Investigation of robust optimization and evidence theory with stochastic expansions for aerospace applications under mixed uncertainty

One of the primary objectives of this research is to develop a method to model and propagate mixed (aleatory and epistemic) uncertainty in aerospace simulations using DSTE. In order to avoid excessive computational cost associated with large scale applications and the evaluation of Dempster Shafer structures, stochastic expansions are implemented for efficient UQ. The mixed UQ with DSTE approach was demonstrated on an analytical example and high fidelity computational fluid dynamics (CFD) study of transonic flow over a RAE 2822 airfoil. Another objective is to devise a DSTE based performance assessment framework through the use of quantification of margins and uncertainties. Efficient uncertainty propagation in system design performance metrics and performance boundaries is achieved through the use of stochastic expansions. The technique is demonstrated on: (1) a model problem with non-linear analytical functions representing the outputs and performance boundaries of two coupled systems and (2) a multi-disciplinary analysis of a supersonic civil transport. Finally, the stochastic expansions are applied to aerodynamic shape optimization under uncertainty. A robust optimization algorithm is presented for computationally efficient airfoil design under mixed uncertainty using a multi-fidelity approach. This algorithm exploits stochastic expansions to create surrogate models utilized in the optimization process. To reduce the computational cost, output space mapping technique is implemented to replace the high-fidelity CFD model by a suitably corrected low-fidelity one. The proposed algorithm is demonstrated on the robust optimization of NACA 4-digit airfoils under mixed uncertainties in transonic flow. --Abstract, page iii

A Mixed Uncertainty Quantification Approach with Evidence Theory and Stochastic Expansions

Uncertainty quantication (UQ) is the process of quantitative characterization and prop-agation of input uncertainties to the response measure of interest in experimental and com-putational models. The input uncertainties in computational models can be either aleatoryi.e. irreducible inherent variations or epistemic i.e. reducible variability which arises fromlack of knowledge. Previously, it has been shown that Dempster-Shafer Theory of Evidence(DSTE) can be applied to model epistemic uncertainty in case of uncertainty informationcoming from multiple sources. The objective of this paper is to model and propagatemixed uncertainty (aleatory and epistemic) using DSTE. In specic, the aleatory vari-ables are modeled as Dempster-Shafer structures by discretizing them into sets of intervalsaccording to their respective probability distributions. In order to avoid excessive compu-tational cost associated with large scale applications, a stochastic response surface basedon point-collocation non-intrusive polynomial chaos has been implemented as the surro-gate model for the response. A convergence study for accurate representation of aleatoryuncertainty in terms of minimum number of subintervals required is presented. The mixedUQ approach is demonstrated on a numerical example and high delity computational uiddynamics study of transonic ow over RAE 2822 airfoil

A Mixed Uncertainty Quantification Approach using Evidence Theory and Stochastic Expansions

Uncertainty quantification (UQ) is the process of quantitative characterization and propagation of input uncertainties to the response measure of interest in experimental and computational models. The input uncertainties in computational models can be either aleatory, i.e., irreducible inherent variations, or epistemic, i.e., reducible variability which arises from lack of knowledge. Previously, it has been shown that Dempster Shafer theory of evidence (DSTE) can be applied to model epistemic uncertainty in case of uncertainty information coming from multiple sources. The objective of this paper is to model and propagate mixed uncertainty (aleatory and epistemic) using DSTE. In specific, the aleatory variables are modeled as Dempster Shafer structures by discretizing them into sets of intervals according to their respective probability distributions. In order to avoid excessive computational cost associated with large scale applications, a stochastic response surface based on point-collocation non-intrusive polynomial chaos has been implemented as the surrogate model for the response. A convergence study for accurate representation of aleatory uncertainty in terms of minimum number of subintervals required is presented. The mixed UQ approach is demonstrated on a numerical example and high fidelity computational fluid dynamics study of transonic flow over RAE 2822 airfoil

