Non-intrusive Spectral Projection Research Articles

UNCERTAINTY QUANTIFICATION IN STEADY STATE SIMULATIONS OF A MOLTEN SALT SYSTEM USING POLYNOMIAL CHAOS EXPANSION ANALYSIS

Uncertainty Quantification (UQ) of numerical simulations is highly relevant in the study and design of complex systems. Among the various approaches available, Polynomial Chaos Expansion (PCE) analysis has recently attracted great interest. It belongs to nonintrusive spectral projection methods and consists of constructing system responses as polynomial functions of the stochastic inputs. The limited number of required model evaluations and the possibility to apply it to codes without any modification make this technique extremely attractive. In this work, we propose the use of PCE to perform UQ of complex, multi-physics models for liquid fueled reactors, addressing key design aspects of neutronics and thermal fluid dynamics. Our PCE approach uses Smolyak sparse grids designed to estimate the PCE coefficients. To test its potential, the PCE method was applied to a 2D problem representative of the Molten Salt Fast Reactor physics. An in-house multi-physics tool constitutes the reference model. The studied responses are the maximum temperature and the effective multiplication factor. Results, validated by comparison with the reference model on 103 Monte-Carlo sampled points, prove the effectiveness of our PCE approach in assessing uncertainties of complex coupled models.

Multilevel Monte Carlo applied for uncertainty quantification in stochastic multiscale systems

AbstractThe aim of this study is to evaluate the performance of multilevel Monte Carlo (MLMC) sampling technique for uncertainty quantification in stochastic multiscale systems. Two systems, a chemical vapor deposition chamber and a catalytic flow reactor, subject to multiple parameter uncertainty, were considered. The distributions of the systems' observables were estimated using standard MC sampling and polynomial chaos expansions (PCE), where the coefficients were calculated by nonintrusive spectral projection. The MLMC technique was used to efficiently sample the two systems and accurately estimate the data necessary for constructing the PCE expressions. The results show that the usage of MLMC improved the precision of identification of PCE versus the traditional heuristic approach and lowered the computational cost of uncertainty quantification compared to standard MC.

Parameter Estimation with Data-Driven Nonparametric Likelihood Functions.

In this paper, we consider a surrogate modeling approach using a data-driven nonparametric likelihood function constructed on a manifold on which the data lie (or to which they are close). The proposed method represents the likelihood function using a spectral expansion formulation known as the kernel embedding of the conditional distribution. To respect the geometry of the data, we employ this spectral expansion using a set of data-driven basis functions obtained from the diffusion maps algorithm. The theoretical error estimate suggests that the error bound of the approximate data-driven likelihood function is independent of the variance of the basis functions, which allows us to determine the amount of training data for accurate likelihood function estimations. Supporting numerical results to demonstrate the robustness of the data-driven likelihood functions for parameter estimation are given on instructive examples involving stochastic and deterministic differential equations. When the dimension of the data manifold is strictly less than the dimension of the ambient space, we found that the proposed approach (which does not require the knowledge of the data manifold) is superior compared to likelihood functions constructed using standard parametric basis functions defined on the ambient coordinates. In an example where the data manifold is not smooth and unknown, the proposed method is more robust compared to an existing polynomial chaos surrogate model which assumes a parametric likelihood, the non-intrusive spectral projection. In fact, the estimation accuracy is comparable to direct MCMC estimates with only eight likelihood function evaluations that can be done offline as opposed to 4000 sequential function evaluations, whenever direct MCMC can be performed. A robust accurate estimation is also found using a likelihood function trained on statistical averages of the chaotic 40-dimensional Lorenz-96 model on a wide parameter domain.

Comparison of the Point-Collocation Non-Intrusive Polynomial (NIPC) and Non-Intrusive Spectral Projection (NISP) Methods for the γ-Rθ Transition Model

In the present work, a comparative study of two major non-intrusive polynomial chaos methods, Point-Collocation Non-Intrusive Polynomial Chaos (NIPC) and Non-Intrusive Spectral Projection (NISP), was conducted for the transitional transitional model. Three multiple model coefficients, Ca2, Ce1, and Ce2 were considered with multiple random inputs with the assumption of uniform distributions with ±10% deviation. The target transitional flows were one around a flat plate and Aerospatiale A-airfoil. Deterministic solutions were obtained by employing the open source software OpenFOAM. The results of two methods were compared to the results of Monte Carlo simulation with 500 runs. The order convergence of the mean value and the standard deviation (STD) were compared in terms of the quantities of interest, drag and lift coefficients. Further, the most effective model coefficient for each transitional flow could be found through the calculation of the Sobol index.

