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At least 1,261 records · Page 70

Propagation of quasifracture in viscoelastic media under low-cycle repeated stressing

The propagation of a craze as quasifracture under repeated cyclic stressing in polymeric systems has been under intensive investigation recently. Based upon a time-dependent crazing theory, the governing differential equation describing the propagation of a single craze as quasifracture in an infinite viscoelastic plate has been solved for sinusoidal stresses. Numerical methods have been employed to obtain the normalized craze length as a function of time. The computed results indicate that the length of a quasifracture may decelerate and decrease indicating that its velocity can reverse. This behavior may be consistent with the observed and much discussed craze healing and the enclosure model in fatigue and fracture of solids.

Liu, X. P.↗

Spline-based Rayleigh-Ritz methods for the approximation of the natural modes of vibration for flexible beams with tip bodies

Rayleigh-Ritz methods for the approximation of the natural modes for a class of vibration problems involving flexible beams with tip bodies using subspaces of piecewise polynomial spline functions are developed. An abstract operator-theoretic formulation of the eigenvalue problem is derived and spectral properties investigated. The existing theory for spline-based Rayleigh-Ritz methods applied to elliptic differential operators and the approximation properties of interpolatory splines are used to argue convergence and establish rates of convergence. An example and numerical results are discussed.

Rosen, I. G.↗

Self-consistent inclusion of space-charge in the traveling wave tube

It is shown how the complete field of the electron beam may be incorporated into the transmission line model theory of the traveling wave tube (TWT). The fact that the longitudinal component of the field due to the bunched beam is not used when formulating the beam-to-circuit coupling equation is not well-known. The fundamental partial differential equation for the traveling wave field is developed and compared with the older (now standard) one. The equation can be solved numerically using the same algorithms, but now the coefficients can be updated continuously as the calculation proceeds down the tube. The coefficients in the older equations are primarily derived from preliminary measurements and some trial and error. The newer coefficients can be found by a recursive method, since each has a well defined physical interpretation and can be calculated once a reasonable first trial solution is postulated. The results of the new expression were compared with those of the older forms, as well as to a field theory model to show the ease in which a reasonable fit to the field prediction is obtained. A complete summary of the existing transmission line modeling of the TWT is given to explain the somewhat vague ideas and techniques in the general area of drifting carrier-traveling circuit wave interactions. The basic assumptions and inconsistencies of the existing theory and areas of confusion in the general literature are examined and hopefully cleared up.

Freeman, Jon C.↗

Interface procedures for overlapping grids

The solution of partial differential equations by means of either implicit or conservative explicit finite-difference methods, on the one hand, or multicomponent overlapping grid schemes on the other, is presently subjected to interpolation at grid boundaries in an attempt to eliminate the time lag. The accuracy of the transient solution of a hyperbolic or parabolic partial differential equation is dependent on the procedures used to transfer information between grids in a composite grid system. The present techniques reduce numerical solution errors.

Mastin, C. Wayne↗

LSENS, a general chemical kinetics and sensitivity analysis code for gas-phase reactions: User's guide

A general chemical kinetics and sensitivity analysis code for complex, homogeneous, gas-phase reactions is described. The main features of the code, LSENS, are its flexibility, efficiency and convenience in treating many different chemical reaction models. The models include static system, steady, one-dimensional, inviscid flow, shock initiated reaction, and a perfectly stirred reactor. In addition, equilibrium computations can be performed for several assigned states. An implicit numerical integration method, which works efficiently for the extremes of very fast and very slow reaction, is used for solving the 'stiff' differential equation systems that arise in chemical kinetics. For static reactions, sensitivity coefficients of all dependent variables and their temporal derivatives with respect to the initial values of dependent variables and/or the rate coefficient parameters can be computed. This paper presents descriptions of the code and its usage, and includes several illustrative example problems.

Radhakrishnan, Krishnan↗

Summary of the Fourth AIAA CFD Drag Prediction Workshop

Results from the Fourth AIAA Drag Prediction Workshop (DPW-IV) are summarized. The workshop focused on the prediction of both absolute and differential drag levels for wing-body and wing-body-horizontal-tail configurations that are representative of transonic transport air- craft. Numerical calculations are performed using industry-relevant test cases that include lift- specific flight conditions, trimmed drag polars, downwash variations, dragrises and Reynolds- number effects. Drag, lift and pitching moment predictions from numerous Reynolds-Averaged Navier-Stokes computational fluid dynamics methods are presented. Solutions are performed on structured, unstructured and hybrid grid systems. The structured-grid sets include point- matched multi-block meshes and over-set grid systems. The unstructured and hybrid grid sets are comprised of tetrahedral, pyramid, prismatic, and hexahedral elements. Effort is made to provide a high-quality and parametrically consistent family of grids for each grid type about each configuration under study. The wing-body-horizontal families are comprised of a coarse, medium and fine grid; an optional extra-fine grid augments several of the grid families. These mesh sequences are utilized to determine asymptotic grid-convergence characteristics of the solution sets, and to estimate grid-converged absolute drag levels of the wing-body-horizontal configuration using Richardson extrapolation.

