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NASA NTRS2004

Dynamics of Drag Free Formations in Earth Orbit

In this paper the translational equations of motion of a formation of n spacecraft in Earth orbit, n(sub f) of which are drag-free spacecraft, are derived in a coordinate-free manner using the balance of linear momentum and direct tensor notation. A drag-free spacecraft consists of a spacecraft bus and a proof mass shielded from external disturbances in an internal cavity. By controlling the spacecraft so that the proof mass remains centered in the cavity, the spacecraft follows a purely gravitational orbit. The results described in this paper provide a first step toward coupling drag-free control technology with formation flying in order to mitigate the effect of differential aerodynamic drag on formation flying missions (e.g., Earth imaging applications) in low Earth orbit.

drag-free technology↗

NASA NTRS1976

Stability of elastic bending and torsion of uniform cantilever rotor blades in hover with variable structural coupling

The stability of elastic flap bending, lead-lag bending, and torsion of uniform, untwisted, cantilever rotor blades without chordwise offsets between the elastic, mass, tension, and areodynamic center axes is investigated for the hovering flight condition. The equations of motion are obtained by simplifying the general, nonlinear, partial differential equations of motion of an elastic rotating cantilever blade. The equations are adapted for a linearized stability analysis in the hovering flight condition by prescribing aerodynamic forces, applying Galerkin's method, and linearizing the resulting ordinary differential equations about the equilibrium operating condition. The aerodynamic forces are obtained from strip theory based on a quasi-steady approximation of two-dimensional unsteady airfoil theory. Six coupled mode shapes, calculated from free vibration about the equilibrium operating condition, are used in the linearized stability analysis. The study emphasizes the effects of two types of structural coupling that strongly influence the stability of hingeless rotor blades. The first structural coupling is the linear coupling between flap and lead-lag bending of the rotor blade. The second structural coupling is a nonlinear coupling between flap bending, lead-lag bending, and torsion deflections. Results are obtained for a wide variety of hingeless rotor configurations and operating conditions in order to provide a reasonably complete picture of hingeless rotor blade stability characteristics.

Hodges, D. H., Roberta.↗

DOE OSTI2021

Mixing Cell Model: A One-Dimensional Numerical Model for Assessment of Water Flow and Contaminant Transport in the Unsaturated Zone

This report describes the Mixing Cell Model code, a one dimensional model for water flow and solute transport in the unsaturated zone under steady state or transient flow conditions. The model is based on the principles and assumptions underlying mixing-cell model formulations. The unsaturated zone is discretized into a series of independent mixing cells. Each cell may have unique hydrologic, lithologic, and sorptive properties. Ordinary differential equations describe the material (water and solute) balance within each cell. Water-flow equations are derived from the continuity equation, assuming that unit gradient conditions exist at all times in each cell. Pressure gradients are considered implicitly through model discretization. Unsaturated hydraulic conductivity and moisture contents are determined by the material specific moisture-characteristic curves. Solute-transport processes include explicit treatment of advective processes, first order chain decay, and linear sorption reactions. Dispersion is addressed through implicit and explicit dispersion. Implicit dispersion is an inherent feature of all mixing-cell models and originates from the formulation of the problem in terms of mass balance around fully mixed volume elements. Expressions are provided that relate implicit dispersion to the physical dispersion of the system. Two FORTRAN codes were developed to solve the water flow and solute-transport equations: (1) the Mixing Cell Model for Flow (MCMF) solves transient water-flow problems and (2) the Mixing Cell Model for Transport (MCMT) solves the solute-transport problem. The transient water-flow problem is typically solved first by estimating the water flux through each cell in the model domain as a function of time using the MCMF code. These data are stored in either ASCII or binary files that are later read by the solute transport code (MCMT). Code output includes solute pore water concentrations, water and solute inventories in each cell and at each specified output time, and water and solute fluxes through each cell and specified output time. Computer run times for coupled transient water flow and solute transport were typically several seconds on a 2 GHz Intel Pentium IV desktop computer. The model was benchmarked against analytical solutions and finite element approximations to the partial differential equations (PDE) describing unsaturated flow and transport. Differences between the maximum solute flux estimated by the mixing cell model and the PDE models were typically less than two percent. This revision includes an option for a fixed concentration lower boundary condition for diffusive fluxes for versions 020321 and later.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS1990

