Polynomial Chaos Expansion Method Research Articles

An advanced Polynomial Chaos Expansion method for sensitivity analysis of aero-engine fuel gear pumps

Global sensitivity analysis plays an important role in robustness optimization design of aero-engine fuel gear pumps to investigate how input parameters uncertainties contribute to performances uncertainties. In this paper, inspired by the Leave-One-Out method, an advanced Polynomial Chaos Expansion (PCE) method equipped with Adaptive design of both training Points and polynomial Order, denoted as APO-PCE, is proposed to efficiently estimate the variance-based sensitivity indices. A novel active learning strategy is developed for identifying the optimal candidate sample point to adaptively design training points and simultaneously update the polynomial order. Input parameters, including rotational speed, inlet pressure, outlet pressure, and gear tip clearance, are reasonably characterized as nine convenient sampling Gaussian variables by probabilistic modeling and Karhuben–Loève expansion. The results show that the proposed APO-PCE method is superior to the classical PCE method, and the uncertainties of rotational speed, inlet pressure, and outlet pressure are the main factors for the uncertainties of supply performances and flow field characteristics, while the gear tip clearance uncertainty only affects the flow loss at lower rotational speed.

An advanced Polynomial Chaos Expansion method for sensitivity analysis of aero-engine fuel gear pumps

Global sensitivity analysis plays an important role in robustness optimization design of aero-engine fuel gear pumps to investigate how input parameters uncertainties contribute to performances uncertainties. In this paper, inspired by the Leave-One-Out method, an advanced Polynomial Chaos Expansion (PCE) method equipped with Adaptive design of both training Points and polynomial Order, denoted as APO-PCE, is proposed to efficiently estimate the variance-based sensitivity indices. A novel active learning strategy is developed for identifying the optimal candidate sample point to adaptively design training points and simultaneously update the polynomial order. Input parameters, including rotational speed, inlet pressure, outlet pressure, and gear tip clearance, are reasonably characterized as nine convenient sampling Gaussian variables by probabilistic modeling and Karhuben–Loève expansion. The results show that the proposed APO-PCE method is superior to the classical PCE method, and the uncertainties of rotational speed, inlet pressure, and outlet pressure are the main factors for the uncertainties of supply performances and flow field characteristics, while the gear tip clearance uncertainty only affects the flow loss at lower rotational speed.

Robust topology optimization for multi-material structures considering material uncertainties

Multi-material structures have attracted more and more attention from scholars and engineers, because they usually have good composite performance. Uncertainty factors are ubiquitous in practice, especially for material uncertainties. Multi-material structures contain many materials with different properties, material uncertainties will affect the allocation of material usage, thereby affecting structural performance. Therefore, it is crucial to consider material uncertainties in the design of multi-material structures. This paper detailedly explores the impacts of elastic modulus uncertainty and Poisson's ratio uncertainty on the design of multi-material structures. The topology optimization is to minimize the compliance subjected to a mass constraint, where multiple materials can be freely allocated during the optimization. Floating projection topology optimization is adopted to search for the structural topologies, and robust sensitivity information is derived. For the quantification of material uncertainties, this paper introduces a non-intrusive polynomial chaos expansion (PCE) method to implicitly quantify them, the computational efficiency and accuracy in PCE is compared with the classical Monte Carlo method. Finally, topology optimization examples are provided to demonstrate the effects of elastic modulus uncertainty, Poisson's ratio uncertainty and hybrid uncertainty on the design of multi-material structures.

