A gradient-based uncertainty optimization framework utilizing dimensional adaptive polynomial chaos expansion

Abstract

To improve the efficiency of solving uncertainty design optimization problems, a gradient-based optimization framework is herein proposed that combines the dimension adaptive polynomial chaos expansion (PCE) and sensitivity analysis. The dimension adaptive PCE is used to quantify the quantities of interest (e.g. reliability, robustness metrics) and sensitivity. The dimension adaptive property is inherited from the dimension adaptive sparse grid, which is used to evaluate the PCE coefficients. Robustness metrics, referred to as statistical moments, and their gradients with respect to design variables are easily derived from the PCE, whereas the evaluation of the reliability and its gradient require integrations. To quantify the reliability, the framework uses Heaviside step function to eliminate the failure domain and computes the integration by Monte Carlo Simulation with the function replaced by PCE. The PCE is further derived to compute the function gradient on the definition domain and is combined with Taylor’s expansion and the finite difference to compute the reliability sensitivity. Since the design vector may affect the sample set determined by dimension adaptive sparse grid, the update of the sample set is controlled by the norm variations of the design vector. The optimization framework is formed by combining reliability, robustness quantification and sensitivity analysis and the optimization module. The accuracy and efficiency of reliability quantification and the reliability sensitivity are verified through canonical examples of a mathematical example，a system of springs ,and a cantilever beam. The effectiveness of the framework for solving optimization problem is verified through a multiple limit states example, a truss optimization example with eight random variables and three design variables, and an airfoil problem with three random variables and eighteen design variables. The results demonstrate that the framework obtains accurate solutions at a manageable computational cost.

More information

Identifiers

ORCID for A Da Ronch: orcid.org/0000-0001-7428-6935

Catalogue record

Export record

Altmetrics