Polynomial Chaos Research Articles

Optimization study of lead-based reactor heat exchangers based on Dakota coupling CFD

The lead-bismuth fast reactor is one of the fourth-generation reactor types.The microchannel heat exchangers are used as intermediate heat exchangers, which significantly influence the efficiency of supercritical carbon dioxide (S–CO2) Brayton cycle power generation. It is necessary to quantify the uncertainties and optimize the design of heat transfer and pressure drop of the microchannel heat exchangers. Due to the high cost and time-consuming problem of numerical simulation, this study employs the Arbitrary polynomial chaos expansion method to generate surrogate models for simulation calculations. Specifically, the microchannel parameters of the heat exchanger, namely flow direction spacing, transverse spacing, staggered spacing, and inlet velocity, are studied for the simulation. The pressure drop and convective heat transfer coefficient of the heat exchanger are considered as the objective functions for both simulation and optimization models. Uncertainty quantification analysis, sensitivity analysis, and optimization design of those parameters are conducted. By coupling DAKOTA with CFD, an optimization objective function is established to obtain optimal design data within specified ranges.

Hierarchical sensitivity study on the aeroelastic stability of the IEA 15 MW reference wind turbine

Increased rotor size and blade flexibility are leading to new stability characteristics for current and future wind turbines. Numerical aeroelastic models are an essential tool to understand these new mechanisms and to design next-generation wind turbines suitably. A comprehensive understanding of how model input parameters influence the stability analysis will enhance the confidence in these simulations. This article presents a study on the sensitivity of uncertain parameters on aeroelastic stability predictions of the IEA 15 MW turbine model in HAWCStab2. It uses a hierarchical approach to handle the large number of investigated model parameters. Relevant uncertainties are identified through one-at-a-time and elementary effects analyses first. The remaining parameters are fed into a variance-based uncertainty quantification (UQ), that employs polynomial chaos expansion representations as surrogates for a full nonlinear description of the sensitivities. A robust post-processing of the stability analysis results is the main challenge of the presented methodology, but it is shown how the UQ process can still be used to explore the parameter space effectively. The presented study shows that the structural blade properties have the highest sensitivity and that a set of relatively small parameter variations can lead to unstable behavior of the reference model.

Uncertainty Quantification Analysis in the Blade Element Momentum Method

The blade element momentum (BEM) method is a widely used non-linear model for the efficient performance evaluation and design of wind turbines. The aim of this paper is to quantify the uncertainty related to BEM model inputs and sub-models, and investigate how these propagate through the model. Uncertainties related to viscous dissipation in the wake, aerofoil force coefficients, and tip-loss models are considered. Global sensitivity to these parameters is analysed using non-intrusive polynomial chaos expansion, which provides a structured method for uncertainty propagation and sensitivity quantification. Sobol indices are employed to rank the relative importance of each factor to the overall uncertainty in the system. By analysing the NREL 5 MW and DTU 10 MW reference wind turbines, we observe that the different rotors may exhibit different levels of sensitivity to input parameters. The effect of viscous mixing in the turbine wake is found to have a significant impact on predicted rotor performance. Uncertainty in tip-loss model coefficients is also found to generally be important, particularly when evaluating spanwise variations in rotor loads. Uncertainty quantification has the potential to improve understanding of BEM modelling, and provide guidance on the importance of sub-model improvements.

Polynomial Chaos Based Statistical Analysis of the Shielding Effectiveness of an Enclosure With Apertures Under Multiple Uncertainties

Polynomial Chaos Based Statistical Analysis of the Shielding Effectiveness of an Enclosure With Apertures Under Multiple Uncertainties

Research on uncertainties in fuel centrifugal pump based on prediction and reconstruction of internal flow field

Geometric machining errors in the blade profile and variable operating conditions in the extreme operating environment are primary factors leading to the uncertainties in pump performance. This paper presents an analysis of uncertainties of fuel centrifugal pumps by modeling the geometry uncertainty in blade machining based on the Karhunen–Loève (KL) expansion and using a polynomial chaos expansion (PCE) model. First, the geometric uncertainty in the blade machining is described by the KL expansion in three sections and a stochastic simulation of the blade geometry is performed. Then, a PCE surrogate model is trained based on the least angle regression method and validated by the bootstrap method to quantify the uncertainties of performance indices. Finally, the influence mechanism and relative importance of each input uncertainty parameter are investigated using a quasi-Monte Carlo simulation method. The results show that the KL expansion of the blade profile uses the random vector perturbation superposition of three stream surface, achieving the dimensional reduction in the blade machining error. The PCE surrogate model, trained with a dataset of 3 × 106 sample points, exhibits excellent fit, and the R-squared and adjusted R-squared for head coefficient and efficiency are both above 80%. The variance of parameter control points of the reconstructed flow field is less than 0.002. The uncertainties in both operating conditions and parameters have an influence on the distribution of the global flow field, while the influence of the uncertainty in machining error on the global flow field mainly concentrates on the power-generating positions of the blade.