Uncertainty Quantification for Direct Aeroacoustic Simulations of Cavity Flows

We investigate the influence of uncertain input parameters on the aeroacoustic feedback of cavity flows. The so-called Rossiter feedback requires a direct numerical computation of the acoustic noise, which solves hydrodynamics and acoustics simultaneously, in order to capture the interaction of acoustic waves and the hydrodynamics of the flow. Due to the large bandwidth of spatial and temporal scales, a high-order numerical scheme with low dissipation and dispersion error is necessary to preserve important small scale information. Therefore, the open-source CFD solver FLEXI, which is based on a high-order discontinuous Galerkin spectral element method, is used to perform the aforementioned direct simulations of an open cavity configuration with a laminar upstream boundary layer. To analyze the precision of the deterministic cavity simulation with respect to random input parameters, we establish a framework for uncertainty quantification (UQ). In particular, a nonintrusive spectral projection method with Legendre and Hermite polynomial basis functions is employed in order to treat uniform and normal probability distributions of the random input. The results indicate a strong, nonlinear dependency of the acoustic feedback mechanism on the investigated uncertain input parameters. An analysis of the stochastic results offers new insights into the noise generation process of open cavity flows and reveals the strength of the implemented UQ framework.

Probabilistic parameter estimation in a 2-step chemical kinetics model for n-dodecane jet autoignition

This paper demonstrates the development of a simple chemical kinetics model designed for autoignition of n-dodecane in air using Bayesian inference with a model-error representation. The model error, i.e. intrinsic discrepancy from a high-fidelity benchmark model, is represented by allowing additional variability in selected parameters. Subsequently, we quantify predictive uncertainties in the results of autoignition simulations of homogeneous reactors at realistic diesel engine conditions. We demonstrate that these predictive error bars capture model error as well. The uncertainty propagation is performed using non-intrusive spectral projection that can also be used in principle with larger scale computations, such as large eddy simulation. While the present calibration is performed to match a skeletal mechanism, it can be done with equal success using experimental data only (e.g. shock-tube measurements). Since our method captures the error associated with structural model simplifications, we believe that the optimised model could then lead to better qualified predictions of autoignition delay time in high-fidelity large eddy simulations than the existing detailed mechanisms. This methodology provides a way to reduce the cost of reaction kinetics in simulations systematically, while quantifying the accuracy of predictions of important target quantities.

Investigation of Biotransport in a Tumor With Uncertain Material Properties Using a Nonintrusive Spectral Uncertainty Quantification Method.

In this study, statistical models are developed for modeling uncertain heterogeneous permeability and porosity in tumors, and the resulting uncertainties in pressure and velocity fields during an intratumoral injection are quantified using a nonintrusive spectral uncertainty quantification (UQ) method. Specifically, the uncertain permeability is modeled as a log-Gaussian random field, represented using a truncated Karhunen-Lòeve (KL) expansion, and the uncertain porosity is modeled as a log-normal random variable. The efficacy of the developed statistical models is validated by simulating the concentration fields with permeability and porosity of different uncertainty levels. The irregularity in the concentration field bears reasonable visual agreement with that in MicroCT images from experiments. The pressure and velocity fields are represented using polynomial chaos (PC) expansions to enable efficient computation of their statistical properties. The coefficients in the PC expansion are computed using a nonintrusive spectral projection method with the Smolyak sparse quadrature. The developed UQ approach is then used to quantify the uncertainties in the random pressure and velocity fields. A global sensitivity analysis is also performed to assess the contribution of individual KL modes of the log-permeability field to the total variance of the pressure field. It is demonstrated that the developed UQ approach can effectively quantify the flow uncertainties induced by uncertain material properties of the tumor.

Uncertainty Quantification at the Molecular–Continuum Model Interface

Non-equilibrium molecular dynamics simulations are widely employed to study transport fluid properties. Observables measured at the atomistic level can serve as inputs for continuum calculations, allowing for improved analysis of phenomena involving multiple scales. In hybrid modelling, uncertainties present in the information transferred across scales can have a significant impact on the final predictions. This work shows the influence of force-field variability on molecular measurements of the shear viscosity of water. In addition, the uncertainty propagation is demonstrated by quantifying the sensitivity of continuum velocity distribution to the particle-based calculations. The uncertainty is modelled with polynomial chaos expansion using a non-intrusive spectral projection strategy. The analysis confirms that low-order polynomial basis are sufficient to calculate the dispersion of observables.

