Numerical Sensitivity and Efficiency in the Treatment of Epistemic and Aleatory Uncertainty

The treatment of both aleatory and epistemic uncertainty by recent methods often requires an high computational effort. In this abstract, we propose a numerical sampling method allowing to lighten the computational burden of treating the information by means of so-called fuzzy random variables

Efficient uncertainty quantification in aerospace analysis and design

The main purpose of this study is to apply a computationally efficient uncertainty quantification approach, Non-Intrusive Polynomial Chaos (NIPC) based stochastic expansions, to robust aerospace analysis and design under mixed (aleatory and epistemic) uncertainties and demonstrate this technique on model problems and robust aerodynamic optimization. The proposed optimization approach utilizes stochastic response surfaces obtained with NIPC methods to approximate the objective function and the constraints in the optimization formulation. The objective function includes the stochastic measures which are minimized simultaneously to ensure the robustness of the final design to both aleatory and epistemic uncertainties. For model problems with mixed uncertainties, Quadrature-Based and Point-Collocation NIPC methods were used to create the response surfaces used in the optimization process. For the robust airfoil optimization under aleatory (Mach number) and epistemic (turbulence model) uncertainties, a combined Point-Collocation NIPC approach was utilized to create the response surfaces used as the surrogates in the optimization process. Two stochastic optimization formulations were studied: optimization under pure aleatory uncertainty and optimization under mixed uncertainty. As shown in this work for various problems, the NIPC method is computationally more efficient than Monte Carlo methods for moderate number of uncertain variables and can give highly accurate estimation of various metrics used in robust design optimization under mixed uncertainties. This study also introduces a new adaptive sampling approach to refine the Point-Collocation NIPC method for further improvement of the computational efficiency. Two numerical problems demonstrated that the adaptive approach can produce the same accuracy level of the response surface obtained with oversampling ratio of 2 using less function evaluations. --Abstract, page iii

Multifidelity Uncertainty Quantification of a Commercial Supersonic Transport

The objective of this work was to develop a multifidelity uncertainty quantification approach for efficient analysis of a commercial supersonic transport. An approach based on non-intrusive polynomial chaos was formulated in which a low-fidelity model could be corrected by any number of high-fidelity models. The formulation and methodology also allows for the addition of uncertainty sources not present in the lower fidelity models. To demonstrate the applicability of the multifidelity polynomial chaos approach, two model problems were explored. The first was supersonic airfoil with three levels of modeling fidelity, each capturing an additional level of physics. The second problem was a commercial supersonic transport. This model had three levels of fidelity that included two different modeling approaches and the addition of physics between the fidelity levels. Both problems illustrate the applicability and significant computational savings of the multifidelity polynomial chaos method

Quantification of uncertainty in aerodynamic heating of a reentry vehicle due to uncertain wall and freestream conditions

The primary focus of this study is to demonstrate an efficient approach for uncertainty quantification of surface heat flux to the spherical non-ablating heatshield of a generic reentry vehicle due to epistemic and aleatory uncertainties that may exist in various parameters used in the numerical solution of hypersonic, viscous, laminar blunt-body flows with thermo-chemical non-equilibrium. Two main uncertainty sources were treated in the computational fluid dynamics (CFD) simulations: (1) aleatory uncertainty in the freestream velocity and (2) epistemic uncertainty in the recombination efficiency for a partially catalytic wall boundary condition. The Second-Order Probability utilizing a stochastic response surface obtained with Point-Collocation Non-Intrusive Polynomial Chaos was used for the propagation of mixed (aleatory and epistemic) uncertainties. The uncertainty quantication approach was validated on a stochastic model problem with mixed uncertainties for the prediction of stagnation point heat transfer with Fay-Riddell relation, which included the comparison with direct Monte Carlo sampling results. In the stochastic CFD problem, the uncertainty in surface heat transfer was obtained in terms of intervals at different probability levels at various locations including the stagnation point and the shoulder region. The mixed uncertainty results were compared to the results obtained with a purely aleatory uncertainty analysis to show the difference between two uncertainty quantication approaches. A global sensitivity analysis indicated that the velocity has a stronger contribution to the overall uncertainty in the stagnation point heat transfer for the range of input uncertainties considered in this study --Abstract, page iii

