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All sourcesNASA NTRSDOE OSTI
DOE OSTI2016

DRSPALL: Impact of the Modification of the Numerical Spallings Model on Waste Isolation Pilot Plant Performance Assessment.

The numerical code DRSPALL (from direct release spallings) is written to calculate the volume of Waste Isolation Pilot Plant (WIPP) solid waste subject to material failure and transport to the surface as a result of a hypothetical future inadvertent drilling intrusion. An error in the implementation of the DRSPALL finite difference equations was discovered as documented in Software Problem Report (SPR) 13-001. The modifications to DRSPALL to correct the finite difference equations are detailed, and verification and validation testing has been completed for the modified DRSPALL code. The complementary cumulative distribution function (CCDF) of spallings releases obtained using the modified DRSPALL is higher compared to that found in previous WIPP performance assessment (PA) calculations. Compared to previous PAs, there was an increase in the number of vectors that result in a nonzero spallings volume, which generally translates to an increase in spallings releases. The overall mean CCDFs for total releases using the modified DRSPALL are virtually unchanged, thus the modification to DRSPALL did not impact WIPP PA calculation results.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
NASA NTRS2001

Numerical Study of Buoyancy and Different Diffusion Effects on the Structure and Dynamics of Triple Flames

Numerical simulations of 2-D triple flames under gravity force have been implemented to identify the effects of gravity on triple flame structure and propagation properties and to understand the mechanisms of instabilities resulting from both heat release and buoyancy effects. A wide range of gravity conditions, heat release, and mixing widths for a scalar mixing layer are computed for downward-propagating (in the same direction with the gravity vector) and upward-propagating (in the opposite direction of the gravity vector) triple flames. Results of numerical simulations show that gravity strongly affects the triple flame speed through its contribution to the overall flow field. A simple analytical model for the triple flame speed, which accounts for both buoyancy and heat release, is developed. Comparisons of the proposed model with the numerical results for a wide range of gravity, heat release and mixing width conditions, yield very good agreement. The analysis shows that under neutral diffusion, downward propagation reduces the triple flame speed, while upward propagation enhances it. For the former condition, a critical Froude number may be evaluated, which corresponds to a vanishing triple flame speed. Downward-propagating triple flames at relatively strong gravity effects have exhibited instabilities. These instabilities are generated without any artificial forcing of the flow. Instead disturbances are initiated by minute round-off errors in the numerical simulations, and subsequently amplified by instabilities. A linear stability analysis on mean profiles of stable triple flame configurations have been performed to identify the most amplified frequency in spatially developed flows. The eigenfunction equations obtained from the linearized disturbance equations are solved using the shooting method. The linear stability analysis yields reasonably good agreements with the observed frequencies of the unstable triple flames. The frequencies and amplitudes of disturbances increase with the magnitude of the gravity vector. Moreover, disturbances appear to be most amplified just downstream of the premixed branches. The effects of mixing width and differential diffusion are investigated and their roles on the flame stability are studied.

Chen, Jyh-Yuan↗
DOE OSTI2014

Deformable Image Registration for Adaptive Radiation Therapy of Head and Neck Cancer: Accuracy and Precision in the Presence of Tumor Changes

Purpose: To compare deformable image registration (DIR) accuracy and precision for normal and tumor tissues in head and neck cancer patients during the course of radiation therapy (RT). Methods and Materials: Thirteen patients with oropharyngeal tumors, who underwent submucosal implantation of small gold markers (average 6, range 4-10) around the tumor and were treated with RT were retrospectively selected. Two observers identified 15 anatomical features (landmarks) representative of normal tissues in the planning computed tomography (pCT) scan and in weekly cone beam CTs (CBCTs). Gold markers were digitally removed after semiautomatic identification in pCTs and CBCTs. Subsequently, landmarks and gold markers on pCT were propagated to CBCTs, using a b-spline-based DIR and, for comparison, rigid registration (RR). To account for observer variability, the pair-wise difference analysis of variance method was applied. DIR accuracy (systematic error) and precision (random error) for landmarks and gold markers were quantified. Time trend of the precisions for RR and DIR over the weekly CBCTs were evaluated. Results: DIR accuracies were submillimeter and similar for normal and tumor tissue. DIR precision (1 SD) on the other hand was significantly different (P<.01), with 2.2 mm vector length in normal tissue versus 3.3 mm in tumor tissue. No significant time trend in DIR precision was found for normal tissue, whereas in tumor, DIR precision was significantly (P<.009) degraded during the course of treatment by 0.21 mm/week. Conclusions: DIR for tumor registration proved to be less precise than that for normal tissues due to limited contrast and complex non-elastic tumor response. Caution should therefore be exercised when applying DIR for tumor changes in adaptive procedures.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2026

