Non-intrusive Polynomial Chaos Method Research Articles

A Single Formulation for Uncertainty Propagation in Turbomachinery: SAMBA PC

This work newly proposes an uncertainty quantification (UQ) method named sparse approximation of moment-based arbitrary polynomial chaos (SAMBA PC) that offers a single solution to many current problems in turbomachinery applications. At the moment, every specific case is characterized by a variety of different input types such as histograms (from experimental data), normal probability density functions (PDFs) (design rules) or fat tailed PDFs (for rare events). Thus, the application of UQ requires the adaptation of ad hoc methods for each individual case. A second problem is that parametric PDFs have to be determined for all inputs. This is difficult if only few samples are available. In gas turbines, however, the collection of statistical information is difficult, expensive, and having scarce information is the norm. A third critical limitation is that if using nonintrusive polynomial chaos (NIPC) methods, the number of required simulations grows exponentially with increasing numbers of input uncertainties: the so-called “curse of dimensionality.” It is shown that the fitting of parametric PDFs to small data sets can lead to large bias and the direct use of the available data is more accurate. This is done by propagating uncertainty through several test functions and the computational fluid dynamics (CFD) simulation of a diffuser, highlighting the impact of different PDF fittings on the output. From the results, it is concluded that the direct propagation of the experimental data set is preferable to the fit of parametric distributions if data is scarce. Thus, the suggested method offers an alternative to the maximum entropy theorem to handle scarce data. SAMBA simplifies the mathematical procedure for many different input types by basing the polynomial expansion on moments. Its moment-based expansion automatically takes care of arbitrary combinations of different input data. It is also numerically efficient compared to other UQ implementations. The relationship between the number of random variables and number of simulation is linear (only 21 simulations for ten input random variables are required). It is shown in this paper that SAMBA's algorithm can propagate a high number of input distributions through a set of nonlinear analytic test functions. Doing this, the code needs a very small number of simulations and preserve a 5% error margin. SAMBA's flexibility to handle different forms of input distributions and a high number of input variables is shown on a low-pressure turbine (LPT) blade-based on H2 profile. The relative importance of manufacturing errors in different location of the blade is analyzed.

Robust aerodynamic shape design based on an adaptive stochastic optimization framework

Optimization techniques combined with uncertainty quantification are computationally expensive for robust aerodynamic optimization due to expensive CFD costs. Surrogate model technology can be used to improve the efficiency of robust optimization. In this paper, non-intrusive polynomial chaos method and Kriging model are used to construct a surrogate model that associate stochastic aerodynamic statistics with airfoil shapes. Then, global search algorithm is used to optimize the model to obtain optimal airfoil fast. However, optimization results always depend on the approximation accuracy of the surrogate model. Actually, it is difficult to achieve a high accuracy of the model in the whole design space. Therefore, we introduce the idea of adaptive strategy to robust aerodynamic optimization and propose an adaptive stochastic optimization framework. The surrogate model is updated adaptively by increasing training airfoils according to historical optimization results to guarantee the accuracy near the optimal design point, which can greatly reduce the number of training airfoils. The proposed method is applied to a robust aerodynamic shape optimization for drag minimization considering uncertainty of Mach number in transonic region. It can be concluded that the proposed method can obtain better optimal results more efficiently than the traditional robust optimization method and global surrogate model method.

Uncertainty Quantification and Sensitivity Analysis of Transonic Aerodynamics with Geometric Uncertainty

Airfoil geometric uncertainty can generate aerodynamic characteristics fluctuations. Uncertainty quantification is applied to compute its impact on the aerodynamic characteristics. In addition, the contribution of each uncertainty variable to aerodynamic characteristics should be computed by the uncertainty sensitivity analysis. In the paper, Sobol’s analysis is used for uncertainty sensitivity analysis and a nonintrusive polynomial chaos method is used for uncertainty quantification and Sobol’s analysis. It is difficult to describe geometric uncertainty because it needs a lot of input parameters. In order to alleviate the contradiction between the variable dimension and computational cost, a principal component analysis is introduced to describe geometric uncertainty of airfoil. Through this technique, the number of input uncertainty variables can be reduced and typical global deformation modes can be obtained. By uncertainty quantification, we can learn that the flow characteristics of shock wave and boundary layer separation are sensitive to the geometric uncertainty in transonic region, which is the main reason that transonic drag is sensitive to the geometric uncertainty. The sensitivity analysis shows that the model can be simplified by eliminating unimportant geometric modes. Moreover, which are the most important geometric modes to transonic aerodynamics can be learnt. This is very helpful for airfoil design.