Theory of biaxial graded-index optical fiber

A biaxial graded-index fiber with a homogeneous cladding is studied. Two methods, wave equation and matrix differential equation, of formulating the problem and their respective solutions are discussed. For the wave equation formulation of the problem it is shown that for the case of a diagonal permittivity tensor the longitudinal electric and magnetic fields satisfy a pair of coupled second-order differential equations. Also, a generalized dispersion relation is derived in terms of the solutions for the longitudinal electric and magnetic fields. For the case of a step-index fiber, either isotropic or uniaxial, these differential equations can be solved exactly in terms of Bessel functions. For the cases of an istropic graded-index and a uniaxial graded-index fiber, a solution using the Wentzel, Krammers and Brillouin (WKB) approximation technique is shown. Results for some particular permittivity profiles are presented. Also the WKB solutions is compared with the vector solution found by Kurtz and Streifer. For the matrix formulation it is shown that the tangential components of the electric and magnetic fields satisfy a system of four first-order differential equations which can be conveniently written in matrix form. For the special case of meridional modes, the system of equations splits into two systems of two equations. A general iterative technique, asymptotic partitioning of systems of equations, for solving systems of differential equations is presented. As a simple example, Bessel's differential equation is written in matrix form and is solved using this asymptotic technique. Low order solutions for particular examples of a biaxial and uniaxial graded-index fiber are presented. Finally numerical results obtained using the asymptotic technique are presented for particular examples of isotropic and uniaxial step-index fibers and isotropic, uniaxial and biaxial graded-index fibers.

Kawalko, Stephen F.↗

DOE OSTI2021

Spectral deferred correction methods for high-order accuracy in poroelastic problems

In this work, we investigate high-order accuracy in time integration by examining two operator splitting methods for poroelastic problems: the two-pass and the spectral deferred correction (SDC) methods. To enhance the order of accuracy, the two-pass method partitions a coupled operator symmetrically, whereas the SDC method corrects truncation errors by establishing an error equation. These high-order methods are applied to underlying solution strategies, i.e., monolithic, fixed-stress sequential, and undrained sequential methods. We observe that semi-discretized systems from spatial discretization have forms similar to those of index-1 differential algebraic equations (DAEs), causing order reduction against the two-pass method when it is used in conjunction with either the monolithic or sequential method. On the other hand, the SDC in conjunction with the monolithic method exhibits the desired second-order accuracy in poroelastic problems while increasing the order of accuracy for index-1 DAEs. However, the SDC in conjunction with either of the two sequential methods does not achieve the desired order of accuracy, and maintains first order because the flow equation for poroelasticity has an additional approximation associated with the volumetric strain rate term, which does not yield exactly the same forms as those of conventional DAEs. Thus, the monolithic SDC method can achieve higher-order accuracy, but may require higher computational costs because it involves solving matrix systems larger than those for the sequential methods.