A comparison of efficient uncertainty quantification techniques for stochastic multiscale systems

The aim of this article is to compare the performance of efficient uncertainty propagation techniques (Polynomial Chaos [PCE] and Power Series [PSE] expansions) for uncertainty quantification in multiscale systems where discrete (molecular) scale is modeled without closed‐form expressions. A multiscale model of thin film formation by chemical vapor deposition was used to study the effects of single parameter and multivariate uncertainty. For the single parameter uncertainty, 2nd order PSE approximations were the most accurate and computationally attractive. For the multivariate uncertainty, PSE performance deteriorated, while 2nd order PCE provided the highest accuracy when its expansion coefficients were calculated using the Least Squares method. However, comparable accuracy was achieved at half the computational cost when the coefficients were calculated using Nonintrusive Spectral Projection (NISP). The response variables were subsequently controlled using robust optimization, and the results obtained using PCE NISP satisfied the optimization constraints more closely than other methods. © 2017 American Institute of Chemical Engineers AIChE J, 63: 3361–3373, 2017

Quantification of uncertainty on the catalytic property of reusable thermal protection materials from high enthalpy experiments

An accurate determination of the catalytic property of thermal protection materials is crucial to design reusable atmospheric entry vehicles. This property is determined by combining experimental measurements and simulations of the reactive boundary layer near the material surface. The inductively-driven Plasmatron facility at the von Karman Institute for Fluid Dynamics provides a test environment to analyze gas-surface interactions under effective hypersonic conditions. In this study, we develop an uncertainty quantification methodology to rebuild values of the gas enthalpy and material catalytic property from Plasmatron experiments. A non-intrusive spectral projection method is coupled with an in-house boundary-layer solver, to propagate uncertainties and provide error bars on the rebuilt gas enthalpy and material catalytic property, as well as to determine which uncertainties have the largest contribution to the outputs of the experiments. We show that the uncertainties computed with the methodology developed are significantly reduced compared to those determined using a more conservative engineering approach adopted in the analysis of previous experimental campaigns.

On the parametric uncertainty quantification of the Rothermel's rate of spread model

Parametric uncertainty quantification of the Rothermel's fire spread model is presented using the Polynomial Chaos expansion method under a Non-Intrusive Spectral Projection (NISP) approach. Several Rothermel's model input parameters have been considered random with an associated prescribed probability density function. Two different vegetation fire scenarios are considered and NISP method results and performance are compared with four other stochastic methodologies: Sensitivity Derivative Enhance Sampling; two Monte Carlo techniques; and Global Sensitivity Analysis. The stochastic analysis includes a sensitivity analysis study to quantify the direct influence of each random parameter on the solution. The NISP approach achieved performance three orders of magnitude faster than the traditional Monte Carlo method. The NISP capability to perform uncertainty quantification associated with fast convergence makes it well suited to be applied for stochastic prediction of fire spread.

Modeling Auto-Ignition Transients in Reacting Diesel Jets

The objective of the present work is to establish a framework to design simple Arrhenius mechanisms for simulation of diesel engine combustion. The goal is to predict auto-ignition over a selected range of temperature and equivalence ratio, at a significantly reduced computational cost, and to quantify the accuracy of the optimized mechanisms for a selected set of characteristics. The methodology is demonstrated for n-dodecane oxidation by fitting the auto-ignition delay time predicted by a detailed reference mechanism to a two-step model mechanism. The pre-exponential factor and activation energy of the first reaction are modeled as functions of equivalence ratio and temperature and calibrated using Bayesian inference. This provides both the optimal parameter values and the related uncertainties over a defined envelope of temperatures, pressures, and equivalence ratios. Nonintrusive spectral projection (NISP) is then used to propagate the uncertainty through homogeneous auto-ignitions. A benefit of the method is that parametric uncertainties can be propagated in the same way through coupled reacting flow calculations using techniques such as large eddy simulation (LES) to quantify the impact of the chemical parameter uncertainty on simulation results.

Uncertainty Quantification of Subcritical Nonlinear Aeroelastic System Using Integrated Interpolation Method and Polynomial Chaos Expansion

The present paper is about analyzing and quantifying parametric uncertainties in a nonlinear aeroelastic system. The response surface of a subcritical system has discontinuities which makes it difficult to analyze using spectral approaches. A quadrature based spectral parametric uncertainty tool called nonintrusive spectral projection (NISP) can capture discontinuities in the response surface only by infinite number of points. As the number of random variable increases, the order of expansion has to increase drastically in order to capture the response accurately. Hence an alternate approach using integrated interpolation scheme which was earlier applied for a thermoacoustic problem is being applied to a classical flutter problem. It is found to be accurate and computationally cheaper than traditional Monte Carlo simulations (MCS) in capturing the response surface. This paper intends to apply the interpolation scheme to the nonlinear aeroelastic model and compares it with quadrature based polynomial chaos expansion (PCE) approach and MCS.

