Abstract Significant axial vibration, loopy orbits, and multiple harmonics with a principal 2X component are the primary methods used to identify misalignment in rotating systems. In this study, a novel method for the modeling and identification of misalignments in a coupled shaft-disk system represented by two nodes of Timoshenko beam finite elements is developed. A typical coupled turbogenerator system comprises two individual rotors interconnected using a coupling and held up by two squeeze film dampers (SFDs). An interaction between both the generator and the turbine rotor systems is modelled mathematically using a coupling model that includes a stiffness matrix with two elements: perpetual coupling stiffness (PCS) and time-dependent coupling stiffness (TCS). The effect of misalignment forcing characteristics is simulated with a suitable direct forcing function for full spectrum fast Fourier transformation (FFT). Active magnetic bearings (AMBs) are auxiliary bearings used in each rotor for vibration reduction and fault detection. The development of the dynamic model is carried further through an identification procedure built using the least-squares fitting approach of inverse analysis. This paper represents the intensity of TCS magnitude over the determination of the system angular misalignment. AMB constants, coupling stiffness coefficients, bearing dynamic parameters, and unbalances are all determined using the least-squares linear estimation approach. Testing has been done on the identification algorithm's sensitivity to skew errors in modeling parameters and signal noise. This study is unique in that it employs the TCS coefficient matrix, which provides precise indications of both the degree and type of misalignment, to characterize and diagnose misalignment.
Image-based computational hemodynamics (ICH) employs medical imaging data to model and simulate patient-specific blood flow. Uncertainties arising from image noise, segmentation inaccuracies, and boundary condition modeling can significantly affect simulation outcomes. This study uses a case study to demonstrate the impact of image segmentation uncertainty and outlet boundary condition variability in ICH of blood flow within a human iliac arterial system reconstructed from CT angiography. To address challenges from high dimensionality and limited image data, the Uncertainty Separation Method is applied to decompose the simulation model into image segmentation and numerical submodels, enabling uncertainty estimations at the average 3D shape and mean numerical inputs. An alignment method is introduced to compute the average 3D anatomy from multiple segmented samples. Results show that this alignment method is essential for statistical analysis of image-based vascular shapes, and that image uncertainty, especially with limited samples, strongly influences the simulation outcomes.
This study applies a version of the predictive capability maturity model adapted for use with scientific machine learning (the SciML-adapted PCMM) to evaluate the credibility of a deep neural network (DNN) surrogate model used to predict aerodynamic coefficients for the NACA 0012 airfoil. The surrogate model is trained on Reynolds-averaged Navier-Stokes (RANS) simulation data across a Latin hypercube sampling (LHS) of Reynolds number, Mach number, and angle of attack. Using the SciML-adapted PCMM framework, we perform a comprehensive credibility assessment across five elements: data representation, domain awareness, explainability, model validation, and uncertainty quantification. The analysis incorporates Shapley additive explanations (SHAP)-based interpretability metrics, comparative validation against experimental data, and quantified numerical and surrogate model run-to-run uncertainties. Credibility levels are assigned based on rigor and maturity of each element. Areas of relative strength and credibility gaps are identified and related to the credibility level. This work demonstrates a path forward in bridging theoretical credibility models with actionable assessment protocols for SciML applications in computational fluid dynamics (CFD), with implications for broader adoption in safety-critical engineering domains.
Uncertainty quantification (UQ) methods are applied to a known equilibrium diffusion thermal radiation wave solution, the Marshak wave, to estimate statistical properties of some diagnostics, the wave breakout time and material temperature profile, in the presence of uncertain material properties and experimental parameters. It is shown that polynomial chaos expansions (PCE) are well suited for representing the analytic breakout time function and moments of the temperature profile, but are a poor choice for calculating percentile values of the temperature profile, which possess sharp wavefronts. The results presented are intended to benchmark uncertainty quantifying radiative transfer codes and provide qualitative insight into the interpretation of experimental results with measurement errors as well as to present an illustrative example of the limitations of polynomial chaos expansions on functions with steep gradients.
