Guide for testing numerical-integration subroutines
Numerical technique has been developed for testing algorithms used to solve differential equations.
Krogh, F. T.↗
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Comment on 'Time-optimal orbit transfer trajectory for solar sail spacecraft'
An analytic proof is presented to show that the orbital transfer times of an earth-to-Mars solar-sail propelled spacecraft trajectory as calculated by Jayaraman (1980) are incorrect. In particular, different boundary conditions are defined, which indicate that a minimization of the Hamiltonian, which Jayaraman used, can yield the wrong stationary solution. Transfer times are calculated using a neighboring extremal algorithm based on numerical differentiation in conjunction with Krogh's variable order, variable step size integrator, resulting in a transfer time of 322 days at 2 mm/sec-sq, with endpoint restraints satisfied to within 1/1 billion. Finally, it is concluded that minimization of flight time is secondary in importance to maximization of delivered payload and minimization of overall mission cost and risk.
Wood, L. J.↗
Grid generation by elliptic partial differential equations for a tri-element Augmentor-Wing airfoil
Two efforts to numerically simulate the flow about the Augmentor-Wing airfoil in the cruise configuration using the GRAPE elliptic partial differential equation grid generator algorithm are discussed. The Augmentor-Wing consists of a main airfoil with a slotted trailing edge for blowing and two smaller airfoils shrouding the blowing jet. The airfoil and the algorithm are described, and the application of GRAPE to an unsteady viscous flow simulation and a transonic full-potential approach is considered. The procedure involves dividing a complicated flow region into an arbitrary number of zones and ensuring continuity of grid lines, their slopes, and their point distributions across the zonal boundaries. The method for distributing the body-surface grid points is discussed.
Sorenson, R. L.↗
Transonic separated solutions for an augmentor-wing
The viscous transonic flow about a multielement airfoil (augmentor-wing) is simulated by coupling full-potential and direct/inverse differential boundary-layer algorithms. Solutions have been obtained for a variety of conditions and are in fair agreement with available experimental data. Typical results from this transonic augmentor-wing code (TAUG-V) require approximately three minutes of CRAY-XMP CPU time. Since this viscous transonic code accounts for most of the important flow physics, yet is still economical, it is a practical tool for the design aerodynamicist.
Flores, J.↗
A numerical solution of Duffing's equations including the prediction of jump phenomena
Numerical methodology for the solution of Duffing's differential equation is presented. Algorithms for the prediction of multiple equilibrium solutions and jump phenomena are developed. In addition, a filtering algorithm for producing steady state solutions is presented. The problem of a rigidly clamped circular plate subjected to cosinusoidal pressure loading is solved using the developed algorithms (the plate is assumed to be in the geometrically nonlinear range). The results accurately predict regions of solution multiplicity and jump phenomena.
Moyer, E. T., Jr.↗
Multistage estimation of received carrier signal parameters under very high dynamic conditions of the receiver
A multistage estimator is provided for the parameters of a received carrier signal possibly phase-modulated by unknown data and experiencing very high Doppler, Doppler rate, etc., as may arise, for example, in the case of Global Positioning Systems (GPS) where the signal parameters are directly related to the position, velocity and jerk of the GPS ground-based receiver. In a two-stage embodiment of the more general multistage scheme, the first stage, selected to be a modified least squares algorithm referred to as differential least squares (DLS), operates as a coarse estimator resulting in higher rms estimation errors but with a relatively small probability of the frequency estimation error exceeding one-half of the sampling frequency, provides relatively coarse estimates of the frequency and its derivatives. The second stage of the estimator, an extended Kalman filter (EKF), operates on the error signal available from the first stage refining the overall estimates of the phase along with a more refined estimate of frequency as well and in the process also reduces the number of cycle slips.
