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At least 1,279 records · Page 71

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DOE OSTI2020

The Differential Equations That Model Diseases

In this brief tutorial, I will describe the simplest way of modeling a deadly infectious disease, such as COVID-19. I will show that at early times, the disease grows exponentially, but at late times, falls off like a Bell curve. I will solve the equations numerically with Python. A common mathematical model is the so-called compartmental model, where diseases can move people between "categories" such as healthy, infectious, or truly sick. This is a large, well established field and there are many good resources on this topic. I started with the SIAM review article The mathematics of infectious diseases. However, there are many excellent textbooks and more modern reviews as well.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2020

Hamiltonian Simulation [Slides]

Given a (qubit) quantum system modeled by a Hamiltonian H, use a quantum computer to obtain time-evolved properties of the system. One way of solving this problem is by actually preparing the evolved quantum state (in some encoded fashion). The goal is to provide a sequence of elementary 1 and 2-qubit gates that would prepare such a state.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Implications of new Reasoning Capabilities for Science and Security: Results from a Quick Initial Study

On Thursday, September 12 OpenAI released “a new series of models designed to spend more time thinking… they can reason through complex tasks and solve harder problems than previous models in science, coding, and math.” These models are referred to as o1-preview and o1-mini and appear to be first results of what had been a closely held project called Strawberry within OpenAI. The models are not described as successors in the earlier GPT series because they provide a qualitatively different type of capability, especially step-by-step reasoning.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Physics-Informed Neural Network Solution of Point Kinetics Equations for a Nuclear Reactor Digital Twin

A digital twin (DT) for nuclear reactor monitoring can be implemented using either a differential equations-based physics model or a data-driven machine learning model. The challenge of a physics-model-based DT consists of achieving sufficient model fidelity to represent a complex experimental system, whereas the challenge of a data-driven DT consists of extensive training requirements and a potential lack of predictive ability. We investigate the performance of a hybrid approach, which is based on physics-informed neural networks (PINNs) that encode fundamental physical laws into the loss function of the neural network. We develop a PINN model to solve the point kinetic equations (PKEs), which are time-dependent, stiff, nonlinear, ordinary differential equations that constitute a nuclear reactor reduced-order model under the approximation of ignoring spatial dependence of the neutron flux. The PINN model solution of PKEs is developed to monitor the start-up transient of Purdue University Reactor Number One (PUR-1) using experimental parameters for the reactivity feedback schedule and the neutron source. The results demonstrate strong agreement between the PINN solution and finite difference numerical solution of PKEs. We investigate PINNs performance in both data interpolation and extrapolation. For the test cases considered, the extrapolation errors are comparable to those of interpolation predictions. Extrapolation accuracy decreases with increasing time interval.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2021

General Quantum Field Theory of Flavor Mixing and Oscillations

We review the canonical transformation in quantum physics known as the Bogoliubov transformation and present its application to the general theory of quantum field mixing and oscillations with an arbitrary number of mixed particles with either boson or fermion statistics. The mixing relations for quantum states are derived directly from the definition of mixing for quantum fields and the unitary inequivalence of the Fock space of energy and flavor eigenstates is shown by a straightforward algebraic method. The time dynamics of the interacting fields is then explicitly solved and the flavor oscillation formulas are derived in a unified general formulation with emphasis on antiparticle content and effect introduced by nontrivial flavor vacuum.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1972

Variable boundary II heat conduction

Computer program for solving both transient and steady-state heat transfer problems is presented. Specific applications of computer program are described. Formulation for individual nodes of solid medium for heat balance is presented. Diffusion equation is solved for all nodes simultaneously at finite increments of time.

Gramer, J.↗

NASA NTRS1971

Advancements in medicine from aerospace research

NASA has taken the lead in implementing the concept of technology utilization, and the Technology Utilization Program is the first vital step in the goal of a technological society to insure maximum benefit from the costs of technology. Experience has shown that the active approach to technology transfer is unique and is well received in the medical profession when appropriate problems are tackled. The problem solving approach is a useful one at the precise time when medicine is recognizing the need for new technology.

