Polynomial Chaos Surrogate Model Research Articles

Application of the polynomial chaos method in global sensitivity analysis of trimaran cross-deck structure

An efficient global sensitivity analysis method based on the Polynomial Chaos (PC) method is applied to investigate the strength of a trimaran cross-deck structure. The PC method can reduce the amount of computation and converge faster than the Monte Carlo (MC) method. For the quantitative investigation of the sensitivity of the stresses in the cross-deck to the changes in the cross-deck plate thickness, Latin Hypercube Sampling (LHS) was used to obtain samples, and a Finite Element Method (FEM) model was used to compute the stress based on a load case prescribed by the Rules for the Classification of Trimarans. The PC surrogate models of different degrees of polynomial were trained and validated from the perspective of bias and variance. The third-degree PC model was selected to approximate the response of the stresses to the plate thickness. The global sensitivity indices were computed on the basis of the PC surrogate model, and the results were compared with those obtained by MC. The third-degree PC surrogate model effectively approximates the response of the stresses to the plate thickness, and the sensitivity indices of PC converges much faster than those of MC with respect to the number of samples. The stress at the corner of the cross-deck is the most sensitive to the thickness of the aft web. Meanwhile, the stress at the main-hull connection is the most sensitive to the thickness of the lower panel. Furthermore, the maximum von Mises stress and shear stress at the side-hull connection are the most sensitive to the thickness of the upper panel, and the maximum principal stress at the side-hull connection is the most sensitive to the thickness of the lower panel. The feasibility and efficiency of the PC method is proven, and the influence of the plate thickness on the stress response is described quantitatively.

Bayesian updating of tall timber building model using modal data

A framework for the probabilistic finite element model updating based on measured modal data is presented. The described framework is applied to a seven-storey building made of cross-laminated timber panels. The experimental estimates based on the forced vibration test are used in the process of model updating. First, a generalized Polynomial Chaos surrogate model is derived representing the map from the model parameters to the eigenfrequencies and the eigenvectors. To overcome the difficulties caused by mode switching, we propose a novel approach to mode tracking based on partitioning an extended and low-rank representation of the k mode shapes resulting from different setups of the finite element model into k clusters by the k-means clustering algorithm. Second, the surrogate model derived with the help of mode pairing is used to efficiently perform sensitivity analysis and uncertainty quantification of the first five frequencies and the corresponding mode shapes. Finally, the surrogate-based Bayesian update of the model parameters is efficiently performed, providing engineers not only with a finite element model that gives a good fit to the experimental modal data, but also a stochastic model that represents the uncertainties originating from the initial model and the uncertainties of measuring modal properties.

Power module heat sink design optimization with ensembles of data-driven polynomial chaos surrogate models

We consider the problem of optimizing the design of a heat sink used for cooling an insulated gate bipolar transistor (IGBT) power module. The thermal behavior of the heat sink is originally estimated using a high-fidelity computational fluid dynamics (CFD) simulation, which renders numerical optimization too computationally demanding. To enable optimization studies, we substitute the CFD simulation model with an inexpensive polynomial surrogate model that approximates the relation between the device’s design features and a relevant thermal quantity of interest. The surrogate model of choice is a data-driven polynomial chaos expansion (DD-PCE), which learns the aforementioned relation by means of polynomial regression. Advantages of the DD-PCE include its applicability in small-data regimes and its easily adaptable model structure. To address the issue of model-form uncertainty and model robustness in view of limited training and test data, ensembles of DD-PCEs are generated based on data re-shuffling. Then, using the full ensemble of surrogate models, the surrogate-based predictions are accompanied by uncertainty metrics such as mean value and variance. Once trained and tested in terms of accuracy and robustness, the ensemble of DD-PCE surrogates replaces the high-fidelity simulation model in optimization algorithms aiming to identify heat sink designs that optimize the thermal behavior of the IGBT under geometrical and operational constraints. Optimized heat sink designs are obtained for a computational cost much smaller than utilizing the original model in the optimization procedure. Due to ensemble modeling, the optimization results can also be assessed in terms of uncertainty and robustness. Comparisons against alternative surrogate modeling techniques illustrate why the DD-PCE should be preferred in the considered setting.