Kumar, Rajendra↗
A mathematical model for simulating noise suppression of lined ejectors
A mathematical model containing the essential features embodied in the noise suppression of lined ejectors is presented. Although some simplification of the physics is necessary to render the model mathematically tractable, the current model is the most versatile and technologically advanced at the current time. A system of linearized equations and the boundary conditions governing the sound field are derived starting from the equations of fluid dynamics. A nonreflecting boundary condition is developed. In view of the complex nature of the equations, a parametric study requires the use of numerical techniques and modern computers. A finite element algorithm that solves the differential equations coupled with the boundary condition is then introduced. The numerical method results in a matrix equation with several hundred thousand degrees of freedom that is solved efficiently on a supercomputer. The model is validated by comparing results either with exact solutions or with approximate solutions from other works. In each case, excellent correlations are obtained. The usefulness of the model as an optimization tool and the importance of variable impedance liners as a mechanism for achieving broadband suppression within a lined ejector are demonstrated.
Watson, Willie R.↗
Multistage estimation of received carrier signal parameters under very high dynamic conditions of the receiver
A multistage estimator is provided for the parameters of a received carrier signal possibly phase-modulated by unknown data and experiencing very high Doppler, Doppler rate, etc., as may arise, for example, in the case of Global Positioning Systems (GPS) where the signal parameters are directly related to the position, velocity and jerk of the GPS ground-based receiver. In a two-stage embodiment of the more general multistage scheme, the first stage, selected to be a modified least squares algorithm referred to as differential least squares (DLS), operates as a coarse estimator resulting in higher rms estimation errors but with a relatively small probability of the frequency estimation error exceeding one-half of the sampling frequency, provides relatively coarse estimates of the frequency and its derivatives. The second stage of the estimator, an extended Kalman filter (EKF), operates on the error signal available from the first stage refining the overall estimates of the phase along with a more refined estimate of frequency as well and in the process also reduces the number of cycle slips.
Kumar, Rajendra↗
Design of Autonomous Medical Response Agent (AMRA) Aggregate Information Dashboard (AID)
Future astronauts in deep space missions will rely on tools and technologies empowering them to self-diagnose and self-treat medical conditions. Given communications delays and limited bandwidth in future long-duration exploration missions (LDEMs), medical decision support technologies must empower the crew to manage routine medical activities, acute medical incidents, as well as emergency medical scenarios independently from ground support.The Autonomous Medical Response Agent (AMRA) is envisioned as a digital tool enabling crew to issue medical complaints and interact with a medical decision support algorithm which develops a differential diagnosis and recommends a treatment protocol for the condition. AMRA will draw from individual crew medical history in addition to crew symptoms to more efficiently identify high-risk medical conditions. A new symptom could be indicative of a chronic condition or a normal adaptation to long-duration spaceflight, but could just as easily be indicative of an adverse vehicle condition affecting the entire crew.While real-time communication with a flight surgeon may not possible, the crew will nonetheless require a means to communicate and document both routine and emergency medical incidents to ground support. Conversely, flight surgeons and medical specialists on the ground will need to understand information such as crew vitals or responses to medical check-ups and examinations within the larger context of crew schedule, mission activities, and vehicle performance. A user interface which establishes communication protocols between an individual crew member and AMRA, as well as ground support to the crew is a significant area of research demanding input and consideration.The design of AMRA AID is intended to: a) represent routine medical activities as well as new (unplanned) medical incidents within the larger context of crew schedule and mission activities, and b) increase confidence between ground support and crew members over the course of LDEMs. Maintaining situation awareness of unplanned medical incidents between ground and crew will be a critical element within LDEMs. Two medical incidents headache and difficulty breathing are being explored within a user interface prototype which captures communications protocols between crew members and mission control, human health monitoring, vehicle or environmental monitoring, as well as crew schedule and mission activities holistically.