Entropy-Conservative Discontinuous Galerkin Methods for the Shallow Water Equations with Uncertainty

In this paper, we develop an entropy-conservative discontinuous Galerkin (DG) method for the shallow water (SW) equation with random inputs. One of the most popular methods for uncertainty quantification is the generalized Polynomial Chaos (gPC) approach which we consider in the following manuscript. We apply the stochastic Galerkin (SG) method to the stochastic SW equations. Using the SG approach in the stochastic hyperbolic SW system yields a purely deterministic system that is not necessarily hyperbolic anymore. The lack of the hyperbolicity leads to ill-posedness and stability issues in numerical simulations. By transforming the system using Roe variables, the hyperbolicity can be ensured and an entropy-entropy flux pair is known from a recent investigation by Gerster and Herty (Commun. Comput. Phys. 27(3): 639–671, 2020). We use this pair and determine a corresponding entropy flux potential. Then, we construct entropy conservative numerical two-point fluxes for this augmented system. By applying these new numerical fluxes in a nodal DG spectral element method (DGSEM) with flux differencing ansatz, we obtain a provable entropy conservative (dissipative) scheme. In numerical experiments, we validate our theoretical findings.

Efficient MAPoD via Least Angle Regression based Polynomial Chaos Expansion Metamodel for Eddy Current NDT

In this article, a metamodeling approach based on non-intrusive polynomial chaos expansion (PCE) with least angle regression (LAR) method is used in boundary element analysis for a model-assisted probability of detection (MAPoD) study of eddy current nondestructive testing (NDT) systems. The LAR-PCE metamodel represents the NDT system model responses by a set of coefficients with the polynomial basis functions in lieu of pure kernel degeneration accelerated boundary element method (KD-BEM) based physical model. Both the computational accuracy and efficiency of the LAR-PCE metamodel over the ordinary least squares (OLS) based PCE metamodel are demonstrated by testing the 3D eddy current NDT benchmarks with different system setups, flaw lengths and widths. The simulation results show two digits accuracy of the PoD metrics compared with the ones achieved by the KD-BEM based physical model as the benchmark. The LAR-PCE metamodel has remarkable improvements in computational efficiency over the OLS-PCE metamodel and accelerates the MAPoD study.

Multi-objective shape optimization of TESLA-like cavities: Addressing stochastic Maxwell's eigenproblem constraints

In this paper, we present a robust formulation to address nonlinear equality-constrained and inequality-constrained multi-objective optimization problems for multi-cell accelerating cavities with TESLA-like geometry under consideration of uncertain parameters. We propose to explore a robust approach that employs the polynomial chaos expansion (PCE) to model the uncertainty. For the uncertainty quantification (UQ), the resulting random-dependent eigenvalue problem is solved using the stochastic collocation method (SCM) combined with PCE. As a result, the shape optimization of the studied multi-cell cavity is defined in terms of statistical moments, which serve as the cost functionals in the multi-objective (MO) steepest descent algorithm. Two versions of this method are developed and implemented. Finally, the optimization results and their implications for the operation of a multi-cell cavity are discussed.

Sampling free iterative PCE filter for state and parameter estimation of nonlinear dynamical systems

We present a novel filter for state and parameter estimation in non-linear dynamical systems, based on a generalised Kalman filter formulation. To achieve a sampling-free implementation, polynomial chaos expansion (PCE) and a Galerkin projection method are utilized for the propagation of uncertainties through the system dynamics. The non-linear dynamics of the system are then linearised by a sequence of Gauss-Newton iterations in combination with linear Kalman updates. Additionally, we introduce a new square root implementation of the PCE-based filter. The proposed filter is evaluated on the Lorenz-63 and Lorenz-84 models for the task of simultaneous state and parameter estimation and is compared with two related approaches. Finally, the computational complexity of our square-root implementation is compared against two existing square root approaches.

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.

