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DOE OSTI2021

Elastic Wave Propagation in Curvilinear Coordinates with Mesh Refinement Interfaces by a Fourth Order Finite Difference Method

In this work, we develop a fourth order accurate finite difference method for the three dimensional elastic wave equation in isotropic media with the piecewise smooth material property. In our model, the material property can be discontinuous at curved interfaces. The governing equations are discretized in second order form on curvilinear meshes by using a fourth order finite difference operator satisfying a summation-by-parts property. The method is energy stable and high order accurate. The highlight is that mesh sizes can be chosen according to the velocity structure of the material so that computational efficiency is improved. At the mesh refinement interfaces with hanging nodes, physical interface conditions are imposed by using ghost points and interpolation. With a fourth order predictor-corrector time integrator, the fully discrete scheme is energy conserving. Numerical experiments are presented to verify the fourth order convergence rate and the energy conserving property.

58 GEOSCIENCES↗

NASA NTRS1985

Modeling the effects of latitudinal gradients in stellar winds, with application to the solar wind

A steady, axisymmetric, quasi-radial, global model previously developed for stellar winds with embedded magnetic fields has been extended to include latitudinal gradient effects on the azimuthal velocity and magnetic field. The linear results at large radii are presented for large-amplitude latitudinal variations in the radial magnetic field, mass loss rate, and radial velocity of the wind. The magnetohydrodynamic (MHD) equations predict meridional flows that develop naturally from internal magnetic stresses. The flows open flux tubes in the star's equatorial plane, redistributing mass and magnetic flux as a function of stellar latitude. The plasma spins up to conserve angular momentum in fields and plasma. The results are generally applicable to stellar winds (including radiatively driven winds), provided that the internal structure is not dominated by rotation. The asymptotic solutions do not explicitly depend on the form of the energy equation, although the assumed O(1) state which drives these solutions depends on the deposition of energy and momentum throughout the wind.

Nerney, S.↗

DOE OSTI2024

Physics-informed State-space Neural Networks for transport phenomena

This work introduces Physics -informed State -space neural network Models (PSMs), a novel solution to achieving real-time optimization, flexibility, and fault tolerance in autonomous systems, particularly in transportdominated systems such as chemical, biomedical, and power plants. Traditional data -driven methods fall short due to a lack of physical constraints like mass conservation; PSMs address this issue by training deep neural networks with sensor data and physics -informing using components' Partial Differential Equations (PDEs), resulting in a physics -constrained, end -to -end differentiable forward dynamics model. Further, through two in silico experiments - a heated channel and a cooling system loop - we demonstrate that PSMs offer a more accurate approach than a purely data -driven model. In the former experiment, PSMs demonstrated significantly lower average root -mean -square errors across test datasets compared to a purely data -driven neural network, with reductions of 44 %, 48 %, and 94 % in predicting pressure, velocity, and temperature, respectively. Beyond accuracy, PSMs demonstrate a compelling multitask capability, making them highly versatile. In this work, we showcase two: supervisory control of a nonlinear system through a sequentially updated state -space representation and the proposal of a diagnostic algorithm using residuals from each of the PDEs. The former demonstrates PSMs' ability to handle constant and time -dependent constraints, while the latter illustrates their value in system diagnostics and fault detection.

42 ENGINEERING↗

DOE OSTI2023

Semiclassical theory and the Koopman-van Hove equation *

Abstract The phase space Koopman-van Hove (KvH) equation can be derived from the asymptotic semiclassical analysis of partial differential equations. Semiclassical theory yields the Hamilton–Jacobi equation for the complex phase factor and the transport equation for the amplitude. These two equations can be combined to form a nonlinear semiclassical version of the KvH equation in configuration space. There is a natural injection of configuration space solutions into phase space and a natural projection of phase space solutions onto configuration space. Hence, every solution of the configuration space KvH equation satisfies both the semiclassical phase space KvH equation and the Hamilton–Jacobi constraint. For configuration space solutions, this constraint resolves the paradox that there are two different conserved densities in phase space. For integrable systems, the KvH spectrum is the Cartesian product of a classical and a semiclassical spectrum. If the classical spectrum is eliminated, then, with the correct choice of Jeffreys–Wentzel–Kramers–Brillouin (JWKB) matching conditions, the semiclassical spectrum satisfies the Einstein–Brillouin–Keller quantization conditions which include the correction due to the Maslov index. However, semiclassical analysis uses different choices for boundary conditions, continuity requirements, and the domain of definition. For example, use of the complex JWKB method allows for the treatment of tunneling through the complexification of phase space. Finally, although KvH wavefunctions include the possibility of interference effects, interference is not observable when all observables are approximated as local operators on phase space. Observing interference effects requires consideration of nonlocal operations, e.g. through higher orders in the asymptotic theory.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

