Stochastic Reduced Order Model Research Articles

Data-informed uncertainty quantification for wave scattering by heterogeneous media

We present an efficient data-driven offline/online Bayesian algorithm for uncertainty quantification (UQ) in the induced scattered field when a time-harmonic incident wave interacts with an uncertain heterogeneous medium. The incident wave of interest need not be known in advance, and the uncertainty is informed by noisy scattering data obtained from other incident waves impinging on the medium. Our UQ algorithm is accelerated by a novel stochastic reduced order model (ROM) based on the T-matrix, and the ROM is independent of both the incident wave, and other incident waves used to generate the data. This important property allows the model to be set up offline.

Representing model uncertainties in brittle fracture simulations

This work focuses on the representation of model-form uncertainties in phase-field models of brittle fracture. Such uncertainties can arise from the choice of the degradation function for instance, and their consideration has been unaddressed to date. The stochastic modeling framework leverages recent developments related to the analysis of nonlinear dynamical systems and relies on the construction of a stochastic reduced-order model. In the latter, a POD-based reduced-order basis is randomized using Riemannian projection and retraction operators, as well as an information-theoretic formulation enabling proper concentration in the convex hull defined by a set of model proposals. The model thus obtained is mathematically admissible in the almost sure sense and involves a low-dimensional hyperparameter, the calibration of which is facilitated through the formulation of a quadratic programming problem. The relevance of the modeling approach is further assessed on one- and two-dimensional applications. It is shown that model uncertainties can be efficiently captured and propagated to macroscopic quantities of interest. An extension based on localized randomization is also proposed to handle the case where the forward simulation is highly sensitive to sample localization. This work constitutes a methodological development allowing phase-field predictions to be endowed with statistical measures of confidence, accounting for the variability induced by modeling choices.

An efficient data-driven multiscale stochastic reduced order modeling framework for complex systems

Suitable reduced order models (ROMs) are computationally efficient tools in characterizing key dynamical and statistical features of nature. In this paper, a systematic multiscale stochastic ROM framework is developed for complex systems with strong chaotic or turbulent behavior. The new ROMs fundamentally differ from the traditional Galerkin ROM (G-ROM) or those deterministic closure ROMs that aim to minimize path-wise errors and apply mainly to laminar systems. Here, the new ROM focuses on recovering the large-scale dynamics to the maximum extent. At the same time, it also exploits cheap but effective conditionally linear functions as the closure terms to capture the statistical features of the medium-scale variables and their feedback to the large scales. The new ROM mimics the model structure of many complex systems, allowing it to be more physically explainable. In addition, physics constraints of energy-conserving nonlinearities are incorporated into the new ROM. One unique feature of the resulting ROM is that it facilitates an efficient and accurate scheme for nonlinear data assimilation, the solution of which is provided by closed analytic formulae that do not require ensemble methods. Such an analytical solvable data assimilation solution significantly accelerates computational efficiency. It allows the new ROM to avoid many potential numerical and sampling issues in recovering the unobserved states from partial observations. The closure term calibration is efficient and systematic via explicit mathematical formulae. The new ROM framework is applied to complex nonlinear systems in which the intrinsic turbulent behavior is either triggered by external random forcing or deterministic nonlinearity. The out-of-sample numerical results show that the new ROM significantly outperforms the G-ROM in both scenarios regarding reproducing the dynamical and statistical features and recovering unobserved states via the efficient data assimilation scheme.

Stochastic analysis of a three phase composite using variational asymptotic based homogenization technique coupled with stochastic reduced order model

The current work describes a stochastic framework to investigate the static response characteristics of periodic and random fiber arrays of reinforced three-phase composites with aleatory mechanical and thermal constituent properties. These properties were randomly modeled via Gaussian distribution using the statistical quantities of first and second moments. The framework is based on Variational Asymptotic Method (VAM) based homogenization technique coupled with Stochastic Reduced Order Model (SROM). When compared against the framework of VAM coupled with the traditional Monte Carlo Simulation (MCS) method, the computational cost of the SROM model with VAM was significantly less. Good agreement with solutions from traditional MCS combined with VAM was also achieved. To demonstrate applicability of the framework, a multivariate stochastic analysis was performed to determine the effect of fiber arrangements on the randomness of mechanical and thermal properties in a three-phase composite. Additionally, univariate stochastic analysis was performed, and the sensitivity of each constituent parameter corresponding to the stochastic homogenized response variable was investigated using the bivariate correlation and linear regression analysis methods.

