Polynomial Chaos Method Research Articles

Convergence Analysis and Adaptive Order Selection for the Polynomial Chaos Approach to Direct Optimal Control under Uncertainties

We consider the use of the polynomial chaos method to approximate optimal control problems with randomly varying uncertain parameters by deterministic surrogate problems. Our focus is on nonlinear problems that require a high expansion order to give meaningful statements about the uncertainty propagation. Their resulting size and complexity pose a computational challenge for traditional optimal control methods. Our contributions include an adaptive optimization strategy which refines the approximation quality separately for each state variable using suitable error estimates. The benefits are twofold: we obtain additional means for solution verification and reduce the computational effort for finding an approximate solution with increased precision, as is highlighted in a numerical case study with two nonlinear real-world problems. The algorithmic contribution is complemented by a convergence proof showing that the optimal control solutions approach the correct solution for increasing expansion orders.

An Adaptive Sparse Anisotropic Polynomial-Chaos Expansion Algorithm Applied to EMC Problems

Various popular methods for Uncertainty Quantification (UQ) rely on the computation of Polynomial-Chaos (PC) expansions, which represent stochastic processes via convergent polynomial series. The computational cost for determining the expansion coefficients in such methods is related to the number of input variables, and is significantly enlarged in high-dimensional problems. On the other hand, the PC coefficients of the basis functions can be considered, in many cases, to form a sparse vector. This property is exploited in this paper to build PC models that are accurate and computationally efficient, by implementing the Orthogonal Matching Pursuit (OMP) as the solver, which can deal with underdetermined linear systems, assuming that their solution is sparse. In order to enhance the computational efficiency even further, the presented PC method is applied adaptively for various PC model orders, and the best solution is chosen according to a leave-one-out criterion. Finally, an optimally-designed multilayer electromagnetic filter for EMC applications is examined, considering ten or twenty stochastic design variables.

Multi-Element Stochastic Galerkin Method Based on Edge Detection for Uncertainty Quantification of Discontinuous Responses

Abstract We propose a new multi-element generalized polynomial chaos (MEgPC) method to minimize the computational costs required for the existing MEgPC to circumvent the Gibbs phenomenon in the presence of discontinuities in a random space. The proposed method uses edge detection to capture the discontinuous behavior of a solution with minimal decomposition. In contrast, the existing MEgPC iterates splitting the random space into two equal parts until achieving a sufficient resolution level. We take advantage of the fact that the stochastic Galerkin (SG) methods facilitate adaptive refinement of the decomposition at every time-step during a computation for the proposed method. The numerical experiments for two-test problems demonstrate the performance of the proposed method. The results show that the proposed method is consistently more accurate than conventional methods for sufficiently high polynomial orders with minimal additional computational costs to capture discontinuities.

Uncertainty quantification on the effects of rain-induced erosion on annual energy production and performance of a Multi-MW wind turbine

Wind turbine blade erosion has risen to the attention of researchers and industry lately in an effort to keep ageing wind farms productive. Although not new, erosion-related blade damage seems to be more severe in recent, particularly off-shore, installations. With the high blade-tip speeds of modern wind turbines, installation in rainy locations can cause significant damage. While all the players in the industry agree that a reduction on Annual Energy Production (AEP) has to be expected, its magnitude remains uncertain, with wide range of variability forecasted in published research. This work proposes a probabilistic framework to assess AEP reductions, allowing for a better understanding of the key mechanism that cause turbine power loss and for a better quantification of AEP losses. The method is tested on the DTU10MW reference case. Erosion-related uncertainties are estimated based on available literature data. Lift and drag coefficients of the airfoils are derived using CFD, and the entire wind turbine is simulated aero-servo-elastically using a Blade Element Momentum code. An arbitrary Polynomial Chaos method is used to estimate the uncertainties associated to key turbine figures due to the erosion inputs. Results show how AEP reductions, while still significant, are lower than most published literature indicates.