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

Uncertainty Analysis of the Adequacy Assessment Model of a Distributed Generation System

Due to the inherent aleatory uncertainties in renewable generators, the reliability/adequacy assessments of distributed generation (DG) systems have been particularly focused on the probabilistic modeling of random behaviors, given sufficient informative data. However, another type of uncertainty (epistemic uncertainty) must be accounted for in the modeling, due to incomplete knowledge of the phenomena and imprecise evaluation of the related characteristic parameters. In circumstances of few informative data, this type of uncertainty calls for alternative methods of representation, propagation, analysis and interpretation. In this study, we make a first attempt to identify, model, and jointly propagate aleatory and epistemic uncertainties in the context of DG systems modeling for adequacy assessment. Probability and possibility distributions are used to model the aleatory and epistemic uncertainties, respectively. Evidence theory is used to incorporate the two uncertainties under a single framework. Based on the plausibility and belief functions of evidence theory, the hybrid propagation approach is introduced. A demonstration is given on a DG system adapted from the IEEE 34 nodes distribution test feeder. Compared to the pure probabilistic approach, it is shown that the hybrid propagation is capable of explicitly expressing the imprecision in the knowledge on the DG parameters into the final adequacy values assessed. It also effectively captures the growth of uncertainties with higher DG penetration levels

Efficient Uncertainty Quantification & Sensitivity Analysis for Hypersonic Flow and Material Response Simulations Under Inherent and Model-Form Uncertainties

Accurate numerical prediction of coupled hypersonic flow fields and ablative TPS material response is challenging due to the complex nature of the physics. The uncertainties associated with various physical models used in high-enthalpy hypersonic flow and material response simulations can have significant effects on the accuracy of the results including the heat-flux and temperature distributions in various layers of ablating TPS material. These uncertainties can arise from the lack of knowledge in physical modeling (model-form or epistemic uncertainty) or inherent variations in the model inputs (aleatory or probabilistic uncertainty). It is important to include both types of uncertainty in the simulations to properly assess the accuracy of the results and to design robust and reliable TPS for reentry or hypersonic cruise vehicles. In addition to the quantification of uncertainties, global sensitivity information for the output quantities of interest play an important role for the ranking of the contribution of each uncertainty source to the overall uncertainty, which may be used for the proper allocation of resources in the improvement of the physical models or reduce the number of uncertain variables to be considered in the uncertainty analysis. The uncertainty quantification for coupled high-fidelity hypersonic flow and material response predictions can be challenging due to the computational expense of the simulations, existence of both model-form and inherent uncertainty sources, large number of uncertain variables, and highly non-linear relations between the uncertain variables and the output response variables. The objective of this talk will be to introduce a computationally efficient and accurate uncertainty quantification (UQ) and global sensitivity analysis approach for potential application to coupled aerothermodynamics and material response simulations, which is being developed to address the aforementioned challenges. The UQ approach to be described is based on the second-order uncertainty quantification theory utilizing a stochastic response surface obtained with non-intrusive polynomial chaos and is capable of efficiently propagating both the inherent and the model-form uncertainties in the physical models. The non-intrusive nature of the UQ approach requires no modification to the deterministic codes, which is a significant benefit for the complex numerical simulation considered in this problem. The global non-linear sensitivity analysis to be introduced is based on variance decomposition, which again utilizes the polynomial chaos expansions. In addition to the description of the UQ approach, the talk will also include the presentation of UQ results from a recent demonstration of the methodology, which included the uncertainty quantification and sensitivity analysis of surface heat-flux on the spherical heat shield of a reentry vehicle (a case selected from CUBRC experimental database). This study involved the use of NASA DPLR code and the treatment of the free-stream velocity (inherent uncertainty), collision integrals for the transport coefficients (model-form uncertainty), and the surface catalysis (model-form uncertainty) as uncertain variables. The talk will also include the description of an adaptive UQ framework being developed as part of a NASA JPL STTR project to quantify the uncertainty in multi-physics spacecraft simulations with large number of uncertain variables

Stochastic simulation methods for structural reliability under mixed uncertainties

