Optimal uncertainty quantification with model uncertainty and legacy data

We present an optimal uncertainty quantification (OUQ) protocol for systems that are characterized by an existing physics-based model and for which only legacy data is available, i.e., no additional experimental testing of the system is possible. Specifically, the OUQ strategy developed in this work consists of using the legacy data to establish, in a probabilistic sense, the level of error of the model, or modeling error, and to subsequently use the validated model as a basis for the determination of probabilities of outcomes. The quantification of modeling uncertainty specifically establishes, to a specified confidence, the probability that the actual response of the system lies within a certain distance of the model. Once the extent of model uncertainty has been established in this manner, the model can be conveniently used to stand in for the actual or empirical response of the system in order to compute probabilities of outcomes. To this end, we resort to the OUQ reduction theorem of Owhadi et al. (2013) in order to reduce the computation of optimal upper and lower bounds on probabilities of outcomes to a finite-dimensional optimization problem. We illustrate the resulting UQ protocol by means of an application concerned with the response to hypervelocity impact of 6061-T6 Aluminum plates by Nylon 6/6 impactors at impact velocities in the range of 5–7 km/s. The ability of the legacy OUQ protocol to process diverse information on the system and its ability to supply rigorous bounds on system performance under realistic—and less than ideal—scenarios demonstrated by the hypervelocity impact application is remarkable.     

PIV-based pressure fluctuations in the turbulent boundary layer

The unsteady pressure field is obtained from time-resolved tomographic particle image velocimetry (Tomo-PIV) measurement within a fully developed turbulent boundary layer at free stream velocity of U _???=?9.3?m/s and Re_???=?2,400. The pressure field is evaluated from the velocity fields measured by Tomo-PIV at 10?kHz invoking the momentum equation for unsteady incompressible flows. The spatial integration of the pressure gradient is conducted by solving the Poisson pressure equation with fixed boundary conditions at the outer edge of the boundary layer. The PIV-based evaluation of the pressure field is validated against simultaneous surface pressure measurement using calibrated condenser microphones mounted behind a pinhole orifice. The comparison shows agreement between the two pressure signals obtained from the Tomo-PIV and the microphones with a cross-correlation coefficient of 0.6 while their power spectral densities (PSD) overlap up to 3?kHz. The impact of several parameters governing the pressure evaluation from the PIV data is evaluated. The use of the Tomo-PIV system with the application of three-dimensional momentum equation shows higher accuracy compared to the planar version of the technique. The results show that the evaluation of the wall pressure can be conducted using a domain as small as half the boundary layer thickness (0.5??₉₉) in both the streamwise and the wall normal directions. The combination of a correlation sliding-average technique, the Lagrangian approach to the evaluation of the material derivative and the planar integration of the Poisson pressure equation results in the best agreement with the pressure measurement of the surface microphones.     

PIV-based investigations of animal flight

An overview is presented of the principles of estimation of fluid forces exerted upon solid bodies, based upon whole-field velocity measurements such as provided by PIV. The focus will be on the range of length and velocity scales characterised by the flight of large insects, birds, bats and small unmanned air vehicles, so that while viscous terms in the Navier–Stokes equations can many times be ignored in the quantitative analysis, understanding and measuring boundary-layer flows, separation and instability will ultimately be critical to predicting and controlling the fluid motions. When properly applied, PIV methods can make accurate estimates of time-averaged and unsteady forces, although even ostensibly simple cases with uncomplicated geometries can prove challenging in detail. Most PIV-based force estimates are embedded in some analytical model of the fluid–structure interaction, and examples of these with varying degrees of complexity are given. In any event, the performance and accuracy of the PIV method in use must be well understood as part of both the overall uncertainty analysis and the initial experimental design.     

薄壁圆筒壳初始几何缺陷不确定性量化的极大熵方法

针对薄壁圆筒壳结构轴压屈曲载荷的缺陷敏感性以及真实几何缺陷的不确定性,提出一种基于实测缺陷数据和极大熵原理的初始缺陷建模与屈曲载荷预测方法。首先,将初始几何缺陷视为二维随机场,并利用实测缺陷数据和Karhunen-Loève展开法将初始缺陷的随机场建模转化为随机向量的建模;其次,利用极大熵方法确定随机向量的概率分布;最后,基于所构建的初始缺陷随机模型,利用MCMC抽样方法和确定性屈曲分析方法,进行随机屈曲分析并给出基于可靠度的屈曲载荷折减因子。数值算例表明,与直接假设随机场相关结构的方法相比,本文方法的结果是对薄壁圆筒壳屈曲载荷的一个更无偏估计。     

Comparison of stochastic and interval methods for uncertainty quantification of metal forming processes

Various sources of uncertainty can arise in metal forming processes, or their numerical simulation, or both, such as uncertainty in material behavior, process conditions, and geometry. Methods from the domain of uncertainty quantification can help assess the impact of such uncertainty on metal forming processes and their numerical simulation, and they can thus help improve robustness and predictive accuracy. In this paper, we compare stochastic methods and interval methods, two classes of methods receiving broad attention in the domain of uncertainty quantification, through their application to a numerical simulation of a sheet metal forming process.     