Computational modeling and simulation (CM&S) is increasingly used to complement or replace traditional bench testing in the design, evaluation, and regulatory assessment of orthopedic devices. For orthopedic bone screws, pullout performance is commonly assessed through ASTM F543 testing, but this process can be costly and time-consuming. Finite element (FE) models, when subjected to rigorous verification, validation, and uncertainty quantification, can serve as surrogates to reduce reliance on experimental testing while supporting regulatory submissions. This study presents an end-to-end methodology for establishing the credibility of a device-agnostic FE model simulating screw pullout in accordance with ASTM F543. The model replicates the mechanical interaction between a bone screw and synthetic bone foam, with the quantity of interest being the maximum pullout force. Its context of use (CoU) is defined as a full surrogate for physical testing, and its associated use risk was determined to be medium-high, based on its role in device safety and performance assessment. Credibility activities were defined using the ASME V&V40 framework and the 2023 Food and Drug Administration (FDA) CM&S credibility guidance. Twenty-five credibility factors were reviewed, and high ratings were achieved for 19, reflecting a high degree of rigor across verification, calibration, validation, and applicability evidence. This work illustrates how a risk-informed credibility framework can be applied to establish a validated computational model of a standardized test in the medical device field, supporting its use as a surrogate for bench testing in both development and regulatory contexts.
In internal combustion engines, access to calibrated physical models of engine subsystems is crucial for building predictive models of engine emissions. Specifically, understanding the temperature and pressure dynamics of the intake manifold (IM) can help characterize the behavior of the cylinder contents, which largely impact the performance. Rather than working with a complex and multidimensional model, we can employ a mean value model (MVM) to accurately capture the IM states. However, in situations where heat transfer affects cannot be ignored, the standard MVM contains unknown physical parameters representing various material properties. This also causes the dynamics to appear in a nonlinear way. When coupled with noisy measurement data, a Bayesian approach is often necessary to reconstruct both the IM states and physical parameters. One would typically apply a variant of the Kalman filter suitable for nonlinear dynamics such as particle filters. For real engine datasets, this is computationally unfeasible, as there are often hundreds of thousands of data points sampled in a matter of minutes. In this work, we demonstrate how information field theory (IFT) can be applied to solve the IM filtering and calibration problem. IFT is a scientific machine learning framework for simultaneous Bayesian calibration of dynamic states and physical parameters. We demonstrate the method across different datasets with varying degrees of fidelity collected from real engines. By informing the IFT before the MVM, the model predictions remain robust even with limited data. Finally, the IFT posterior also quantifies the uncertainty about the IM material properties.
The assessment of the modeling error of computational simulations (validation) requires the comparison of quantities of interest obtained from simulations and experiments, which are supposed to be obtained for similar conditions (geometry, boundary conditions, material properties, heat transfer coefficients, etc.). It is not unusual to find validation exercises assuming two-dimensional geometries. Such assumption implies that it is not possible to strictly satisfy the similarity of experimental and simulations conditions. In the recent AVT-349 project, a two-dimensional validation exercise was proposed for boundary-layers developing on the Virginia Tech wind-tunnel wall with pressure gradients imposed by a rectangular wing with a NACA 0012 airfoil section in the middle of the tunnel at angles of attack between - 10 deg and 12 deg . Modifications of the wind-tunnel geometry and boundary conditions were proposed for the simulations to take into account the effect of the displacement thickness of the "side walls" boundary-layers. Furthermore, no information was available to specify inflow and outflow boundary conditions, and so the length of the domain for the simulations is extended to allow the use of simplified boundary conditions. In this paper, we use the Virginia Tech test case for an angle of attack of - 10 deg to illustrate the effects of changing geometry and boundary conditions in two-dimensional validation exercises. The mathematical/computational models are the Reynolds-averaged Navier-Stokes (RANS) equations using four different turbulence models (three eddy-viscosity models and a Reynolds-stress model). Sets of geometrically similar grids are used to allow the estimation of the quantities of interest that include the pressure and friction coefficients, displacement thickness, momentum thickness, and mean velocity profiles of the boundary-layer on the bottom wall of the wind-tunnel. Input uncertainties generated by the approximate boundary conditions used in the simulations are estimated using sensitivity coefficients. We also address the consequences of the postprocessing techniques used to determine the quantities of interest, especially the shear-stress at the wall, displacement thickness, and momentum thickness of the bottom wall boundary-layer. The ASME V&V20 point-wise validation metric and a multivariate metric, which take into account experimental, numerical, and input uncertainties, are used to quantify the modeling error of the selected mathematical/computational models.
