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At least 505 records · Page 28
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DOE OSTI2021

Uncovering turbulent plasma dynamics via deep learning from partial observations

One of the most intensely studied aspects of magnetic confinement fusion is edge plasma turbulence which is critical to reactor performance and operation. Drift-reduced Braginskii two-fluid theory has for decades been widely applied to model boundary plasmas with varying success. Towards better understanding edge turbulence in both theory and experiment, we demonstrate that a novel multi-network physics-informed deep learning framework constrained by partial differential equations can accurately learn turbulent fields consistent with the two-fluid theory from partial observations of electron pressure which is not otherwise possible using conventional equilibrium models. This technique presents a novel paradigm for the advanced design of plasma diagnostics and validation of magnetized plasma turbulence theories in challenging thermonuclear environments.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1973

A sequence solution to the Fokker-Planck equation.

In this paper a new approach to nonlinear system analysis based on the Fokker-Planck equation is developed. The development begins as a modification of the parametrix method of partial differential equation theory and provides a sequence solution to the multidimensional time-varying Fokker-Planck equation. This sequence is simplified for the case where only a steady-state solution or a solution for the variance of a particular combination of states is desired. Equations are presented in a form suitable for obtaining numerical results using a digital computer.

Mayfield, W. W.↗
NASA NTRS1978

A three-dimensional model of corotating streams in the solar wind. I - Theoretical foundations

The paper is concerned with the development of the theoretical and mathematical background pertinent to the study of steady, corotating solar wind structure in all three spatial dimensions. The dynamical evolution of the plasma in interplanetary space (defined as the region beyond roughly 35 Rs where the flow is supersonic) is approximately described by the nonlinear, single-fluid, polytropic magnetohydrodynamic or hydrodynamic equations. Efficient numerical techniques are outlined for solving this complex system of coupled, hyperbolic partial differential equations. The present formulation is inviscid and nonmagnetic, but the methods used allow for the potential inclusion of both features with only modest modifications. A simple, highly idealized hydrodynamic model stream is examined to illustrate the fundamental processes involved in the three-dimensional dynamics of stream evolution. It is found that spatial variations in the rotational stream interaction mechanism produce small nonradial flows on a global scale that lead to the transport of mass, energy, and momentum away from regions of relative compression and into regions of relative rarefaction. Comparison with simpler models demonstrates the essential nonlinear, multidimensional nature of the interplanetary dynamics.

Pizzo, V.↗
NASA NTRS1982

Magnetohydrodynamic equilibrium. II - General integrals of the equations with one ignorable coordinate

The steady equations of hydromagnetics for the isentropic or nonisentropic flow of an inviscid magnetofluid of high electrical conductivity, with one ignorable coordinate in a general orthogonal system, are treated. Several integrals of the equations are established thereafter reducing them to a scalar, quasi-linear, second order, partial differential equation for the magnetic potential. Simple solutions of this final equation are presented. The result, together with a similar treatment of helically symmetric hydromagnetic flows presented in a subsequent paper, allows a unified and systematic approach to the solution of problems involving steady hydromagnetic fields with a topological invariance in various curvilinear coordinates.

Tsinganos, K. C.↗
NASA NTRS1989

Semidiscrete Galerkin modelling of compressible viscous flow past a circular cone at incidence

A numerical study of the laminar and compressible boundary layer, about a circular cone in a supersonic free stream, is presented. It is thought that if accurate and efficient numerical schemes can be produced to solve the boundary layer equations, they can be joined to numerical codes that solve the inviscid outer flow. The combination of these numerical codes is competitive with the accurate, but computationally expensive, Navier-Stokes schemes. The primary goal is to develop a finite element method for the calculation of 3-D compressible laminar boundary layer about a yawed cone. The proposed method can, in principle, be extended to apply to the 3-D boundary layer of pointed bodies of arbitrary cross section. The 3-D boundary layer equations governing supersonic free stream flow about a cone are examined. The 3-D partial differential equations are reduced to 2-D integral equations by applying the Howarth, Mangler, Crocco transformations, a linear relation between viscosity, and a Blasius-type of similarity variable. This is equivalent to a Dorodnitsyn-type formulation. The reduced equations are independent of density and curvature effects, and resemble the weak form of the 2-D incompressible boundary layer equations in Cartesian coordinates. In addition the coordinate normal to the wall has been stretched, which reduces the gradients across the layer and provides high resolution near the surface. Utilizing the parabolic nature of the boundary layer equations, a finite element method is applied to the Dorodnitsyn formulation. The formulation is presented in a Petrov-Galerkin finite element form and discretized across the layer using linear interpolation functions. The finite element discretization yields a system of ordinary differential equations in the circumferential direction. The circumferential derivatives are solved by an implicit and noniterative finite difference marching scheme. Solutions are presented for a 15 deg half angle cone at angles of attack of 5 and 10 deg. The numerical solutions assume a laminar boundary layer with free stream Mach number of 7. Results include circumferential distribution of skin friction and surface heat transfer, and cross flow velocity distributions across the layer.