Shadow Lagrangian dynamics for superfluidity

Motivated by a similar approach for Born-Oppenheimer molecular dynamics, this paper proposes an extended "shadow" Lagrangian density for quantum states of superfluids. The extended Lagrangian contains an additional field variable that is forced to follow the wave function of the quantum state through a rapidly oscillating extended harmonic oscillator. By considering the adiabatic limit for large frequencies of the harmonic oscillator, we can derive the two equations of motions, a Schrödinger-type equation for the quantum state and a wave equation for the extended field variable. The equations are coupled in a nonlinear way, but each equation individually is linear with respect to the variable that it defines. The computational advantage of this new system is that it can be easily discretized using linear time stepping methods, where we propose to use a Crank-Nicolson-type approach for the Schrödinger equation and an extended leapfrog scheme for the wave equation. Furthermore, the difference between the quantum state and the extended field variable defines a consistency error that should go to zero if the frequency tends to infinity. By coupling the time-step size in our discretization to the frequency of the harmonic oscillator we can extract an easily computable consistency error indicator that can be used to estimate the numerical error without additional costs. The findings are illustrated in numerical experiments.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

NASA NTRS1979

Remote sensing as a tool for estimating soil erosion potential

The Universal Soil Loss Equation is a frequently used methodology for estimating soil erosion potential. The Universal Soil Loss Equation requires a variety of types of geographic information (e.g. topographic slope, soil erodibility, land use, crop type, and soil conservation practice) in order to function. This information is traditionally gathered from topographic maps, soil surveys, field surveys, and interviews with farmers. Remote sensing data sources and interpretation techniques provide an alternative method for collecting information regarding land use, crop type, and soil conservation practice. Airphoto interpretation techniques and medium altitude, multi-date color and color infrared positive transparencies (70mm) were utilized in this study to determine their effectiveness for gathering the desired land use/land cover data. Successful results were obtained within the test site, a 6136 hectare watershed in Dane County, Wisconsin.

Morris-Jones, D. R.↗

NASA NTRS1979

Plasma property and performance prediction for mercury ion thrusters

The discharge chambers of mercury ion thrusters are modelled so the principal effects and processes which govern discharge plasma properties and thruster performance are described. The conservation relations for mass, charge and energy when applied to the Maxwellian electron population in the ion production region yield equations which may be made one-dimensional by the proper choice of coordinates. Solutions to these equations with the appropriate boundary conditions give electron density and temperature profiles which agree reasonably well with measurements. It is then possible to estimate plasma properties from thruster design data and those operating parameters which are directly controllable. By varying the operating parameter inputs to the computer code written to solve these equations, perfromance curves are obtained which agree quite well with measurements.

Longhurst, G. R.↗

NASA NTRS1982

Numerical study of terrain-induced mesoscale motions in a mixed layer

Numerical integrations using a potential enstrophy-conserving scheme are presented for the flow within a mixed layer over hilly terrain using the hydrostatic shallow-water equations with a quadratic drag law. The mesoscale area treated is 150 km on a side; cyclic lateral boundary conditions are used. It is found that for the idealized conditions treated (no surface heating, no entrainment and no pressure adjustments aloft), the topography quickly induces a steady state flow pattern by means of surface friction. Unsteadiness does not occur unless a surface-friction Reynolds number is greater than approximately 100. Effects of varying the Rossby number, Froude number and terrain-height parameter are examined.

Han, Y.-J.↗

NASA NTRS1989

Solution of steady-state one-dimensional conservation laws by mathematical programming

Solution techniques for a class of steady-state scalar conservation laws are developed analytically. Discretization by finite-volume formulas is employed to obtain an overdetermined system of algebraic equations, which are then perturbed nonsingularly (with perturbation coefficient = epsilon) and solved using the l(1) mathematical-programming algorithm of Seneta and Steiger (1984); this approach limits the matrix bandwidth to two, so that an explicit solution can be found efficiently. It is shown that, for small values of epsilon, the l(1) solutions exhibit sharp correctly located shocks and are nonoscillatory O(epsilon) approximations of the physically relevant solutions.

