and accuracy for large-scale nonlinear systems. Thus, this talk presents a new version of the DR method, referred to as the enhanced DR (eDR) method. The eDR method has two new features: one-dimensional response approximation and numerical integration scheme. The Stepwise Moving Least Squares (SMLS) method is employed to accurately approximate the responses by adaptively selecting the optimal basis set for one-dimensional response approximation. In aid of approximated responses, there is more flexibility to conduct numerical integration for calculating statistical moments of system responses, since any numerical integration scheme can be used with accurate but free response values at any integration points. Instead of Gauss-Legendre and Gauss-Hermite quadratures, any numerical integration methods can be used to increase accuracy for numerical integration. Both stability and accuracy shortcomings are thus overcome in the eDR method without requiring additional computational cost. Results for some engineering examples indicate that the eDR method is more accurate and/or stable than the DR method, perturbation method, Monte Carlo Simulation (MCS), etc. in estimating statistical moments of system responses. Furthermore, probabilistic sensitivity analysis based on the eDR method is proposed for uncertainty quantification and Reliability-Based Design Optimization (RBDO). Two case studies are used to show the effectiveness of the eDR method applied in RBDO. Bayesian introduction, Bayesian reliability analysis and design Last decade, considerable advances have been made in reliability-based design optimization. One strong hypothesis in reliability-based design optimization (RBDO) is to know complete information of input uncertainties. However, the hypothesis is not true in practical engineering applications, since very few input uncertainties have been adequately modeled. It is mainly due to a lack of resources, such as facilities, labors, expenses, knowledge, etc. Possibility and evidence theories have been proposed to deal with situations in having epistemic uncertainties with insufficient data. But, when involving uncertainties with both sufficient and insufficient data sizes (limited to sufficient) simultaneously, it is hard to deal with those data consistently in predictive modeling and design. To properly handle both sufficient and insufficient data for random inputs, this talk presents an integration of Bayesian approach to RBDO. When a design problem involves input uncertainties with both sufficient and insufficient information, reliability must be uncertain and subjective. Bayesian inference is used to model uncertain and subjective reliability. So, uncertain and subjective reliability can be modeled with a Beta distribution in a Bayesian sense [4], where the Beta distribution is bounded in 0 (= 0% reliability) and 1 (= 100% reliability). For design optimization, two requirements must be considered: 1) reliability must be uniquely defined and 2) reliability must be greater than target reliability when an exact reliability is realized. The exact reliability means the reliability evaluated with a complete data for input uncertainties. To satisfy both requirements, Bayesian reliability will be defined as the median value of the extreme distribution of the smallest reliability value. For the design optimization, continuum sensitivity for Bayesian reliability is also discussed in the talk. Then, Bayesian RBDO will be conducted in aid of Bayesian reliability and its sensitivity. The methodology will be demonstrated with some examples.CSRI POC: Rich Field, (505) 284-4060 |