Adaptive sparse interpolation for accelerating nonlinear stochastic reduced-order modeling with time-dependent bases

Stochastic reduced-order modeling based on time-dependent bases (TDBs) has proven successful for extracting and exploiting low-dimensional manifold from stochastic partial differential equations (SPDEs). The nominal computational cost of solving a rank-r reduced-order model (ROM) based on time-dependent basis, a.k.a. TDB-ROM, is roughly equal to that of solving the full-order model for r random samples. As of now, this nominal performance can only be achieved for linear or quadratic SPDEs—at the expense of a highly intrusive process. On the other hand, for problems with non-polynomial nonlinearity, the computational cost of solving the TDB evolution equations is the same as solving the full-order model. In this work, we present an adaptive sparse interpolation algorithm that enables stochastic TDB-ROMs to achieve nominal computational cost for generic nonlinear SPDEs. Our algorithm constructs a low-rank approximation for the right hand side of the SPDE using the discrete empirical interpolation method (DEIM). The presented algorithm does not require any offline computation and as a result the low-rank approximation can adapt to any transient changes of the dynamics on the fly. We also propose a rank-adaptive strategy to control the error of the sparse interpolation. Our algorithm achieves computational speedup by adaptive sampling of the state and random spaces. We illustrate the efficiency of our approach for two test cases: (1) one-dimensional stochastic Burgers’ equation, and (2) two-dimensional compressible Navier–Stokes equations subject to one-hundred-dimensional random perturbations. In all cases, the presented algorithm results in orders of magnitude reduction in the computational cost.

Non‐Stationary Probabilistic Tsunami Hazard Assessments Compounding Tides and Sea Level Rise

AbstractTides are often the largest source of sea levels fluctuations. Two new probabilistic tsunami hazard assessments (PTHA) methods are proposed to combine the tidal phase uncertainty at the moment of tsunami occurrence with other sources of uncertainty. The first method adopts a Stochastic Reduced Order Model (SROM) producing sets of tidal phase samples to be used in tsunami simulations. The second method uses tsunami simulations with prescribed collocation tidal phases and tide probability distributions to model the uncertainty. The methods are extended to non‐stationary probabilistic tsunami hazard assessment, compounding tsunamis, tides and sea level rise (SLR). As an illustration, these methods are applied for assessing tsunamis generated in the Manila Subduction Zone, on the coasts of Kao Hsiung and Hong Kong. While the SROM‐based method is faster solving for the PTHA if only tides are considered, the collocation‐based method is faster when both SLR and tides are considered. For the illustration case, tides have a relevant impact on PTHA results, however, the SLR within an exposure time of 100 years has stronger impact. PTHA curves of the maximum tsunami elevation are affected by tides and SLR differently. While tides and SLR increase the dispersion of PTHA hazard curve distributions, the latter also produces a translation toward higher elevations. The development of formulations based on SROM or collocation tides is the key to establishing a method which feasibly can be applied to other regions for comprehensive analysis at a global scale.

Predictive and stochastic reduced-order modeling of wind turbine wake dynamics

Abstract. This article presents a reduced-order model of the highly turbulent wind turbine wake dynamics. The model is derived using a large eddy simulation (LES) database, which cover a range of different wind speeds. The model consists of several sub-models: (1) dimensionality reduction using proper orthogonal decomposition (POD) on the global database, (2) projection in modal coordinates to get time series of the dynamics, (3) interpolation over the parameter space that enables the prediction of unseen cases, and (4) stochastic time series generation to generalize the modal dynamics based on spectral analysis. The model is validated against an unseen LES case in terms of the modal time series properties as well as turbine performance and aero-elastic responses. The reduced-order model provides LES accuracy and comparable distributions of all channels. Furthermore, the model provides substantial insights about the underlying flow physics, how these change with respect to the thrust coefficient CT, and whether the model is constructed for single wake or deep array conditions. The predictive and stochastic capabilities of the reduced-order model can effectively be viewed as a generalization of a LES for statistically stationary flows, and the model framework can be applied to other flow cases than wake dynamics behind wind turbines.