Uncertainty Quantification of Numerical Simulation of Flows around a Cylinder Using Non-intrusive Polynomial Chaos**Supported by the National Natural Science Foundation of China under Grant No 11371069, the Young Foundation of Institute of Applied Physics and Computational Mathematics under Grant No ZYSZ1518-13, and the Science Foundation of China Academy of Engineering Physics under Grant No 2013A0101004.

The uncertainty quantification of flows around a cylinder is studied by the non-intrusive polynomial chaos method. Based on the validation with benchmark results, discussions are mainly focused on the statistic properties of the peak lift and drag coefficients and base pressure drop over the cylinder with the uncertainties of viscosity coefficient and inflow boundary velocity. As for the numerical results of flows around a cylinder, influence of the inflow boundary velocity uncertainty is larger than that of viscosity. The results indeed demonstrate that a five-order degree of polynomial chaos expansion is enough to represent the solution of flow in this study.

Simulation-based determination of systematic errors of flow meters due to uncertain inflow conditions

Computational fluid dynamics (CFD) provides well-established tools for the prediction of the velocity profiles in turbulent pipe flows. As far as industrial pipe and district heating systems are concerned, combinations of elbows are the most common pipe assemblies. Among the different pipe combinations, double elbows out-of-plane are of special interest, since they introduce strong disturbances into the flow profile and have a strong influence on many common types of flow meters. In front of a double elbow there is often another flow-disturbing installation. As a result the upstream conditions are unknown and an investigation of the resulting systematic bias on the measurement of the flow rate and the associated contribution to its measurement uncertainty is necessary. We demonstrate here that this can be achieved by a variation of the inlet profile in terms of swirls and asymmetry components. In particular, an ultrasonic and an electromagnetic flow meter are modeled in order to quantify the systematic errors stemming from uncertain inflow conditions. For this purpose, a generalized non-intrusive polynomial chaos method has been used in conjunction with a commercial CFD code. As the most influential parameters on the measured volume flow, the distance between the double elbow and the flow meter as well as the orientation of the flow meter are considered as random variables in the polynomial chaos approach. This approach allowed us to obtain accurate prediction of the systematic error for the ultrasonic and electromagnetic meter as functions of the distance to the double elbow. The resulting bias in the flow rate has been found to be in the range of 1.5–4.5% (0.1–0.5%) with a systematic uncertainty contribution of 2–2.4% (0.6–0.7%) for the ultrasonic (electromagnetic) flow meter if the distance to the double elbow is smaller than 40 pipe diameters. Moreover, it is demonstrated that placing the flow meters in a Venturi constriction leads to substantial decrease of the bias and the contribution to the measurement uncertainty stemming from the uncertain inflow condition.

A Non-Intrusive Polynomial Chaos Method to Efficiently Quantify Uncertainty in an Aircraft T-Tail

The problem of developing robust methods for uncertainty quantification (UQ) is of major interest in the engineering and scientific community. To quantify uncertainty, probabilistic models have been developed where traditionally Monte Carlo (MC) methods were used to capture uncertainty bounds. In the engineering context, UQ methods can be practically implemented to limit the amount of prototype redesigns. However MC methods are computationally inefficient due to the large number of samples required to obtain an accurate solution. Polynomial Chaos (PC) methods have recently emerged as an efficient method of probabilistic quantification in lower dimensions compared to MC. This paper will show the ability of a non-intrusive PC method to efficiently quantify uncertainty through first and second order statistics. This approach will lend itself to the treatment of a finite element T-Tail model, using Nastran as a black box around which PC curves can be fit based on its outputs.