Quantification of Margins and Mixed Uncertainties Using Evidence Theory and Stochastic Expansions

The objective of this paper is to implement Dempster–Shafer Theory of Evidence (DSTE) in the presence of mixed (aleatory and multiple sources of epistemic) uncertainty to the reliability and performance assessment of complex engineering systems through the use of quantification of margins and uncertainties (QMU) methodology. This study focuses on quantifying the simulation uncertainties, both in the design condition and the performance boundaries along with the determination of margins. To address the possibility of multiple sources and intervals for epistemic uncertainty characterization, DSTE is used for uncertainty quantification. An approach to incorporate aleatory uncertainty in Dempster–Shafer structures is presented by discretizing the aleatory variable distributions into sets of intervals. In view of excessive computational costs for large scale applications and repetitive simulations needed for DSTE analysis, a stochastic response surface based on point-collocation non-intrusive polynomial chaos (NIPC) has been implemented as the surrogate for the model response. The technique is demonstrated on a model problem with non-linear analytical functions representing the outputs and performance boundaries of two coupled systems. Finally, the QMU approach is demonstrated on a multi-disciplinary analysis of a high speed civil transport (HSCT)

Quantification of Margins and Mixed Uncertainties using Evidence Theory and Stochastic Expansions

The objective of this paper is to implement Dempster-Shafer Theory of Evidence (DSTE) in the presence of mixed (aleatory and epistemic) uncertainty to the Quantication of Margins and Uncertainties (QMU) of aerospace simulations. This study focuses on quantifying the simulation uncertainties, both in the design condition and the performance boundaries along with the determination of margins. To address the possibility of multiple sources and intervals for epistemic uncertainty characterization, DSTE is used for uncertainty quantification. An approach to incorporate aleatory uncertainty in Dempster-Shafer structures is presented by discretizing the probability distributions into sets of intervals. In view of excessive computational costs for large scale applications and repetitive simulations needed for DSTE analysis, a stochastic response surface based on point-collocation non-intrusive polynomial chaos (NIPC) has been implemented as the surrogate for the model response. The technique is demonstrated on a model problem with non-linear analytical functions representing the outputs and performance boundaries of two coupled systems. Finally, the QMU approach is demonstrated on a multi-disciplinary analysis of a high speed civiltransport (HSCT)

Multi-Fidelity Robust Aerodynamic Design Optimization under Mixed Uncertainty

The objective of this paper is to present a robust optimization algorithm for computationally efficient airfoil design under mixed (inherent and epistemic) uncertainty using a multi-fidelity approach. This algorithm exploits stochastic expansions derived from the Non-Intrusive Polynomial Chaos (NIPC) technique to create surrogate models utilized in the optimization process. A combined NIPC expansion approach is used, where both the design and the mixed uncertain parameters are the independent variables of the surrogate model. To reduce the computational cost, the high-fidelity Computational Fluid Dynamics (CFD) model is replaced by a suitably corrected low-fidelity one, the latter being evaluated using the same CFD solver but with a coarser mesh. The model correction is implemented to the low-fidelity CFD solutions utilized for the construction of stochastic surrogate by using multi-point Output Space Mapping (OSM) technique. The proposed algorithm is applied to the design of NACA 4-digit airfoils with four deterministic design variables (the airfoil shape parameters and the angle of attack), one aleatory uncertain variable (the Mach number) and one epistemic variable (a geometry parameter) to demonstrate robust optimization under mixed uncertainties. In terms of computational cost, the proposed technique outperforms the conventional approach that exclusively uses the high-fidelity model to create the surrogates. The design cost reduces to only 34 equivalent high-fidelity model evaluations versus 168 obtained with the conventional method

Multi-Fidelity Robust Aerodynamic Design Optimization under Mixed Uncertainty

The objective of this paper is to present a robust optimization algorithm for computationally efficient airfoil design under mixed (inherent and epistemic) uncertainty using a multi-fidelity approach