Data assimilation method and application of shear stress transport turbulence model for complex separation of internal shock boundary layer flow

Data assimilation method and application of shear stress transport turbulence model for complex separation of internal shock boundary layer flow

Active Learning of Ensemble Polynomial Chaos Expansion Method for Global Sensitivity Analysis

Global sensitivity analysis (GSA) techniques, especially Sobol indices, have gained prominence in assessing the impact of uncertainty across all input variables on the variation of model response. However, GSA can incur intensive computational costs, particularly when dealing with time-consuming models. To address this issue, an ensemble polynomial chaos expansion (EPCE) surrogate model based on an active learning hybrid (ALH) criterion is proposed. The proposed method integrates a weighted combination of multiple sparse PCE surrogate models to formulate the EPCE method. To enhance the performance, the ALH criterion, which balances local exploitation and global exploration, is developed to sequentially select samples for updating the EPCE model. In terms of local exploitation, the model correlation and sensitivity pursuit criteria are derived to identify the local region displaying both the largest discrepancy among multiple sparse PCE methods and the most substantial contribution to the total variance. The ALH criterion also incorporates a minimum distance-based measure for global exploration. The performance of the proposed method is evaluated by four benchmark functions and an engineering application. The numerical studies demonstrate that the proposed method exhibits favorable performance for conducting GSA.

An efficient sparse surrogate model for aerodynamic characteristics of a supersonic compressor cascade with uncertain geometric deformations

The increasing demand for high-load compressors necessitates research and design of supersonic compressor blades. Geometric deformations of the blades resulting from manufacturing errors introduce uncertainties in their aerodynamic characteristics. For uncertainty analysis, Polynomial Chaos Expansion (PCE) has gained popularity among engineers in various disciplines. However, as the number of random variables used to describe uncertainties increases, the efficiency of the PCE method diminishes. To overcome this limitation, an Efficient Sparse Surrogate Model (ESSM) was first proposed. The ESSM combines PCE to approximate global characteristics and Gaussian process modeling to capture local variability, resulting in a highly accurate model. Additionally, by utilizing an iteratively diffeomorphic modulation under observable response preserving homotopy, important PCE basis functions are adaptively selected, leading to a sparse surrogate model that significantly reduces the required training samples. Then using the ESSM, efficient Uncertainty Quantification (UQ) and global sensitivity analysis were conducted to evaluate the impact of uncertain geometric deformations on the aerodynamic characteristics of a supersonic compressor cascade. Principal Component Analysis (PCA) was employed in conjunction with an Improved Class Shape Transformation (ICST) to characterize geometric deformations caused by manufacturing errors. The proposed ICST method exhibits enhanced accuracy in fitting near the leading and trailing edges compared to the CST method. PCA enables a 50% reduction in the number of random variables needed to describe geometric deformations. The UQ results establish a quantitative correlation between geometric deformations and supersonic aerodynamics, and the sensitivity analysis identifies some specific forms of geometric deformations that significantly impact the aerodynamics. Finally, the study investigated the flow mechanisms associated with typical geometric deformation forms, further providing recommendations for manufacturing and testing of supersonic blades to prevent performance deterioration.

A comprehensive study on seismic dynamic responses of stochastic structures using sparse grid-based polynomial chaos expansion

The seismic response is a critical aspect of structural engineering, as it directly influences the design and safety evaluation of such systems. Structural seismic response becomes more complicated when unavoidable uncertainties in the structure are considered. Therefore, solving and analyzing the stochastic dynamic response of structures efficiently and accurately is of great significance. Various uncertainty quantification methods such as the stochastic finite element method, Monte Carlo method, and polynomial chaos expansion method have made significant progress in this field. However, these methods do not provide a comprehensive study of the seismic response of stochastic structures and efficiency problems rapidly emerge as the number of random variables increases. This paper delivers a comprehensive study of the seismic dynamic response of space truss and reticulated shell structure considering uncertainties in material properties, geometry, and loading. To this end, the finite element model of the structure was first modeled and the deterministic solutions were obtained. Then, polynomial surrogates are modeled for the natural frequency, mode participation factor, mode superposition response, and transient response of the structure under stochastic uncertainties. The sparse grid technique is used to select sparse quadrature points, which overcomes the curse of dimensionality caused by full grid integration. The results demonstrate that the sparse grid-based polynomial chaos expansion is more efficient. Uncertainty quantification reveals the uncertainty of modal analysis, response spectrum analysis, and transient response analysis of space truss and reticulated shell under seismic excitation. In summary, this study provides valuable insights and an efficient and accurate uncertainty quantification tool for the seismic dynamic analysis of stochastic structures, ultimately contributing to safer and more robust structural designs.