The method of nearby problems (MNP) is a solution verification technique that does not require the use of multiple spatial grids. To estimate spatial discretization error without requiring a high-fidelity spatial grid, an analytical curve fit is interpolated from the numerical solution. The residual between the curve fit solution and numerical solution is calculated and added as an additional source term to the governing equation. The nearby solution is estimated using the updated source term and boundary conditions to remain consistent with the curve fit interpolation. The nearby solution can be compared to the curve fit solution as a discretization error estimation while using a single spatial grid. Without the use of higher fidelity spatial grids, the MNP is able to approximate the spatial discretization error, a facet of solution verification. The application of the method of nearby problems is presented for one- and two-dimensional neutron transport problems for both fixed source and criticality problems on the spatial variable. The fixed source results demonstrate the effectiveness of nearby problems for spatial error identification using the discrete ordinates method. Criticality results are shown to identify area of high spatial error for the C5G7 problem as well as for the discrete ordinates solver. A novel approach of combining the capabilities of Monte Carlo with the discrete ordinates nearby problems is presented for one- and two-dimensional fixed source problems. The MNP demonstrates its effectiveness at identifying spatial error on a single structured grid with a wide variety of neutron transport problems.
Hypersonic aerodynamics models exist across a range of physics fidelities with associated computational expenses. These models may be run independently or in a multifidelity framework that leverages their complementary strengths of speed for lower-fidelity and accuracy for higher-fidelity models. This work presents applied code verification of two lower-fidelity models contained within the Sandia hypersonic aerodynamics code. Each model has a different form that requires individualized verification approaches, including comparison to analytical solutions as well as manufactured solutions with order-of-accuracy testing. Results of this effort include the identification and resolution of code errors and shortcomings, as well as the demonstration of code correctness and consistency for both models.
Regression is a fundamental problem in statistics and machine learning, where the true output is a continuous and stochastic function of the input. Regression methods model the output variables using training datasets composed of inputoutput pairs derived from real-world scenarios. Inputs are broadly categorized as known or unknown, where unknown inputs are often represented as additive noise. Most existing models rely on two key assumptions: Gaussian residuals and homoscedasticity. While foundational, these assumptions often limit the applicability of traditional machine learning methods. To address these limitations, this study employs deep neural networks, leveraging the Universal Approximation Theorem, which ensures their ability to approximate any continuous function given sufficient hidden units. Conventional neural network- based regression methods typically use MSE or MAE as their objective functions, adhering to Gaussian or uniform residual assumptions. However, such methods remain constrained by homoscedasticity and fixed residual probability density functions. Inspired by classification tasks, this study proposes an innovative approach that discretizes continuous regression outputs into bins and models the probability of each bin. The framework utilizes Softmax as the activation function in the output layer and cross-entropy as the objective function, akin to multi-class classification. A comparative analysis against linear regression, Gaussian processes, and conventional MSEbased DNNs is conducted on synthetic datasets with varying complexity and combined cycle power plant dataset. Results demonstrate that the proposed Softmax-based DNN outperforms aforementioned regression methods by effectively estimating conditional probability density functions without having to assume Gaussian or homoscedasticity residuals. The Softmaxbased DNNs pave the way for a new class of regression models that can more effectively handle the complexities of real-world data.