A framework to quantify uncertainty in simulations of oil transport in the ocean

AbstractAn uncertainty quantification framework is developed for the DeepC Oil Model based on a nonintrusive polynomial chaos method. This allows the model's output to be presented in a probabilistic framework so that the model's predictions reflect the uncertainty in the model's input data. The new capability is illustrated by simulating the far‐field dispersal of oil in a Deepwater Horizon blowout scenario. The uncertain input consisted of ocean current and oil droplet size data and the main model output analyzed is the ensuing oil concentration in the Gulf of Mexico. A 1331 member ensemble was used to construct a surrogate for the model which was then mined for statistical information. The mean and standard deviations in the oil concentration were calculated for up to 30 days, and the total contribution of each input parameter to the model's uncertainty was quantified at different depths. Also, probability density functions of oil concentration were constructed by sampling the surrogate and used to elaborate probabilistic hazard maps of oil impact. The performance of the surrogate was constantly monitored in order to demarcate the space‐time zones where its estimates are reliable.

Non-intrusive polynomial chaos for efficient uncertainty analysis in parametric roll simulations

Monte Carlo analyses are generally considered the standard for uncertainty analysis. While accurate, these analyses can be expensive computationally. Recently, polynomial chaos has been proposed as an alternative approach to the estimation of uncertainty distributions (Hosder et al. A non-intrusive polynomial chaos method for uncertainty propagation in CFD simulations. In: 44th AIAA aerospace sciences meeting and exhibit, Reno, Nevada, 2006; Wu et al. Uncertainty analysis for parametric roll using non-intrusive polynomial chaos. In: Proceedings of the 12th international ship stability workshop, Washington, DC, USA, 2011). This approach works by representing the function as a series of orthogonal polynomials; the weights for which can be calculated via several methods. Previous studies have demonstrated the usefulness of this technique for comparatively simple systems such as parametric roll modeled by the Mathieu equation with normally distributed parameter values (Wu et al. Uncertainty analysis for parametric roll using non-intrusive polynomial chaos. In: Proceedings of the 12th international ship stability workshop, Washington, DC, USA, 2011). In the present work, a polynomial chaos method is applied to a nonlinear computational ship dynamics model with normally distributed input parameters. Test cases were selected where parametric roll was expected to potentially occur. The resulting probability distributions are compared to the results of a Monte Carlo analysis. In general, these results demonstrate good agreement between Monte Carlo simulation and polynomial chaos in the absence of capsize with significant computation gains found with polynomial chaos. Overall, we conclude that polynomial chaos is an effective tool for reducing simulation time costs when studying parametric roll, and potentially other ship dynamics phenomena, particularly in the absence of capsize-like bifurcations.

Uncertainty Analysis in Transcranial Magnetic Stimulation Using Nonintrusive Polynomial Chaos Expansion

We propose a framework of nonintrusive polynomial chaos methods for transcranial magnetic stimulation (TMS) to investigate the influence of the uncertainty in the electrical conductivity of biological tissues on the induced electric field. The conductivities of three different tissues, namely, cerebrospinal fluid, gray matter (GM), and white matter, are modeled as uniformly distributed random variables. The investigations are performed on a simplified model of a cortical gyrus/sulcus structure. The statistical moments are calculated by means of a generalized polynomial chaos expansion using a regression and cubature approach. Furthermore, the results are compared with the solutions obtained by stochastic collocation. The accuracy of the methods to predict random field distributions was compared by applying different grids and orders of expansion. An investigation on the convergence of the expansion showed that in the present framework, an order 4 expansion is sufficient to determine results with an error of <;1%. The results indicate a major influence of the uncertainty in electrical conductivity on the induced electric field. The standard deviation exceeds values of 20%-40% of the mean induced electric field in the GM. A sensitivity analysis revealed that the uncertainty in electrical conductivity of the GM affects the solution the most. This paper outlines the importance of exact knowledge of the electrical conductivities in TMS in order to provide reliable numerical predictions of the induced electric field. Furthermore, it outlines the performance and the applicability of spectral methods in the framework of TMS for future studies.