Deterministic and stochastic flexural behaviors of laminated composite thin-walled I-beams using a sinusoidal higher-order shear deformation theory

In this study, a novel sinusoidal higher-order shear deformation thin-walled beam theory is presented to examine the effects of material properties and external load uncertainty on static responses of laminated composite thin-walled beams with open sections. The solution for the deterministic flexural analysis is based on Hamilton’s principle and Ritz-type exponential shape function series. Several mechanical parameters of laminated composite materials are randomized and plugged into the beam solver to investigate the thin-walled beam’s stochastic flexural behaviors. The computational cost and accuracy of the polynomial chaos expansion (PCE) method with both projection and linear regression approaches are presented and evaluated by comparing its results with crude Monte Carlo simulation (MCS). This comparison allows for a thorough assessment of the PCE method’s performance. Additionally, a sensitivity analysis is conducted to compare the relative significance of the uncertainty in material properties and loads on the stochastic responses. The supervised training of the artificial neural network based on the MCS beam data is also conducted and compared to the PCE and MCS methods. The findings about the stochastic outputs are introduced in various statistical metrics and illustrations to demonstrate the influences of material properties’ randomness on different laminated composite thin-walled beam configurations.

Inverse analysis of deformation moduli for high arch dams using the displacement reconstruction technique and multi‐objective optimization

AbstractThe inverse analysis of the deformation moduli of high arch dams based on displacement monitoring data is essential for structural safety assessment. In traditional inverse analysis methods, the deformation moduli are identified based on the single‐objective optimization and the hydrostatic component derived from the statistical model. This type of method has two main shortcomings: First, it treats the essential multi‐objective optimization problem as a single‐objective problem; second, the extracted hydrostatic component may be biased due to the multicollinearity of variables in the statistical model. This paper presents a methodology for the inverse analysis of the deformation moduli of high arch dams under a multi‐objective optimization strategy. The methodology employs empirical mode decomposition to extract the aging component from displacement monitoring data. Then, thermomechanical analysis is used to reconstruct the remaining hydrostatic and temperature components, thereby avoiding the biases encountered in solving the statistical model. The adaptive polynomial chaos expansion method is embedded in the NSGA‐III algorithm to establish and solve multi‐objective functions in the inverse analysis. Additionally, a composite decision index considering errors and test information is proposed to determine acceptable deformation moduli from the Pareto solution set. A high arch dam is selected to illustrate this methodology with static and dynamic monitoring data. The results show that the identified deformation moduli have errors of 3.8% and 7.2% in displacement and acceleration, respectively. The proposed methodology can yield deformation modulus values that are more consistent with the physical implications than those of the single‐objective optimization method.

Generative adversarial surrogate modeling framework for aerospace engineering structural system reliability design

To effectively realize the reliability design of engineering structural system, a generative adversarial surrogate modeling (GASM) concept is proposed by innovating generative adversarial theory into surrogate modeling methods, which realize the aerospace engineering structural system reliability design adversarial modeling. In this concept, the surrogate model is adopted as a basis function to describe the functional relationship between input variables and output response; the generative adversarial theory is employed to obtain hyper-parameters of basis functions by continuous confrontation and evolution between the generation model and discrimination model. Under the GASM concept, the generative adversarial polynomial chaos expansion (GAPCE) method is developed to achieve aerospace engineering structural system reliability design. The effectiveness of presented GAPCE is verified by three examples including the nonlinear functions approximation, reliability design of flap deflection angle, and reliability estimation for turbine blisk radial deformation. The reliability level of flap deflection angle and turbine blisk radial deformation are 0.9998 and 0.9984, when the allowable values are 9° and 1.9222 × 10−3 m. Besides, the introduced approach possesses advantages of modeling performance (i.e., modeling accuracy and efficiency) and simulation properties (i.e., simulation precision and efficiency) by comparing with various methods. The efforts of this paper can provide a valuable reference for the operational safety of flap systems and engine structures and the development of reliability design theory.