Thermal batteries are crucial for supplying power to high-consequence engineering applications such as rockets. Computational simulations have been developed to predict thermal battery behavior, but these simulations often suffer from modeling errors, including model form uncertainty. Addressing this uncertainty can be achieved by quantifying either the model discrepancy in the output or the model form error (MFE) in the governing equation. MFE is particularly valuable as it can be better extrapolated beyond observed outputs, which is essential for predictions involving changes in external system loading, system configuration and geometry, or output quantities. This paper employs a state estimation approach to estimate MFE using experimental data and then utilizes machine learning (ML) to model its relationship with state variables. A nonintrusive technique is used to estimate MFE in a black-box thermal battery heat transfer simulation. The trained machine learning model for MFE is then applied to correct simulation predictions under extrapolated initial conditions and battery configurations. The methodology's performance is evaluated using additional experimental data, demonstrating its effectiveness in improving prediction accuracy.
A mixed least-squares finite element model with spectral/hp approximations was developed for a three-dimensional non-isothermal flows of a generalized Newtonian fluid, which obeys the Power-law constitutive model. The finite element model consists of velocity, pressure, viscous stress, temperature, and heat flux as the field variables. A least-squares formulation provides a variational framework for the Navier--Stokes equations and does not require compatibility of the approximation spaces for field variables. Also, using spectral/$hp$ elements in conjunction with the least-squares formulation, various forms of locking can be avoided, which often appear in low-order least-squares finite element models for incompressible viscous flows; thus, accurate results are obtained with exponential convergence in least-squares functionals. The method of manufactured solutions is used to verify the present code.
A history is described on roll decay analysis for experiments with surface-ship scale models at the Naval Warfare Center Carderock Division (NSWCCD), a naval hydrodynamic facility known as David Taylor Model Basin (DTMB). The earliest roll decay analysis for a model test is from a report in 1923 that compares roll decay analysis for bilge keels off and on. A time history plot provides a measurement of roll period. The roll damping is indicated graphically by a curve fit of the peaks. With modern methods, the damping coefficient is computed with a curve fit of exponential damping. An early example of the estimate of roll damping is by log decrement of the ratio of successive roll peak pairs in 1976. More recently, both damping coefficient and period are computed from a curve fit of exponentially decaying cosine function, which is the solution of a second-order ordinary differential equation with constant coefficients. The largest uncertainty in damping coefficient is by log decrement, and lowest by the exponential cosine with the exponential fit of peaks in between. For the log decrement method, roll period must be computed independently. The roll period is calculated from the time between zero crossings in the time series, the time between peaks, or the peak in the power spectrum of the roll angle.
Vehicle models have a long history of research and, as of today, are able to model the involved physics in a reasonable manner. However, each new vehicle has its own unique characteristics or parameters. Identifying these is the main task of an engineer. Validating whether the correct parameter set has been chosen is a tedious task and often can only be performed by experts. Metrics commonly used in literature can compare different results under certain aspects. However, they fail to answer the question: Are the models accurate enough? In this article, we propose the usage of a custom metric trained on expert knowledge to tackle this problem. Our approach involves three main steps: first, the formalized collection of subject matter experts' opinions on the question: Having seen the measurement and simulation time-series in comparison, is the model quality sufficient? From this step, we obtain a dataset that quantifies the sufficiency of a simulation result based on a comparison to corresponding experimental data. In the second step, we compute common model metrics on the measurement and simulation time-series and use these model metrics as features in a regression model. Third, we fit a regression model to the experts' opinions. This regression model, i.e., our custom metric, can then predict the sufficiency of a new simulation result and provide a confidence level on this prediction.