A Stochastic Reduced-Order Model for Statistical Microstructure Descriptors Evolution

Abstract Integrated computational materials engineering (ICME) models have been a crucial building block for modern materials development, relieving heavy reliance on experiments and significantly accelerating the materials design process. However, ICME models are also computationally expensive, particularly with respect to time integration for dynamics, which hinders the ability to study statistical ensembles and thermodynamic properties of large systems for long time scales. To alleviate the computational bottleneck, we propose to model the evolution of statistical microstructure descriptors as a continuous-time stochastic process using a non-linear Langevin equation, where the probability density function (PDF) of the statistical microstructure descriptors, which are also the quantities of interests (QoIs), is modeled by the Fokker–Planck equation. We discuss how to calibrate the drift and diffusion terms of the Fokker–Planck equation from the theoretical and computational perspectives. The calibrated Fokker–Planck equation can be used as a stochastic reduced-order model to simulate the microstructure evolution of statistical microstructure descriptors PDF. Considering statistical microstructure descriptors in the microstructure evolution as QoIs, we demonstrate our proposed methodology in three integrated computational materials engineering (ICME) models: kinetic Monte Carlo, phase field, and molecular dynamics simulations.

Stochastic reduced order modelling and analysis of rotating bladed discs

A computational finite-element (FE)-based technique is proposed for developing a stochastic reduced order model for rotating bladed disc with spatial random inhomogeneities. The spatial inhomogeneities imply the system to be randomly mistuned. The formulation assumes the availability of a high fidelity FE model for the tuned system. The corresponding FE matrices are antisymmetric on account of the Coriolis forces due to rotation. The spatial inhomogeneities, available from limited point measurements on the blades, are modelled as non-Gaussian random fields with arbitrary distributions. A low order stochastic computational model is developed by projecting the FE model onto a reduced dimensional state space defined in terms of specified observable nodal points and expressing the stochasticity through an arbitrary polynomial chaos (aPC) basis. This model enables probabilistic quantification of the variabilities in the system response and estimating failure probabilities. The methodology enables drastic reduction in the state space and stochastic dimensions, addresses the practical difficulties with having limited measurable data points, antisymmetric FE matrices, aPC representation in complex irregular geometries and carrying out probabilistic analyses on industrial systems, at significantly reduced computational costs. The methodology is illustrated through an academic rotor and an industrial rotor blade.

Convergence Determination of EMC Uncertainty Simulation Based on the Improved Mean Equivalent Area Method

Uncertainty analysis plays a significant role in electromagnetic compatibility (EMC) simulation, but suffers from convergence determination thereby reducing simulation accuracy and computational efficiency. In this paper, an improved mean equivalent area method is proposed to enhance calculation accuracy. It shows that, using a benchmark example, the proposed method successfully achieves the convergence determination of the stochastic reduced order models (SROMs), and realizes further promotion of uncertainty analysis method.

Static condensation based reduced order modelling of stochastically parametered large ordered systems

A surrogate stochastic reduced order model is developed for the analysis of randomly parametered structural systems with complex geometries. It is assumed that the mathematical model is available in terms of large ordered finite element (FE) matrices. The structure material properties are assumed to have spatial random inhomogeneities and are modelled as non-Gaussian random fields. A polynomial chaos expansion (PCE) based framework is developed for modelling the random fields directly from measurements and for uncertainty quantification of the response. Difficulties in implementing PCE due to geometrical complexities are circumvented by adopting PCE on a geometrically regular domain that bounds the physical domain and are shown to lead to mathematically equivalent representation. The static condensation technique is subsequently extended for stochastic cases based on PCE formalism to obtain reduced order stochastic FE models. The efficacy of the method is illustrated through two numerical examples.