A Machine Learning based Approach of Estimating Equivalent Circuit Model Parameters at Different SoCs of Li-ion Batteries from Voltage Relaxation

Abstract: In this study, an approach of estimating the equivalent circuit model (ECM) parameters for Li-ion batteries (LIBs) is proposed based on the voltage value at different intervals while relaxing the LIB after discharge. The typical approach for estimating ECM parameters of a LIB is to conduct electrochemical impedance spectroscopy (EIS) measurements at different frequencies and fit them to a predefined circuit model, which requires additional measuring arrangements and specialized devices. The proposed methodology utilizes four different voltages at 0s, 60s, 360s, and 1800s alongside the specific state of charge (SoC) value for a specific constant discharge current value of ~1C until the relaxation stage to train and evaluate three regression-based machine learning models— Support Vector Regression (SVR), Extreme Gradient Boosting (XGBoost), and Gaussian Process Regression (GPR)—for estimating the ECM parameters of the selected model. Bayesian optimization is employed for hyperparameter tuning to achieve optimal performance for all the regressor models, among which, the GPR provided the best performance with the root-mean-squared error (RMSE) of less than 4x10-4 on average for the resistive components and less than 0.27 for capacitive components with excellent R2 scores. The simplicity of the approach enables it to eliminate the need for sophisticated measuring equipment and computation power.

Sagar, Md. Samiul [The University of Alabama (UA)]↗
DOE OSTI2015

WE-G-207-06: 3D Fluoroscopic Image Generation From Patient-Specific 4DCBCT-Based Motion Models Derived From Physical Phantom and Clinical Patient Images

Purpose: Respiratory-correlated cone-beam CT (4DCBCT) images acquired immediately prior to treatment have the potential to represent patient motion patterns and anatomy during treatment, including both intra- and inter-fractional changes. We develop a method to generate patient-specific motion models based on 4DCBCT images acquired with existing clinical equipment and used to generate time varying volumetric images (3D fluoroscopic images) representing motion during treatment delivery. Methods: Motion models are derived by deformably registering each 4DCBCT phase to a reference phase, and performing principal component analysis (PCA) on the resulting displacement vector fields. 3D fluoroscopic images are estimated by optimizing the resulting PCA coefficients iteratively through comparison of the cone-beam projections simulating kV treatment imaging and digitally reconstructed radiographs generated from the motion model. Patient and physical phantom datasets are used to evaluate the method in terms of tumor localization error compared to manually defined ground truth positions. Results: 4DCBCT-based motion models were derived and used to generate 3D fluoroscopic images at treatment time. For the patient datasets, the average tumor localization error and the 95th percentile were 1.57 and 3.13 respectively in subsets of four patient datasets. For the physical phantom datasets, the average tumor localization error and the 95th percentile were 1.14 and 2.78 respectively in two datasets. 4DCBCT motion models are shown to perform well in the context of generating 3D fluoroscopic images due to their ability to reproduce anatomical changes at treatment time. Conclusion: This study showed the feasibility of deriving 4DCBCT-based motion models and using them to generate 3D fluoroscopic images at treatment time in real clinical settings. 4DCBCT-based motion models were found to account for the 3D non-rigid motion of the patient anatomy during treatment and have the potential to localize tumor and other patient anatomical structures at treatment time even when inter-fractional changes occur. This project was supported, in part, through a Master Research Agreement with Varian Medical Systems, Inc., Palo Alto, CA. The project was also supported, in part, by Award Number R21CA156068 from the National Cancer Institute.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2019

Vector Calibration for Vertical Tests of Superconducting RF Cavities

We propose procedure for vector calibration of Q 0 vs E acc curve measurement for superconducting resonators. The proposed vector calibration procedure and hardware has several advantages over the techniques currently used for vertical cavity tests. It provides high level of automation and reduces manpower and preparation time for the test; it provides in situ, real time control of hardware parameters and allows to detect hardware failure and operator's errors; it allows to control and exclude the most significant systematic uncertainties, measured values of Q 0 (E acc ) can be made traceable to national standards.