Lavery, J. E.↗

NASA NTRS1991

Time domain convergence properties of Lyapunov stable penalty methods

Linear hyperbolic partial differential equations are analyzed using standard techniques to show that a sequence of solutions generated by the Liapunov stable penalty equations approaches the solution of the differential-algebraic equations governing the dynamics of multibody problems arising in linear vibrations. The analysis does not require that the system be conservative and does not impose any specific integration scheme. Variational statements are derived which bound the error in approximation by the norm of the constraint violation obtained in the approximate solutions.

Kurdila, A. J.↗

NASA NTRS2002

New Boundary Constraints for Elliptic Systems used in Grid Generation Problems

This paper discusses new boundary constraints for elliptic partial differential equations as used in grid generation problems in generalized curvilinear coordinate systems. These constraints, based on the principle of local conservation of thermal energy in the vicinity of the boundaries, are derived using the Green's Theorem. They uniquely determine the so called decay parameters in the source terms of these elliptic systems. These constraints' are designed for boundary clustered grids where large gradients in physical quantities need to be resolved adequately. It is observed that the present formulation also works satisfactorily for mild clustering. Therefore, a closure for the decay parameter specification for elliptic grid generation problems has been provided resulting in a fully automated elliptic grid generation technique. Thus, there is no need for a parametric study of these decay parameters since the new constraints fix them uniquely. It is also shown that for Neumann type boundary conditions, these boundary constraints uniquely determine the solution to the internal elliptic problem thus eliminating the non-uniqueness of the solution of an internal Neumann boundary value grid generation problem.

Kaul, Upender K.↗

DOE OSTI2023

The MOOSE fluid properties module

The fluid properties module in MOOSE~\cite{lindsay2022moose} serves a variety of fluid simulation applications based on MOOSE, including the MOOSE Navier Stokes module~\cite{moose_ns}, Pronghorn~\cite{pgh}, the MOOSE thermal hydraulics module, SAM~\cite{sam}, RELAP-7~\cite{relap7}, Sockeye~\cite{sockeye}, Pronghorn-subchannel~\cite{subchannel} and the MOOSE porous flow module~\cite{porous}. These applications are used to solve coarse mesh multi-dimensional thermal-hydraulics~\cite{pgh}, 1D systems analysis~\cite{sam,relap7} in nuclear reactor analysis, heat pipe modeling~\cite{sockeye} and porous flow simulations~\cite{porous} for underground gas storage and water seepage. The use of consistent fluid properties across fluid flow applications facilitates coupled flow simulations~\cite{anl_sam_pgh,osti_1889653}. Each application has historically driven the implementation of several fluid properties, which were later extended to be compatible with other applications. The unique diversity of applications of the module, due to its presence in MOOSE, has driven its expansion to new fluids, such as advanced nuclear reactor coolants and, more recently, arbitrary functions or tables-based property definitions, as detailed in section~\ref{content}, as well as numerous thermophysical properties and variable sets, as detailed in subsection~\ref{sec:prop}. The need for different discretizations of flow equations based on the compressibility of the fluid has motivated support for both primitive (pressure- and temperature-based) and conservative (internal energy- and specific volume-based) flow variables in the module. Thermodynamic relations are used to convert between these two formulations, as needed. The module serves a dual purpose of providing fluid properties for direct use in flow simulations and facilitating the implementation of user-specific fluid properties. Contributions of new properties for existing fluids or new fluids are strongly encouraged.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

NASA NTRS2003

On a Non-Reflecting Boundary Condition for Hyperbolic Conservation Laws

A non-reflecting boundary condition (NRBC) for practical computations in fluid dynamics and aeroacoustics is presented. The technique is based on the hyperbolicity of the Euler equation system and the first principle of plane (simple) wave propagation. The NRBC is simple and effective, provided the numerical scheme maintains locally a C(sup 1) continuous solution at the boundary. Several numerical examples in ID, 2D and 3D space are illustrated to demonstrate its robustness in practical computations.

