Non-intrusive Polynomial Chaos Research Articles

Application of the NIPC-based uncertainty quantification in prediction of ship maneuverability

The importance of the uncertainty quantification in ship maneuverability prediction is expounded. An efficient method for the uncertainty quantification problem, the Non-Intrusive Polynomial Chaos (NIPC) method, is introduced. A 3-DOF MMG model is adopted to carry out simulations of 35° turning circle tests and 10°/10° zig-zag tests for a KCS container ship model. The effects of the uncertainty of the hull-related hydrodynamic derivatives in the MMG model on the maneuverability parameters, i.e., the advance, the tactical diameter, the initial turning time and the overshoot angles are analyzed quantitatively, and the sensitivities of these parameters to the hydrodynamic derivatives are determined. The results are compared with those obtained by the Monte Carlo (MC) method. It is shown that the NIPC method is much better than the MC method in terms of the convergence with respect to the number of samples, and the computation accuracy of the NIPC method is higher than that of the MC method under the same number of samples; the considered maneuverability parameters are most sensitive to the uncertainty of the yaw moment related hydrodynamic derivatives, and least sensitive to that of the longitudinal force related ones. The feasibility of the NIPC method in the uncertainty quantification of ship maneuverability prediction is proved, which provides a novel approach to analyze the influence of the accuracy of the hydrodynamic derivatives on ship maneuverability prediction.

Geometric uncertainty propagation in laminar flows solved by RBF-FD meshless technique

The Non-Intrusive Polynomial Chaos method is employed to analyze incompressible and laminar fluid flows in presence of geometric uncertainties on the boundaries, which are described by stochastic variables with known probability distribution. Non-Intrusive methods allow the use of existing deterministic solvers, which are treated as black boxes. Therefore the quantification of the fluid flow uncertainties is based on a set of deterministic response evaluations. The required thermo-fluid dynamics solutions over the deterministic geometries are obtained through a Radial Basis Function-generated Finite Differences (RBF-FD) meshless method. The validation of the presented approach is carried out through analytical test cases (isothermal flow between non-parallel walls) with one geometric uncertainty. The applicability of the presented approach to practical problems is then presented through the prediction of geometric uncertainty effects on the non-isothermal flow over a heated backward-facing step.

Analysis of geometric uncertainties in CFD problems solved by RBF-FD meshless method

In order to analyze incompressible and laminar fluid flows in presence of geometric uncertainties on the boundaries, the Non-Intrusive Polynomial Chaos method is employed, which allows the use of a deterministic fluid dynamic solver. The quantification of the fluid flow uncertainties is based on a set of deterministic response evaluations, which are obtained through a Radial Basis Function-generated Finite Differences meshless method. The use of such deterministic solver represents the key point of the analysis, thanks to the computational efficiency and similar accuracy over the traditional mesh-based numerical methods. The validation of the proposed approach is carried out through the solution of the flow past a 2D spinning cylinder near a moving wall and the flow over a backward-facing step, in presence of stochastic geometries. The applicability to practical problems is demonstrated through the investigation of geometric uncertainty effects on the forced convection of Al2O3-water nanofluid laminar flow in a grooved microchannel.

Aerodynamic Design Optimization of Transonic Natural-Laminar-Flow Airfoil at Low Reynolds Number

The position and size of laminar separation bubble on airfoil surfaces exert a profound impact on the efficiency of transonic natural-laminar-flow airfoil at low Reynolds number. Based on the particle swarm algorithm, an optimization methodology in the current work would be established with the aim of designing a high and robust performance transonic natural-laminar-flow airfoil at low Reynolds number. This methodology primarily includes two design processes: a traditional deterministic optimization at on-design point and a multi-objective of uncertainty-based optimization. First, a multigroup cooperative particle swarm optimization was used to obtain the optimal deterministic solution. The crowing distance multi-objective particle swarm optimization and the non-intrusive polynomial chaos expansion method were then adopted to determinate the Pareto-optimal front of uncertainty-based optimization. Additionally, the γ−Re¯θt transition model was employed to predict the laminar-turbulent transition. Regarding to the established optimization methodology, a propeller tip airfoil of solar energy unmanned aerial vehicle was finally designed. During optimization processes, the minimized pressure drag was particularly chosen as the optimization objective, while the friction drag increment served as a constraint condition. The deterministic results indicate that the optimized airfoil has a good ability to control the separation and reattachment positions, and the pressure drag can be greatly reduced when the laminar separation bubble is weakened. The multi-objective results show that the uncertainty-based optimized airfoil possesses a significant robust performance by considering the uncertainty of Mach number. The findings evidently demonstrate that the proposed optimization methodology and mathematical model are valuable tools to design a high-efficiency airfoil for the propeller tip.