Moment-Based Hybrid Polynomial Chaos Method for Interval and Random Uncertain Analysis of Periodical Composite Structural-Acoustic System with Multi-Scale Parameters

An interval and random moment-based arbitrary polynomial chaos method (IRMAPCM) is proposed in this paper for the analysis of periodical composite structural-acoustic systems with multi-scale uncertain-but-bounded parameters. In IRMAPCM, the response of structural-acoustic system is approximated as moment-based arbitrary polynomial chaos (maPC) expansion. IRMAPCM can construct the polynomial basis according to the moment of the random variable without knowing the Probability Density Function (PDF), which can avoid the errors introduced by estimating the PDF. Numerical examples of a hexahedral box and an automobile passenger compartment are given to investigate the effectiveness of IRMAPCM for the prediction of the sound pressure response of structural-acoustic systems.

Reverse jet parameters study on aerodynamic thermal uncertainty of a blunt body

Adding head opposing jets is a common flow control approach to reduce the aerodynamic heating of hypersonic vehicles with blunt-shape head. However, interests of present studies are mostly focused on surface heating reduction originate from opposing jets only on single-and-determinate free incoming conditions, rather than that under existence of freestream perturbations, to the best of knowledge. In this work, two-layers research structure is formed to investigate the jet design parameters on surface thermal uncertainty with occurrence of free incoming perturbations. In the inner layer, the Polynomial Chaos method is applied for quantifying the surface thermal uncertainty, including the mean value and the standard deviation, under free incoming perturbations. In the outer layer, the Variance Analysis method is employed to describe the jet geometry and flow parameters on surface thermal uncertainty. As a consequence, the parameters impact on the mean surface heating under perturbations are found similar to that in baseline freestream condition. Moreover, only jet entry diameter and total-pressure ratio of jet-to-freestream are important when it comes to the standard deviation of surface heating. This research is anticipated to be available of robust thermal-reduction optimization on hypersonic vehicles with blunt-shape head in a wide range of free incoming conditions for reference.

Robust optimization of the NASA C3X gas turbine vane under uncertain operational conditions

The aim of the current paper is the robust optimization of an internally cooled gas turbine vane by increasing the cooling performance and decreasing the sensitivity of the performance against operational uncertainties. The basic geometry of the C3X vane cooling system consists of ten circular channels for reducing the heat load. The numerical analysis is performed using the conjugate heat transfer methodology and the v2−f turbulence model to minimize the simulation error. The operational conditions are considered to be uncertain with Beta probability distribution functions. For quantification of the uncertainties, the polynomial chaos method is used. The main objective of the present study is to increase the blade life span through the minimization of the vane maximum temperature and maximum temperature gradient. To this end, both deterministic and robust optimizations are carried out via a hybrid evolutionary algorithm. The deterministic optimum blade yields lower maximum temperature and temperature gradients. The optimization results clearly show that the robust optimum design is less sensitive to the operational uncertainties, and the maximum blade temperature and temperature gradient are still remarkably lower than the corresponding values of the baseline C3X vane configuration.

High-dimensional uncertainty quantification for Mars atmospheric entry using adaptive generalized polynomial chaos

The probabilistic uncertainties in Mars atmospheric entry degrade the entry guidance performance. The propagation law of high-dimensional uncertainty during Mars atmospheric entry is still an open problem that should be investigated. The current work aims to examine the uncertainty propagation during Mars atmospheric entry due to uncertain initial state and model parameters, with introducing the generalized polynomial chaos method into Mars atmospheric entry dynamics simulations. For more efficient and accurate, generalized polynomial chaos is modified through spectral decomposition and random space decomposition. First, stochastic dynamics are modeled and transformed into equivalent deterministic dynamics in a higher-dimensional space and are updated adaptively when the statistic characteristic of the system state changes greatly. Second, the random space is decomposed when the relative error in variance becomes larger than the predefined threshold. In each random sub-domain, the updated generalized polynomial chaos is employed. Finally, the adaptive generalized polynomial chaos is used to quantify the uncertainty propagation in Mars atmospheric entry dynamics. Comparison studies are also performed with traditional generalized polynomial chaos and Monte-Carlo simulations. The influence levels and the evolution profiles of the initial and parametric uncertainties are revealed through numerical simulations.