Vassberg, John C.↗

Laser Assisted Free-Free Transition in Electron - Atom Collision

Free-free transition is studied for electron-Hydrogen atom system in ground state at very low incident energies in presence of an external homogeneous, monochromatic and linearly polarized laser field. The incident electron is considered to be dressed by the laser in a non perturbative manner by choosing the Volkov solutions in both the channels. The space part of the scattering wave function for the electron is solved numerically by taking into account the effect of electron exchange, short range as well as of the long range interactions. Laser assisted differential as well as elastic total cross sections are calculated for single photon absorption/emission in the soft photon limit, the laser intensity being much less than the atomic field intensity. A strong suppression is noted in the laser assisted cross sections as compared to the field free situations. Significant difference is noted in the singlet and the triplet cross sections.

Sinha, C.↗

The CMC:3DPNS computer program for prediction of three-dimensional, subsonic, turbulent aerodynamic juncture region flow. Volume 1: Theoretical

An order-of-magnitude analysis of the subsonic three dimensional steady time averaged Navier-Stokes equations, for semibounded aerodynamic juncture geometries, yields the parabolic Navier-Stokes simplification. The numerical solution of the resultant pressure Poisson equation is cast into complementary and particular parts, yielding an iterative interaction algorithm with an exterior three dimensional potential flow solution. A parabolic transverse momentum equation set is constructed, wherein robust enforcement of first order continuity effects is accomplished using a penalty differential constraint concept within a finite element solution algorithm. A Reynolds stress constitutive equation, with low turbulence Reynolds number wall functions, is employed for closure, using parabolic forms of the two-equation turbulent kinetic energy-dissipation equation system. Numerical results document accuracy, convergence, and utility of the developed finite element algorithm, and the CMC:3DPNS computer code applied to an idealized wing-body juncture region. Additional results document accuracy aspects of the algorithm turbulence closure model.

Baker, A. J.↗

Advances in Entry Modeling for Impact Risk Assessment

A summary of recent advancements in the detailed modeling of asteroid atmospheric entry processes made through NASA’s Asteroid Threat Assessment Project (ATAP) is presented. Understanding, and accurately modeling these processes and their associated uncertainties is critical to predicting all downstream impact effects, such as blast wave and thermal damage footprints. Furthermore, there is (perhaps thankfully) a dearth of empirical data for large impactors of the kind that would pose a threat to human populations, on which to anchor and/or validate models used in risk assessments. Therefore, we must rely heavily on detailed theoretical and numerical modeling to develop robust assessments for decision makers. To that end, ATAP has made some significant progress in advancing the capabilities in this area. Two areas in particular are highlighted in the present work: meteoroid ablation mechanisms, and bolide luminosity. The first research area – meteoroid ablation mechanisms – has focused on performing novel high-enthalpy wind tunnel experiments on meteorites and meteorite analogs, and utilizing the resulting data to develop high-fidelity models for impactor mass loss at scale. These efforts have resulted in several insights. Of note, these data suggest a differential vaporization process where volatiles are liberated preferentially when the asteroidal material is subject to high heat, while refractory components remain in the molten layer on the surface. A numerical model has been developed which considers this phenomena, and its effect on the bulk impactors effective heat of ablation is examined. The second research thrust that is discussed is focused on accurate modeling of impactor radiation phenomena. While another submission to the conference will discuss the application of our approach to thermal ground damage modeling, here, we present an overview of our extensive efforts to utilize available ground- and space-based observations of large bolides (~1m diameter, and above) to inform and validate our detailed modeling approaches. These methods have been shown to accurately reconstruct the detailed spectra for the Benesov bolide, as well as approximate the burn footprint for the Tunguska event. Recent effort has focused on reconciling light curve data from multiple sources (all-sky camera networks, GLM, US government sensors), using our validated model, and providing a calibrated model for luminous efficiency which can then be utilized to infer impactor properties such as shape, mass, and composition. This work will be demonstrated through an exemplar case study focusing on a large bolide event with observational data (e.g. Chelyabinsk, Flensburg, etc.). Finally, our team’s assessment on the current maturity of atmospheric entry models, and priorities for future research will be provided.