Wooten, F. T.↗

NASA NTRS1972

Numerical method for the solution of large systems of differential equations of the boundary layer type

A numerical method for the solution of large systems of nonlinear differential equations of the boundary-layer type is described. The method is a modification of the technique for satisfying asymptotic boundary conditions. The present method employs inverse interpolation instead of the Newton method to adjust the initial conditions of the related initial-value problem. This eliminates the so-called perturbation equations. The elimination of the perturbation equations not only reduces the user's preliminary work in the application of the method, but also reduces the number of time-consuming initial-value problems to be numerically solved at each iteration. For further ease of application, the solution of the overdetermined system for the unknown initial conditions is obtained automatically by applying Golub's linear least-squares algorithm. The relative ease of application of the proposed numerical method increases directly as the order of the differential-equation system increases. Hence, the method is especially attractive for the solution of large-order systems. After the method is described, it is applied to a fifth-order problem from boundary-layer theory.

Green, M. J.↗

NASA NTRS1972

Contributions from space technology to central power generation

The central power crisis, and the present and relatively near-time contributions that aerospace technology is making to help solve this crisis are discussed. The principal emphasis is placed on the prospects of aerospace derived magnetohydrodynamic (MHD) large scale power generation. The strides that the Soviet Union is making in this field with the startup of the new U-25 plant near Moscow, having a total power capability of 75 MW, are reviewed. A much smaller program in the U.S. is outlined, and prospects of future benefits are discussed.

Dicks, J. B., Jr.↗

NASA NTRS1972

Normal evaporation of binary alloys

In the study of normal evaporation, it is assumed that the evaporating alloy is homogeneous, that the vapor is instantly removed, and that the alloy follows Raoult's law. The differential equation of normal evaporation relating the evaporating time to the final solute concentration is given and solved for several important special cases. Uses of the derived equations are exemplified with a Ni-Al alloy and some binary iron alloys. The accuracy of the predicted results are checked by analyses of actual experimental data on Fe-Ni and Ni-Cr alloys evaporated at 1600 C, and also on the vacuum purification of beryllium. These analyses suggest that the normal evaporation equations presented here give satisfactory results that are accurate to within an order of magnitude of the correct values, even for some highly concentrated solutions. Limited diffusion and the resultant surface solute depletion or enrichment appear important in the extension of this normal evaporation approach.

Li, C. H.↗

NASA NTRS1973

Remote haze monitoring by satellite: A study to demonstrate the feasibility and determine the optimum method

The author has identified the following significant results. A general purpose flexible computer program for the analysis of ERTS MSS data has been written for the Control Data 7600 computer. By putting all of the analysis operations into a separate subroutine the user may concentrate on the analysis. The problems of getting the information into the computer in convenient format is solved once and need not be tampered with each time a new analysis procedure is desired. The problems of getting information written by the XDS computers in 8 bit byte format into a convenient format for analysis on the 60 bit word format CDC 7600 computer are not trivial. A general purpose program to accomplish this has been written and is available from the investigators upon request.

Rogers, E. H.↗

NASA NTRS1973

Application of digital computer APU modeling techniques to control system design.

Study of the required controls for a H2-O2 auxiliary power unit (APU) technology program for the Space Shuttle. A steady-state system digital computer program was prepared and used to optimize initial system design. Analytical models of each system component were included. The program was used to solve a nineteen-dimensional problem, and then time-dependent differential equations were added to the computer program to simulate transient APU system and control. Some system parameters were considered quasi-steady-state, and others were treated as differential variables. The dynamic control analysis proceeded from initial ideal control modeling (which considered one control function and assumed the others to be ideal), stepwise through the system (adding control functions), until all of the control functions and their interactions were considered. In this way, the adequacy of the final control design over the required wide range of APU operating conditions was established.

Bailey, D. A.↗

NASA NTRS1975

The SAR arc event observed during the December 1971 magnetic storm

Observations from five separate experiments on the Isis 2 spacecraft are used to study the atmosphere/ionosphere during the magnetic storm of December 16-20, 1971. The data are most complete in the midlatitude region, permitting a study of the SAR arc (subauroral red arc) which developed during the night of December 17-18. Ion composition and temperature, electron temperature, electron-density height profiles from the spacecraft to the F region, and the intensity of the 6300-A oxygen emission are all presented for the region of interest. It is found that the H(+) concentration had sharp gradients near the SAR arc and that the plasma temperature was significantly enhanced over typical nighttime values, reaching nearly 7000 K at 1400 km on the field line which intersected the arc. A system of time-dependent equations for atmospheric/ionospheric composition and temperature is solved using boundary conditions which were selected so that the solutions are in agreement with the observations. From these solutions, an assessment is made of the influence of (1) the efflux of plasma from the ionosphere to the magnetosphere, (2) the decrease in O/N2 at the turbopause, (3) the increase in the loss coefficient as a result of an increase in the vibrational temperature of N2, and (4) the conduction of thermal energy into the ionosphere.