Uncertainty Quantification of Differential Algebraic Equations Using Polynomial Chaos

Abstract The focus of this paper is on the use of polynomial chaos (PC) for developing surrogate models for differential algebraic equations (DAEs) with time-invariant uncertainties. Intrusive and nonintrusive approaches to synthesize PC surrogate models are presented including the use of Lagrange interpolation polynomials as basis functions. Unlike ordinary differential equations (ODEs), if the algebraic constraints are a function of the stochastic variable, some initial conditions of the DAEs are also random. A benchmark RLC circuit which is used as a benchmark for linear models is used to illustrate the development of a PC-based surrogate model. A nonlinear example of a simple pendulum also serves as a benchmark to illustrate the potential of the proposed approach. Statistics of the results of the PC models are validated using Monte Carlo (MC) simulations in addition to estimating the evolving probably density functions (PDFs) of the states of the pendulum.

Data-driven inverse uncertainty quantification in the transverse tensile response of carbon fiber reinforced composites

Uncertainty quantification is critical for the full exploitation of composite materials’ potential. Inverse methods offer the possibility of indirectly characterizing the uncertainty of microscopic parameters by employing data sets from standard structural tests in higher scales. Two crucial requirements though, are the efficient modeling especially for the nonlinear prediction, and the measurement error availability from the tests which affects the updated scatter. This study employs effective stiffness and strength experimental data in order to quantify uncertainties of a carbon fiber UD composite in the microscale. A polynomial chaos surrogate model is trained from finite element simulations, able to efficiently predict the homogenized stiffness and strength for the uncertainty quantification procedure. The random parameters which are influential enough to be updated, are identified via a variance-based global sensitivity analysis. The inverse problem is solved with the Bayesian inference method, which updates any prior estimation of the probability models of the input parameters, based on output observations from the tests. Results show significant uncertainty reduction in comparison with typically used variance values in the literature and can be used to enrich the composite material databases. The proposed methodology is applied for the transverse tensile load case, although its non-intrusive nature allows applications for more load cases and various setups.

Shell Buckling With Polymorphic Uncertain Surface Imperfections and Sensitivity Analysis

Abstract In a probabilistic design approach for cylindrical shells, Gaussian random fields are used to simulate geometric imperfections. The shape of imperfections depends, among others, on the autocorrelation properties of the random field. Underlying uncertainties such as a small sample size or imprecise measurements make it practically impossible to define a crisp correlation function. For a more realistic description of the imprecise correlation structure, the classical probabilistic approach is extended to a fuzzy stochastic approach. More exactly, the polymorphic uncertainty approach is used taking into account natural variability and incompleteness. Consequently, geometric imperfections are represented as fuzzy probability based random fields. Therefore, the required correlation parameters are described as polymorphic uncertain parameters. The quantification of uncertainties is demonstrated on real data. Furthermore, the polynomial chaos surrogate model is used for the alpha-level optimization in the fuzzy analysis. The sensitivity indices as a by-product of the surrogate model show the influence of the input parameters on the statistical parameters of the critical buckling load factor. The main purpose of this paper is to show how the presented methods can support the design process of cylindrical shells.

Optimization of a higher-order sandwich composite beam under uncertainties

The present paper investigates the reliable and robust optimum design of a higher-order sandwich composite beam under the effect of uncertainty in material properties. The sandwich beam is modeled using the extended higher-order sandwich panel theory. The optimization approaches employed in this work are reliability-based design optimization, robust design optimization, and a hybrid reliability-based robust design optimization. The efficiency of the optimization process is enhanced by using a novel time domain spectral element method-based polynomial chaos surrogate model. The performance of the surrogate model is further improved by using Sobol indices based sensitivity analysis. The optimization procedure is performed using an accelerated particle swarm optimization. The numerical results of the reliable and robust optimal design are presented for both the soft and stiff core sandwich beam. Furthermore, the effect of load density and allowable deflection on the optimal design is also examined.

A Predictor–Corrector Algorithm for Fast Polynomial Chaos-Based Uncertainty Quantification of Multi-Walled Carbon Nanotube Interconnects

In this article, a predictor–corrector algorithm for the fast polynomial chaos (PC)-based uncertainty quantification (UQ) of multi-walled carbon nanotube (MWCNT) interconnect networks is presented. The proposed algorithm intelligently combines the numerical efficiency of the approximate equivalent single conductor (ESC) model of the MWCNT interconnect network with the rigor and accuracy of a multi-conductor circuit (MCC) model. Consequently, this algorithm significantly accelerates the generation of the PC surrogate models (or metamodels) of the network responses for minimal loss in accuracy. These metamodels can be probed efficiently and repeatedly to quantify the impact of manufacturing and fabrication process uncertainty on the MWCNT network responses.