43 PARTICLE ACCELERATORS↗
DOE OSTI2026

PIP-II LLRF Master Oscillator and Precision Reference Line - Station Level Design and Testing

The PIP-II superconducting linac at Fermilab requires a highly stable RF Reference Line to maintain phase syn- chronization throughout the accelerator. Temperature- induced changes in the electrical length of long coaxial cables can introduce phase drift and measurement errors. The reference-line architecture mitigates these effects by phase averaging the forward and reflected RF signals, while a phase-locked loop anchors the system to the mas- ter oscillator. This work focuses on the characterization, validation, and mechanical integration of station-level RF assemblies using CAD modeling, vector network analyzer measurements, and spectrum analyzer testing. PID-controlled thermal plates will stabilize critical RF components and further reduce temperature-dependent phase and amplitude variations. These methods sup- port repeatable, standardized designs that can be reliably integrated across the different reference-line stations.

Mosher, Alexander [U. Illinois, Chicago]↗
NASA NTRS1981

Structure of Jupiter's magnetopause - Pioneer 10 and 11 observations

The magnetic structure of Jupiter's magnetopause, as observed by the space probes Pioneer 10 and 11, is compared with terrestrial magnetopause structures from the OGO 5 mission. The Jovian magnetopause thickness, deduced from a rapid triple crossing, is found to lie in the range 3500-5200 km, while the adjoining plasma boundary layer was 5600-8400 km thick. Comparison with the terrestrial situation suggests that the dayside magnetopause thickness in both cases is a few times the ion gyroradius and that the dayside boundary layer is also of about the same width. The magnetopause normal vector and normal magnetic field component are determined for each crossing by use of minimum variance analysis. The results indicate a blunt and floppy magnetopause surface, for the most part with an insignificant normal magnetic field component. Only for three of the 14 crossings did this component exceed 1.5 times its error estimate. In two of these cases, the magnetic field tangential to the magnetopause displayed the characteristic features of a rotational discontinuity.

Sonnerup, B. U. O.↗
NASA NTRS1989

Satellite Data Processing System (SDPS) users manual V1.0

SDPS is a menu driven interactive program designed to facilitate the display and output of image and line-based data sets common to telemetry, modeling and remote sensing. This program can be used to display up to four separate raster images and overlay line-based data such as coastlines, ship tracks and velocity vectors. The program uses multiple windows to communicate information with the user. At any given time, the program may have up to four image display windows as well as auxiliary windows containing information about each image displayed. SDPS is not a commercial program. It does not contain complete type checking or error diagnostics which may allow the program to crash. Known anomalies will be mentioned in the appropriate section as notes or cautions. SDPS was designed to be used on Sun Microsystems Workstations running SunView1 (Sun Visual/Integrated Environment for Workstations). It was primarily designed to be used on workstations equipped with color monitors, but most of the line-based functions and several of the raster-based functions can be used with monochrome monitors. The program currently runs on Sun 3 series workstations running Sun OS 4.0 and should port easily to Sun 4 and Sun 386 series workstations with SunView1. Users should also be familiar with UNIX, Sun workstations and the SunView window system.

Caruso, Michael↗
NASA NTRS1993

Visual information processing II; Proceedings of the Meeting, Orlando, FL, Apr. 14-16, 1993

Various papers on visual information processing are presented. Individual topics addressed include: aliasing as noise, satellite image processing using a hammering neural network, edge-detetion method using visual perception, adaptive vector median filters, design of a reading test for low-vision image warping, spatial transformation architectures, automatic image-enhancement method, redundancy reduction in image coding, lossless gray-scale image compression by predictive GDF, information efficiency in visual communication, optimizing JPEG quantization matrices for different applications, use of forward error correction to maintain image fidelity, effect of peanoscanning on image compression. Also discussed are: computer vision for autonomous robotics in space, optical processor for zero-crossing edge detection, fractal-based image edge detection, simulation of the neon spreading effect by bandpass filtering, wavelet transform (WT) on parallel SIMD architectures, nonseparable 2D wavelet image representation, adaptive image halftoning based on WT, wavelet analysis of global warming, use of the WT for signal detection, perfect reconstruction two-channel rational filter banks, N-wavelet coding for pattern classification, simulation of image of natural objects, number-theoretic coding for iconic systems.