Parameterized coefficient fine-tuning-based polynomial chaos expansion method for sphere-biconic reentry vehicle reliability analysis and design

Parameterized coefficient fine-tuning-based polynomial chaos expansion method for sphere-biconic reentry vehicle reliability analysis and design

Uncertain sound transmission loss of composite laminated plates with embedded viscoelastic damping layer

Laminated composite structures have been widely applied in the engineering field due to their excellent capacities such as high specific stiffness, high specific strengths and tailorable anisotropic elastic properties. Embedding the viscoelastic damping layers into the laminated composites to construct co-cured composite damping structures provides an effective way to improve the sound insulation capability. In this paper, a semi-analytical model is proposed to investigate the sound transmission loss (STL) of composite laminated structure with embedded viscoelastic damping layer considering its frequency-dependent characteristic. Furthermore, Polynomial Chaos Expansion (PCE) method in conjunction with the semi-analytical model is employed for uncertainty propagation analysis of sound transmission loss of the composite laminated plates subjected to uncertain parameters of frequency-dependent interleaved viscoelastic damping layer. Moreover, numerical simulations are conducted to demonstrate the impacts of the uncertain parameters of the embedded viscoelastic damping layer on the STL of the composite laminated plate. Finally, experiments are carried out to investigate the uncertain acoustic insulation performance of composite laminated plates with embedded damping layer. Key words: Laminated composite structure; Sound transmission loss; Uncertainty analysis; Damping layer; Frequency-dependent properties

Sparse polynomial chaos expansion for high-dimensional nonlinear damage mechanics

Finite Element Simulations in solid mechanics are nowadays common practice in engineering. However, considering uncertainties based on this powerful method remains a challenging task, especially when nonlinearities and high stochastic dimensions have to be taken into account. Although Monte Carlo Simulation (MCS) is a robust method, the computational burden is high, especially when a nonlinear finite element analysis has to be performed behind each sample. To overcome this burden, several “model-order reduction” techniques have been discussed in the literature. Often, these studies are limited to quite smooth responses (linear or smooth nonlinear models and moderate stochastic dimensions).This paper presents systematic studies of the promising Sparse Polynomial Chaos Expansion (SPCE) method to investigate the capabilities and limitations of this approach using MCS as a benchmark. A nonlinear damage mechanics problem serves as a reference, which involves random fields of material properties. By this, a clear limitation of SPCE with respect to the stochastic dimensionality could be shown, where, as expected, the advantage over MCS disappears.As part of these investigations, options to optimise SPCE have been studied, such as different error measures and optimisation algorithms. Furthermore, we have found that combining SPCEs with sensitivity analysis to reduce the stochastic dimension improves accuracy in many cases at low computational cost.

Chance-constrained optimal dispatch of integrated energy systems based on data-driven sparse polynomial chaos expansion

In integrated energy systems, stochastic variations of different energy types of loads and the increasing penetration of renewable energy generation have resulted in considerable uncertainties. These uncertainties pose significant challenges to the economics and safe operation of integrated energy systems. Conventional deterministic methods of optimal dispatch overlook the effects of uncertainties, while stochastic optimization, though accounting for uncertainties, often yields conservative solutions that may adversely affect the economic operations. Chance-constrained optimization can effectively deal with uncertainties and provide more flexibility in balancing operational risks and benefits by expanding the feasible region. Hence, this paper proposes a chance-constrained optimal dispatch method for integrated energy systems and employs data-driven sparse polynomial chaos expansion method to enhance solving efficiency. First, the proposed chance-constrained optimization aims to ensure the optimal economic operation with an affordable security confidence level that balances safety and economics compared with deterministic and stochastic optimizations. The introduced data-driven sparse polynomial chaos expansion method enables the fast computing of the output response using only historical data, i.e., without knowledge pertaining to the distribution functions. Moreover, an improved iterative verification structure is proposed, which further improves the convergence speed and accuracy. Finally, the advantages and feasibility of the proposed method are verified using a test case and compared with those of deterministic and stochastic optimization. Results show that the proposed method successfully reduces the operational cost and controls violation probability.