02 PETROLEUM↗

NASA NTRS1998

Analysis and Testing of Plates with Piezoelectric Sensors and Actuators

Piezoelectric material inherently possesses coupling between electrostatics and structural dynamics. Utilizing linear piezoelectric theory results in an intrinsically coupled pair of piezoelectric constitutive equations. One equation describes the direct piezoelectric effect where strains produce an electric field and the other describes the converse effect where an applied electrical field produces strain. The purpose of this study is to compare finite element analysis and experiments of a thin plate with bonded piezoelectric material. Since an isotropic plate in combination with a thin piezoelectric layer constitutes a special case of a laminated composite, the classical laminated plate theory is used in the formulation to accommodated generic laminated composite panels with multiple bonded and embedded piezoelectric layers. Additionally, the von Karman large deflection plate theory is incorporated. The formulation results in laminate constitutive equations that are amiable to the inclusion of the piezoelectric constitutive equations yielding in a fully electro-mechanically coupled composite laminate. Using the finite element formulation, the governing differential equations of motion of a composite laminate with embedded piezoelectric layers are derived. The finite element model not only considers structural degrees of freedom (d.o.f.) but an additional electrical d.o.f. for each piezoelectric layer. Comparison between experiment and numerical prediction is performed by first treating the piezoelectric as a sensor and then again treating it as an actuator. To assess the piezoelectric layer as a sensor, various uniformly distributed pressure loads were simulated in the analysis and the corresponding generated voltages were calculated using both linear and nonlinear finite element analyses. Experiments were carried out by applying the same uniformly distributed loads and measuring the resulting generated voltages and corresponding maximum plate deflections. It is found that a highly nonlinear relationship exists between maximum deflection and voltage versus pressure loading. In order to assess comparisons of predicted and measured piezoelectric actuation, sinusoidal excitation voltages are simulated/applied and maximum deflections are calculated/measured. The maximum deflection as a function of time was determined using the linear finite elements analysis. Good correlation between prediction and measurement was achieved in all cases.

Bevan, Jeffrey S.↗

DOE OSTI2024

Sparsified time-dependent Fourier neural operators for fusion simulations

This paper presents a sparsified Fourier neural operator for coupled time-dependent partial differential equations (ST-FNO) as an efficient machine learning surrogate for fluid and particle-based fusion codes such as NIMROD (Non-Ideal Magnetohydrodynamics with Rotation - Open Discussion) and GTC (Gyrokinetic Toroidal Code). ST-FNO leverages the structures in the governing equations and utilizes neural operators to represent Green's function-like numerical operators in the corresponding numerical solvers. Once trained, ST-FNO can rapidly and accurately predict dynamics in fusion devices compared with first-principle numerical algorithms. In general, ST-FNO represents an efficient and accurate machine learning surrogate for numerical simulators for multi-variable nonlinear time-dependent partial differential equations, with the proposed architectures and loss functions. The efficacy of ST-FNO has been demonstrated using quiescent H-mode simulation data from NIMROD and kink-mode simulation data from GTC. The ST-FNO H-mode results show orders of magnitude reduction in memory and central processing unit usage in comparison with the numerical solvers in NIMROD when computing fields over a selected poloidal plane. The ST-FNO kink-mode results achieve a factor of 2 reduction in the number of parameters compared to baseline FNO models without accuracy loss.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1993

A new flux conserving Newton's method scheme for the two-dimensional, steady Navier-Stokes equations

A new numerical method is developed for the solution of the two-dimensional, steady Navier-Stokes equations. The method that is presented differs in significant ways from the established numerical methods for solving the Navier-Stokes equations. The major differences are described. First, the focus of the present method is on satisfying flux conservation in an integral formulation, rather than on simulating conservation laws in their differential form. Second, the present approach provides a unified treatment of the dependent variables and their unknown derivatives. All are treated as unknowns together to be solved for through simulating local and global flux conservation. Third, fluxes are balanced at cell interfaces without the use of interpolation or flux limiters. Fourth, flux conservation is achieved through the use of discrete regions known as conservation elements and solution elements. These elements are not the same as the standard control volumes used in the finite volume method. Fifth, the discrete approximation obtained on each solution element is a functional solution of both the integral and differential form of the Navier-Stokes equations. Finally, the method that is presented is a highly localized approach in which the coupling to nearby cells is only in one direction for each spatial coordinate, and involves only the immediately adjacent cells. A general third-order formulation for the steady, compressible Navier-Stokes equations is presented, and then a Newton's method scheme is developed for the solution of incompressible, low Reynolds number channel flow. It is shown that the Jacobian matrix is nearly block diagonal if the nonlinear system of discrete equations is arranged approximately and a proper pivoting strategy is used. Numerical results are presented for Reynolds numbers of 100, 1000, and 2000. Finally, it is shown that the present scheme can resolve the developing channel flow boundary layer using as few as six to ten cells per channel width, depending on the Reynolds number.