Huck, Friedrich O.↗
DOE OSTI2018

Unsupervised Parsing of Gaze Data with a Beta-process Vector Auto-regressive Hidden Markov Model

The first stage of analyzing eye-tracking data is commonly to code the data into sequences of fixations and saccades. This process is usually automated using simple, predetermined rules for classifying ranges of the time series into events, such as “if the dispersion of gaze samples is lower than the threshold, then code as a fixation; otherwise code as a saccade.” More recent approaches incorporate additional eye-movement categories in automated parsing algorithms, particularly glissades, by using time-varying, data-driven thresholds. We describe an alternative approach using the beta-process auto-regressive hidden Markov model (BP-AR-HMM). The BP-AR-HMM offers two main advantages over existing frameworks. First, it provides a statistical model for eye movement classification rather than a single estimate. Second, the BP-AR-HMM uses a latent process to model the number and nature of the types of eye-movements and hence is not constrained to predetermined categories. We applied the BP-AR-HMM both to high-sampling rate gaze data from Andersson, Larsson, Holmqvist, Stridh, and Nyström (2016) and to low-sampling rate data from the DIEM project (Mital, Smith, Hill, & Henderson, 2011). Driven by the data properties, the BP-AR-HMM identified clusters of samples resembling saccades, fixations and smooth pursuit, and a category that approximately corresponds to glissades (as well as blinks, errors, etc.). The BP-AR-HMM serves as an effective algorithm for data-driven event parsing alone or as an initial step in exploring the characteristics of gaze data sets.

eye tracking, fixation, fixation, saccade, smooth ↗
DOE OSTI2024

Power generation forecasting for solar plants based on Dynamic Bayesian networks by fusing multi-source information

A Dynamic Bayesian network (DBN) model for solar power generation forecasting in solar plants is proposed in this paper. The key idea is to fuse sensor data, operational indicators, meteorological data, lagged output power information, and model errors for more accurate short-term (e.g., hours) and mid-term (e.g., days to weeks) power generation forecasting. The proposed DBN augments automated data-driven structure learning with expert knowledge encoding using continuous and categorical data given constraints to represent causal relationships within a solar inverter system. Additionally, an error compensation mechanism is proposed to capture temporal fluctuation. The effectiveness of the DBN on solar power generation forecasting was evaluated by rolling window analysis with one-year testing data collected from a local solar plant. The proposed DBN is compared with four state-of-art methods including support-vector regression (SVR), k-nearest neighbors (kNN), artificial neural network (ANN), and long short-term memory (LSTM) models. The result show that the proposed DBN achieves better accuracy in general, and it is not as data-hungry as some neural network-based models. The proposed DBN is also shown to have robust and consistent forecasting power with different forecasting horizons. The accuracy is 92% - 95% from one hour to one week ahead forecasting.

14 SOLAR ENERGY↗
DOE OSTI2016

Short-Term Distribution System State Forecast Based on Optimal Synchrophasor Sensor Placement and Extreme Learning Machine

This paper proposes an approach for distribution system state forecasting, which aims to provide an accurate and high speed state forecasting with an optimal synchrophasor sensor placement (OSSP) based state estimator and an extreme learning machine (ELM) based forecaster. Specifically, considering the sensor installation cost and measurement error, an OSSP algorithm is proposed to reduce the number of synchrophasor sensor and keep the whole distribution system numerically and topologically observable. Then, the weighted least square (WLS) based system state estimator is used to produce the training data for the proposed forecaster. Traditionally, the artificial neural network (ANN) and support vector regression (SVR) are widely used in forecasting due to their nonlinear modeling capabilities. However, the ANN contains heavy computation load and the best parameters for SVR are difficult to obtain. In this paper, the ELM, which overcomes these drawbacks, is used to forecast the future system states with the historical system states. The proposed approach is effective and accurate based on the testing results.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2018

A Hierarchical Multi-Agent Based Anomaly Detection for Wide-Area Protection in Smart Grid