Acoustic Performance of a Metamaterial Coating with Uncertainty in Design Parameters

The effect of uncertainty in design parameters on the acoustic performance of a viscoelastic metamaterial coating is investigated. The coating is composed of a layer of hard inclusions embedded in a soft host medium. The layer of inclusions in the soft medium is modelled as a homogenised layer using effective medium approximation theory. Stochastic models based on the polynomial chaos expansion (PCE) method and Monte Carlo (MC) simulations are developed. The effect of uncertainty in geometric and material parameters are quantified by examining the mean and envelopes of the absorption coefficient. Uncertainty in geometric and material parameters produce similar levels of variation in the absorption coefficient of the coating, particularly around the peaks of the absorption coefficient spectrum.

Radial Basis Function Surrogates for Uncertainty Quantification and Aerodynamic Shape Optimization under Uncertainties

This paper investigates the adequacy of radial basis function (RBF)-based models as surrogates in uncertainty quantification (UQ) and CFD shape optimization; for the latter, problems with and without uncertainties are considered. In UQ, these are used to support the Monte Carlo, as well as, the non-intrusive, Gauss Quadrature and regression-based polynomial chaos expansion methods. They are applied to the flow around an isolated airfoil and a wing to quantify uncertainties associated with the constants of the γ−R˜eθt transition model and the surface roughness (in the 3D case); it is demonstrated that the use of the RBF-based surrogates leads to an up to 50% reduction in computational cost, compared with the same UQ method that uses CFD computations. In shape optimization under uncertainties, solved by stochastic search methods, RBF-based surrogates are used to compute statistical moments of the objective function. In applications with geometric uncertainties which are modeled through the Karhunen–Loève technique, the use on an RBF-based surrogate reduces the turnaround time of an evolutionary algorithm by orders of magnitude. In this type of applications, RBF networks are also used to perform mesh displacement for the perturbed geometries.

Uncertainty Quantification for the Noah‐MP Land Surface Model: A Case Study in a Grassland and Sandy Soil Region

AbstractLand surface models rely on a multitude of parameters to simulate land‐atmosphere interactions, but the parameter uncertainty can limit the reliability of model predictions. This study utilizes a parameter uncertainty quantification (UQ) framework to quantify and reduce the parameter uncertainty of the Noah‐MP land surface model in a grassland and sandy soil region in the Midwest of the USA. First, the sparse polynomial chaos expansion method which can quantify the interaction effect of parameters, is employed. A relatively small parameter sample size (i.e., 20 times of the number of parameters) was sufficient to identify the sensitive parameters; an additional sensitive parameter, the saturated soil hydraulic conductivity, was screened out compared to previous study. Then, based on the selected sensitive parameters, the weighted multi‐objective adaptive surrogate modeling‐based optimization algorithm is used as the parameter optimization method. The optimization results showed that the root mean square error of flux of latent heat (FLH) on about 82% of the total grids was reduced, and the number was about 57% for gross primary production (GPP) compared to the results using the original parameter settings, indicating that the Pareto parameter set by the UQ framework improved the Noah‐MP model in simulating FLH and GPP in a grassland and sandy soil region in the Midwest of the USA.

