Serhat Hosder

Point-Collocation Nonintrusive Polynomial Chaos Method for Stochastic Computational Fluid Dynamics

This paper describes a point-collocation nonintrusive polynomial chaos technique used for uncertainty propagation in computational fluid dynamics simulations. The application of point-collocation nonintrusive polynomial chaos to stochastic computational fluid dynamics is demonstrated with two examples: 1) a stochastic expansion-wave problem with an uncertain deflection angle (geometric uncertainty) and 2) a stochastic transonic-wing case with uncertain freestream Mach number and angle of attack. For each problem, input uncertainties are propagated with both the nonintrusive polynomial chaos method and Monte Carlo techniques to obtain the statistics of various output quantities. Confidence intervals for Monte Carlo statistics are calculated using the bootstrap method. For the expansion-wave problem, a fourth-degree polynomial chaos expansion, which requires five deterministic computational fluid dynamics evaluations, has been sufficient to predict the statistics within the confidence interval of 10,000 crude Monte Carlo simulations. In the transonic-wing case, for various output quantities of interest, it has been shown that a fifth-degree point-collocation nonintrusive polynomial chaos 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. Overall, the examples demonstrate that the point-collocation nonintrusive polynomial chaos has a promising potential as an effective and computationally efficient uncertainty propagation technique for stochastic computational fluid dynamics simulations.

Polynomial Response Surface Approximations for the Multidisciplinary Design Optimization of a High Speed Civil Transport

Surrogate functions have become an important tool in multidisciplinary design optimization to deal with noisy functions, high computational cost, and the practical difficulty of integrating legacy disciplinary computer codes. A combination of mathematical, statistical, and engineering techniques, well known in other contexts, have made polynomial surrogate functions viable for MDO. Despite the obvious limitations imposed by sparse high fidelity data in high dimensions and the locality of low order polynomial approximations, the success of the panoply of techniques based on polynomial response surface approximations for MDO shows that the implementation details are more important than the underlying approximation method (polynomial, spline, DACE, kernel regression, etc.). This paper selectively surveys some of the ancillary techniques—statistics, global search, parallel computing, variable complexity modeling—that augment the construction and use of polynomial surrogates.

Uncertainty and Sensitivity Analysis for Reentry Flows with Inherent and Model-form Uncertainties

The objective of this paper is to introduce a computationally efficient methodology for the quantification of mixed (inherent and model-form) uncertainties and global sensitivity analysis (SA) in hypersonic reentry flow computations. The uncertainty-quantification (UQ) approach is based on the second-order UQ theory, using a stochastic response surface obtained with nonintrusive polynomial chaos. The global nonlinear SA is based on Sobol variance decomposition, which uses polynomial chaos expansions. The methodology was used to quantify the uncertainty and sensitivity information for surface heat flux to the spherical nonablating heat shield of a reentry vehicle at an angle of attack of 0 deg. Three uncertainty sources were treated in computational fluid dynamics simulations: inherent uncertainty in the freestream velocity, model-form uncertainty in the recombination efficiency used in partially catalytic wall-boundary condition, and model-form uncertainty in the binary-collision integrals. The SA showed that the velocity and recombination efficiency were the major contributors to the heat-flux uncertainty for the reentry case considered. The UQ and SA were performed with three different levels of input uncertainty in velocity, which revealed the importance of characterizing the velocity with well-defined uncertainty levels in the study of reentry flows because the variations in this quantity can drastically impact the accuracy of the heat-flux prediction.

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).

Efficient Uncertainty Quantification in Multidisciplinary Analysis of a Reusable Launch Vehicle

The objective of this study was to apply a recently developed uncertainty quantification framework to the multidisciplinary analysis of a reusable launch vehicle (RLV). This particular framework is capable of efficiently propagating mixed (inherent and epistemic) uncertainties through complex simulation codes. The goal of the analysis was to quantify uncertainty in various output parameters obtained from the RLV analysis, including the maximum dynamic pressure, cross-range, range, and vehicle takeoff gross weight. Three main uncertainty sources were treated in the simulations: (1) reentry angle of attack (inherent uncertainty), (2) altitude of the initial reentry point (inherent uncertainty), and (3) the Young’s Modulus (epistemic uncertainty). The Second-Order Probability Theory utilizing a stochastic response surface obtained with Point-Collocation Non-Intrusive Polynomial Chaos was used for the propagation of the mixed uncertainties. This particular methodology was applied to the RLV analysis, and the uncertainty in the output parameters of interested was obtained in terms of intervals at various probability levels. The preliminary results have shown that there is a large amount of uncertainty associated with the vehicle takeoff gross weight. Furthermore, the study has demonstrated the feasibility of the developed uncertainty quantification framework for efficient propagation of mixed uncertainties in the analysis of complex aerospace systems.