Seismic analysis of gravity dam-foundation systems using stochastic spectral finite element method

Seismic analysis of dam-foundation systems is inherently complex due to uncertainties in material properties. This study investigates the effects of material randomness and heterogeneity on the seismic response of gravity dam-foundation systems using the stochastic spectral finite element method (SSFEM) based on the Karhunen Loève expansion and the polynomial chaos expansion. The Pine Plane dam is taken as an analysis example. First, the appropriate Karhunen Loève expansion modes and polynomial chaos order are determined for SSFEM by comparing them with the Monte-Carlo simulations. Subsequently, the seismic response of the dam is analyzed in the stochastic and deterministic models, respectively. Results indicate that material randomness and heterogeneity may reduce the seismic response of gravity dams, and the deterministic model may overestimate dynamic responses. Subsequently, the parametric analysis of the correlation length, which measures the spatial correlation between variables in different positions and indicates the size of heterogeneity material patches, is conducted. It is demonstrated that the correlation length would not significantly affect the mean responses, but a larger correlation length leads to greater dispersions in responses. Finally, the effects of randomness in the foundation are investigated. The heterogeneities in the foundation affect the response of the dam in the high-frequency range. Therefore, the uncertainty in material properties should be fully considered in a structural assessment and risk analysis.

Novel Numerical Solutions to the Fractional-Stochastic Systems

Fractional-stochastic differential equations are widely used tools to simulate a wide range of engineering and scientific phenomena. In this paper, the applicability of the approach of indeterminate coefficients to various fractional-stochastic models is examined. These models have a fractional white noise term and are mostly produced by fractional-order derivative operators. We also look into the application of a polynomial chaos algorithm to stochastic Lotka-Volterra and Benney systems. Fractional-stochastic equations are entirely novel systems that have the potential to function as models for a wide range of scientific and engineering phenomena. It is noted that fractional-order systems with uncertainty or a noise term can benefit from the effective use of Galerkin type approaches used in this article. Keywords: Galerkin method, Numerical solutions, Stochastic-fractional systems

Stochastic finite element analysis using polynomial chaos on a flexible rotor with contact nonlinearity

Stochastic finite element analysis using polynomial chaos on a flexible rotor with contact nonlinearity

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.

Dynamic characteristics analysis of multi-link mechanism with clearances considering uncertain parameters

As a typical multi-body system, multi-link mechanisms are widely used in industry. Inevitably, there are joint clearances in multi-link mechanisms. Besides under the conditions of manufacturing, assembly and wear, the parameters of the multi-link mechanism can create uncertainty, and joint clearances and uncertain parameters of the multi-link mechanisms deteriorate the dynamic characteristics, this directly leads to a reduction in working accuracy of the mechanisms. At present, the research focus on multi-link mechanism with clearance considering certain parameters, and relatively few research on the multi-link mechanism with clearance considering uncertain parameters, the existing research on uncertain parameters mainly focus on the uncertainty of single parameter, while the research on the dynamic response of the mechanism with multi-clearances considering the uncertainty of multiple parameters is very less. The main purpose of this article is to provide a novel feasible method for studying the dynamic characteristics of multi-link mechanisms with joint clearances under the consideration of interval uncertain parameters and random parameters. Therefore, the dynamic characteristics of mechanism with multiple clearances considering uncertain parameters, which includes interval parameters and random parameters, are investigated in this article. Taking the 6-bar mechanism as the research object, considering clearances, Poisson’s ratios, stiffness coefficients, and driving speed as the uncertain parameters, the dynamic models of 6-link mechanism with multiple clearances considering interval parameters and random parameters are developed respectively. Using Chebyshev polynomial to solve interval parameters and polynomial chaos to solve random parameters, the effects of uncertain parameters such as clearances, Poisson’s ratios, stiffness coefficients, and driving speed on the dynamic response of the mechanism are analyzed, respectively. This research can not only provide a basis for studying the essential dynamic characteristics of multi-link mechanism with uncertain parameters, but also lay the foundation for dynamic optimization design of the real mechanisms.

Uncertainty quantification and global sensitivity analysis of bearing capacity of push-out specimen considering randomness in bond-slip behaviour

As an artificial composite material, the uncertainty in mechanical property of concrete significantly impacts the bond-slip behaviour of the shaped steel concrete structure. To investigate the effect of the uncertainty of concrete mechanical properties on the bearing capacity of push-out specimens, based on the data of a small number of push-out test with identical design parameters, the improved Bootstrap method is utilized to execute sample augmentation for the bond stress-slip curves, and the stochastic bond slip constitutive model is established after considering five parameters (K1, …, K5) in the deterministic bond-slip model as random variables. Simultaneously, considering the randomness in elastic modulus of concrete (Ec), uncertainty quantification (UQ) of the bearing capacities of the push-out specimens is conducted using the arbitrary polynomial chaos (aPC) method. Subsequently, the global sensitivity analysis of the ultimate and residual bearing capacities is accomplished through the aPC surrogate model. The results indicate that the probability distribution types of the parameters in the stochastic bond-slip constitutive model can be arbitrary when considering the randomness of concrete material, and the aPC method with at least a third-order expansion is recommended to execute UQ for the push-out test due to the strong nonlinearty between the bearing capacities and the random parameters. The ultimate bearing capacity is primarily affected by the parameter K2, followed by K4, K3, and K5. Whereas the residual bearing capacity is predominantly influenced by the parameter K4, followed by K5, the remaining parameters exhibit negligible influence. In contrast, K1 and Ec have the greatest influence on equivalent plastic strain of the concrete near the loading end of the push-out specimen.