Design sensitivity analysis with polynomial chaos for robust optimization

Polynomial chaos (PC) methods with Gauss-type quadrature formulae have been widely applied for robust design optimization. During the robust optimization, gradient-based optimization algorithms are commonly employed, where the sensitivities of the mean and variance of the output response with respect to design variables are calculated. For robust optimization with computationally expensive response functions, although the PC method can significantly reduce the computational cost, the direct application of the classical finite difference method for the analysis of the design sensitivity is impractical with a limited computational budget. Therefore, in this paper, a semi-analytical design sensitivity analysis method based on the PC method is proposed, in which the sensitivity is directly derived based on the Gauss-type quadrature formula without additional function evaluations. Comparative studies conducted on several mathematical examples and an aerodynamic robust optimization problem revealed that the proposed method can reduce the computational cost of robust optimization to a certain extent with comparable accuracy compared with the finite difference-based PC method.

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.

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.

A data-driven polynomial chaos method considering correlated random variables

Variable correlation commonly exists in practical engineering applications. However, most of the existing polynomial chaos (PC) approaches for uncertainty propagation (UP) assume that the input random variables are independent. To address variable correlation, an intrusive PC method has been developed for dynamic system, which however is not applicable to problems with black-box-type functions. Therefore, based on the existing data-driven PC method, a new non-intrusive data-driven polynomial chaos approach that can directly consider variable correlation for UP of black-box computationally expensive problems is developed in this paper. With the proposed method, the multivariate orthogonal polynomial basis corresponding to the correlated input random variables is conveniently constructed by solving the moment-matching equations based on the correlation statistical moments to consider the variable correlation. A comprehensive comparative study on several numerical examples of UP and design optimization under uncertainty with correlated input random variables is conducted to verify the effectiveness and advantage of the proposed method. The results show that the proposed method is more accurate than the existing data-driven PC method with Nataf transformation when the variable distribution is known, and it can produce accurate results with unknown variable distribution, demonstrating its effectiveness.

Stochastic constitutive modeling of elastic-plastic materials with uncertain properties

A new deterministic constitutive model is developed that has a form that is efficient for use in a stochastic constitutive model in which the material properties and the stress and stiffness fields are considered uncertain and modeled as random variables. Simplifications are made to the deterministic model to allow for efficient propagation of uncertainty of model inputs using the Polynomial Chaos (PC) method to quantify the effect of uncertainty in the model parameters on the system response. With large uncertainties in the model parameters that are typical in soil models, the use of a simplified model is justified because the effect of the simplification will be masked by the uncertainties in the model parameters. Each component of the stress and stiffness tensor is expanded along the PC basis, and a small number of PC coefficients are updated along loading/unloading increments. The simplified deterministic model gives similar results to other traditional models, but the equations it involves are computationally much more efficient when extended to the stochastic model and solved with PC. The results are PC coefficients of each stress and stiffness component along the loading/unloading parts of the curve, from which probability distributions of the stress and stiffness can be reconstructed. Compared with traditional Monte-Carlo simulation, the PC method gives similar results while being several orders of magnitude faster computationally. The PC expansion method can also be extended to the non-linear stochastic finite element method.

