Uncertainty Quantification Techniques Research Articles

Analytical probabilistic modeling of RBE-weighted dose for ion therapy

Particle therapy is especially prone to uncertainties. This issue is usually addressed with uncertainty quantification and minimization techniques based on scenario sampling. For proton therapy, however, it was recently shown that it is also possible to use closed-form computations based on analytical probabilistic modeling (APM) for this purpose. APM yields unique features compared to sampling-based approaches, motivating further research in this context.This paper demonstrates the application of APM for intensity-modulated carbon ion therapy to quantify the influence of setup and range uncertainties on the RBE-weighted dose. In particular, we derive analytical forms for the nonlinear computations of the expectation value and variance of the RBE-weighted dose by propagating linearly correlated Gaussian input uncertainties through a pencil beam dose calculation algorithm. Both exact and approximation formulas are presented for the expectation value and variance of the RBE-weighted dose and are subsequently studied in-depth for a one-dimensional carbon ion spread-out Bragg peak. With V and B being the number of voxels and pencil beams, respectively, the proposed approximations induce only a marginal loss of accuracy while lowering the computational complexity from order to for the expectation value and from to for the variance of the RBE-weighted dose. Moreover, we evaluated the approximated calculation of the expectation value and standard deviation of the RBE-weighted dose in combination with a probabilistic effect-based optimization on three patient cases considering carbon ions as radiation modality against sampled references. The resulting global γ-pass rates (2 mm,2%) are 99.15% for the expectation value and 94.95% for the standard deviation of the RBE-weighted dose, respectively. We applied the derived analytical model to carbon ion treatment planning, although the concept is in general applicable to other ion species considering a variable RBE.

Stereo-DIC Uncertainty Estimation Using the Epipolar Constraint and Optimized Three Camera Triangulation

Even though stereo-DIC is widely used in the field of experimental mechanics, uncertainty quantification techniques lag behind. Camera motion for example is often neglected, even though it is present and it can introduce bias. We propose to estimate the error caused by camera motion, based on the epipolar distance, by using a 3-camera system. This solves the problem of the lack of sensitivity along the epipolar line by placing a third camera perpendicular, providing sensitivity in both directions. The third camera provides modest improvement on the results with an optimized triangulation.

Uncertainty quantification and management in additive manufacturing: current status, needs, and opportunities

One of the major barriers that hinder the realization of significant potential of metal-based additive manufacturing (AM) techniques is the variation in the quality of the manufactured parts. Uncertainty quantification (UQ) and uncertainty management (UM) can resolve this challenge based on the modeling and simulation of the AM process. This paper reviews the research state of the art and discusses needs and opportunities in the UQ/UM of the AM processes, with a focus on laser powder bed fusion AM. The major methods and models of laser powder bed fusion AM process are summarized first. The current research work in UQ of AM processes is then reviewed. Based on the review of AM process models and current UQ approaches for the AM process, this paper presents insights into how the current state of the art UQ and UM techniques can be applied to AM to improve the product quality. Future research needs in UQ and UM of AM are also discussed. Laser sintering of metal nanoparticles, which is part of the micro-AM process, is used as an example to illustrate the application of UQ and UM in the AM.

Uncertainty Footprint: Visualization of Nonuniform Behavior of Iterative Algorithms Applied to 4D Cell Tracking

Research on microscopy data from developing biological samples usually requires tracking individual cells over time. When cells are three-dimensionally and densely packed in a time-dependent scan of volumes, tracking results can become unreliable and uncertain. Not only are cell segmentation results often inaccurate to start with, but it also lacks a simple method to evaluate the tracking outcome. Previous cell tracking methods have been validated against benchmark data from real scans or artificial data, whose ground truth results are established by manual work or simulation. However, the wide variety of real-world data makes an exhaustive validation impossible. Established cell tracking tools often fail on new data, whose issues are also difficult to diagnose with only manual examinations. Therefore, data-independent tracking evaluation methods are desired for an explosion of microscopy data with increasing scale and resolution. In this paper, we propose the uncertainty footprint, an uncertainty quantification and visualization technique that examines nonuniformity at local convergence for an iterative evaluation process on a spatial domain supported by partially overlapping bases. We demonstrate that the patterns revealed by the uncertainty footprint indicate data processing quality in two algorithms from a typical cell tracking workflow - cell identification and association. A detailed analysis of the patterns further allows us to diagnose issues and design methods for improvements. A 4D cell tracking workflow equipped with the uncertainty footprint is capable of self diagnosis and correction for a higher accuracy than previous methods whose evaluation is limited by manual examinations.