An Anisotropic A posteriori Error Estimator for CFD

In this article, a robust anisotropic adaptive algorithm is presented, to solve compressible-flow equations using a stabilized CFD solver and automatic mesh generators. The association includes a mesh generator, a flow solver, and an a posteriori error-estimator code. The estimator was selected among several choices available (Almeida et al. (2000). Comput. Methods Appl. Mech. Engng , 182 , 379-400; Borges et al. (1998). "Computational mechanics: new trends and applications". Proceedings of the 4th World Congress on Computational Mechanics , Bs.As., Argentina) giving a powerful computational tool. The main aim is to capture solution discontinuities, in this case, shocks, using the least amount of computational resources, i.e. elements, compatible with a solution of good quality. This leads to high aspect-ratio elements (stretching). To achieve this, a directional error estimator was specifically selected. The numerical results show good behavior of the error estimator, resulting in strongly-adapted meshes in few steps, typically three or four iterations, enough to capture shocks using a moderate and well-distributed amount of elements.     

Assessment of PIV-based analysis of water entry problems through synthetic numerical datasets

The phenomenon of hull-slamming, that is, the sudden impact of a solid body on the water surface, is critical in the design of naval structures. Thus, the development and validation of schemes to predict the slamming load and elucidate energy exchange during water entry are of fundamental importance in a wide range of engineering applications. Recent studies have demonstrated the possibility of using direct flow measurements from particle image velocimetry (PIV) to investigate the kinetics of water entry. Specifically, these efforts have contributed a first characterization of the hydrodynamic loading on impacting wedges and of the energy imparted to the water pile-up and the spray jets. Here, we seek to provide a thorough assessment of such a PIV-based approach through synthetic datasets, in which PIV parameters, such as the camera acquisition rate and the size of the interrogation area, are systematically varied, without experimental confounds. We implement a direct computational framework to study the two-dimensional flow physics generated during the water entry of a rigid wedge. Water and air are treated as immiscible phases and their relative motion is utilized to track the free surface dynamics. Our results show that the PIV-based methodology allows for an accurate reconstruction of the pressure field from the measured velocity field, except for early stages of the impact and for a small region close to the free surface. We also demonstrate that the reconstruction is only marginally affected by the spatial resolution, while a sufficiently high acquisition frequency is required to correctly predict the pressure field in the pile-up region. The proposed computational framework can also find application in the analysis of less studied aspects of water entry problems, such as cycling loading, flow transitions and separation, and formation of spray jets.     

A posteriori error bounds in boundary layer theory

The subject of this paper is the two-dimensional steady laminar boundary layer of an incompressible medium at a wall. For a given approximate solution of the Prandtl equation, error bounds are computed with the aid of parabolic inequalities. Also, bounds for the wall shear stress, for the displacement thickness, for the momentum loss thickness, and for the energy loss thickness are given. The bounds converge toward zero if the residual error and the initial error of the approximation both vanish.This research was sponsored by the United States Army under Contract No.: DA-31-124-ARO-D-462. This paper was written while the author was a member of the Mathematics Research Center, University of Wisconsin, Madison, Wisconsin, USA. It has appeared previously as MRC Technical Summary Report #1143, April 16, 1971. Some parts of it were presented at the Second International Conference on Numerical Methods in Fluid Dynamics at the University of California, Berkeley, California, 1970, and are published in the Proceedings of this symposium [6].     

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.     

Robust optimization and uncertainty quantification in the nonlinear mechanics of an elevator brake system

Meccanica - This paper deals with nonlinear mechanics of an elevator brake system subjected to uncertainties. A deterministic model that relates the braking force with uncertain parameters is...     