A Novel Weakly-Intrusive Non-linear Multiresolution Framework for Uncertainty Quantification in Hyperbolic Partial Differential Equations

In this paper, a novel multiresolution framework, namely the Truncate and Encode (TE) approach, previously proposed by some of the authors (Abgrall et al. in J Comput Phys 257:19---56, 2014. doi:10.1016/j.jcp.2013.08.006), is generalized and extended for taking into account uncertainty in partial differential equations (PDEs). Innovative ingredients are given by an algorithm permitting to recover the multiresolution representation without requiring the fully resolved solution, the possibility to treat a whatever form of pdf and the use of high-order (even non-linear, i.e. data-dependent) reconstruction in the stochastic space. Moreover, the spatial-TE method is introduced, which is a weakly intrusive scheme for uncertainty quantification (UQ), that couples the physical and stochastic spaces by minimizing the computational cost for PDEs. The proposed scheme is particularly attractive when treating moving discontinuities (such as shock waves in compressible flows), even if they appear during the simulations as it is common in unsteady aerodynamics applications. The proposed method is very flexible since it can easily coupled with different deterministic schemes, even with high-resolution features. Flexibility and performances of the present method are demonstrated on various numerical test cases (algebraic functions and ordinary differential equations), including partial differential equations, both linear and non-linear, in presence of randomness. The efficiency of the proposed strategy for solving stochastic linear advection and Burgers equation is shown by comparison with some classical techniques for UQ, namely Monte Carlo or the non-intrusive polynomial chaos methods.

Nonintrusive-Polynomial-Chaos-Based Kinematic Reliability Analysis for Mechanisms with Mixed Uncertainty

Due to the scarcity of statistical data, epistemic uncertainties are inevitable in the mechanism. As a promising uncertainty quantification technique, polynomial chaos has advantages over other methods in terms of accuracy and efficiency. In this paper, an improved nonintrusive polynomial chaos method is proposed for the kinematic reliability analysis of the mechanism with fuzzy and random variables as well as fuzzy failure/safety states. Klir log-scale transformation is applied to unify the fuzzy and random variables. Meanwhile, the polynomial-chaos-based probability formula of the fuzzy event is developed to characterize the fuzzy failure/safety states. The proposed method is applied to the reliability analysis of a retractable mechanical system.The results show good accuracy and efficiency of the proposed method when compared with the response surface method (RSM), Kriging method, and Monte Carlo simulation (MCS).

Non-intrusive polynomial chaos methods and its application in the parameters assessment of explosion product JWL

A non-intrusive polynomial chaos method was introduced, and the main procedure of uncertainty quantification for JWL-EOS parameters was given. The method was implemented for the uncertainty quantification of the input parametersR1 and R2 of JWL-EOS to the detonation of plane and divergence. The results show that the methods of non-intrusive polynomial chaos can provide a valuable tool for the simulation of propagation of uncertainties, and uncertainty quantification for modeling and simulation in complex engineering.

Dynamic system uncertainty propagation using polynomial chaos

The classic polynomial chaos method (PCM), characterized as an intrusive methodology, has been applied to uncertainty propagation (UP) in many dynamic systems. However, the intrusive polynomial chaos method (IPCM) requires tedious modification of the governing equations, which might introduce errors and can be impractical. Alternative to IPCM, the non-intrusive polynomial chaos method (NIPCM) that avoids such modifications has been developed. In spite of the frequent application to dynamic problems, almost all the existing works about NIPCM for dynamic UP fail to elaborate the implementation process in a straightforward way, which is important to readers who are unfamiliar with the mathematics of the polynomial chaos theory. Meanwhile, very few works have compared NIPCM to IPCM in terms of their merits and applicability. Therefore, the mathematic procedure of dynamic UP via both methods considering parametric and initial condition uncertainties are comparatively discussed and studied in the present paper. Comparison of accuracy and efficiency in statistic moment estimation is made by applying the two methods to several dynamic UP problems. The relative merits of both approaches are discussed and summarized. The detailed description and insights gained with the two methods through this work are expected to be helpful to engineering designers in solving dynamic UP problems.