Bayesian and variational Bayesian approaches for flows in heterogeneous random media

In this paper, we study porous media flows in heterogeneous stochastic media. We propose an efficient forward simulation technique that is tailored for variational Bayesian inversion. As a starting point, the proposed forward simulation technique decomposes the solution into the sum of separable functions (with respect to randomness and the space), where each term is calculated based on a variational approach. This is similar to Proper Generalized Decomposition (PGD). Next, we apply a multiscale technique to solve for each term (as in [1]) and, further, decompose the random function into 1D fields. As a result, our proposed method provides an approximation hierarchy for the solution as we increase the number of terms in the expansion and, also, increase the spatial resolution of each term. We use the hierarchical solution distributions in a variational Bayesian approximation to perform uncertainty quantification in the inverse problem. We conduct a detailed numerical study to explore the performance of the proposed uncertainty quantification technique and show the theoretical posterior concentration.

Thermochemical ablation modeling forward uncertainty analysis—Part II: Application to plasma wind-tunnel testing

The coordination between modeling, simulations, and experiments is critical for the design of the thermal protections of space vehicles. This paper illustrates a proof-of-concept of the coupling between a thermo-chemical ablation model and modern uncertainty quantification techniques with the aim of rebuilding the ablative material tests performed in the inductively coupled Plasmatron facility at the von Karman Institute. In Part I of this two-part work we presented the thermo-chemical ablation model and studied the effect of its uncertain parameters on the simulation outcomes. The present analysis is devoted to understand the effects of the uncertain estimates of the plasma wind-tunnel test conditions on the final results of the ablation model. A two-step uncertainty analysis is performed: the impact of the experimental uncertainties on the plasma flow rebuilding is first analyzed; then, the obtained mean, variance and distribution of the free-stream conditions are used as input, together with the model uncertain parameters, for the uncertainty analysis of the ablation model. Our results show that the experimental uncertainties have a substantial impact on the ablation model output when surface nitridation is not considered among the surface reactions.

Thermochemical ablation modeling forward uncertainty analysis—Part I: Numerical methods and effect of model parameters

Next generation spacecraft will bring back heavier payloads from explored planets. Advance in the modeling of the thermo-chemical ablation of carbon-based thermal protection system materials is fundamental to improve the design capabilities of these vehicles. Computational fluid dynamic approaches are extensively used to model the gas-surface interaction phenomena over ablative materials. The advantage of such kind of approaches is the accurate description of the aerothermal environment obtained through the full resolution of the mechanical, thermal, and chemical boundary layers that develop over an ablative surface when exposed to a high-enthalpy flow. This paper is devoted to the assessment of the uncertainties of such kind of thermo-chemical ablation model and to study their effect on the model final outcomes. A sphere of non-pyrolyzing carbon-based material, exposed to conditions similar to those of a typical plasma wind tunnel test, is the selected test case for the analysis. Two forward non-intrusive uncertainty quantification techniques are used to analyze the influence of the defined set of uncertain parameters on the estimate of steady-state mass blowing flux and surface temperature. Our results show that for the selected conditions, and uncertainty ranges, the surface nitridation reaction probability has the strongest impact on the model outcomes.

Estimating yield-strain via deformation-recovery simulations

In computational materials science, predicting the yield strain of crosslinked polymers remains a challenging task. A common approach is to identify yield as the first critical point of stress-strain curves simulated by molecular dynamics (MD). However, in such cases the underlying data can be excessively noisy, making it difficult to extract meaningful results. In this work, we propose an alternate method for identifying yield on the basis of deformation-recovery simulations. Notably, the corresponding raw data (i.e. residual strains) produce a sharper signal for yield via a transition in their global behavior. We analyze this transition by non-linear regression of computational data to a hyperbolic model. As part of this analysis, we also propose uncertainty quantification techniques for assessing when and to what extent the simulated data is informative of yield. Moreover, we show how the method directly tests for yield via the onset of permanent deformation and discuss recent experimental results, which compare favorably with our predictions.