ATAP↗

Fully quantum algorithm for mesoscale fluid simulations with application to partial differential equations

Fluid flow simulations marshal our most powerful computational resources. In many cases, even this is not enough. Quantum computers provide an opportunity to speed up traditional algorithms for flow simulations. We show that lattice-based mesoscale numerical methods can be executed as efficient quantum algorithms due to their statistical features. This approach revises a quantum algorithm for lattice gas automata to reduce classical computations and state preparation at every time step. For this, the algorithm approximates the qubit relative phases and subtracts them at the end of each time step. Phases are evaluated using the iterative phase estimation algorithm and subtracted using single-qubit rotation phase gates. Further, this method optimizes the quantum resource required and makes it more appropriate for near-term quantum hardware. We also demonstrate how the checkerboard deficiency that the D1Q2 scheme presents can be resolved using the D1Q3 scheme. The algorithm is validated by simulating two canonical partial differential equations: the diffusion and Burgers' equations on different quantum simulators. We find good agreement between quantum simulations and classical solutions for the presented algorithm.

97 MATHEMATICS AND COMPUTING↗

Numerical Investigation of Hot Gas Ingestion by STOVL Aircraft

This report compiles the various research activities conducted under the auspices of the NASA Grant NAG3-1026, "Numerical Investigation of Hot Gas Ingestion by STOVL Aircraft" during the period of April 1989 to April 1994. The effort involved the development of multigrid based algorithms and computer programs for the calculation of the flow and temperature fields generated by Short Take-off and Vertical Landing (STOVL) aircraft, while hovering in ground proximity. Of particular importance has been the interaction of the exhaust jets with the head wind which gives rise to the hot gas ingestion process. The objective of new STOVL designs to reduce the temperature of the gases ingested into the engine. The present work describes a solution algorithm for the multi-dimensional elliptic partial-differential equations governing fluid flow and heat transfer in general curvilinear coordinates. The solution algorithm is based on the multigrid technique which obtains rapid convergence of the iterative numerical procedure for the discrete equations. Initial efforts were concerned with the solution of the Cartesian form of the equations. This algorithm was applied to a simulated STOVL configuration in rectangular coordinates. In the next phase of the work, a computer code for general curvilinear coordinates was constructed. This was applied to model STOVL geometries on curvilinear grids. The code was also validated in model problems. In all these efforts, the standard k-Epsilon model was used.

Vanka, S. P.↗

Numerical simulation of the world ocean circulation

A multi-level model, based on the primitive equations, is developed for simulating the temperature and velocity fields produced in the world ocean by differential heating and surface wind stress. The model ocean has constant depth, free slip at the lower boundary, and neglects momentum advection; so that there is no energy exchange between the barotropic and baroclinic components of the motion, although the former influences the latter through temperature advection. The ocean model was designed to be coupled to the UCLA atmospheric general circulation model, for the study of the dynamics of climate and climate changes. But here, the model is tested by prescribing the observed seasonally varying surface wind stress and the incident solar radiation, the surface air temperature and humidity, cloudiness and the surface wind speed, which, together with the predicted ocean surface temperature, determine the surface flux of radiant energy, sensible heat and latent heat.

Takano, K.↗

Elastic Differential Cross Sections

The eikonal, partial wave (PW) Lippmann-Schwinger, and three-dimensional Lippmann-Schwinger (LS3D) methods are compared for nuclear reactions that are relevant for space radiation applications. Numerical convergence of the eikonal method is readily achieved when exact formulas of the optical potential are used for light nuclei (A less than or equal to 16) and the momentum-space optical potential is used for heavier nuclei. The PW solution method is known to be numerically unstable for systems that require a large number of partial waves, and, as a result, the LS3D method is employed. The effect of relativistic kinematics is studied with the PW and LS3D methods and is compared to eikonal results. It is recommended that the LS3D method be used for high energy nucleon- nucleus reactions and nucleus-nucleus reactions at all energies because of its rapid numerical convergence and stability.

Werneth, Charles M.↗

Quantifying local and global mass balance errors in physics-informed neural networks

Physics-informed neural networks (PINN) have recently become attractive for solving partial differential equations (PDEs) that describe physics laws. By including PDE-based loss functions, physics laws such as mass balance are enforced softly in PINN. This paper investigates how mass balance constraints are satisfied when PINN is used to solve the resulting PDEs. We investigate PINN’s ability to solve the 1D saturated groundwater flow equations (diffusion equations) for homogeneous and heterogeneous media and evaluate the local and global mass balance errors. We compare the obtained PINN’s solution and associated mass balance errors against a two-point finite volume numerical method and the corresponding analytical solution. We also evaluate the accuracy of PINN in solving the 1D saturated groundwater flow equation with and without incorporating hydraulic heads as training data. We demonstrate that PINN’s local and global mass balance errors are significant compared to the finite volume approach. Tuning the PINN’s hyperparameters, such as the number of collocation points, training data, hidden layers, nodes, epochs, and learning rate, did not improve the solution accuracy or the mass balance errors compared to the finite volume solution. Mass balance errors could considerably challenge the utility of PINN in applications where ensuring compliance with physical and mathematical properties is crucial.