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.

A Hybrid Methodology for Jitter and Eye Estimation in High-Speed Serial Channels Using Polynomial Chaos Surrogate Models

Estimation of jitter and eye diagram in high-speed serial channels can be challenging. The existing methods might fail to show inter-symbol interference (ISI) and data dependent jitter because they are either excessively time consuming or only applicable to linear time-invariant (LTI) systems. Therefore, this paper suggests a hybrid methodology to develop surrogate models of the system from a short transient simulation and using them to efficiently determine the eye diagram and related parameters. This paper is based on uncertainty quantification techniques that use polynomial chaos (PC) expansions to estimate the output. Necessary modifications are applied to PC modeling to make it applicable to high-speed channels. Finally, the performance of the proposed approach is shown through numerical examples.

Uncertainty Propagation in Coupled Atmosphere–Wave–Ocean Prediction System: A Study of Hurricane Earl (2010)

This study focuses on understanding the evolution of Hurricane Earl (2010) with respect to random perturbations in the storm’s initial strength, size, and asymmetry in wind distribution. We rely on the Unified Wave Interface-Coupled Model (UWIN-CM), a fully coupled atmosphere–wave–ocean system to generate a storm realization ensemble, and use polynomial chaos (PC) expansions to build surrogate models for time evolution of both the maximum wind speed and minimum sea level pressure in Earl. The resulting PC surrogate models provide statistical insights on probability distributions of model responses throughout the simulation time span. Statistical analysis of rapid intensification (RI) suggests that initial perturbations having intensified and counterclockwise-rotated winds are more likely to undergo RI. In addition, for the range of initial conditions considered RI seems mostly sensitive to azimuthally averaged maximum wind speed and asymmetry orientation, rather than storm size and asymmetry magnitude; this is consistent with global sensitivity analysis of PC surrogate models. Finally, we combine initial condition perturbations with a stochastic kinetic energy backscatter scheme (SKEBS) forcing in the UWIN-CM simulations and conclude that the storm tracks are substantially influenced by the SKEBS forcing perturbations, whereas the perturbations in initial conditions alone had only limited impact on the storm-track forecast.

AN ADAPTIVE MULTIFIDELITY PC-BASED ENSEMBLE KALMAN INVERSION FOR INVERSE PROBLEMS

The ensemble Kalman inversion (EKI), as a derivative-free methodology, has been widely used in the parameter estimation of inverse problems. Unfortunately, its cost may become moderately large for systems described by high dimensional nonlinear PDEs, as EKI requires a relatively large ensemble size to guarantee its performance. In this paper, we propose an adaptive multi-fidelity polynomial chaos (PC) based EKI technique to address this challenge. Our new strategy combines a large number of low-order PC surrogate model evaluations and a small number of high-fidelity forward model evaluations, yielding a multi-fidelity approach. Especially, we present a new approach that adaptively constructs and refines a multi-fidelity PC surrogate during the EKI simulation. Since the forward model evaluations are only required for updating the low-order multi-fidelity PC model, whose number can be much smaller than the total ensemble size of the classic EKI, the entire computational costs are thus significantly reduced. The new algorithm was tested through the two-dimensional time fractional inverse diffusion problems and demonstrated great effectiveness in comparison with PC based EKI and classic EKI.

Variance‐based simplex stochastic collocation with model order reduction for high‐dimensional systems

SummaryIn this work, an adaptive simplex stochastic collocation method is introduced in which sample refinement is informed by variability in the solution of the system. The proposed method is based on the concept of multi‐element stochastic collocation methods and is capable of dealing with very high‐dimensional models whose solutions are expressed as a vector, a matrix, or a tensor. The method leverages random samples to create a multi‐element polynomial chaos surrogate model that incorporates local anisotropy in the refinement, informed by the variance of the estimated solution. This feature makes it beneficial for strongly nonlinear and/or discontinuous problems with correlated non‐Gaussian uncertainties. To solve large systems, a reduced‐order model (ROM) of the high‐dimensional response is identified using singular value decomposition (higher‐order SVD for matrix/tensor solutions) and polynomial chaos is used to interpolate the ROM. The method is applied to several stochastic systems of varying type of response (scalar/vector/matrix) and it shows considerable improvement in performance compared to existing simplex stochastic collocation methods and adaptive sparse grid collocation methods.