Efficient Bayesian inference of subsurface flow models using nested sampling and sparse polynomial chaos surrogates

An efficient Bayesian calibration method based on the nested sampling (NS) algorithm and non-intrusive polynomial chaos method is presented. Nested sampling is a Bayesian sampling algorithm that builds a discrete representation of the posterior distributions by iteratively re-focusing a set of samples to high likelihood regions. NS allows representing the posterior probability density function (PDF) with a smaller number of samples and reduces the curse of dimensionality effects. The main difficulty of the NS algorithm is in the constrained sampling step which is commonly performed using a random walk Markov Chain Monte-Carlo (MCMC) algorithm. In this work, we perform a two-stage sampling using a polynomial chaos response surface to filter out rejected samples in the Markov Chain Monte-Carlo method. The combined use of nested sampling and the two-stage MCMC based on approximate response surfaces provides significant computational gains in terms of the number of simulation runs. The proposed algorithm is applied for calibration and model selection of subsurface flow models.

A one-time truncate and encode multiresolution stochastic framework

In this work a novel adaptive strategy for stochastic problems, inspired from the classical Hartenʼs framework, is presented. The proposed algorithm allows building, in a very general manner, stochastic numerical schemes starting from a whatever type of deterministic schemes and handling a large class of problems, from unsteady to discontinuous solutions. Its formulations permits to recover the same results concerning the interpolation theory of the classical multiresolution approach, but with an extension to uncertainty quantification problems. The present strategy permits to build numerical scheme with a higher accuracy with respect to other classical uncertainty quantification techniques, but with a strong reduction of the numerical cost and memory requirements. Moreover, the flexibility of the proposed approach allows to employ any kind of probability density function, even discontinuous and time varying, without introducing further complications in the algorithm. The advantages of the present strategy are demonstrated by performing several numerical problems where different forms of uncertainty distributions are taken into account, such as discontinuous and unsteady custom-defined probability density functions. In addition to algebraic and ordinary differential equations, numerical results for the challenging 1D Kraichnan–Orszag are reported in terms of accuracy and convergence. Finally, a two degree-of-freedom aeroelastic model for a subsonic case is presented. Though quite simple, the model allows recovering some physical key aspect, on the fluid/structure interaction, thanks to the quasi-steady aerodynamic approximation employed. The injection of an uncertainty is chosen in order to obtain a complete parameterization of the mass matrix. All the numerical results are compared with respect to classical Monte Carlo solution and with a non-intrusive Polynomial Chaos method.

A one-dimensional polynomial chaos method in CFD–Based uncertainty quanti-fication for ship hydrodynamic performance

A one-dimensional non-intrusive Polynomial Chaos (PC) method is applied in Uncertainty Quantification (UQ) studies for CFD–based ship performances simulations. The uncertainty properties of Expected Value (EV) and Standard Deviation (SD) are evaluated by solving the PC coefficients from a linear system of algebraic equations. The one-dimensional PC with the Legendre polynomials is applied to: (1) stochastic input domain and (2) Cumulative Distribution Function (CDF) image domain, allowing for more flexibility. The PC method is validated with the Monte-Carlo benchmark results in several high-fidelity, CFD-based, ship UQ problems, evaluating the geometrical, operational and environmental uncertainties for the Delft Catamaran 372. Convergence is studied versus PC order P for both EV and SD, showing that high order PC is not necessary for present applications. Comparison is carried out for PC with/without the least square minimization when solving the PC coefficients. The least square minimization, using larger number of CFD samples, is recommended for current test cases. The study shows the potentials of PC method in Robust Design Optimization (RDO) and Reliability-Based Design Optimization (RBDO) of ship hydrodynamic performances.