Uncertainty Quantification for CFD Simulation of Stochastic Drag Flow Based on Non-Intrusive Polynomial Chaos Method

Uncertainty Quantification for CFD Simulation of Stochastic Drag Flow Based on Non-Intrusive Polynomial Chaos Method

Drag Prediction of Nasa Common Research Model Under Stochastic Inflow Conditions

Numerical investigations of NASA Common Research Model under stochastic inflow conditions are analysed in this paper by Non-intrusive Polynomial Chaos (NIPC) method. Reynolds number, Mach number and temperature of inflow conditions are assumed to be independent stochastic variables. It is found that predicted drag mainly depends on Mach number of incoming flow. The contribution of Reynolds number and temperature to the variance of drag is negligible. The mean value of drag shows consistent convergence with grid refinement. The investigation of this paper quantitates the uncertainty induced by stochastic inflow conditions to drag prediction and recognizes the most significant input variable. This will help the validation of numerical methods with experiment.

Stochastic Characteristic Analysis of Lid-Driven Cavity Flow

The stochastic characteristic of lid-driven cavity flow under the condition of Re=100 is investigated in this paper by a non-intrusive polynomial chaos method. The lid-driven velocity and viscosity coefficient are assumed to be two independent stochastic input variables with Gaussian distributions. The velocities at every grid points are the response of interest and no longer deterministic but stochastic. Their stochastic characteristics are represented by Hermite polynomials with interaction effects between two input variables considered. The statistical results, including mean value, variance and the relative contribution of each input variable to the variance of outputs are analysed. It is found that the variance of velocity shows similar structures with its magnitude of mean value and the correlation coefficient is larger than 0.96. The nearly linear correlation coefficient shows that the lid-driven velocity is the main factor affecting the velocity distribution and the influence of viscosity coefficient on most regions is unimportant. The analysis of stochastic flow field helps understand the inherent mechanism in the lid-driven cavity flow.

Point-Collocation NIPC 기법 기반 천이 난류 모델 계수의 베이지안 추론 연구

In the present work, Bayesian inference based on the point-collocation Non-intrusive Polynomial Chaos was conducted in transitional flow around the flat plate using Menter’s γ-Re<SUB>θ</SUB> transition model. Three model coefficients, C<SUB>a2</SUB>, C<SUB>e1</SUB> and C<SUB>e2</SUB> were considered for random variables based on the sensitivity analysis of the present turbulence model and quantity of interest was set to the drag coefficient, C<SUB>d</SUB>. With the assumption of uniform distribution of three model coefficients within ±10%, the surrogate model was obtained based on the general Polynomial Chaos expansion. The simulation results were used for observation to calculate the likelihood function and MCMC sampling algorithm was adopted for Bayesian inference. The correlation between C<SUB>e1</SUB> and C<SUB>d</SUB> was predicted with highest value than other two model coefficients. Based on the original Bayesian inference result, the effect of number of observation data was studied. Also the 2nd order of gPC for surrogate model was adequate through comparison with results of 3rd order gPC. Finally, two prior distributions of input random variables for Bayesian inference were considered and the differences of posterior distributions of input and output were investigated.

Damage mechanics based probabilistic high‐cycle fatigue life prediction for Al 2024‐T3 using non‐intrusive polynomial chaos

AbstractThis paper proposes a low‐cost method for predicting probabilistic high‐cycle fatigue life for Al 2024‐T3 based on continuum damage mechanics and non‐intrusive polynomial chaos (NIPC). To randomize Lemaitre's two scale fatigue damage model, parameters S and s are regarded as random variables. Based on small sample of test life, inverse analysis is performed to obtain samples of the two parameters. Statistic characteristics of the two parameters are calculated analytically through coefficients of NIPC. Fatigue test of aluminum alloy 2024‐T3 standard coupon and plate with hole under different spectrum loading shows that the proposed method is effective.