Loh, Ching Y.↗

NASA NTRS1993

An automatic multigrid method for the solution of sparse linear systems

An automatic version of the multigrid method for the solution of linear systems arising from the discretization of elliptic PDE's is presented. This version is based on the structure of the algebraic system solely, and does not use the original partial differential operator. Numerical experiments show that for the Poisson equation the rate of convergence of our method is equal to that of classical multigrid methods. Moreover, the method is robust in the sense that its high rate of convergence is conserved for other classes of problems: non-symmetric, hyperbolic (even with closed characteristics) and problems on non-uniform grids. No double discretization or special treatment of sub-domains (e.g. boundaries) is needed. When supplemented with a vector extrapolation method, high rates of convergence are achieved also for anisotropic and discontinuous problems and also for indefinite Helmholtz equations. A new double discretization strategy is proposed for finite and spectral element schemes and is found better than known strategies.

Shapira, Yair↗

NASA NTRS2006

Stochastic Representation of Chaos Using Terminal Attractors

A nonlinear version of the Liouville equation based on terminal attractors is part of a mathematical formalism for describing postinstability motions of dynamical systems characterized by exponential divergences of trajectories leading to chaos (including turbulence as a form of chaos). The formalism can be applied to both conservative systems (e.g., multibody systems in celestial mechanics) and dissipative systems (e.g., viscous fluids). The development of the present formalism was undertaken in an effort to remove positive Lyapunov exponents. The means chosen to accomplish this is coupling of the governing dynamical equations with the corresponding Liouville equation that describes the evolution of the flow of error probability. The underlying idea is to suppress the divergences of different trajectories that correspond to different initial conditions, without affecting a target trajectory, which is one that starts with prescribed initial conditions.

Zak, Michail↗

DOE OSTI2024

Predictions for a Low-mass Cutoff for the Primordial Black Hole Mass Spectrum

In this note, we outline how a modest violation in the conservation of mass during the merger of two PBHs affects the PBH mass spectrum that we previously obtained using a Boltzmann equation model for the evolution of the mass spectrum with no mass loss. We find that if the initial cosmological redshift is of the order of 10 12 , then the fraction of primordial holes with masses greater than 10 3 solar masses appears to be close to what is required to provide the seeds for galaxies. In addition, we note that as a result of rapid collisions and strong coupling to electromagnetic radiation for temperatures > GeV, there will be an effective low-mass cutoff in the mass spectrum for PBH masses less than a certain PBH mass less than 0.1M ⊙ . We also point out that this cutoff in the mass spectrum below ~ 0.1M ⊙ can be confirmed by combining future microlensing observations from the Roman Space Telescope and the Vera C. Rubin Observatory with astrometric observations.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1971

Kinetic theory of a two-dimensional magnetized plasma.

Several features of the equilibrium and nonequilibrium statistical mechanics of a two-dimensional plasma in a uniform dc magnetic field are investigated. The charges are assumed to interact only through electrostatic potentials. The problem is considered both with and without the guiding-center approximation. With the guiding-center approximation, an appropriate Liouville equation and BBGKY hierarchy predict no approach to thermal equilibrium for the spatially uniform case. For the spatially nonuniform situation, a guiding-center Vlasov equation is discussed and solved in special cases. For the nonequilibrium, nonguiding-center case, a Boltzmann equation, and a Fokker-Planck equation are derived in the appropriate limits. The latter is more tractable than the former, and can be shown to obey conservation laws and an H-theorem, but contains a divergent integral which must be cut off on physical grounds. Several unsolved problems are posed.

Vahala, G.↗

NASA NTRS1981

A numerical method for solving the equations of compressible viscous flow

Although much progress has already been made in solving problems in aerodynamic design, many new developments are still needed before the equations for unsteady compressible viscous flow can be solved routinely. This paper describes one such development. A method for solving these equations has been devised that: (1) is second-order accurate in space and time; (2) is unconditionally stable; (3) preserves conservation form; (4) requires no block or scalar tridiagonal inversions; (5) is simple and straightforward to program (estimated 10% modification for the update of many existing programs); (6) is more efficient than present methods; and (7) should easily adapt to current and future computer architectures. Computational results for laminar and turbulent flows at Reynolds numbers from 300,000 to 3 x 10 to the 7th and at CFL numbers as high as 1000 are compared with theory and experiment.

Maccormack, R. W.↗