Much of the effort in improving the accuracy of computational fluid dynamics (CFD) simulations is focused on mesh refinement and adaptation although studies have shown that the use of high-order methods are more efficient in improving accuracy. Stability issues, complexity of implementation, and demand of computational resources are some of the key factors hindering the use of high-order methods in commercial CFD solvers. This paper demonstrates an improvement in the order of accuracy of finite volume solutions on unstructured meshes without using a high-order solver. Defect correction and the error transport equation method are the techniques discussed, along with the method for obtaining an appropriate estimate of the truncation error, which is crucial in both these techniques. Methods to obtain high-order interpolation of the control volume averages and high-order integral functionals along curved boundaries are also discussed. Third-order accurate results are obtained for a variety of problems, including the 2D Euler equations, without using a third-order discretization scheme.
One of the major applications of the adjoint method is the improvement in the order of accuracy of integral quantities obtained from CFD simulations. Although the theory requires the use of a smooth interpolation of the solution, this has seldom been used with unstructured finite volume solvers. In this paper, the adjoint based correction is applied to output functionals obtained using finite volume method on unstructured meshes. A smoothing spline based on a C1 continuous representation of the discrete solution is employed to reduce the random noise in the solution and to improve the rate of convergence of the derivatives. Tests performed on randomly perturbed meshes in 1-D showed fourth-order convergence of output functionals obtained from second-order solution, corrected using the truncation error obtained using the smoothing spline and second-order accurate adjoint solution. The extension of this method to 2-D problems showed superconvergence for output functionals and improvements over existing results.
In the present paper, we focus on the simulation of viscous flows at high Reynolds numbers using the Reynolds-averaged Navier-Stokes (RANS) equations. Time-averaging is used to define the mean flow properties and only deterministic simulations are considered. Therefore, numerical errors are a consequence of round-off, iterative and discretization errors. In carefully performed simulations, round-off and iterative errors are reduced to negligible levels when compared to the discretization error and so the numerical error is dominated by the contribution of the discretization error. The use of grid refinement studies is one of the most flexible and popular techniques for the estimation of discretization errors for steady simulations. Several methods have been proposed in the open literature and most of them share common features. The discretization error of a quantity of interest is described as a function of the typical cell size by power series expansions. The estimation of the exact solution requires numerical solutions in more than one grid and so a family of (nearly) geometrical similar grids needs to be generated. The requirement of grid similarity is a consequence of the definition of the typical cell size. In the numerical solution of the RANS equations, the determination of the shear-stress at the wall tau(w) can be performed in two alternative ways: directly from its definition, or using wall functions. The grid refinement strategy required by each case is significantly different. In the first option, the near-wall cell must be systematically refined as all the remaining grid cells. When wall functions are used, the size of the near-wall cell size should remain fixed. In this paper, we present the consequences of using the wrong refinement strategy, i.e., by keeping the size of the near-wall cell fixed when tau(w) is calculated from its definition and by refining the near-wall cell when tau(w) is determined from wall functions. The selected test case is the flow over a flat plate at Reynolds numbers of 10(7) and 10(9). The results show that using the wrong grid refinement strategy can lead to misleading results that exhibit reasonable orders of grid convergence.
When stress concentration factors are not available in handbooks, finite element analysis has become the predominant method for determining their values. For such determinations, there is a need to know if they have sufficient accuracy. Tuned test problems can provide a way of assessing the accuracy of stress concentration factors found with finite elements. Previously, we offered a means of constructing such test problems for stress concentrations within boundaries that have local constant radii of curvature, and are subjected to tensile loading. Here we extend the means of construction for the same local geometry, but now with the stress concentrations being induced by shear loading. These new test problems are tuned to their originating shear applications by sharing the same global geometries and having slightly higher peak stresses. Consequently, they are a little more challenging to analyze with finite elements than their originating shear applications. These test problems also have exact solutions. Thus a precise determination can be made of the errors incurred in their finite element analysis. Given the increased challenge of these test problems, their errors are likely to serve as upper bounds on the errors incurred when their originating configurations are analyzed with the same set of finite element meshes.