Yashar, M.↗
Application of Sparse Identification of Nonlinear Dynamics for Physics-Informed Learning
Advances in machine learning and deep neural networks has enabled complex engineering tasks like image recognition, anomaly detection, regression, and multi-objective optimization, to name but a few. The complexity of the algorithm architecture, e.g., the number of hidden layers in a deep neural network, typically grows with the complexity of the problems they are required to solve, leaving little room for interpreting (or explaining) the path that results in a specific solution. This drawback is particularly relevant for autonomous aerospace and aviation systems, where certifications require a complete understanding of the algorithm behavior in all possible scenarios. Including physics knowledge in such data-driven tools may improve the interpretability of the algorithms, thus enhancing model validation against events with low probability but relevant for system certification. Such events include, for example, spacecraft or aircraft sub-system failures, for which data may not be available in the training phase. This paper investigates a recent physics-informed learning algorithm for identification of system dynamics, and shows how the governing equations of a system can be extracted from data using sparse regression. The learned relationships can be utilized as a surrogate model which, unlike typical data-driven surrogate models, relies on the learned underlying dynamics of the system rather than large number of fitting parameters. The work shows that the algorithm can reconstruct the differential equations underlying the observed dynamics using a single trajectory when no uncertainty is involved. However, the training set size must increase when dealing with stochastic systems, e.g., nonlinear dynamics with random initial conditions.
Corbetta, Matteo↗
Characterization of Pinhole Collimators for High-Resolution Gamma Imaging of Irradiated Fuel
Post-irradiation examination (PIE) of nuclear fuels requires imaging tools capable of resolving isotopic and spatial features with high throughput. This project contributes to a proof-of-concept effort aimed at advancing gamma emission tomography (GET) by evaluating novel fine-aperture pinhole collimators. Two Rose’s metal collimators, 100 µm (20° acceptance angle) and 350 µm (30° acceptance angle), were prototyped and characterized for their effectiveness in transporting gamma rays through the pinhole aperture. To support data collection, a Python-based data acquisition system was developed to coordinate a rotation stage, linear stage, and CZT detector, reducing latency in high-rate gamma event logging to one second per acquisition. Queue-based file writing enabled seamless real-time data capture for count rates up to 35,000 counts per second (cps). List-mode parsing algorithms were implemented to differentiate single and simultaneous gamma interactions for future tomographic reconstruction. Detector response was evaluated in both spectroscopy and list mode acquisition methods across varying source distances to confirm absolute and collimator efficiencies. Preliminary efficiency figures suggest effective collimation of gamma-rays with energies below 700 keV, with ~4% residual intensity through the aperture for Cs-137. The impact of collimator geometry on image quality is currently being evaluated. This groundwork supports the ongoing development of a sub mm resolution cone-beam CT system for imaging fuel phantoms, an essential step toward improving GET efficiency and accelerating nuclear fuel qualification efforts.
46 - INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AN↗
Spectral identification of minerals using imaging spectrometry data: Evaluating the effects of signal to noise and spectral resolution using the tricorder algorithm
The rapid development of sophisticated imaging spectrometers and resulting flood of imaging spectrometry data has prompted a rapid parallel development of spectral-information extraction technology. Even though these extraction techniques have evolved along different lines (band-shape fitting, endmember unmixing, near-infrared analysis, neural-network fitting, and expert systems to name a few), all are limited by the spectrometer's signal to noise (S/N) and spectral resolution in producing useful information. This study grew from a need to quantitatively determine what effects these parameters have on our ability to differentiate between mineral absorption features using a band-shape fitting algorithm. We chose to evaluate the AVIRIS, HYDICE, MIVIS, GERIS, VIMS, NIMS, and ASTER instruments because they collect data over wide S/N and spectral-resolution ranges. The study evaluates the performance of the Tricorder algorithm, in differentiating between mineral spectra in the 0.4-2.5 micrometer spectral region. The strength of the Tricorder algorithm is in its ability to produce an easily understood comparison of band shape that can concentrate on small relevant portions of the spectra, giving it an advantage over most unmixing schemes, and in that it need not spend large amounts of time reoptimizing each time a new mineral component is added to its reference library, as is the case with neural-network schemes. We believe the flexibility of the Tricorder algorithm is unparalleled among spectral-extraction techniques and that the results from this study, although dealing with minerals, will have direct applications to spectral identification in other disciplines.
Swayze, Gregg A.↗
Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. I - The dynamics of time discretization and its implications for algorithm development in computational fluid dynamics
Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit.