Uncertainty quantification techniques applied to rotating systems: A comparative study

The modeling of mechanical systems involves different parameters that are susceptible to uncertainties. Commonly, the variations result from the mathematical difficulty of representing the peculiarities of the dynamic systems and the lack of knowledge about the physical properties of the materials used in a given application. In this context, the analysis of uncertainties that affect the performance of the system is an important design issue. Uncertainty analysis of dynamic systems has been studied by applying techniques based on stochastic and fuzzy logic approaches. The fuzzy logic technique seems to be more appropriate when the stochastic process that characterizes the uncertainties is unknown. Therefore, in the present contribution, the inherent uncertainties affecting the performance of a horizontal rotating machine are modeled by using both stochastic and fuzzy logic-based analysis. These methodologies have been compared through numerical simulations in terms of the dynamic behavior of the system as represented by rotor orbits, unbalance responses, and frequency response functions. The proposed uncertainty analysis provides additional information that can be useful for design and maintenance purposes.

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

Numerical characterization under uncertainties of a piston expander for exhaust heat recovery on heavy commercial vehicles

While nearly 30 percent of the fuel energy is lost as waste heat in the form of hot exhaust gases, exhaust heat recovery promises one of the biggest fuel economy potential regarding the technologies available in the next decade. Applied to heavy commercial vehicles (HCVs), buses or off road vehicles, a bottoming Rankine Cycle (RC) on exhaust heat shows a great potential in recovering the exhaust gases energy, even for part loads. The objective of this paper is to illustrates the interest in assessing the uncertainty of this kind of systems for getting a robust prediction of the associated numerical model. In particular, the focus here is on the simulation of a piston expander for exhaust heat recovery.Uncertainties associated to the experimental measurements are propagated through the numerical code by means of uncertainty quantification techniques. Several sources of uncertainties are taken into account at the same time, thus yielding various indications concerning the most predominant parameters, and their influence on several quantities of interest, such as the mechanical power, the mass flow and the exhaust temperature.

Computation of higher‐order moments of generalized polynomial chaos expansions

SummaryBecause of the complexity of fluid flow solvers, non‐intrusive uncertainty quantification techniques have been developed in aerodynamic simulations in order to compute the quantities of interest required in an optimization process, for example. The objective function is commonly expressed in terms of moments of these quantities, such as the mean, standard deviation, or even higher‐order moments. Polynomial surrogate models based on polynomial chaos expansions have often been implemented in this respect. The original approach of uncertainty quantification using polynomial chaos is however intrusive. It is based on a Galerkin‐type formulation of the model equations to derive the governing equations for the polynomial expansion coefficients. Third‐order, indeed fourth‐order moments of the polynomials are needed in this analysis. Besides, both intrusive and non‐intrusive approaches call for their computation provided that higher‐order moments of the quantities of interest need to be post‐processed. In most applications, they are evaluated by Gauss quadratures and eventually stored for use throughout the computations. In this paper, analytical formulas are rather considered for the moments of the continuous polynomials of the Askey scheme, so that they can be evaluated by quadrature‐free procedures instead. Matlab© codes have been developed for this purpose and tested by comparisons with Gauss quadratures. Copyright © 2017 John Wiley & Sons, Ltd.

On the Accuracy of Polynomial Models in Stochastic Computational Electromagnetics Simulations Involving Dielectric Uncertainties

Uncertainties inherent in the measured electromagnetic properties of most materials give rise to statistical uncertainty in field values obtained in computational electromagnetics (CEM) simulations. A number of recently proposed techniques for quantifying uncertainty in CEM simulations use polynomial expansions to approximate these stochastic field variations. We investigate the accuracy of polynomial approximations using three canonical scattering scenarios with dielectric uncertainties. Analytical expressions for field values are used to assess the accuracy of the standard deviation of the field quantities derived from the polynomial approximations. The accuracy of the polynomial approximations is shown to be strongly dependent on the mean dielectric properties of the scatterers. Extremely high-order expansions are necessary to achieve accurate approximations in some instances. The computational cost of high-order polynomial approximations limits the utility of polynomial basis function expansions in uncertainty quantification techniques for CEM.