NIPC-based uncertainty analysis of infrared radiation from rocket exhaust plumes caused by nozzle exit conditions

In this study, an uncertainty analysis for the infrared radiation characteristics of rocket exhaust plumes at representative trajectory points is performed. Owing to the instability of the rocket motor’s working characteristics, numerical rocket plume infrared radiation predictions possess significant uncertainties. In this study, four epistemic uncertain variables (freestream velocity, nozzle exit pressure, temperature, and velocity) are considered for uncertainty and sensitivity analyses. Based on the infrared signature analysis tool, the response surface of statistical samples is established through the point collocation nonintrusive polynomial chaos expansion method. Polynomial chaos expansion coefficients are solved using the quadrature method to calculate the statistical characteristics and uncertainty of random input variables. The tensor-product quadrature sparse grid method is utilized to reduce the number of samples for multiple input variables. Based on these models, the uncertainty quantification of infrared radiation for Atlas-IIA rocket plumes is analyzed, including the flows, radiation images, spectra, and radiance. The results show that the uncertainty mainly results from afterburning at low altitude, and the nozzle exit velocity has a significant influence on the radiation intensity of the plume. With an increase in altitude, the uncertainty of infrared radiation owing to the afterburning effect decreases, and the influence of the freestream velocity increases. In addition, the proportion of radiation intensity in the 4.3-μm band is higher than that in the 2.7-μm band, and the corresponding uncertainty band is gradually widened. The nozzle exit temperature is the dominant factor that affects the radiation characteristics of the plume at high altitudes. These results of uncertainty and sensitivity analyses are helpful for improving numerical models of the plume infrared signature.

Acoustic Performance of a Periodically Voided Viscoelastic Medium With Uncertainty in Design Parameters

Abstract The effect of uncertainties in material and geometric parameters on the acoustic performance of a viscoelastic coating is investigated. The model of the coating comprises a structure conventionally used in underwater applications, namely a soft elastic matrix embedded with periodic arrangements of voids. To investigate the effect of uncertainties on the acoustic performance of the coating, stochastic models based on the non-intrusive polynomial chaos expansion (PCE) method and Monte Carlo (MC) simulations are developed. The same analytical formulation of the acoustic coating is employed in both stochastic models. In the PCE method, the analytical model is transformed into a computationally efficient surrogate model using stochastic collocation. The effect of uncertainty in an individual geometric or material parameter on the acoustic performance of the coating is investigated by examining the mean, envelopes, and probability distribution of the monopole resonance frequency and sound transmission through the coating. The effect of variation in combinations of geometric and material parameters is then examined. Uncertainty in the geometric parameters is observed to have greater impact on the resonance frequency of the voids and sound transmission through the coating compared to uncertainty in the material properties.

Efficient Bayesian inversion for shape reconstruction of lithography masks

Background: Scatterometry is a fast, indirect and non-destructive optical method for quality control in the production of lithography masks. To solve the inverse problem in compliance with the upcoming need for improved accuracy, a computationally expensive forward model has to be defined which maps geometry parameters to diffracted light intensities. Aim: To quantify the uncertainties in the reconstruction of the geometry parameters, a fast to evaluate surrogate for the forward model has to be introduced. Approach: We use a non-intrusive polynomial chaos based approximation of the forward model which increases speed and thus enables the exploration of the posterior through direct Bayesian inference. Additionally, this surrogate allows for a global sensitivity analysis at no additional computational overhead. Results: This approach yields information about the complete distribution of the geometry parameters of a silicon line grating, which in return allows to quantify the reconstruction uncertainties in the form of means, variances and higher order moments of the parameters. Conclusion: The use of a polynomial chaos surrogate allows to quantify both parameter influences and reconstruction uncertainties. This approach is easy to use since no adaptation of the expensive forward model is required.

Uncertainty Quantification of an Explicitly Coupled Multiphysics Simulation of In-Situ Pyrolysis by Radio Frequency Heating in Oil Shale