54 ENVIRONMENTAL SCIENCES↗

A low dimensional dynamical system for the wall layer

Low dimensional dynamical systems which model a fully developed turbulent wall layer were derived.The model is based on the optimally fast convergent proper orthogonal decomposition, or Karhunen-Loeve expansion. This decomposition provides a set of eigenfunctions which are derived from the autocorrelation tensor at zero time lag. Via Galerkin projection, low dimensional sets of ordinary differential equations in time, for the coefficients of the expansion, were derived from the Navier-Stokes equations. The energy loss to the unresolved modes was modeled by an eddy viscosity representation, analogous to Heisenberg's spectral model. A set of eigenfunctions and eigenvalues were obtained from direct numerical simulation of a plane channel at a Reynolds number of 6600, based on the mean centerline velocity and the channel width flow and compared with previous work done by Herzog. Using the new eigenvalues and eigenfunctions, a new ten dimensional set of ordinary differential equations were derived using five non-zero cross-stream Fourier modes with a periodic length of 377 wall units. The dynamical system was integrated for a range of the eddy viscosity prameter alpha. This work is encouraging.

Aubry, N.↗

SubTSBR to tackle high noise and outliers for data-driven discovery of differential equations

Data-driven discovery of differential equations has been an emerging research topic. We propose a novel algorithm subsampling-based threshold sparse Bayesian regression (SubTSBR) to tackle high noise and outliers. The subsampling technique is used for improving the accuracy of the Bayesian learning algorithm. It has two parameters: subsampling size and the number of subsamples. When the subsampling size increases with fixed total sample size, the accuracy of our algorithm goes up and then down. When the number of subsamples increases, the accuracy of our algorithm keeps going up. We demonstrate how to use our algorithm step by step and compare our algorithm with threshold sparse Bayesian regression (TSBR) for the discovery of differential equations. We show that our algorithm produces better results. We also discuss the merits of discovering differential equations from data and demonstrate how to discover models with random initial and boundary condition as well as models with bifurcations. The numerical examples are: (1) predator-prey model with noise, (2) shallow water equations with outliers, (3) heat diffusion with random initial and boundary condition, and (4) fish-harvesting problem with bifurcations.

97 MATHEMATICS AND COMPUTING↗

Bayesian Optimization Framework for Imperfect Data or Models

Conventional Bayesian optimization methods implicitly assume that the data and model being optimized are “perfect.” This assumption leads to inaccurate posterior probability distribution functions (PDFs) when applied to “imperfect” data or models. The new Bayesian optimization framework presented in this report provides a way to parameterize the effect of imperfections usually encountered in a prior PDF of generalized data or a model on the posterior PDF. The effects of imperfections are parameterized by a set of constraints imposed on the posterior expectation values of deviations between the data and the model and on their covariance matrix elements. A particular set of values for these constraints conveys an evaluator’s best estimate of the effect of imperfections on the corresponding posterior expectation values. When a prior PDF of generalized data is assumed to be normal, an expression for a posterior PDF satisfying an arbitrary set of constraints is derived analytically for linear models. An analogous iterative algorithm is given for nonlinear models. The corresponding posterior PDF should be used to estimate any posterior expectation values in the presence of imperfections parameterized by that set of constraints. A posterior PDF of a conventional Bayesian optimization method is recovered analytically when all evaluator-specified constraints are set to zero (i.e., in the absence of any imperfections). The analytical expressions derived in this report for normal PDFs and linear models were verified numerically by a Metropolis–Hastings Monte Carlo method. The methods presented herein could be applied to any kind of data or models, including differential cross-section data or integral benchmark experiments.

97 MATHEMATICS AND COMPUTING↗

Taylor wave solution for a general equation of state

This document describes a solution procedure for calculating the Taylor wave behind an unsupported Chapman–Jouguet (CJ) detonation in planar, cylindrical, and spherical geometries given a general equation of state. The resulting semi-analytic solution can be utilized to examine new equation of state models for detonation products and during the verification of hydrodynamic codes. The governing partial differential equations are reduced to ordinary differential equations in both characteristic and self-similar forms. The first-order systems corresponding to each geometry are amenable to solution numerically using commonly available methods. A difficulty arises at the CJ point in radial coordinates where the similarity equations become singular. Two separate strategies are proposed to integrate the first-order system. The first one uses an asymptotic approximation near the CJ point that can be used to perturb the boundary conditions. The second one applies a change of variables which removes the singularity at the expense of an additional equation to be integrated. A test problem is provided for the Davis products equation of state to illustrate the qualitative features of the Taylor wave in each geometric configuration and compared with a Lagrangian hydrodynamics research code. A Python code listing gives an implementation using the SciPy library to assists users in generating the results.

97 MATHEMATICS AND COMPUTING↗