On the uncertainty quantification of the unsteady aerodynamics of 2D free falling plates

The aim of this paper is to conduct a statistical analysis of the effects of the fillet radii on the dynamics of the falling plate using the nonintrusive spectral projection (NISP) method. The free fall of two-dimensional cards immersed in a fluid was studied using a deterministic and stochastic numerical approach. The motion is characterized by the fluid-body interaction described by coupling the Navier–Stokes and rigid body dynamic equations. The model’s predictions have been validated using both experimental and numerical data available in the literature. In the stochastic simulations, the fillet radius of the plate was considered a random variable characterized by a uniform probability density function introducing, in this way, some uncertainties in the plate’s trajectory. To take into account the uncertainties, we employed the NISP method based on polynomial chaos expansion. The analysis was focused on finding the ensemble mean trajectory and error bar for a confidence interval of 95 % for both tumbling and fluttering regimes.

Simulation and uncertainty quantification in high temperature microwave heating

Microwave heating processes are highly sensitive to variability in the operating frequency and usually there are high uncertainties associated with measurements of the material's imaginary permittivity. Therefore, the harmonic Maxwell's equations and time dependent heat equation are considered for the simulation of high temperature ceramic microwave heating with uncertain operating frequency and dielectric constants. To analyze the uncertainty of the system, the Non-Intrusive Spectral Projection Method (NISP) is used based on the Polynomial Chaos expansion. The conclusions of this work indicate that 13 Gauss collocation points and a polynomial degree of 11 must be used with the NISP in order to obtain accurate results. Furthermore it also adds that uncertainty in frequency leads to high values of variability in the stochastic solution. The uncertainty in the material's imaginary part of permittivity has only a significant impact for high temperatures of the material. Temperature probability density functions, error bars and standard deviations characterize the material heating up to high temperatures.

Sensitivity analysis and variance reduction in a stochastic non-destructive testing problem

In this paper, we present a framework to deal with uncertainty quantification in case where the ranges of variability of the random parameters are ill-known. Namely the physical properties of the corrosion product (magnetite) which frequently clogs the tube support plate of steam generator, which is inaccessible in nuclear power plants. The methodology is based on polynomial chaos (PC) for the direct approach and on Bayesian inference for the inverse approach. The direct non-intrusive spectral projection (NISP) method is first employed by considering prior probability densities and therefore constructing a PC surrogate model of the large-scale non-destructive testing finite element model. To face the prohibitive computational cost underlying the high-dimensional random space, an adaptive sparse grid technique is applied on NISP resulting in drastic time reduction. The PC surrogate model, with reduced dimensionality, is used as a forward model in the Bayesian procedure. The posterior probability densities are then identified by inferring from few noisy experimental data. We demonstrate effectiveness of the approach by identifying the most influential parameter in the clogging detection as well as a variability range reduction.

Large scale applicability of a Fully Adaptive Non-Intrusive Spectral Projection technique: Sensitivity and uncertainty analysis of a transient

Since the early years of reactor physics the most prominent sensitivity and uncertainty (S&U) analysis methods in the nuclear community have been adjoint based techniques. While these are very effective for pure neutronics problems due to the linearity of the transport equation, they become complicated when coupled non-linear systems are involved. With the continuous increase in computational power such complicated multi-physics problems are becoming progressively tractable, hence affordable and easily applicable S&U analysis tools also have to be developed in parallel.For reactor physics problems for which adjoint methods are prohibitive Polynomial Chaos (PC) techniques offer an attractive alternative to traditional random sampling based approaches. At TU Delft such PC methods have been studied for a number of years and this paper presents a large scale application of our Fully Adaptive Non-Intrusive Spectral Projection (FANISP) algorithm for performing the sensitivity and uncertainty analysis of a Gas Cooled Fast Reactor (GFR) Unprotected Loss Of Flow (ULOF) transient. The transient was simulated using the Cathare 2 code system and a fully detailed model of the GFR2400 reactor design that was investigated in the European FP7 GoFastR project. Several sources of uncertainty were taken into account amounting to an unusually high number of stochastic input parameters (42) and numerous output quantities were investigated.The results show consistently good performance of the applied adaptive PC methods, being superior to standard Monte Carlo sampling both in terms of accuracy and computational cost. This demonstrates that such PC techniques can provide a viable alternative to random sampling even for larger scale systems, which is especially appealing for the S&U analysis of problems using legacy codes common in the nuclear field.