Uncertainty quantification (UQ) has been widely recognized as one of the most important, yet challenging task in both structural engineering and system engineering, and the current researches are mainly on the proper treatment of different types of uncertainties, resulting from either natural randomness or lack of information, in all related sub-problems of UQ such as uncertainty characterization, uncertainty propagation, sensitivity analysis, model updating, model validation, risk and reliability analysis, etc. It has been widely accepted that those uncertainties can be grouped as either aleatory uncertainty or epistemic uncertainty, depending on whether they are reducible or not. For dealing with the above challenge, many non-traditional uncertainty characterization models have been developed, and those models can be grouped as either imprecise probability models (e.g., probability-box model, evidence theory, second-order probability model and fuzzy probability model) or non-probabilistic models (e.g., interval/convex model and fuzzy set theory). This thesis concerns the efficient numerical propagation of the three kinds of uncertainty characterization models, and for simplicity, the precise probability model, the distribution probability-box model, and the interval model are taken as examples. The target is to develop efficient numerical algorithms for learning the functional behavior of the probabilistic responses (e.g., response moments and failure probability) with respect to the epistemic parameters of model inputs, which is especially useful for making reliable decisions even when the available information on model inputs is imperfect. To achieve the above target, my thesis presents three main developments for improving the Non-intrusive Imprecise Stochastic Simulation (NISS), which is a general methodology framework for propagating the imprecise probability models with only one stochastic simulation. The first development is on generalizing the NISS methods to the problems with inputs including both imprecise probability models and non-probability models. The algorithm is established by combining Bayes rule and kernel density estimation. The sensitivity indices of the epistemic parameters are produced as by-products. The NASA Langley UQ challenge is then successfully solved by using the generalized NISS method. The second development is to inject the classical line sampling to the NISS framework so as to substantially improve the efficiency of the algorithm for rare failure event analysis, and two strategies, based on different interpretations of line sampling, are developed. The first strategy is based on the hyperplane approximations, while the second-strategy is derived based on the one-dimensional integrals. Both strategies can be regarded as post-processing of the classical line sampling, while the results show that their resultant NISS estimators have different performance. The third development aims at further substantially improving the efficiency and suitability to highly nonlinear problems of line sampling, for complex structures and systems where one deterministic simulation may take hours. For doing this, the active learning strategy based on Gaussian process regression is embedded into the line sampling procedure for accurately estimating the interaction point for each sample line, with only a small number of deterministic simulations. The above three developments have largely improved the suitability and efficiency of the NISS methods, especially for real-world engineering applications. The efficiency and effectiveness of those developments are clearly interpreted with toy examples and sufficiently demonstrated by real-world test examples in system engineering, civil engineering, and mechanical engineering

Advancements in uncertainty quantification with stochastic expansions applied to supersonic and hypersonic flows

The primary objective of this study was to develop improved methodologies for efficient and accurate uncertainty quantification with stochastic expansions and apply them to problems in supersonic and hypersonic flows. Methods introduced included approaches for efficient dimension reduction, sensitivity analysis, and sparse approximations. These methods and procedures were demonstrated on multiple stochastic models of hypersonic, planetary entry flows, which included high-fidelity, computational fluid dynamics models of radiative heating on the surface of hypersonic inflatable aerodynamic decelerators during Mars and Titan entry. For these stochastic problems, construction of an accurate surrogate model was achieved with as few as 10% of the number of model evaluations needed to construct a full dimension, total order expansion. Another objective of this work was to introduce methodologies used for further advancement of a quantification of margins and uncertainties framework. First, the use of stochastic expansions was introduced to efficiently quantify the uncertainty in system design performance metrics and performance boundaries. Then, procedures were defined to measure margin and uncertainty metrics for systems subject to multiple types of uncertainty in operating conditions and physical models. To demonstrate the new quantification of margins and uncertainties methodologies, two multi-system, multi-physics stochastic models were investigated: (1) a model for reentry dynamics, control, and convective heating and (2) a model of ground noise prediction of low-boom, supersonic aircraft configurations. Overall the methods and results of this work have outlined many effective approaches to uncertainty quantification of large-scale, high-dimension, aerospace problems containing both epistemic and inherent uncertainty. The methods presented showed significant improvement in the efficiency and accuracy of uncertainty analysis capability when stochastic expansions were used for uncertainty quantification. --Abstract, page iii