Efficient Sampling for Non-Intrusive Polynomial Chaos Applications with Multiple Uncertain Input Variables
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The accuracy and the computational efficiency of a Point-Collocation Non-Intrusive Polynomial Chaos (NIPC) method applied to stochastic problems with multiple uncertain input variables has been investigated. Two stochastic model problems with multiple uniform random variables were studied to determine the effect of different sampling methods (Random, Latin Hypercube, and Hammersley) for the selection of the collocation points. The effect of the number of collocation points on the accuracy of polynomial chaos expansions were also investigated. The results of the stochastic model problems show that all three sampling methods exhibit a similar performance in terms of the the accuracy and the computational efficiency of the chaos expansions. It has been observed that using a number of collocation points that is twice more than the minimum number required gives a better approximation to the statistics at each polynomial degree. This improvement can be related to the increase of the accuracy of the polynomial coefficients due to the use of more information in their calculation. The results of the stochastic model problems also indicate that for problems with multiple random variables, improving the accuracy of polynomial chaos coefficients in NIPC approaches may reduce the computational expense by achieving the same accuracy level with a lower order polynomial expansion. To demonstrate the application of Point-Collocation NIPC to an aerospace problem with multiple uncertain input variables, a stochastic computational aerodynamics problem which includes the numerical simulation of steady, inviscid, transonic flow over a three-dimensional wing with an uncertain free-stream Mach number and angle of attack has been studied. For this study, a 5th degree Point-Collocation NIPC expansion obtained with Hammersley sampling was capable of estimating the statistics at an accuracy level of 1000 Latin Hypercube Monte Carlo simulations with a significantly lower computational cost.