Robust design of a low cost flat plate collector under uncertain design parameters

The high expenses of electrical energy needed for heating water can be significantly decreased by using flat plate collectors. Therefore, the design of flat plate collectors having optimal performance has received a lot of attention in recent research projects. However, due to the existence of uncertainties in material properties, geometry and environmental conditions, it is imperative to incorporate uncertainty analysis into the design optimization to obtain reliable results. In this work, a multi-criteria robust design based on multi-objective optimization of the flat plate collector system is developed. Firstly, a determinist optimization (non-consideration of uncertain design parameters in this optimization strategy) of flat plate collector design is conducted. The obtained results showed that the determinist optimization leads to a flat plate collector design with 30% efficiency sensitivity and 27% total cost sensitivity to design parameters uncertainty. As an alternative, the multi-criteria robust design based on multi-objective optimization is developed. For this goal, a combined multi-objective colonial competitive algorithm and polynomial chaos expansion method is developed and used. This multi-criteria robust optimization considers simultaneously four objectives functions including the flat plate collector efficiency, the flat plate collector total cost, and their sensitivities to uncertain design parameters. It is noted that the multi-criteria robust design based on multi-objective optimization offers a robust flat plate collector with performances sensitivity to design parameters uncertainty less than 4.5%. Furthermore, when compared to the literature results, the performances sensitivities of flat plate collector design, obtained by the multi-criteria robust design based on multi-objective optimization, were reduced until 64%.

Evaluation of uncertainty in direct measurement for parameterization of pyrolysis models, Part I: Thermal analysis

A comprehensive uncertainty analysis is an important part of any engineering calculation because it allows stakeholders to assess the confidence in the conclusions that come from the calculation. Without this necessary step in hypothesis testing and model validation, the true accuracy and confidence of predictions cannot be completely quantified. An experimental and analytical approach is presented in this work to determine the kinetics and energetics of pyrolysis which form the foundation for larger scale models of material burning. A detailed uncertainty quantification which utilized the generalized polynomial chaos expansion method is presented to propagate uncertainty through the pyrolysis model representation of a thermal analysis experiment. Using this methodology, the uncertainty in the predictions was found to be comparable to the uncertainty in the experimental thermal analysis data within calibration conditions. Sensitivity analyses revealed the largest contributor to uncertainty in heat flow rate predictions was the specific heat capacity of all components. Comparison of the uncertainty quantification and sensitivity analyses to well-known results validated the method of uncertainty propagation for more complex scenarios.

Probabilistic analysis of width‐limited 3D slope in spatially variable soils: UBLA enhanced with efficiency‐improved discretization of horn‐like failure mechanism

AbstractReliability analysis of earth slopes considering soil spatial variability has garnered significant attention from researchers. However, previous studies have predominantly focused on long slopes with infinite or ample width, such as dams and subgrades, but rare research was dedicated to slopes with width restricted by boundary constraints. In this regard, this paper proposes an efficient reliability framework suitable for width‐limited slopes in spatially variable soils. The proposed framework employed spatial discretization‐based upper bound limit analysis (UBLA) to conduct the deterministic slope stability analysis. The established failure mechanism can well capture the width‐limited three‐dimensional slip surface characteristics and satisfies the kinematically admissible conditions in spatially variable soils. Two innovative strategies are proposed to boost the efficiency of the discretized mechanism. They reduce the computational time to determine safety factors from over 20 min to less than 2 min. For the uncertainty modeling and slope reliability computing procedure, the Sparse Polynomial Chaos Expansion (SPCE) and Monte Carlo Simulation (MCS) methods are combined to generate the distribution of safety factors and failure probabilities. This combination addresses the high‐dimensional stochastic problem arising from 3D spatial variability. Using the proposed framework, a parametric analysis of a 3D spatially variable slope is conducted to investigate the impact of various factors. The findings of this paper are beneficial to the risk evolution of width‐limited slopes, and the efficiency enhancement strategies have the potential to expedite the limit analysis of other geo‐structures, such as tunnels, retaining walls, and foundations.