Long duration response evaluation of linear structural system with random system properties using time dependent polynomial chaos

Polynomial Chaos (PC) is one of the popular methods for approximate evaluation of responses of stochastic structural mechanics problems. In the case of a structural system under dynamic loading, the PC method is combined with a time integration scheme. The PC method with Galerkin projection converts the stochastic PDE to a set of simultaneous deterministic PDE, which can be solved using any standard time integration scheme. However, the convergence of responses is known to fail for long duration time integration problem due to the development of unacceptable error in calculated response. The probability density functions of the responses are not of constant type as the time progresses. However, in the case of generalized PC (gPC), these are approximated using initially considered gPC, thus loses its efficiency. Time-dependent general polynomial chaos (TDgPC) can be considered to overcome these difficulties, where polynomials are generated on the fly to match with the pdf of the responses. The present study considered an investigation of structural mechanics responses with both Gaussian and non-Gaussian random system properties under dynamic loading using TDgPC. An adaptive updation scheme based on the RMS value of the coefficients of PC expansion is investigated for a proper updation of PC expansion. The curse of dimensionality of PC expansion is addressed by considering only the dominant components of the responses evaluated using KL expansion. Further, starting with lower-order PC, the order of PC expansion is increased only after the first updation, thus reducing computational complexities.

Environmental consequence analysis of oil spills from onshore pipelines with parametric uncertainty

Due to spatial heterogeneities, measurement errors and inherent errors of parameter estimation, hydrogeology parameters are not exactly known. Therefore, the accuracy of environmental consequence prediction for oil spills from onshore pipeline accidents is subject to the uncertainty of soil parameters. However, due to the time-consuming calculation of uncertainty analysis and the difficulty of modeling the cross-correlation among soil parameters, the correlativity is often overlooked in the current researches involving the impacts of uncertain soil parameters on the environmental consequence estimation. To address above concerns, this study introduces a novel methodological framework for investigating the contribution of correlated soil parameters on the environmental consequence caused by spilled oil from onshore oil pipeline accidents. The proposed methodological framework combines a polynomial chaos expansion(PCE) surrogate model and methods from Uncertainty Analysis (UA) and PAWN-based Global Sensitivity Analysis (GSA). In addition, the practical application of this framework is illustrated with a case study for estimating the contaminated range of spilled oil from onshore pipeline accidents.First of all, the correlation of soil parameters is modeled by the copula function, and a PCE surrogate model has been constructed in order to alleviate the computational burden of the following UA and PAWN-based GSA. Then, UA is performed to quantify the impacts of correlativity by means of Monte Carlo (MC) method. Results reveal the correlativity of soil parameters introduces a substantial variation in the estimation of contaminated depth of spilled oil. In addition, the UA results show that selecting different copula functions to describe the correlation of soil parameters may lead to different variations of contaminated depth. Moreover, the uncertainty of the calculation results has been reduced with the consideration of correlated soil parameters, which may make the calculation results more reliable. Finally, the PAWN-based GSA is performed to identify the most influential parameter on the variation of contaminated depth. Results of the GSA show that saturated permeability and scale parameter α are primarily responsible for the variability of the model output, while the porosity exerts a smaller influence. This finding also suggests the possibility of replacing the less influential parameters with its average value, thus appreciably reducing the computation cost of the problem. The results of GSA not only offer a better understanding of correlativity to the contaminated range of spilled oil, but also identify the parameters for which additional effort needs to be invested to reduce their uncertainty and, as a result, the uncertainty associated with environmental consequence estimation of onshore pipeline accidents.

Uncertainty Quantification of Waveguide Dispersion Using Sparse Grid Stochastic Testing

To quantify waveguide dispersion uncertainty, we propose a stochastic analysis technique based on the polynomial chaos method, where Smolyak sparse grids are used as sample spaces for the recently developed nonintrusive stochastic testing (ST) technique. This strategy allows for a smaller set of candidate collocation nodes in the ST algorithm compared with the computationally expensive tensor grid approach. The method is tailored to the analysis of waveguide structures with a statistically spatially varying dielectric-property profile, which is then captured by a discrete set of uncorrelated random variables by means of Karhunen–Loeve transforms. In this way, the propagation properties of realistic stochastic waveguides can be predicted, which is of major importance in the electronic design process. The presented method, which is constructed around a second-order full-wave deterministic finite element solver, is illustrated through the uncertainty quantification of two waveguide dispersion problems and is validated using commercial electromagnetic field solvers and the Monte Carlo method.