The Effect of Grid Resolution and Reaction Models in Simulation of a Fluidized Bed Gasifier Through Nonintrusive Uncertainty Quantification Techniques

To improve quality of numerical models used in simulations of a fluidized bed gasifier at any scale, the sources of uncertainty in the simulation have to be identified and quantified. There are several sources of uncertainty that can affect any simulation result and scale up process such as uncertainty in the model input values, uncertainty in the reaction models and kinetic rates, uncertainty in selection of the appropriate numerical models affecting the hydrodynamics, uncertainty in selection of adequate computational grid resolution (uncertainty due to discretization error), uncertainty in the selection of proper numerical techniques required for solution of the discretized conservation equations, and many more. The current study aims to investigate the effect that reaction models for gasification, char oxidation, carbon monoxide oxidation, and water gas shift will have on the syngas composition at different grid resolution, along with bed temperature, which affects the reactions. The global sensitivity analysis conducted showed that among various reaction models employed for water gas shift, gasification, char oxidation, the choice of reaction model for water gas shift has the greatest influence on syngas composition, with gasification reaction model being second. Syngas composition also shows a small sensitivity to temperature of the bed. The hydrodynamic behavior of the bed did not change beyond grid spacing of 18 times the particle diameter. However, the syngas concentration continued to be affected by the grid resolution as low as 9 times the particle diameter. This is due to a better resolution of the phasic interface between the gas and solid that leads to stronger heterogeneous reactions.

Uncertainty quantification in prediction of material properties during additive manufacturing

Based on our experience gained from uncertainty quantification (UQ) of traditional manufacturing, this paper discusses UQ for additive manufacturing with a focus on the prediction of material properties. Applications of UQ methods in traditional manufacturing are briefly summarized first. Based on that, we investigate how the state of the art UQ techniques can be applied to AM process to quantify the uncertainty in the material properties due to various sources of uncertainty. The UQ of ultimate tensile strength of a structure obtained from laser sintering of nanoparticles is used as an example to illustrate the proposed UQ framework.

Application of an Adaptive Polynomial Chaos Expansion on Computationally Expensive Three-Dimensional Cardiovascular Models for Uncertainty Quantification and Sensitivity Analysis.

When applying models to patient-specific situations, the impact of model input uncertainty on the model output uncertainty has to be assessed. Proper uncertainty quantification (UQ) and sensitivity analysis (SA) techniques are indispensable for this purpose. An efficient approach for UQ and SA is the generalized polynomial chaos expansion (gPCE) method, where model response is expanded into a finite series of polynomials that depend on the model input (i.e., a meta-model). However, because of the intrinsic high computational cost of three-dimensional (3D) cardiovascular models, performing the number of model evaluations required for the gPCE is often computationally prohibitively expensive. Recently, Blatman and Sudret (2010, "An Adaptive Algorithm to Build Up Sparse Polynomial Chaos Expansions for Stochastic Finite Element Analysis," Probab. Eng. Mech., 25(2), pp. 183-197) introduced the adaptive sparse gPCE (agPCE) in the field of structural engineering. This approach reduces the computational cost with respect to the gPCE, by only including polynomials that significantly increase the meta-model's quality. In this study, we demonstrate the agPCE by applying it to a 3D abdominal aortic aneurysm (AAA) wall mechanics model and a 3D model of flow through an arteriovenous fistula (AVF). The agPCE method was indeed able to perform UQ and SA at a significantly lower computational cost than the gPCE, while still retaining accurate results. Cost reductions ranged between 70-80% and 50-90% for the AAA and AVF model, respectively.

On the accuracy of RANS simulations with DNS data.

