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.     

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.     

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...     

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.     

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...     

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

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A split Hopkinson pressure bar technique to determine compressive stress-strain data for rock materials

This paper presents a split Hopkinson pressure bar technique to obtain compressive stress-strain data for rock materials. This technique modifies the conventional split Hopkinson bar apparatus by placing a thin copper disk on the impact surface of the incident bar. When the striker bar impacts the copper disk, a nondispersive ramp pulse propagates in the incident bar and produces a nearly constant strain rate in a rock sample. Data from experiments with limestone show that the samples are in dynamic stress equilibrium and have constant strain rates over most of the test durations. In addition, the ramp pulse durations can be controlled such that samples are unloaded just prior to failure. Thus, intact samples that experience strains beyond the elastic region and postpeak stresses can be retrieved for microstructural evaluations. The paper also presents analytical models that predict the time durations for sample equilibrium and constant strain rate. Model predictions are in good agreement with measurements.     

On the pressure correction of capillary melt rheology data

In this paper, the importance of a pressure correction of viscosity data obtained in capillary melt rheology is demonstrated. A linear polycarbonate has been chosen as a highly pressure-sensitive material for which data obtained by rotational rheometry does not overlap with capillary data. This apparent problem with the Cox–Merz relation is attributed to the existence of a mean pressure inside the capillary which is significantly different from atmospheric conditions. Different methods to determine the pressure coefficient of polycarbonate have been evaluated based on experiments performed with a capillary rheometer equipped with a pressure chamber. It is demonstrated that the pressure coefficient obtained at constant shear stress and the pressure coefficient obtained by the superposition method represent accurate pressure coefficient values. Two approaches are proposed to correct the original capillary data. In the direct methodology, the pressure coefficient is used to rescale the mean pressure inside the capillary to atmospheric conditions. The indirect approach consists of first constructing a mastercurve at a certain reference pressure using capillary data obtained with a pressure chamber. The resulting mastercurve can then be rescaled to atmospheric conditions. It is shown that both methods lead to viscosity curves on which both rotational and capillary data overlap, hence confirming the Cox–Merz relationship for polycarbonate. The indirect method is proven to be advantageous since it opens the possibility to significantly extend the shear rate window in which viscosities can be measured.     

Towing tank PIV measurement system, data and uncertainty assessment for DTMB Model 5512

 A towed PIV system designed by DANTEC Measurement Technology for the Iowa Institute of Hydraulic Research towing tank is commissioned by measuring the mean velocity and Reynolds stresses at the nominal-wake plane of a model-scale ship. The mean velocities are compared with previous 5-hole pitot probe data. Uncertainty assessment following standard procedures is used to quantify the comparisons and reach conclusions regarding the quality of the data. The PIV results are analyzed with regard to axial velocity defects, axial vorticity, and level, pattern and anisotropy of turbulence. Quantitative comparisons with 5-hole pitot data shows that PIV uncertainties are about 1% lower than those for 5-hole pitot. However, data differences are larger than RSS of PIV and 5-hole pitot uncertainties for the axial and vertical velocity components, indicating unaccounted for bias and precision limits. The bias error of the pitot probe in a shear flow is discussed, and a method for reducing the bias error is suggested. The data is being used for verification and validation of RANS simulation of DTMB Model 5512. Received: 1 September 1999/Accepted: 20 March 2001