Yee, H. C.↗
Numerical methods of solving a system of multi-dimensional nonlinear equations of the diffusion type
The principles of conservation and stability of difference schemes achieved using the iteration control method were examined. For the schemes obtained of the predictor-corrector type, the conversion was proved for the control sequences of approximate solutions to the precise solutions in the Sobolev metrics. Algorithms were developed for reducing the differential problem to integral relationships, whose solution methods are known, were designed. The algorithms for the problem solution are classified depending on the non-linearity of the diffusion coefficients, and practical recommendations for their effective use are given.
Agapov, A. V.↗
A General Event Location Algorithm with Applications to Eclispe and Station Line-of-Sight
A general-purpose algorithm for the detection and location of orbital events is developed. The proposed algorithm reduces the problem to a global root-finding problem by mapping events of interest (such as eclipses, station access events, etc.) to continuous, differentiable event functions. A stepping algorithm and a bracketing algorithm are used to detect and locate the roots. Examples of event functions and the stepping/bracketing algorithms are discussed, along with results indicating performance and accuracy in comparison to commercial tools across a variety of trajectories.
Parker, Joel J. K.↗
A General Event Location Algorithm with Applications to Eclipse and Station Line-of-Sight
A general-purpose algorithm for the detection and location of orbital events is developed. The proposed algorithm reduces the problem to a global root-finding problem by mapping events of interest (such as eclipses, station access events, etc.) to continuous, differentiable event functions. A stepping algorithm and a bracketing algorithm are used to detect and locate the roots. Examples of event functions and the stepping/bracketing algorithms are discussed, along with results indicating performance and accuracy in comparison to commercial tools across a variety of trajectories.
Parker, Joel J. K.↗
Global Free-tropospheric NO2 Abundances Derived Using a Cloud Slicing Technique from AURA OMI
We derive free-tropospheric NO2 volume mixing ratios (VMRs) by applying a cloud-slicing technique to data from the Ozone Monitoring Instrument (OMI) on the Aura satellite. In the cloud-slicing approach, the slope of the above-cloud NO2 column versus the cloud scene pressure is proportional to the NO2 VMR. In this work, we use a sample of nearby OMI pixel data from a single orbit for the linear fit. The OMI data include cloud scene pressures from the rotational-Raman algorithm and above-cloud NO2 vertical column density (VCD) (defined as the NO2 column from the cloud scene pressure to the top of the atmosphere) from a differential optical absorption spectroscopy (DOAS) algorithm. We compare OMI-derived NO2 VMRs with in situ aircraft profiles measured during the NASA Intercontinental Chemical Transport Experiment Phase B (INTEX-B) campaign in 2006. The agreement is generally within the estimated uncertainties when appropriate data screening is applied. We then derive a global seasonal climatology of free-tropospheric NO2 VMR in cloudy conditions. Enhanced NO2 in the free troposphere commonly appears near polluted urban locations where NO2 produced in the boundary layer may be transported vertically out of the boundary layer and then horizontally away from the source. Signatures of lightning NO2 are also shown throughout low and middle latitude regions in summer months. A profile analysis of our cloud-slicing data indicates signatures of lightning-generated NO2 in the upper troposphere. Comparison of the climatology with simulations from the global modeling initiative (GMI) for cloudy conditions (cloud optical depth less than10) shows similarities in the spatial patterns of continental pollution outflow. However, there are also some differences in the seasonal variation of free-tropospheric NO2 VMRs near highly populated regions and in areas affected by lightning-generated NOx.
NO2↗
Efficient numerical algorithm for multi-level ionization of high-atomic-number gases
An efficient numerical algorithm for laser driven multi-level ionization of high-atomic-number gases is proposed and implemented in an electromagnetic particle-in-cell code SPACE. The algorithm is based on analytical solutions to the system of differential equations describing ionization evolution. Using analytical solutions resolves the multiscale issue of ionization due to different characteristic time scales of ionization processes and the main code time step. Algorithm efficiency and memory requirements are significantly improved by using a locally reduced system of differential equations. The algorithm also assigns proper orbital quantum numbers and their projections to ionization states. The algorithm is verified and validated using experimental data.
Cheng, A. (ORCID:000000021945282X)↗