Simulation results conducted for incompressible planar wall-bounded turbulent flows with the Reynolds-Averaged Navier-Stokes (RANS) equations with no modeling involved are presented. Instead, all terms but the molecular diffusion are represented by the data from direct numerical simulation (DNS). In simulations, the transport equations for velocity moments through the second order (and the fourth order where the data are available) are solved in a zero-pressure gradient boundary layer over a flat plate and in a fully-developed channel flow in a wide range of Reynolds numbers using DNS data from Sillero et al. (2013), Lee & Moser (2015), and Jeyapaul et al. (2015). The results obtained demonstrate that DNS data are the significant and dominant source of uncertainty in such simulations (hereafter, RANS-DNS simulations). Effects of the Reynolds number, flow geometry, and the velocity moment order as well as an uncertainty quantification technique used to collect the DNS data on the results of RANS-DNS simulations are analyzed. New criteria for uncertainty quantification in statistical data collected from DNS are proposed to guarantee the data accuracy sufficient for their use in RANS equations and for the turbulence model validation.

Special Issue on Uncertainties in Tsunami Effects

Special Issue on Uncertainties in Tsunami Effects

Analysis of the performance of a H-Darrieus rotor under uncertainty using Polynomial Chaos Expansion

Due to the growing importance of wind energy, improving the efficiency of energy conversion is essential. Horizontal Axis Wind Turbines are the most well-spread, but H-Darrieus turbines are becoming popular as well due to their simple design and easier integration. Due to the high efficiency of existing wind turbines, further improvements require numerical optimization. One important aspect is to find a better configuration that is also robust, i.e., a configuration that retains its performance under uncertainties. For this purpose, forward uncertainty propagation has to be applied. In the present work, an Uncertainty Quantification (UQ) method, Polynomial Chaos Expansion, is applied to transient, turbulent flow simulations of a variable-speed H-Darrieus turbine, taking into account uncertainty in the preset pitch angle and in the angular velocity. The resulting uncertainty of the performance coefficient and of the quasi-periodic torque curve are quantified. In the presence of stall the instantaneous torque coefficients tend to show asymmetric distributions, meaning that error bars cannot be correctly reconstructed using only mean value and standard deviation. The expected performance was always found to be smaller than in computations without UQ techniques, corresponding to up to 10% of relative losses for λ = 2.5.

On Uncertainty Quantification of Thermal Aspects of Lithium-Ion Battery

Thermal management of lithium-ion battery pack in hybrid electric vehicles (HEVs) and electric vehicles (EVs) have become a research focus in recent years because performance, safety and life time of battery have significant dependency on temperature distribution throughout the cell. As the first step in designing a reliable thermal strategy, heat generation of battery must be computed accurately. But the most effective physical properties involved in heat source may not be exactly known, possibly because of intrinsic variability which cannot be measured directly in practice. In any simulation process, some uncertainties appear whose understanding and quantification is critical to assess the differences between the numerical predictions and actual system behavior. For this purpose, these uncertainties must be quantified and propagated through energy equation and heat flux to roll up the uncertainties of interest in order to determine range of temperature distribution, and consequently improve the safety and increase the life-cycle of lithium-ion battery. Commonly, the uncertain model parameters are represented by random variables/processes which are known as probabilistic techniques. In the present study, stochastic spectral methods based on polynomial chaos (PC) expansions is employed due to their advantages over traditional uncertainty quantification (UQ) techniques, i.e., perturbation-based and Monte Carlo sampling (MCS) methods. In PC-based method, the coefficients of the solution expansion are computed intrusively or non-intrusively. Non-intrusive methods depend on individual realizations to determine the stochastic model response to random inputs and consume less time in comparison with intrusive methods. Since governing equations of lithium-ion battery are very complex, non-intrusive method is chosen in the present study. Numerical simulation results are verified by discharge process at different C-rates which show good agreement with previous studies. Moreover, PC-based model is implemented to study the effects of parametric model uncertainties on the heat generation of cell and temperature distribution. The obtained numerical results can be used to design more efficient thermal management strategy. Moreover, the numerical results show that the proposed UQ method can accurately compute the variability in the output quantity of interest such as the heat generation and temperature distribution within a wide range of heat convection coefficients for lithium-ion battery.