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At least 1,279 records · Page 71

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.↗

A fast Fourier transform-based mesoscale field dislocation mechanics study of grain size effects and reversible plasticity in polycrystals

A numerical implementation of a non-local polycrystal plasticity theory based on a mesoscale version of the field dislocation mechanics theory (MFDM) of Acharya and Roy (2006) is presented using small-strain elasto-viscoplastic fast Fourier transform-based (EVPFFT) algorithm developed by Lebensohn et al. (2012). In addition to considering plastic flow and hardening only due to SSDs (statistically stored dislocations) as in the classic EVPFFT framework, the proposed method accounts for the evolution of GND (geometrically necessary dislocations) densities solving a hyperbolic-type partial differential equation, and GND effects on both plastic flow and hardening. This allows consideration of an enhanced strain-hardening law that includes the effect of the GND density tensor. The numerical implementation of a reduced version of the MFDM is presented in the framework of the FFT-based augmented Lagrangian procedure of Michel et al. (2001). A Finite Differences scheme combined with discrete Fourier transforms is applied to solve both incompatibility and equilibrium equations. The numerical procedure named MFDM-EVPFFT is used to perform full field simulations of polycrystal plasticity considering different grain sizes and their mechanical responses during monotonic tensile and reversible tension-compression tests. Using Voronoi tessellation and periodic boundary conditions, voxelized representative volume elements (RVEs) with different grain sizes are generated. With MFDM-EVPFFT, a Hall-Petch type scaling law is obtained in contrast with the conventional crystal plasticity EVPFFT. In the case of reversible plasticity, a stronger Bauschinger effect is observed with the MFDM-EVPFFT approach in comparison with conventional EVPFFT. Here, the origin of these differences is analyzed in terms of heterogeneity, GND density and stress evolutions during the compression stage.

36 MATERIALS SCIENCE↗

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↗

Impulsive model for reactive collisions

A simple classical mechanical model of the reactive scattering of a structureless atom A and a quasi-diatomic BC is developed which takes full advantage of energy, linear and angular momentum conservation relations but introduces a minimum of further assumptions. These are as follows: (1) the vibrational degree of freedom of the reactant (BC) and product (AB) molecules is suppressed, so the change in vibrational energy is simply a parameter; (2) straight-line trajectories are assumed outside of a reaction shell; (3) within this zone, momentum transfer occurs impulsively (essentially instantaneously) following mass transfer; (4) the impulse, which may be either positive or negative, is directed along the BC axis, which may, however, assume all orientations with respect to the incident relative velocity. The model yields differential and total cross sections and product rotational energy distributions for a given collision exoergicity Q, or for any known distribution over Q. Numerical results are presented for several prototype reactions whose dynamics have been well-studied.

Marron, M. T.↗

A semi-analytical method of computation of oceanic tidal perturbations in the motion of artificial satellites

The method of expansion of the satellite's perturbations, as caused by the oceanic tides, into Fourier series is discussed. The coefficients of the expansion are purely numerical and peculiar to each particular satellite. Such a method is termed as semi-analytical in celestial mechanics. Gaussian form of the differential equations for variation of elements, with the right hand sides averaged over the orbit of the satellite, is convenient to use with the semi-analytical expansion.

Musen, P.↗

Quasi-static extension of a tensile crack contained in a viscoelastic-plastic solid

Final stretch criterion of failure is applied to the problem of quasi-static extension of a crack embedded in an elastic-plastic or viscoelastic-plastic matrix. The slow growth under subcritical conditions in a rate-sensitive Tresca solid is shown to be a superposition of creep rupture and McClintock's ductile growth. This type of growth occurs at subcritical magnitude of the imposed K-factor and can be accounted for only through a recognition of inelastic properties of solids. In the subcritical range there is no unique value for K sub c independent of geometrical configuration and flaw size. Not only the produced states of stress and strain are dependent on the loading path, but also the material resistance to fracture turns out to be a function of the history of loading that precedes catastrophic failure. A nonlinear integro-differential equation of motion is derived for a crack progressing through a viscoelastic medium with some limited ability to plastic flow. Examples of numerical integration are given incorporating both monotonic and cyclic loading programs.

Wnuk, M. P.↗

Estimation of material properties of a nonlinearly elastic bar

A method of determining certain characteristic flexural rigidities and elastic properties of nonlinearly elastic materials is presented. An estimation method utilizing perturbation methods and a least squares fitting technique is used to solve the nonlinear differential equation derived from the moment curvature relation, subject to boundary values representing deflections of the bar at discrete points. Deflection data from numerical simulations of a nonlinearly elastic, prismatic bar are used to demonstrate the estimation method. Numerical experiments relating the accuracy of the identification to the number and accuracy of the boundary values are presented. Conclusions based on the numerical experiments are included.

Baker, B. E.↗

Error behavior of multistep methods applied to unstable differential systems

The problem of modeling a dynamic system described by a system of ordinary differential equations which has unstable components for limited periods of time is discussed. It is shown that the global error in a multistep numerical method is the solution to a difference equation initial value problem, and the approximate solution is given for several popular multistep integration formulas. Inspection of the solution leads to the formulation of four criteria for integrators appropriate to unstable problems. A sample problem is solved numerically using three popular formulas and two different stepsizes to illustrate the appropriateness of the criteria.

Brown, R. L.↗

Error behaviour of multistep methods applied to unstable differential systems

The problem of modelling a dynamic system described by a system of ordinary differential equations which has unstable components for limited periods of time is discussed. It is shown that the global error in a multistep numerical method is the solution to a difference equation initial value problem, and the approximate solution is given for several popular multistep integration formulae. Inspection of the solution leads to the formulation of four criteria for integrators appropriate to unstable problems. A sample problem is solved numerically using three popular formulae and two different stepsizes to illustrate the appropriateness of the criteria.

Brown, R. L.↗

Two-Dimensional Grids About Airfoils and Other Shapes

GRAPE computer program generates two-dimensional finite-difference grids about airfoils and other shapes by use of Poisson differential equation. GRAPE can be used with any boundary shape, even one specified by tabulated points and including limited number of sharp corners. Numerically stable and computationally fast, GRAPE provides aerodynamic analyst with efficient and consistant means of grid generation.

Sorenson, R.↗