Future smart grid capabilities provide assurance to expand the advanced information and communication technologies to evolve into densely interconnected cyber physical system. Remedial Action Scheme (RAS), widely used for wide-area protection, relies on the interconnected networks and data sharing devices, which are exposed to the multitude of vulnerabilities. This paper presents our proposed approach to developing multi-agent based RAS scheme against the system-aware stealthy cyber-attacks. Specifically, we propose the two-level hierarchical architecture which consists of distributed local RAS controllers (RAScs) as local agents, operating at different zones/ areas, which are constantly monitored by an overseer, the central agent. The local controllers receive local and randomly changing outside zonal measurements and cyclically forwards to the overseer. The overseer identifies the corrupted controller using the anomaly detection algorithm which processes the measurements coming from the local controllers, compute measurement errors using local and outside zonal measurements, perform validation checks, and finally detect anomalies based on the two-step verification. Next, as a proof of concept, we have implemented and validated the proposed methodology in cyber physical environment at Iowa State’s PowerCyber testbed. We have also implemented the coordinated attack vectors which involve corrupting the local controller and later performing stealthy attacks on the system’s generator. We have evaluated its performance during the online testing in terms of detection rate and Iatency. The experimental results show that it is efficient in detecting different classes of attacks, including ramp and pulse attacks.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2016

WE-H-207A-02: Attenuation Correction in 4D-PET Using a Single-Phase Attenuation Map

Purpose: 4D-PET imaging has been proposed as a potential solution to the respiratory motion effect in thoracic region. CT-based attenuation correction (AC) is an essential step toward quantitative imaging for PET. However, due to the temporal difference of 4D-PET and a single breath-hold CT, motion artifacts are observed in the attenuation-corrected PET images that can lead to error in tumor shape and uptake. We introduce a practical method for aligning single-phase CT to all other 4D-PET phases using a penalized non-rigid demons registration. Methods: Individual 4D-PET frames were reconstructed without AC. Non-rigid Demons registration was used to derive deformation vector fields (DVFs) between the PET matched with CT phase and other 4D-PET images. While attenuated PET images provide enough useful data for organ borders such as lung and liver, tumors are not distinguishable from background due to loss of contrast. To preserve tumor shape in different phases, from CT image an ROI covering tumor was excluded from non-rigid transformation. Mean DVF of the central region of the tumor was assigned to all voxels in the ROI. This process mimics a rigid transformation of tumor along with a non-rigid transformation of other organs. 4D XCAT phantom with spherical tumors in lung with diameters ranging from 10 to 40 mm was used to evaluate the algorithm. Results: Motion related induced artifacts in attenuation-corrected 4D-PET images were significantly reduced. For tumors smaller than 20 mm, non-rigid transformation was capable to provide quantitative results. However, for larger tumors, where tumor self-attenuation is considerable, our combined method yields superior results. Conclusion: We introduced a practical method for deforming a single CT to match all 4D-PET images for accurate AC. Although 4D-PET data include insignificant anatomical information, we showed that they are still useful to estimate DVFs for aligning attenuation map and accurate AC.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2022

Development of Short-Term Forecasting Models Using Plant Asset Data and Feature Selection

Nuclear power plants collect and store large volumes of heterogeneous data from various components and systems. With recent advances in machine learning (ML) techniques, these data can be leveraged to develop diagnostic and short-term forecasting models to better predict future equipment condition. Maintenance operations can then be planned in advance whenever degraded performance is predicted, thus resulting in fewer unplanned outages and the optimization of maintenance activities. This enables lower maintenance costs and improves the overall economics of nuclear power. This paper focuses on developing a short-term forecasting process that leverages a feature selection process to distill large volumes of heterogeneous data and predict specific equipment parameters. A variety of feature selection methods, including Shapley Additive Explanations (SHAP) and variance inflation factor (VIF), were used to select the optimal features as inputs for three ML methods: long short-term memory (LSTM) networks, support vector regression (SVR), and random forest (RF). Each combination of model and input features was used to predict a pump bearing temperature both 1 and 24 hours in advance, based on actual plant system data. The optimal inputs for the LSTM and SVR were selected using the SHAP values, while the optimal input for the RF consisted solely of the response variable itself. Each model produced similar 1-hour-ahead predictions, with root mean square errors (RMSEs) of roughly 0.006. For the 24-hour-ahead predictions, differences could be seen between LSTM, SVR, and RF, as reflected by model performances of 0.036 +- 0.014, 0.0026 +- 0, and 0.063 +- 0.004 RMSE, respectively. As big data and continuous online monitoring become more widely available, the proposed feature selection process can be used for many applications beyond the prediction of process parameters within nuclear infrastructure.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
NASA NTRS1999

