Propagation Of Input Uncertainties Research Articles

Using stochastic analysis to capture unstable equilibrium in natural convection

A stabilized stochastic finite element implementation for the natural convection system of equations under Boussinesq assumptions with uncertainty in inputs is considered. The stabilized formulations are derived using the variational multiscale framework assuming a one-step trapezoidal time integration rule. The stabilization parameters are shown to be functions of the time-step size. Provision is made for explicit tracking of the subgrid-scale solution through time. A support-space/stochastic Galerkin approach and the generalized polynomial chaos expansion (GPCE) approach are considered for input–output uncertainty representation. Stochastic versions of standard Rayleigh–Bènard convection problems are used to evaluate the approach. It is shown that for simulations around critical points, the GPCE approach fails to capture the highly non-linear input uncertainty propagation whereas the support-space approach gives fairly accurate results. A summary of the results and findings is provided.

High-Fidelity Computational Optimization for 3-D Flexible Wings: Part II—Effect of Random Geometric Uncertainty on Design

The effect of geometric uncertainty due to statistically independent, random, normally distributed shape parameters is demonstrated in the computational design of a 3-D flexible wing. A first-order second-moment statistical approximation method is used to propagate the assumed input uncertainty through coupled Euler CFD aerodynamic/finite element structural codes for both analysis and sensitivity analysis. First-order sensitivity derivatives obtained by automatic differentiation are used in the input uncertainty propagation. These propagated uncertainties are then used to perform a robust design of a simple 3-D flexible wing at supercritical flow conditions. The effect of the random input uncertainties is shown by comparison with conventional deterministic design results. Sample results are shown for wing planform, airfoil section, and structural sizing variables.

Adaptive probability analysis using an enhanced hybrid mean value method

This paper proposes an adaptive probability analysis method that can effectively generate the probability distribution of the output performance function by identifying the propagation of input uncertainty to output uncertainty. The method is based on an enhanced hybrid mean value (HMV+) analysis in the performance measure approach (PMA) for numerical stability and efficiency in search of the most probable point (MPP). The HMV+ method improves numerical stability and efficiency especially for highly nonlinear output performance functions by providing steady convergent behavior in the MPP search. The proposed adaptive probability analysis method approximates the MPP locus, and then adaptively refines this locus using an a posteriori error estimator. Using the fact that probability levels can be easily set a priori in PMA, the MPP locus is approximated using the interpolated moving least-squares method. For refinement of the approximated MPP locus, additional probability levels are adaptively determined through an a posteriori error estimator. The adaptive probability analysis method will determine the minimum number of necessary probability levels, while ensuring accuracy of the approximated MPP locus. Several examples are used to show the effectiveness of the proposed adaptive probability analysis method using the enhanced HMV+ method.

Approach for Input Uncertainty Propagation and Robust Design in CFD Using Sensitivity Derivatives1

An implementation of the approximate statistical moment method for uncertainty propagation and robust optimization for quasi 1-D Euler CFD code is presented. Given uncertainties in statistically independent, random, normally distributed input variables, first-and second-order statistical moment procedures are performed to approximate the uncertainty in the CFD output. Efficient calculation of both first- and second-order sensitivity derivatives is required. In order to assess the validity of the approximations, these moments are compared with statistical moments generated through Monte Carlo simulations. The uncertainties in the CFD input variables are also incorporated into a robust optimization procedure. For this optimization, statistical moments involving first-order sensitivity derivatives appear in the objective function and system constraints. Second-order sensitivity derivatives are used in a gradient-based search to successfully execute a robust optimization. The approximate methods used throughout the analyses are found to be valid when considering robustness about input parameter mean values.