Uncertainty quantification of pure and mixed mode interlaminar fracture of fibre-reinforced composites via a stochastic reduced order model

A comprehensive stochastic analysis of pure and mixed mode interlaminar fracture of fibre-reinforced polymer (FRP) composites, using different uncertainty quantification approaches, is presented. The primary aim of this work is to evaluate the accuracy and computational cost of stochastic reduced order models in the finite element modelling of FRP composite fracture. The Monte Carlo method, with different sampling methods was considered as a reference method for comparison purposes. By comparing the descriptive statistics of uncertain quantities of interest, the advantages and drawbacks of these methods is revealed. Double Cantilever Beam (DCB), End Notch Flexure (ENF) and Mixed Mode Bending (MMB) tests were simulated using a stochastic cohesive zone model. Fundamental characteristics of the cohesive zone model, such as the interlaminar fracture toughness and cohesive strength, were assumed as uncertain sources, and crack extension and critical force were considered as uncertain quantities of interest. The use of a stochastic reduced order model combined with a surrogate model is shown to be computationally efficient, for a given level of accuracy, even with a limited number of samples.

Robust topology optimization under loading uncertainties via stochastic reduced order models

AbstractAn efficient approach for topology optimization under uncertainty is presented. Stochastic reduced order models (SROMs) are leveraged for the modeling and propagation of uncertainties within a robust topology optimization (RTO) formulation. The SROM approach provides an alternative to existing uncertainty quantification methods and yields a substantial improvement in efficiency over a classical Monte Carlo based approach while retaining similar accuracy when representing the uncertainty in system parameters. In particular, random input parameters can be discrete or continuous and specified either analytically (standard distributions) or numerically (dataset samples). Furthermore, multiple random quantities need not be treated as uncorrelated; an SROM can seamlessly model random vectors with arbitrary correlation structure. The nonintrusive nature of the SROM method yields an implementation that can be seen as a drop‐in replacement for a simple RTO approach that leverages Monte Carlo simulation and is therefore straightforward to implement in existing topology optimization software. The proposed approach is demonstrated in the context of structural topology optimization with uncertainty in applied loads. Several numerical results are presented, covering a range of uncertainty distributions that illustrate the flexibility afforded by the general SROM method, while highlighting the efficiency and accuracy achieved in uncertainty propagation.

Deciphering the optimal exergy field in closed-wet cooling towers using Bi-level reduced-order models

This paper introduces a bi-level reduced-order models (ROMs) approach for quickly deciphering the optimal exergy fields in closed wet cooling towers (CWCTs) with consideration of weather variations. First, an efficient sampling method based on stochastic reduced-order model is performed for the approximation of the multivariate probability distributions by generating a finite set of samples. The uncertainty associated with input variables is propagated via multi-sample CFD simulations of the CWCT model for each of the samples. The results of the state and output variables stored in the CFD solutions are used to construct the data-driven and physics-based ROMs by combining principal component analysis and artificial neural network methods. The constructed data-driven ROM is embedded in a sampling-based stochastic optimization model that seeks the maximization of the expected exergy efficiency ratio. The physics-based ROM is used to visualize the optimal field profiles of the thermal-, mechanical-, and chemical-exergy fluxes. Finally, the results of a case study demonstrate that the main strengths of the proposed approach is to simultaneously obtain the optimal exergy efficiency ratios and the exergy field profiles of the CWCT system in a computationally efficient manner.