Space‐time stochastic Galerkin boundary elements for acoustic scattering problems

SummaryAcoustic emission or scattering problems naturally involve uncertainties about the sound sources or boundary conditions. This article initiates the study of time domain boundary elements for such stochastic boundary problems for the acoustic wave equation. We present a space‐time stochastic Galerkin boundary element method which is applied to sound‐hard, sound‐soft and absorbing scatterers. Uncertainties in both the sources and the boundary conditions are considered using a polynomial chaos expansion. The numerical experiments illustrate the performance and convergence of the proposed method in model problems and present an application to a problem from traffic noise.

Effects of compositional uncertainties in cracked NH3/biosyngas fuel blends on the combustion characteristics and performance of a combined-cycle gas turbine: A numerical thermokinetic study

Blending of partially cracked ammonia with biosyngas is an attractive strategy for improving NH3 combustion. In practice, products of biomass gasification and those of thermo-catalytic cracking of NH3 are subject to some compositional uncertainties. Despite their practical importance, so far, the effects of such uncertainties on combustion systems remained largely unexplored. Hence, this paper quantifies the effects of small compositional uncertainties of reactants upon combustion of partially cracked NH3/syngas/air mixtures. An uncertainty quantification method, based on polynomial chaos expansion and a data-driven model, is utilised to investigate the effects of uncertainty in fuel composition on the laminar flame speed (SL) and adiabatic flame temperature (Tad) at different inlet pressures (Pi). The analysis is then extended to the power output of a combined-cycle gas turbine fuelled by the reactants. It is found that 1.5% fuel compositional uncertainty can cause 12–21% of SL uncertainty depending on the inlet pressure. Furthermore, the effect of compositional uncertainty on Tad increases at higher ratios of H2 to NH3. Sensitivity analysis reveals that the uncertainty of CO contribution to SL uncertainty is higher than that of NH3, while the trend is reversed for the Tad uncertainty. In addition, the power output from the combined-cycle gas turbine system varies between 4 and 6% with 1.5% of fuel compositional uncertainty. This become more noticeable at elevated Pi [5–10 atm], particularly when the fuel mixture contains high H2 which is the main contributor to Tad variability.

Efficient global sensitivity analysis method for dynamic models in high dimensions

AbstractDynamic models generating time‐dependent model predictions are typically associated with high‐dimensional input spaces and high‐dimensional output spaces, in particular if time is discretized. It is computationally prohibitive to apply traditional global sensitivity analysis (SA) separately on each time output, as is common in the literature on multivariate SA. As an alternative, we propose a novel method for efficient global SA of dynamic models with high‐dimensional inputs by combining a new polynomial chaos expansion (PCE)‐driven partial least squares (PLS) algorithm with the analysis of variance. PLS is used to simultaneously reduce the dimensionality of the input and output variables spaces, by identifying the input and output latent variables that account for most of their joint variability. PCE is incorporated into the PLS algorithm to capture the non‐linear behavior of the physical system. We derive the sensitivity indices associated with each output latent variable, based on which we propose generalized sensitivity indices that synthesize the influence of each input on the variance of entire output time series. All sensitivities can be computed analytically by post‐processing the coefficients of the PLS‐PCE representation. Hence, the computational cost of global SA for dynamic models essentially reduces to the cost for estimating these coefficients. We numerically compare the proposed method with existing methods by several dynamic models with high‐dimensional inputs. The results show that the PLS‐PCE method can obtain accurate sensitivity indices at low computational cost, even for models with strong interaction among the inputs.

Efficient Verilog-A based time-domain variability analysis method for passive photonic integrated circuits

The performance of silicon photonic integrated circuits (PICs), especially wavelength filters, can be highly sensitive to variations in the fabrication process due to the large refractive index contrast of the silicon on insulator platform. This paper proposes an easy-to-implement and efficient time-domain variability analysis method for passive PICs. The method utilizes the polynomial chaos expansion technique to construct Verilog-A based models for estimating the statistical information of stochastic passive PICs. In comparison to existing methods, this approach is considerably easy to implement, efficient, and exhibits superior scalability, particularly as the numbers of ports and random parameters in the studied PICs increase. The technique is demonstrated via the time-domain variability analysis of a ring-resonator-based wavelength filter and a Mach-Zehnder interferometer-based demultiplexer filter.