Non-intrusive Polynomial Chaos Methods for Uncertainty Quantification in Fluid Dynamics

This paper examines uncertainty quantification in computational fluid dynamics (CFD) with non-intrusive polynomial chaos (NIPC) methods which require no modification to the existing deterministic models. The NIPC methods have been increasingly used for uncertainty propagation in high-fidelity CFD simulations due to their non-intrusive nature and strong potential for addressing the computational efficiency and accuracy requirements associated with large-scale complex stochastic simulations. We give the theory and description of various NIPC methods used for non-deterministic CFD simulations. We also present several stochastic fluid dynamics examples to demonstrate the application and effectiveness of NIPC methods for uncertainty quantification in fluid dynamics. These examples include stochastic computational analysis of a laminar boundary layer flow over a flat plate, supersonic expansion wave problem, and inviscid transonic flow over a three-dimensional wing with rigid and aeroelastic assumptions.

Uncertainty Quantification of Hypersonic Reentry Flows with Sparse Sampling and Stochastic Expansions

The objective of this study was to introduce a combined sparse sampling and stochastic expansion approach for efficient and accurate uncertainty quantification. The new techniques are applied to high-fidelity, hypersonic reentry flow simulations, which may contain large numbers of aleatory and epistemic uncertainties. Stochastic expansion coefficients were obtained using the point-collocation nonintrusive polynomial chaos technique with a number of samples less than the minimum number required for a total-order expansion. This study introduced two methods of measuring the accuracy of the expansion coefficients, as well as their convergence with iteratively increasing sample size. The newly developed approaches were demonstrated on two model problems. The first was a model for stagnation point, convective heat transfer in hypersonic flow. Mixed uncertainty quantification analysis results showed that accurate expansion coefficients could be obtained with half of the minimum number of samples required for a ...

Uncertainty Analysis of Mars Entry Flows over a Hypersonic Inflatable Aerodynamic Decelerator

A detailed uncertainty analysis for high-fidelity flowfield simulations over a fixed aeroshell of hypersonic inflatable aerodynamic decelerator scale for Mars entry is presented for fully laminar and turbulent flows at peak stagnation-point heating conditions. This study implements a sparse-collocation approach based on stochastic expansions for efficient and accurate uncertainty quantification under a large number of uncertainty sources in the computational model. The convective and radiative heating and shear stress uncertainties are computed over the hypersonic inflatable aerodynamic decelerator surface and are shown to vary due to a small fraction of 65 flowfield and radiation modeling parameters considered in the uncertainty analysis. The main contributors to the convective heating uncertainty near the stagnation point are the CO2–CO2, CO2–O, and CO–O binary collision interactions, freestream density, and freestream velocity for both boundary-layer flows. In laminar flow, exothermic recombination rea...

Aerodynamic Design Optimization: Physics-Based Surrogate Approaches for Airfoil and Wing Design

The aerodynamic optimization community has recently started an effort to develop benchmark problems suitable for exercising aerodynamic optimization methods in a constrained design space. In the first round, four problems have been developed, two involving two-dimensional airfoils and the other two three-dimensional wings. In this paper, we address the two-dimensional problems which involve optimization of the NACA 0012 in inviscid transonic flow, as well as optimization of the RAE 2822 in viscous transonic flow. We solve the problems using a computationally efficient physics-based surrogate approach exploiting space mapping. Our results indicate that by shifting the computational burden to fast low-fidelity models, significant performance improvements can be achieved at the cost of a few evaluations of the expensive computational fluid dynamic models. In our approach, a commercial package FLUENT is used as the high-fidelity fluid flow solver with a hyperbolic C-mesh, whereas the versatile viscous-inviscid solver MSES is utilized as the low-fidelity model. The PARSEC parameterization method is used to describe the airfoil shapes with up to 10 design variables.

Robust Airfoil Optimization under Inherent and Model-form Uncertainties Using Stochastic Expansions

The objective of this paper was to introduce a computationally e cient approach for robust aerodynamic optimization under aleatory (inherent) and epistemic (model-form) uncertainties using stochastic expansions that are based on Non-Intrusive Polynomial Chaos method. The stochastic surfaces were used as surrogates in the optimization process. To create the surrogates, a combined non-intrusive polynomial chaos expansion approach was utilized, which is a function of both the design and the uncertain variables. In this paper, two stochastic optimization formulations were given: (1) optimization under pure aleatory uncertainty and (2) optimization under mixed (aleatory and epistemic) uncertainty. The formulations were demonstrated for the drag minimization of NACA 4-digit airfoils described with three geometric design variables over the range of uncertainties at transonic ow conditions. The deterministic CFD simulations were performed to solve steady, 2-D, compressible, turbulent RANS equations. The pure aleatory uncertainty case included the Mach number as the uncertain variable. For the mixed uncertainty case, a k factor which is multiplied with the turbulent eddy-viscosity coe cient is introduced to the problem as the epistemic uncertain variable. The results of both optimization cases con rmed the e ectiveness of the robust optimization approach with stochastic expansions by giving the optimum airfoil shape that has the minimum drag over the range of aleatory and epistemic uncertainties. The optimization under pure aleatory uncertainty case required 90 deterministic CFD evaluations, whereas the optimization under mixed uncertainty case required 126 CFD evaluations to create the stochastic response surfaces, which show the computational e ciency of the proposed stochastic optimization approach. The stochastic optimization methodology described in this paper is general in the sense that it can be applied to aerodynamic optimization problems that utilize di erent shape parameterization techniques.