Grid and basis adaptive polynomial chaos techniques for sensitivity and uncertainty analysis

The demand for accurate and computationally affordable sensitivity and uncertainty techniques is constantly on the rise and has become especially pressing in the nuclear field with the shift to Best Estimate Plus Uncertainty methodologies in the licensing of nuclear installations. Besides traditional, already well developed methods – such as first order perturbation theory or Monte Carlo sampling – Polynomial Chaos Expansion (PCE) has been given a growing emphasis in recent years due to its simple application and good performance.This paper presents new developments of the research done at TU Delft on such Polynomial Chaos (PC) techniques. Our work is focused on the Non-Intrusive Spectral Projection (NISP) approach and adaptive methods for building the PCE of responses of interest. Recent efforts resulted in a new adaptive sparse grid algorithm designed for estimating the PC coefficients. The algorithm is based on Gerstnerʼs procedure for calculating multi-dimensional integrals but proves to be computationally significantly cheaper, while at the same it retains a similar accuracy as the original method.More importantly the issue of basis adaptivity has been investigated and two techniques have been implemented for constructing the sparse PCE of quantities of interest. Not using the traditional full PC basis set leads to further reduction in computational time since the high order grids necessary for accurately estimating the near zero expansion coefficients of polynomial basis vectors not needed in the PCE can be excluded from the calculation. Moreover the sparse PC representation of the response is easier to handle when used for sensitivity analysis or uncertainty propagation due to the smaller number of basis vectors.The developed grid and basis adaptive methods have been implemented in Matlab as the Fully Adaptive Non-Intrusive Spectral Projection (FANISP) algorithm and were tested on four analytical problems. These show consistent good performance both in terms of the accuracy of the resulting PC representation of quantities and the computational costs associated with constructing the sparse PCE. Basis adaptivity also seems to make the employment of PC techniques possible for problems with a higher number of input parameters (15–20), alleviating a well known limitation of the traditional approach. The prospect of larger scale applicability and the simplicity of implementation makes such adaptive PC algorithms particularly appealing for the sensitivity and uncertainty analysis of complex systems and legacy codes.

Exploring emerging manycore architectures for uncertainty quantification through embedded stochastic Galerkin methods

We explore approaches for improving the performance of intrusive or embedded stochastic Galerkin uncertainty quantification methods on emerging computational architectures. Our work is motivated by the trend of increasing disparity between floating-point throughput and memory access speed on emerging architectures, thus requiring the design of new algorithms with memory access patterns more commensurate with computer architecture capabilities. We first compare the traditional approach for implementing stochastic Galerkin methods to non-intrusive spectral projection methods employing high-dimensional sparse quadratures on relevant problems from computational mechanics, and demonstrate the performance of stochastic Galerkin is reasonable. Several reorganizations of the algorithm with improved memory access patterns are described and their performance measured on contemporary manycore architectures. We demonstrate these reorganizations can lead to improved performance for matrix–vector products needed by iterative linear system solvers, and highlight further algorithm research that might lead to even greater performance.

Uncertainty quantification in the catalytic partial oxidation of methane

This work focuses on uncertainty quantification of eight random parameters required as input for 1D modelling of methane catalytic partial oxidation within a highly dense foam reactor. Parameters related to geometrical properties, reactor thermophysics and catalyst loading are taken as uncertain. A widely applied 1D heterogeneous mathematical model that accounts for proper transport and surface chemistry steps is considered for the evaluation of deterministic samples. The non-intrusive spectral projection approach based on polynomial chaos expansion is applied to determine the stochastic temperature and species profiles along the reactor axial direction as well as their ensemble mean and error bars with a confidence interval of 95%. Probability density functions of relevant variables in specific reactor sections are also analysed. A different contribution is noticed from each random input to the total uncertainty range. Porosity, specific surface area and catalyst loading appear as the major sources of uncertainty to bulk gas and surface temperature and species molar profiles. Porosity and the mean pore diameter have an important impact on the pressure drop along the whole reactor as expected. It is also concluded that any trace of uncertainty in the eight input random variables can be almost dissipated near the catalyst outlet section for a long-enough catalyst, mainly due to the approximation to thermodynamic equilibrium.