Impact of density stratification on the global mode in a swirling jet: Stochastic modelling and Lagrangian coherent structures

In an experimental investigation, the stochastic dynamics of the global mode in a turbulent swirling jet are considered. From the application of the swirling jet in gas turbine combustors, it was observed that a specific density gradient in the flow leads to a suppression of the global mode. This phenomenon was replicated in a generic swirling jet using an electrical heating coil placed inside the breakdown bubble. In the present investigation, the dynamics of the global mode obtained from PIV and pressure measurements are analysed using a stochastic reduced-order model to describe the instability. The stochastic model is necessary to explain the interaction between the deterministic dynamics of the global mode and the perturbations by the background turbulence. The calibration of the stochastic model provides the amplification rate of the global mode that defines the transition of the flow, dependent on the swirling strength and the density difference. The spatial structure of the global mode is further investigated from Lagrangian coherent structures of the flow field which are computed from the 3D time-resolved velocity field reconstruction based on planar PIV measurements. The Lagrangian visualisations and schlieren visualisations are used to explain the absence of the density effects on the global mode at larger Reynolds numbers. The analysis gives a detailed view of the stochastic dynamics of a hydrodynamic instability in a turbulent flow.

Sustainable retrofit of petrochemical energy systems under multiple uncertainties using the stochastic optimization method

We report a multi-objective stochastic mixed-integer non-linear programming (MINLP) framework for sustainable retrofit and capability expansion of traditional energy systems in petrochemical complexes. Multiple uncertainties including energy demands, solar radiation and wind speeds are considered in the optimization framework, these are characterized by historical data or normal distributions which are pre-defined with assumed mean values and standard variations. A stochastic reduced order model approach is introduced to describe the uncertainties by a small number of scenarios and their individual probabilities. The optimization framework further accounts for system configuration selection and sizing of the candidate energy conversion equipment, such as thermal storage units, gas turbines, boilers, steam turbines, as well as their operating capacities in each time period. A case study is investigated to demonstrate the performance of the proposed strategy and then the optimization results under three modes (deterministic, stochastic and semi-stochastic programs) are compared.

Probing Sensitivity of Discharge Characteristics to Model Selection Using Uncertainty Quantification in an Aprotic Li-Oxygen Battery

Currently, there are several models in the literature, such as kinetic, microstructural, and mass transport, that describe a Li-oxygen battery’s discharge behavior. Many of these models are calibrated and tested at low current densities and are not easily transferable to higher current densities (local current density exceeding 1μA/cm2). Even at low current densities, there is no quantitative method for a researcher to choose a reaction kinetic model such as classical Butler-Volmer and its derivatives, or modified Marcus-Hush-Chidsey, a resistance model for lithium peroxide such as electron transport via tunneling or linear resistivity, a surface coverage model (lithium peroxide growth) such as partial coverage or full coverage, and mass transport model (a review of these models is discussed in Ref. [1]). Also, it is time-consuming to test different models at high current density (≥1C) due to a lack of well-tested models and well-calibrated model parameters. For this presentation, we will develop an analytical model, which acts as a surrogate model for a full finite element model to predict discharge time and discharge voltage. Next, we use an uncertainty quantifying technique called reduced-order stochastic optimization [2, 3] to determine the uncertainty in model parameters for rate kinetics, lithium peroxide resistivity, and parasitic resistance. Finally, a finite element simulation is performed to determine the error introduced by the surrogate model (from various assumptions and simplifications) and its influence on the uncertainty in the model parameters.Our preliminary polarization analysis (overvoltage vs. current density) shows that the resistance models, coupled with Butler-Volmer kinetics, cannot describe the voltage polarization observed in the experiments at high current densities. In order to quantify and rank models describing the lithium peroxide deposition, first, an analytical model is developed to account for discharge product distribution, concentration polarization, and voltage polarization as a function of time, based on the previous work on aqueous electrolytes, presented in Ref. [4]. The analytical model considers reaction kinetics, oxygen mass transport, voltage polarization from lithium peroxide deposition as a conformal coating, and the change in porosity, lithium peroxide thickness, and the active surface is assumed to be uniform throughout the cathode.In order to rank the kinetic and resistance models, the uncertainty in the model parameters is assumed using a probability distribution function, and the resulting statistics are gathered using the Stochastic Reduced Order Model (SROM) approach. SROM is a computationally lite approach that uses significantly less number of samples compared to Monte Carlo to represent the input distribution. Since the number of samples required for Monte Carlo increases with model complexity, which significantly increases the computational time and resources required for full finite element simulations. The SROM approach can reduce the computational requirement by at least ten times without introducing algorithm-based error into the analysis.Although the analytical model incorporates the important aspects to simulate a battery, it neglects the complexity of spatial distribution and temporal variation of the discharge product and the complicated relationship between microstructure and mass transport of oxygen and lithium ions. These neglected interactions introduce a model error in the final parameter estimation and need to be accounted for in order to rank various models. The finite element model (FEM) is used to quantify the error in the surrogate model because of its low fidelity. The FEM model is implemented in COMSOL and considers porous electrode theory for the cathode, concentrated theory for the electrolyte, a resistance model for the discharge products, a kinetic model, Fickian diffusion for the oxygen transport, and an oxygen dissolution model [5].