Spatial Orientation and Balance Control Changes Induced by Altered Gravito-Inertial Force Vectors

Seventeen healthy and eight vestibular deficient subjects were exposed to an interaural centripetal acceleration of 1 G (resultant 45 deg roll tilt of 1.4 G) on a 0.8 meter radius centrifuge for a period of 90 minutes in the dark. The subjects sat with head fixed upright, except every 4 of 10 minutes when instructed to rotate their head so that their nose and eyes pointed towards a visual point switched on every 3 to 5 seconds at random places (within +/- 30 deg) in the Earth horizontal plane. Motion sickness caused some subjects to limit their head movements during significant portions of the 90 minute period, and led three normal subjects to stop the test earlier. Eye movements, including directed saccades for subjective Earth- and head-referenced planes, were recorded before, during, and immediately after centrifugation using electro-oculography. Postural stability measurements were made before and within ten minutes after centrifugation. In normal subjects, postural sway and multisegment body kinematics were gathered during an eyes-closed head movement cadence (sway-referenced support platform), and in response to translational/rotational platform perturbations. A significant increase in postural sway, segmental motion amplitude and hip frequency was observed after centrifugation. This effect was short-lived, with a recovery time of several postural test trials. There were also asymmetries in the direction of post-centrifugation center of sway and head tilt which depended on the subject's orientation during the centrifugation adaptation period (left ear or right ear out). To delineate the effect of the magnitude of the gravito-inertial vector versus its direction during the adaptive centrifugation period, we tilted eight normal subjects in the roll axis at a 45 deg angle in the dark for 90 minutes without rotational motion. Their postural responses did not change following the period of tilt. Based on verbal reports, normal subjects overestimated roll-tilt during 90 minutes of both tilt and centrifugation stimuli. Subjective estimates of head-horizontal, provided by directed saccades, revealed significant errors after approximately 30 minutes that tended to increase only in the group who underwent centrifugation. Immediately after centrifugation, subjects reported feeling tilted on average 10 degrees in the opposite direction, which was in agreement with the direction of their earth-directed saccades. In vestibular deficient (VD) subjects, postural sway was measured using a sway-referenced or earth-fixed support surface, and with or without a head movement sequence. 'Me protocol was selected for each patient during baseline testing, and corresponded to the most challenging condition in which the patient was able to maintain balance with eyes closed. Bilaterally VD subjects showed no postural decrement after centrifugation, while unilateral VD subjects had varying degrees of decrement. Unilateral VD subjects were tested twice; they underwent centrifugation both with right ear out and left ear out. Their post-centrifuation center of sway shifted at right angles depending on the centrifuge GIF orientation. Bilateral VD subjects bad shifts as well, but no consistent directional trend. VD subjects underestimated roll-tilt during centrifugation, These results suggest that orientation of the gravito-inertial vector and its magnitude arc both used by the central nervous system for calibration of multiple orientation systems. A change in the background gravito-inertial force (otolith input) can rapidly initiate postural and perceptual adaptation in several sensorimotor systems, independent of a structured visual surround.

Kaufman, Galen D.↗
DOE OSTI2018

Solid Mechanics Code Verification: Contrasting Classical and Manufactured Reference Solutions

Verification results for Sierra/SM using inexact reference solutions have often exhibited unsatisfactory convergence behavior. With an understanding of the convergence behavior for these types of tests, one can avoid falsely attributing pathologies of the test with incorrectness of the code. Simple theoretical results highlight that for an inexact reference solution two conditions must be met to observe asymptotic convergence. These conditions, and the resulting types of convergence behaviors, are further illustrated with graphical examples depicting the exact, inexact reference, and sequence of numerical solutions as vectors (in a function space). A stress concentration problem is adopted to contrast convergence behaviors when using inexact (classical linear elastic) and exact (manufactured) reference solutions. Convergence is not initially attained with the classical solution. Convergence with the manufactured solution indicates the convergence failure with the classical reference did not result from code error and provides insight on how for this problem asymptotic convergence could be attained with the classical reference solution by modifying the computational models.

42 ENGINEERING↗