Scott, James R.↗

NASA NTRS2007

Enhanced Elliptic Grid Generation

An enhanced method of elliptic grid generation has been invented. Whereas prior methods require user input of certain grid parameters, this method provides for these parameters to be determined automatically. "Elliptic grid generation" signifies generation of generalized curvilinear coordinate grids through solution of elliptic partial differential equations (PDEs). Usually, such grids are fitted to bounding bodies and used in numerical solution of other PDEs like those of fluid flow, heat flow, and electromagnetics. Such a grid is smooth and has continuous first and second derivatives (and possibly also continuous higher-order derivatives), grid lines are appropriately stretched or clustered, and grid lines are orthogonal or nearly so over most of the grid domain. The source terms in the grid-generating PDEs (hereafter called "defining" PDEs) make it possible for the grid to satisfy requirements for clustering and orthogonality properties in the vicinity of specific surfaces in three dimensions or in the vicinity of specific lines in two dimensions. The grid parameters in question are decay parameters that appear in the source terms of the inhomogeneous defining PDEs. The decay parameters are characteristic lengths in exponential- decay factors that express how the influences of the boundaries decrease with distance from the boundaries. These terms govern the rates at which distance between adjacent grid lines change with distance from nearby boundaries. Heretofore, users have arbitrarily specified decay parameters. However, the characteristic lengths are coupled with the strengths of the source terms, such that arbitrary specification could lead to conflicts among parameter values. Moreover, the manual insertion of decay parameters is cumbersome for static grids and infeasible for dynamically changing grids. In the present method, manual insertion and user specification of decay parameters are neither required nor allowed. Instead, the decay parameters are determined automatically as part of the solution of the defining PDEs. Depending on the shape of the boundary segments and the physical nature of the problem to be solved on the grid, the solution of the defining PDEs may provide for rates of decay to vary along and among the boundary segments and may lend itself to interpretation in terms of one or more physical quantities associated with the problem.

Kaul, Upender K.↗

DOE OSTI2022

A new moment-based general-relativistic neutrino-radiation transport code: Methods and first applications to neutron star mergers

ABSTRACT We present a new moment-based energy-integrated neutrino transport code for neutron star merger simulations in general relativity. In the merger context, ours is the first code to include Doppler effects at all orders in υ/c, retaining all non-linear neutrino–matter coupling terms. The code is validated with a stringent series of tests. We show that the inclusion of full neutrino–matter coupling terms is necessary to correctly capture the trapping of neutrinos in relativistically moving media, such as in differentially rotating merger remnants. We perform preliminary simulations proving the robustness of the scheme in simulating ab-initio mergers to black hole collapse and long-term neutron star remnants up to ${\sim }70\,$ ms. The latter is the longest dynamical space-time, 3D, general relativistic simulations with full neutrino transport to date. We compare results obtained at different resolutions and using two different closures for the moment scheme. We do not find evidences of significant out-of-thermodynamic equilibrium effects, such as bulk viscosity, on the post-merger dynamics or gravitational wave emission. Neutrino luminosities and average energies are in good agreement with theory expectations and previous simulations by other groups using similar schemes. We compare dynamical and early wind ejecta properties obtained with M1 and with our older neutrino treatment. We find that the M1 results have systematically larger proton fractions. However, the differences in the nucleosynthesis yields are modest. This work sets the basis for future detailed studies spanning a wider set of neutrino reactions, binaries, and equations of state.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1993