SummaryIn-situ pyrolysis provides an enhanced oil recovery (EOR) technique for exploiting oil and gas from oil shale by converting in-place solid kerogen into liquid oil and gas. Radio-frequency (RF) heating of the in-place oil shale has previously been proposed as a method by which the electromagnetic energy gets converted to thermal energy, thereby heating in-situ kerogen so that it converts to oil and gas. In order to numerically model the RF heating of the in-situ oil shale, a novel explicitly coupled thermal, phase field, mechanical, and electromagnetic (TPME) framework is devised using the finite element method in a 2D domain. Contemporaneous efforts in the commercial development of oil shale by in-situ pyrolysis have largely focused on pilot methodologies intended to validate specific corporate or esoteric EOR strategies. This work focuses on addressing efficient epistemic uncertainty quantification (UQ) of select thermal, oil shale distribution, electromagnetic, and mechanical characteristics of oil shale in the RF heating process, comparing a spectral methodology to a Monte Carlo (MC) simulation for validation. Attempts were made to parameterize the stochastic simulation models using the characteristic properties of Green River oil shale. The geologic environment being investigated is devised as a kerogen-poor under- and overburden separated by a layer of heterogeneous yet kerogen-rich oil shale in a target formation. The objective of this work is the quantification of plausible oil shale conversion using TPME simulation under parametric uncertainty; this, while considering a referenced conversion timeline of 1.0 × 107 seconds. Nonintrusive polynomial chaos (NIPC) and MC simulation were used to evaluate complex stochastically driven TPME simulations of RF heating. The least angle regression (LAR) method was specifically used to determine a sparse set of polynomial chaos coefficients leading to the determination of summary statistics that describe the TPME results. Given the existing broad use of MC simulation methods for UQ in the oil and gas industry, the combined LAR and NIPC is suggested to provide a distinguishable performance improvement to UQ compared to MC methods.

Uncertainty quantification of sensitivities of time-average quantities in chaotic systems

We consider time-average quantities of chaotic systems and their sensitivity to system parameters. When the parameters are random variables with a prescribed probability density function, the sensitivities are also random. The central aim of the paper is to study and quantify the uncertainty of the sensitivities; this is useful to know in robust design applications. To this end, we couple the nonintrusive polynomial chaos expansion (PCE) with the multiple shooting shadowing (MSS) method, and apply the coupled method to two standard chaotic systems, the Lorenz system and the Kuramoto-Sivashinsky equation. The method leads to accurate results that match well with Monte Carlo simulations (even for low chaos orders, at least for the two systems examined), but it is costly. However, if we apply the concept of shadowing to the system trajectories evaluated at the quadrature integration points of PCE, then the resulting regularization can lead to significant computational savings. We call the new method shadowed PCE (sPCE).

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 concept. An approach based on point-collocation, nonintrusive polynomial chaos was formulated in which a low-fidelity model could be corrected using multiple higher-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 and potential computational savings of the multifidelity polynomial chaos approach, two model problems were explored. The first was a supersonic airfoil with three levels of modeling fidelity, each capturing a gradual increase in modeling of the underlying flow physics. As much as 50% computational cost reduction was observed using the multifidelity approach, while predicting nearly the same amount of uncertainty in drag. 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. Results of this analysis yielded nearly a 70% computational savings to predict a comparable amount of uncertainty in ground noise. Both problems illustrate the applicability and significant computational savings of the multifidelity method for efficient and accurate uncertainty quantification.

Uncertainty quantification of hydrodynamic forces on the DTC model in shallow water waves using CFD and non-intrusive polynomial chaos method

In this paper, uncertainty quantification of hydrodynamic forces on the DTC model advancing in shallow water waves is studied. First, verification and validation (V&V) and uncertainty quantification (UQ) in uncertainty analysis for CFD simulation are compared. A state-of-the-art method for UQ problems, the non-intrusive polynomial chaos (NIPC) method is introduced and validated to be effective by studying a stochastic function, together with the Monte-Carlo (MC) method. Then NIPC method is applied to quantify the uncertainty of the resistance, heave motion and pitch motion of the DTC model in shallow water waves. Computations are performed with the CFD software STAR-CCM+. The uncertainty induced by waves treated as stochastic variables with Gaussian distribution is studied. UQ results obtained by MC method and NIPC method are compared with the benchmark data. The results obtained by NIPC method show better agreement with the benchmark data than those by MC method.

Small Sample-Based Fatigue Reliability Analysis Using Non-Intrusive Polynomial Chaos

Based on small sample of fatigue test data, a new method to obtain p-S-N curve for fatigue reliability analysis using non-intrusive polynomial chaos (NIPC) is proposed to lower test cost. Parameter C in Basquin S-N model is regarded as random variable. Samples of C are calculated through inverse analysis based on small sample of fatigue test life. Then non-intrusive polynomial chaos expansions of C with respect to fatigue life are constituted under different stress levels. Statistics of C can be calculated directly by polynomial coefficients. A fast large-sample of C can be obtained based on NIPC and probability distribution type can be determined through EDF test. Then samples of C under the stress levels can be obtained and substituted into S-N model to calculate corresponding fatigue life samples. The fatigue life under different reliabilities are calculated for fitting p-S-N curve. Fatigue test of Al 2024-T3 plate with hole is performed. p-S-N curves are obtained by proposed method and compared with that obtained by linear regression based on least square method. Almost all relative errors are less than 5%, which show that the proposed method can predict p-S-N curve effectively.