Maier, E. J.↗

NASA NTRS1977

Non-linear analysis of vibrations of irregular plates

A numerical perturbation method is used to investigate the forced vibrations of irregular plates. Nonlinear terms associated with the midplane stretching are retained in the analysis. The numerical part of the method involves the use of linear, finite element techniques to determine the free oscillation mode shapes and frequencies and to obtain the linear midplane stress resultants caused by the midplane stretching. Representing the solution as an expansion in terms of these linear mode shapes, these modes and the resultants are used to determine the equations governing the time-dependent coefficients of this expansion. These equations are solved by using the method of multiple scales. Specific solutions are given for the main-resonant vibrations of an elliptical plate in the presence of internal resonances. The results indicate that modes other than the driven mode can be drawn into the steady state response. Though the excitation is composed of a single harmonic, the response may not be periodic. Moreover, the particular types of responses that can occur are highly dependent on the mode being excited and are sensitive to small geometrical changes.

Lobitz, D. W.↗

NASA NTRS1978

Turbulent viscous shock layer solutions for Jovian entry at small angles of attack

The equations governing the laminar and turbulent flows of reacting gas mixtures in chemical equilibrium over axially symmetric blunt bodies at small angles of attack are developed and presented in the unsteady conservative form. Solutions are obtained in the planes of symmetry of the flow field for the typical conditions encountered by a probe entering the Jovian atmosphere at small angles of attack. The eddy-viscosity is approximated by a two-layer model. The shock wave is treated as a discontinuity across which the shock relations are used to compute the flow conditions behind the shock. A time-asymptotic finite-difference method is used to solve the equations. The zero angle of attack results for a hyperboloid are compared with the existing results but neither experimental nor computational results are available for comparison with the present results at angle of attack.

Jumar, A.↗

NASA NTRS1979

A wave-envelope of sound propagation in nonuniform circular ducts with compressible mean flows

An acoustic theory is developed to determine the sound transmission and attenuation through an infinite, hard-walled or lined circular duct carrying compressible, sheared, mean flows and having a variable cross section. The theory is applicable to large as well as small axial variations, as long as the mean flow does not separate. The technique is based on solving for the envelopes of the quasi-parallel acoustic modes that exist in the duct instead of solving for the actual wave, thereby reducing the computation time and the round-off error encountered in purely numerical techniques. The solution recovers the solution based on the method of multiple scales for slowly varying duct geometry. A computer program was developed based on the wave-envelope analysis for general mean flows. Results are presented for the reflection and transmission coefficients as well as the acoustic pressure distributions for a number of conditions: both straight and variable area ducts with and without liners and mean flows from very low to high subsonic speeds are considered.

Nayfeh, A. H.↗

NASA NTRS1979

Numerical solution for perpendicular sonic hydrogen injection into a ducted supersonic airstream

This note discusses a computer program being developed to study the flow field near opposing perpendicular fuel injectors in scramjets. The MacCormack time-split, finite difference relaxation technique was used to solve the full two-dimensional compressible Navier-Stokes equations along with energy and species equations. By using this technique, a program was developed to consider the turbulent nonreacting flow of hydrogen and air in a rectangular duct. A damping term, proportional to the second derivative of pressure and temperature, was used to produce a stable solution behind the hydrogen jet in the neighborhood of the recompression shock. A case using actual conditions encountered in current scramjet design was analyzed, with results agreeing qualitatively with experimental observations.

Drummond, J. P.↗

NASA NTRS1980

Transmission of sound through nonuniform circular ducts with compressible mean flows

An acoustic theory is developed to determine the sound transmission and attenuation through an infinite hard-walled or lined circular duct carrying compressible, sheared mean flows and having a variable cross section. The theory is applicable to large as well as small axial variations, as long as the mean flow does not separate. The technique is based on solving for the envelopes of the quasi-parallel acoustic modes that exist in the duct instead of solving for the actual wave, thereby reducing the computation time and the round-off error encountered in purely numerical techniques. A number of test cases that demonstrate the flexibility of the program are included. Convergence of the transmission coefficients and the acoustic pressure profiles with an increasing number of modes is illustrated.

Nayfeh, A. H.↗