Multistep Uncertainty Quantification Approach Applied to Hypersonic Reentry Flows

The objective of this study was to introduce and demonstrate a computationally efficient, multistep uncertainty quantification approach for high-fidelity, hypersonic reentry flow simulations, which may include large numbers of aleatory and epistemic uncertainties. The multistep uncertainty quantification approach included several key components, including a sensitivity-based dimension reduction process that used a local sensitivity analysis at selected sample locations to approximate global sensitivities. Other components included a method to update existing deterministic samples after dimension reduction and a modified point-collocation nonintrusive polynomial chaos method that incorporates existing local sensitivity information. The multistep uncertainty quantification approach was demonstrated on two model problems. The first was a model for stagnation point convective heat transfer in hypersonic flow. Mixed uncertainty quantification analysis results in reduced dimensions compared well with Monte Carlo simulations. The second problem was a high-fidelity, computational fluid dynamics model for stagnation point radiative heat flux on a Hypersonic Inflatable Aerodynamic Decelerator during a Mars entry. The model consisted of 93 uncertain parameters, coming from both flowfield and radiation modeling. The model was reduced to ten and five uncertain variables, accounting for 95 and 90% of the total output variance, respectively. Pure aleatory, epistemic, and mixed uncertainty quantification analyses were in agreement with previous work, proving the potential and applicability of the multistep uncertainty quantification process for complex hypersonic reentry flow models.

Robust Design Optimization Under Mixed Uncertainties With Stochastic Expansions

The objective of this paper is to introduce a computationally efficient and accurate approach for robust optimization under mixed (aleatory and epistemic) uncertainties using stochastic expansions that are based on nonintrusive polynomial chaos (NIPC) method. This 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 a weighted sum of the stochastic measures, which are minimized simultaneously to ensure the robustness of the final design to both inherent and epistemic uncertainties. The optimization approach is demonstrated on two model problems with mixed uncertainties: (1) the robust design optimization of a slider-crank mechanism and (2) robust design optimization of a beam. The stochastic expansions are created with two different NIPC methods, Point-Collocation and Quadrature-Based NIPC. The optimization results are compared to the results of another robust optimization technique that utilizes double-loop Monte Carlo sampling (MCS) for the propagation of mixed uncertainties. The optimum designs obtained with two different optimization approaches agree well in both model problems; however, the number of function evaluations required for the stochastic expansion based approach is much less than the number required by the Monte Carlo based approach, indicating the computational efficiency of the optimization technique introduced.

Uncertainty‐based robust aerodynamic optimization of rotor blades

SUMMARYThe aerodynamic performance of a compressor is highly sensitive to uncertain working conditions. This paper presents an efficient robust aerodynamic optimization method on the basis of nondeterministic computational fluid dynamic (CFD) simulation and multi‐objective genetic algorithm (MOGA). A nonintrusive polynomial chaos method is used in conjunction with an existing well‐verified CFD module to quantify the uncertainty propagation in the flow field. This method is validated by comparing with a Monte Carlo method through full 3D CFD simulations on an axial compressor (National Aeronautics and Space Administration rotor 37). On the basis of the validation, the nondeterministic CFD is coupled with a surrogate‐based MOGA to search for the Pareto front. A practical engineering application is implemented to the robust aerodynamic optimization of rotor 37 under random outlet static pressure. Two curve angles and two sweep angles at tip and hub are used as design variables. Convergence analysis shows that the surrogate‐based MOGA can obtain the Pareto front properly. Significant improvements of both mean and variance of the efficiency are achieved by the robust optimization. The comparison of the robust optimization results with that of the initial design, and a deterministic optimization demonstrate that the proposed method can be applied to turbomachinery successfully. Copyright © 2012 John Wiley &amp; Sons, Ltd.

Stochastic response surfaces based on non-intrusive polynomial chaos for uncertainty quantification

This paper gives an overview of computationally efficient stochastic response surface techniques based on non-intrusive polynomial chaos (NIPC) for uncertainty quantification in numerical models. The results of uncertainty analysis can be used in robust and reliability-based design and optimisation studies as well as the assessment of the accuracy of the simulation results. In particular, the uncertainty quantification with NIPC methods which require no modification to the existing deterministic models are demonstrated on computational fluid dynamics (CFD) simulations in this paper. 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. The theory and description of various NIPC methods used for non-deterministic CFD simulations are presented, which can be applied to any other computational models used in analysis and optimisation problems. Several stochastic fluid dynamics examples are given 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.