Flexure and the role of inplane force around coronae on Venus

Large coronae on Venus, such as Artemis and Latona, are rimmed by conspicuous trenches and associated outer rises. Sandwell and Schubert have observed that these systems resemble terrestrial subduction zones in planform and have succeeded in fitting an elastic plate bending equation to the inferred flexural topography. However, the first zero crossing bending moments required are -2.5 x 10(exp 17) N for Artemis and -5.0 x 10(exp 16) N for Latona. Since these moments are similar in magnitude to those of subducting slabs on Earth, a rollback subduction mechanism was proposed to explain the flexure around the largest coronae, although a differential thermal subsidence model is sufficient to account for the topography around some coronae. The purpose is to investigate the effect of inplane force as a possible alternative to large applied moments in producing flexure at Artemis and Latona. The close correlation of gravity to topography on Venus implies the absence of a low viscosity zone and the strong coupling of the lithosphere to mantle convection. If coronae are the surface manifestations of mantle plumes, they may be the sites of active convective stress coupling. As the upwelling reaches the lithosphere, it spreads radially outward, inducing shear tractions on the base of the plate. In addition, the hot, expanding corona may load the surrounding plate horizonally. Both the basal shear stresses and radial loading can be treated as an equivalent compressive inplane force in the mechanical lithosphere, which contributes to the bending of the outlying plate. Using a model that relates inplane force to the measured gravity anomalies, a rough value of the inplane force at Artemis was calculated. Recent Pioneer Venus spherical harmonic gravity models indicate a geoid anomaly of about 75 m over Artemis, which corresponds to an estimated inplane force on the order of -1x10(exp 13) N/m. The gravity model is unable to resolve Latona, but an inplane force of similar dimensions is assumed. The maximum possible inplane force based on the expected rheology can be constrained by using the approximate 5 K/km thermal gradient inferred from the best fit 30 km elastic plate at Artemis and Latona. For a dry olivine flow law in the upper mantle, the compressional load limit of the 60 km thick mechanical lithosphere is -4 x 10(exp 13) N/m. This value is equivalent to a load of -8 x 10(exp 13) N/m on a 30 km thick elastic plate.

Brown, C. David↗

DOE OSTI2020

Learning and meta-learning of stochastic advection–diffusion–reaction systems from sparse measurements

Physics-informed neural networks (PINNs) were recently proposed in [18] as an alternative way to solve partial differential equations (PDEs). A neural network (NN) represents the solution, while a PDE-induced NN is coupled to the solution NN, and all differential operators are treated using automatic differentiation. Here, we first employ the standard PINN and a stochastic version, sPINN, to solve forward and inverse problems governed by a non-linear advection–diffusion–reaction (ADR) equation, assuming we have some sparse measurements of the concentration field at random or pre-selected locations. Subsequently, we attempt to optimise the hyper-parameters of sPINN by using the Bayesian optimisation method (meta-learning) and compare the results with the empirically selected hyper-parameters of sPINN. In particular, for the first part in solving the inverse deterministic ADR, we assume that we only have a few high-fidelity measurements, whereas the rest of the data is of lower fidelity. Hence, the PINN is trained using a composite multi-fidelity network, first introduced in [12], that learns the correlations between the multi-fidelity data and predicts the unknown values of diffusivity, transport velocity and two reaction constants as well as the concentration field. For the stochastic ADR, we employ a Karhunen–Loève (KL) expansion to represent the stochastic diffusivity, and arbitrary polynomial chaos (aPC) to represent the stochastic solution. Correspondingly, we design multiple NNs to represent the mean of the solution and learn each aPC mode separately, whereas we employ a separate NN to represent the mean of diffusivity and another NN to learn all modes of the KL expansion. For the inverse problem, in addition to stochastic diffusivity and concentration fields, we also aim to obtain the (unknown) deterministic values of transport velocity and reaction constants. The available data correspond to 7spatial points for the diffusivity and 20 space–time points for the solution, both sampled 2000 times. We obtain good accuracy for the deterministic parameters of the order of 1–2% and excellent accuracy for the mean and variance of the stochastic fields, better than three digits of accuracy. In the second part, we consider the previous stochastic inverse problem, and we use Bayesian optimisation to find five hyper-parameters of sPINN, namely the width, depth and learning rate of two NNs for learning the modes. Here, we obtain much deeper and wider optimal NNs compared to the manual tuning, leading to even better accuracy, i.e., errors less than 1% for the deterministic values, and about an order of magnitude less for the stochastic fields.

97 MATHEMATICS AND COMPUTING↗