On Stochastic Reduced-Order and LES-based Models of Offshore Wind Turbine Wakes

In this paper, the primary objective is to investigate flow structures in the wake of wind turbines based on applying a truncated Proper Orthogonal Decomposition (POD) approach. This scheme decomposes the three-dimensional velocity fields produced by the high-fidelity PArallelized LES Model (PALM) into a number of orthogonal spatial modes and time-dependent weighting coefficients. PALM has been combined with an actuator disk model with rotation to incorporate the effects of a turbine array. The time-dependent deterministic weights from applying the POD scheme are replaced by stochastic weights, estimated from two independent stochastic techniques that aim to account for unresolved small-scale features for a number of POD modes. We then reconstruct the flow field by a small number of stochastic modes to investigate how well the applied stochastic methodologies can reproduce the flow field compared to the original LES results.

An improvement of tsunami hazard analysis in Central Chile based on stochastic rupture scenarios

ABSTRACT Central Chile is exposed to tsunami hazard, and large earthquakes and tsunamis have occurred over the last 500 years. Tsunami hazard analysis in Chile has been traditionally implemented by means of a deterministic approach, which is based on historical events and uniform slip distribution. The objective of the present study is to improve tsunami hazard analysis in central Chile (30°S to 38°S). To encompass the purpose, stochastic earthquake scenarios of magnitude 8.8 to 9.2 were generated. Two different sets of stochastic tsunami scenarios were selected by means of the Stochastic Reduced Order Model (SROM), which were applied to Quintero bay to perform a Probabilistic Tsunami Hazard Analysis (PTHA). The results showed that PTHA of Quintero bay from stochastic tsunami scenarios agrees with paleotsunami records in the bay, while a deterministic tsunami scenario underestimated the hazard. Two sets (50 and 100 scenarios, respectively) give similar results when smaller return periods are analyzed. However, for larger return periods (2000 yr) the set of 100 scenarios show better results consistent with previous paleoseismological findings. The methodology implemented here can be replicated in other seismic regions in Chile as well as in other active subduction zones, thus, both near field and far field events can be analyzed.

Radiated Susceptibility Analysis of Single-wire Transmission Lines by Means of Modified Stochastic Reduced-order Modeling

This work investigates the statistical radiatedsusceptibility of an electrically-short transmission line (TL)consisting of a single wire over a ground plane. The angularparameters of the impinging plane wave and the height ofthe wire over the ground plane are modelled as randomvariables with Gaussian/Uniform distributions. Thestatistical properties of the current in the TL terminations(i.e., mean value, standard deviation, and cumulativedistribution function) are derived through a properlydefined numerical methodology consisting in a modifiedversion of the conventional Stochastic Reduced-OrderModel (SROM) approach. The proposed methodologyconsists in a straightforward approximation of the inputcontinuous random variables with small-size discreterandom variables. In contrast with conventional SROM, nonumerical pre-processing is needed. The modified SROMproposed in this paper demonstrates high efficiency whencompared with classical Monte Carlo approach. Theproposed technique can be applied to the statistical analysisof much more complex systems whose input/outputrelationship requires a huge computational burden and forwhich the conventional Monte Carlo approach is notsuitable