A Two-Phase Monte Carlo Simulation/Non-Intrusive Polynomial Chaos (MSC/NIPC) Method for Quantification of Margins and Mixed Uncertainties (QMMU) in Flutter Analysis

A two-phase Monte Carlo Simulation/Non-intrusive Polynomial Chaos (MCS/NIPC) method for quantification of margins and mixed uncertainties (aleatory and epistemic uncertainties) is proposed in this paper for the flutter speed boundary analysis. Compared with the traditional MCS/MCS method which needs lots of numerical simulations, the MCS/NIPC method can reduce the computational cost without losing accuracy due to the use of point collocation non-intrusive polynomial chaos in the inner loop. Based on the results of uncertainty quantification, a novel practical quantification of margins and mixed uncertainties (QMMU) framework is proposed considering both aleatory and epistemic uncertainties such as the parametric uncertainties in complicated models. A two-dimensional airfoil and a three-dimensional benchmark wing, the AGARD 445.6 wing, are both employed to illustrate the practical application of the proposed methods for flutter speed computation in the presence of mixed uncertainties arising from the material properties and flight conditions. Within the analysis, aleatory and mixed uncertainty quantification are conducted to prove the effectiveness and efficiency of the MCS/NIPC method. Then the QMMU metrics are accomplished for the flutter speed boundary analysis of the two airfoil models. The results demonstrate the potential of the proposed QMMU framework to analyze the stability of engineering systems.

Comparison of intrusive and nonintrusive polynomial chaos expansion-based approaches for high dimensional parametric uncertainty quantification and propagation

We present an uncertainty quantification (UQ) algorithm using the intrusive generalized polynomial chaos (gPC) expansion in combination with dimension reduction techniques and compare the UQ accuracy and computational efficiency of the intrusive gPC-based UQ algorithm to other sampling-based nonintrusive methods. The successful application of intrusive gPC-based UQ is associated with the stochastic Galerkin (SG) projection, which yields a family of models described by several coupled equations of gPC coefficients. Using these coefficients, the evolution of uncertainty in a dynamic system can be quickly determined when there is probabilistic uncertainty in the system. While elegant, when dealing with models that involve complex functions (e.g., nonpolynomial terms) and larger numbers of uncertainties, SG projection becomes computationally intractable and cannot be applied directly to solve gPC coefficients in real-time. To address this issue, the generalized dimension reduction method (gDRM) is used to convert a high-dimensional integral involved in the SG projection into several lower-dimensional integrals that can be easily solved. To show the accuracy of UQ, the algorithm in this work is compared to sampling-based approaches such as the nonintrusive stochastic collocation (SC) and Monte Carlo (MC) simulations using three cases: a nonlinear algebraic benchmark, a penicillin manufacturing process, and autocrine signalling networks of living cells.

Uncertainty and sensitivity analysis of inflow parameters for HyShot II scramjet numerical simulaiton

The flowfield inside a combustor of scramjet such as Hyshot II system is highly complicated and many factors have effect on its numerical prediction. Uncertainty and sensitivity analysis of aerodynamics prediction on Hyshot II intake configuration is performed in this work. The inflow conditions of intake configuration are of vital crucial since they directly determine the inflow conditions of combustor configuration, which in turn have a remarkable influence on the combustion and mixing process. Meanwhile, the varied flight conditions may also introduce great and direct impact on the performance in actual flight tests. It is significant to evaluate the uncertainties and sensitivities and to identify the critical factors of inflow conditions for the intake configuration. Five input parameters are selected as the epistemic uncertain variables. The flowfield prediction is conducted via using Reynolds-averaged Navier-Stokes simulations as the numerical study approach with standard Menter's shear stress transport turbulence model and species transport process for Hyshot II configuration. Stochastic expansions based on point collocation non-intrusive polynomial chaos, which require 42 deterministic simulations in this study, are employed to propagate and estimate the uncertainties. Meanwhile, Sobol indices are adopted to rank and indicate the individual contribution and to identify the critical inflow parameters. The results demonstrate that freestream Mach number and temperature are the dominated contributing parameters among the performance parameters. In addition, as observed from the wall-normal profile of related variables extracted at 355 mm in x coordinate direction in the simulation results, the freestream pressure plays a dominant role in determining the pressure distribution and the freestream temperature is the most prominent contributor to the distribution of temperature and velocity.