Bayesian inference of earthquake rupture models using polynomial chaos expansion

Abstract. In this paper, we employed polynomial chaos (PC) expansions to understand earthquake rupture model responses to random fault plane properties. A sensitivity analysis based on our PC surrogate model suggests that the hypocenter location plays a dominant role in peak ground velocity (PGV) responses, while elliptical patch properties only show secondary impact. In addition, the PC surrogate model is utilized for Bayesian inference of the most likely underlying fault plane configuration in light of a set of PGV observations from a ground-motion prediction equation (GMPE). A restricted sampling approach is also developed to incorporate additional physical constraints on the fault plane configuration and to increase the sampling efficiency.

Probabilistic modeling and global sensitivity analysis for CO2 storage in geological formations: a spectral approach

• Comprehensive analysis of impact of parametric uncertainties on simulation of CO 2 storage in porous media is performed. • A non-isothermal two-phase two-component flow system is considered. • Rigorous numerical study of the accuracy of the polynomial chaos surrogate model is performed. • A comprehensive global sensitivity analysis is conducted. This work focuses on the simulation of CO 2 storage in deep underground formations under uncertainty and seeks to understand the impact of uncertainties in reservoir properties on CO 2 leakage. To simulate the process, a non-isothermal two-phase two-component flow system with equilibrium phase exchange is used. Since model evaluations are computationally intensive, instead of traditional Monte Carlo methods, we rely on polynomial chaos (PC) expansions for representation of the stochastic model response. A non-intrusive approach is used to determine the PC coefficients. We establish the accuracy of the PC representations within a reasonable error threshold through systematic convergence studies. In addition to characterizing the distributions of model observables, we compute probabilities of excess CO 2 leakage. Moreover, we consider the injection rate as a design parameter and compute an optimum injection rate that ensures that the risk of excess pressure buildup at the leaky well remains below acceptable levels. We also provide a comprehensive analysis of sensitivities of CO 2 leakage, where we compute the contributions of the random parameters, and their interactions, to the variance by computing first, second, and total order Sobol’ indices.

Uncertainty quantification and global sensitivity analysis of complex chemical processes with a large number of input parameters using compressive polynomial chaos

Uncertainties are ubiquitous and unavoidable in process design and modeling while they can significantly affect safety, reliability, and economic decisions. The large number of uncertainties in complex chemical processes make the well-known Monte-Carlo and polynomial chaos approaches for uncertainty quantification computationally expensive and even infeasible. This study focused on the uncertainty quantification and sensitivity analysis of complex chemical processes with a large number of uncertainties. An efficient method was proposed using a compressed sensing technique to overcome the computational limitations for complex and large scale systems. In the proposed method, compressive sparse polynomial chaos surrogates were constructed and applied to quantify the uncertainties and reflect their propagation effect on process design. Rigorous case studies were provided by the interface between MATLAB™ and Aspen HYSYS™ for a propylene glycol production process and lean dry gas processing plant. The proposed methodology was compared with traditional Monte-Carlo/Quasi Monte-Carlo sampling-based and standard polynomial chaos approaches to highlight its advantages in terms of computational efficiency. The proposed approach could mitigate the simulation costs significantly using an accurate, efficient-to-evaluate polynomial chaos that can be used in place of expensive simulations. In addition, the global sensitivity indices, which show the relative importance of uncertain inputs on the process output, could be derived analytically from the obtained polynomial chaos surrogate model.