Transonic velocity fluctuations simulated using extremum diminishing uncertainty quantification based on inverse distance weighting

For reliable computational predictions of transonic flows, it is important to resolve the significant effects of physical variations on the shock wave locations. The resulting discontinuities in probability space require extremum diminishing uncertainty quantification to avoid overshoots and undershoots in the response surface approximation. In this paper, the extremum diminishing concept in probability space is extended to infinite parameter domains using inverse distance weighting interpolation of deterministic samples. Based on results for three analytical test functions, the combination of Halton sampling and power parameter limit value c → ∞ is selected. The approach is employed to model spatial-free-stream velocity fluctuations in the highly sensitive transonic AGARD 445.6 wing test case in an up to ten-dimensional probability space. The 0.5% input variations are amplified to a coefficient of variation for the wave drag of cv_D?=?9.58% in combination with an increase of the mean drag by 1.75% compared to the deterministic value.     

Rapid uncertainty quantification for non-linear and stochastic wind excited structures: a metamodeling approach

Meccanica - The application of performance-based design (PBD) requires the modeling of the dynamic response of the system beyond the elastic limit. If probabilistic PBD is considered, this implies...     

Nonlinear constitutive force model selection,update and uncertainty quantification for periodically sequential impact applications

Nonlinear Dynamics - Contact–impact events frequently occur between solid contact interfaces in complex dynamic mechanical systems and engineering applications. Impact force models are used...     

混合不确定度量化方法及其在计算流体动力学迎风格式中的应用

针对流体力学数值求解间断问题时,初始状态含有偶然和认知混合型的不确定性,将认知不确定度作为外层,偶然不确定度作为内层,分别使用非嵌入多项式混沌方法(non-intrusive polynomial chaos,NIPC)和概率盒(P-box)理论处理偶然不确定度和认知不确定度,发展了流体力学数值求解过程中,初始状态含有混合不确定度传播量化的一种方法。以迎风格式和黎曼解法器求解Sod问题为例,评估了左状态密度(偶然不确定度)和理想气体多方指数(认知不确定度)对模型输出结果的影响,有效验证了该方法的可行性。     

A posteriori error estimate of the DSD method for first-order hyperbolic equations

ＩｎｔｒｏｄｕｃｔｉｏｎＩｔ’ｓｋｎｏｗｎｔｈａｔｉｎｎｕｍｅｒｉｃａｌａｐｐｒｏｘｉｍａｔｉｏｎｏｆｆｉｒｓｔ＿ｏｒｄｅｒｈｙｐｅｒｂｏｌｉｃｅｑｕａｔｉｏｎｓ,ｔｈｅｕｓｅｏｆａｄａｐｔｉｖｅｆｉｎｉｔｅｅｌｅｍｅｎｔｍｅｔｈｏｄｓ (ｓｅｅ [1 ] )ｈａｓｂｅｅｎｅｘｐａｎｄｅｄｔｏｍａｎｙｆｉｅｌｄｓｓｕｃｈａｓｃｏｍｐｕｔａｔｉｏｎａｌｆｌｏｗｍｅｃｈａｎｉｃｓ,ｔｈｅｒｍａｌａｎａｌｙｓｅｓ,ｅｌｅｃｔｒｉｃａｌｅｎｇｉｎｅｅｒｉｎｇ ,ｅｔｃ.Ｔｈｅｈ＿ｖｅｒｓｉｏｎａｄａｐｔｉｖｅｆｉｎｉ…     

Real-time Bayesian data assimilation with data selection,correction of model bias,and on-the-fly uncertainty propagation

The work introduces new advanced numerical tools for data assimilation in structural mechanics. Considering the general Bayesian inference context, the proposed approach performs real-time and robust sequential updating of selected parameters of a numerical model from noisy measurements, so that accurate predictions on outputs of interest can be made from the numerical simulator. The approach leans on the joint use of Transport Map sampling and PGD model reduction into the Bayesian framework. In addition, a procedure for the dynamical and data-based correction of model bias during the sequential Bayesian inference is set up, and a procedure based on sensitivity analysis is proposed for the selection of the most relevant data among a large set of data, as encountered for instance with full-field measurements coming from digital image/volume correlation (DIC/DVC) technologies. The performance of the overall numerical strategy is illustrated on a specific example addressing structural integrity on damageable concrete structures, and dealing with the prediction of crack propagation from a damage model and DIC experimental data.     

Optimal solution error quantification in variational data assimilation involving imperfect models

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition. If the model is ‘perfect,’ the optimal solution (analysis) error rises because of the presence of the input data errors (background and observation errors). Then, this error is quantified by the covariance matrix, which can be approximated by the inverse Hessian of an auxiliary control problem. If the model is not perfect, the optimal solution error includes an additional component because of the presence of the model error. In this paper, we study the influence of the model error on the optimal solution error covariance, considering strong and weak constraint data assimilation approaches. For the latter, an additional equation describing the model error dynamics is involved. Numerical experiments for the 1D Burgers equation illustrate the presented theory. Copyright © 2016 John Wiley & Sons, Ltd.