Quantifying initial and wind forcing uncertainties in the Gulf of Mexico

This study aims at analyzing the combined impact of uncertainties in initial conditions and wind forcing fields in ocean general circulation models (OGCM) using polynomial chaos (PC) expansions. Empirical orthogonal functions (EOF) are used to formulate both spatial perturbations to initial conditions and space-time wind forcing perturbations, namely in the form of a superposition of modal components with uniformly distributed random amplitudes. The forward deterministic HYbrid Coordinate Ocean Model (HYCOM) is used to propagate input uncertainties in the Gulf of Mexico (GoM) in spring 2010, during the Deepwater Horizon oil spill, and to generate the ensemble of model realizations based on which PC surrogate models are constructed for both localized and field quantities of interest (QoIs), focusing specifically on sea surface height (SSH) and mixed layer depth (MLD). These PC surrogate models are constructed using basis pursuit denoising methodology, and their performance is assessed through various statistical measures. A global sensitivity analysis is then performed to quantify the impact of individual modes as well as their interactions. It shows that the local SSH at the edge of the GoM main current—the Loop Current—is mostly sensitive to perturbations of the initial conditions affecting the current front, whereas the local MLD in the area of the Deepwater Horizon oil spill is more sensitive to wind forcing perturbations. At the basin scale, the SSH in the deep GoM is mostly sensitive to initial condition perturbations, while over the shelf it is sensitive to wind forcing perturbations. On the other hand, the basin MLD is almost exclusively sensitive to wind perturbations. For both quantities, the two sources of uncertainty have limited interactions. Finally, the computations indicate that whereas local quantities can exhibit complex behavior that necessitates a large number of realizations, the modal analysis of field sensitivities can be suitably achieved with a moderate size ensemble.

Stochastic Galerkin Formulations for $$\hbox {CO}_2$$ Transport in Aquifers: Numerical Solutions with Uncertain Material Properties

Abstract A stochastic Galerkin formulation for the transport of $$\hbox {CO}_2$$ CO 2 in a tilted aquifer with uncertain heterogeneous properties is presented. We consider a simplified physics model assuming capillary pressure to be negligible compared to hydrostatic and viscous pressure. The flow is dominated by buoyancy and capillary trapping. We assume a stochastic permeability field and a stochastic model for the uncertain relative permeabilities. We prove that the proposed stochastic Galerkin formulation results in a hyperbolic system of equations, and we devise a numerical method that captures the expected solution discontinuities. The shock-capturing solver for the flux function is combined with an adaptive quadrature method for discontinuous isosurfaces that is used to compute the discontinuous stochastic accumulation coefficient. The stochastic solver is validated against Monte Carlo sampling of an analytical solution for the deterministic problem. The sharp features of the statistics of the solution are accurately captured by the numerical solver. The polynomial chaos framework admits low-cost post-processing of the output to obtain statistics of interest. By construction of an accurate polynomial chaos surrogate model of the output, fast sampling admits calculation of risk for leakage and failure probabilities.

Towards predictive data-driven simulations of wildfire spread – Part I: Reduced-cost Ensemble Kalman Filter based on a Polynomial Chaos surrogate model for parameter estimation

Abstract. This paper is the first part in a series of two articles and presents a data-driven wildfire simulator for forecasting wildfire spread scenarios, at a reduced computational cost that is consistent with operational systems. The prototype simulator features the following components: an Eulerian front propagation solver FIREFLY that adopts a regional-scale modeling viewpoint, treats wildfires as surface propagating fronts, and uses a description of the local rate of fire spread (ROS) as a function of environmental conditions based on Rothermel's model; a series of airborne-like observations of the fire front positions; and a data assimilation (DA) algorithm based on an ensemble Kalman filter (EnKF) for parameter estimation. This stochastic algorithm partly accounts for the nonlinearities between the input parameters of the semi-empirical ROS model and the fire front position, and is sequentially applied to provide a spatially uniform correction to wind and biomass fuel parameters as observations become available. A wildfire spread simulator combined with an ensemble-based DA algorithm is therefore a promising approach to reduce uncertainties in the forecast position of the fire front and to introduce a paradigm-shift in the wildfire emergency response. In order to reduce the computational cost of the EnKF algorithm, a surrogate model based on a polynomial chaos (PC) expansion is used in place of the forward model FIREFLY in the resulting hybrid PC-EnKF algorithm. The performance of EnKF and PC-EnKF is assessed on synthetically generated simple configurations of fire spread to provide valuable information and insight on the benefits of the PC-EnKF approach, as well as on a controlled grassland fire experiment. The results indicate that the proposed PC-EnKF algorithm features similar performance to the standard EnKF algorithm, but at a much reduced computational cost. In particular, the re-analysis and forecast skills of DA strongly relate to the spatial and temporal variability of the errors in the ROS model parameters.

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.