Meade, Andrew James, Jr.↗
NASA NTRS1991

Asynchronous and corrected-asynchronous numerical solutions of parabolic PDES on MIMD multiprocessors

A major problem in achieving significant speed-up on parallel machines is the overhead involved with synchronizing the concurrent process. Removing the synchronization constraint has the potential of speeding up the computation. The authors present asynchronous (AS) and corrected-asynchronous (CA) finite difference schemes for the multi-dimensional heat equation. Although the discussion concentrates on the Euler scheme for the solution of the heat equation, it has the potential for being extended to other schemes and other parabolic partial differential equations (PDEs). These schemes are analyzed and implemented on the shared memory multi-user Sequent Balance machine. Numerical results for one and two dimensional problems are presented. It is shown experimentally that the synchronization penalty can be about 50 percent of run time: in most cases, the asynchronous scheme runs twice as fast as the parallel synchronous scheme. In general, the efficiency of the parallel schemes increases with processor load, with the time level, and with the problem dimension. The efficiency of the AS may reach 90 percent and over, but it provides accurate results only for steady-state values. The CA, on the other hand, is less efficient, but provides more accurate results for intermediate (non steady-state) values.

Amitai, Dganit↗
NASA NTRS2004

A Study of Supersonic Surface Sources: The Ffowcs Williams-Hawkings Equation and the Kirchhoff Formula

In this paper we address the mathematical problem of noise generation from high speed moving surfaces. The problem we are solving is the linear wave equation with sources on a moving surface. The Ffowcs Williams-Hawkings (FW-H) equation as well as the govern- ing equation for deriving the Kirchhoff formula for moving surfaces are both this type of partial differential equation. We give a new exact solution of this problem here in closed form which is valid for subsonic and supersonic motion of the surface but it is particularly suitable for supersonically moving surfaces. This new solution is the simplest of all high speed formulations of Langley and is denoted formulation 4 following the tradition of numbering of our major results for the prediction of the noise of rotating blades. We show that for a smooth surface moving at supersonic speed, our solution has only removable singularities. Thus it can be used for numerical work.

Farassat, F.↗
NASA NTRS1981

Stability analysis of numerical boundary conditions and implicit difference approximations for hyperbolic equations

Implicit, noniterative, finite difference schemes were recently developed by several authors for multidimensional systems of nonlinear hyperbolic partial differential equations. When applied to linear model equations with periodic boundary conditions those schemes are unconditionally stable (A-stable). As applied in practice the algorithms often face a severe time step restriction. A major source of the difficulty is the treatment of the numerical boundary conditions. One conjecture was that unconditional stability requires implicit numerical boundary conditions. An apparent counter example was the space time extrapolation considered by Gustafsson, Kreiss, and Sunstrom. Spatial (implicit) and space time (explicit) extrapolation using normal mode analysis for a finite and infinite number of spatial mesh intervals are examined. The results indicate that for unconditional stability with a finite number of spatial mesh intervals, the numerical boundary conditions must be implicit.

Beam, R. M.↗
NASA NTRS1988

Stochastic time-optimal control problems

Two types of stochastic time-optimal controls in a one-dimensional setting are considered. Multidimensional problems, in the case of complete state information available and the system modeled by stochastic differential equations, are studied under the formulation of minimizing the expected transient-response time. The necessary condition of optimality is the satisfaction for the value function of a parabolic partial differential equation with boundary conditions. The sufficient condition of optimality is also provided, based on Dynkin's formula. Finally, three examples are given.

Zhang, W.↗
NASA NTRS1997

[Development of New Mathematical Methodology in Air Traffic Control for the Analysis of Hybrid Systems]

The aim of this research is to develop new mathematical methodology for the analysis of hybrid systems of the type involved in Air Traffic Control (ATC) problems. Two directions of investigation were initiated. The first used the methodology of nonlinear generalized functions, whose mathematical foundations were initiated by Colombeau and developed further by Oberguggenberger; it has been extended to apply to ordinary differential. Systems of the type encountered in control in joint work with the PI and M. Oberguggenberger. This involved a 'mixture' of 'continuous' and 'discrete' methodology. ATC clearly involves mixtures of two sorts of mathematical problems: (1) The 'continuous' dynamics of a standard control type described by ordinary differential equations (ODE) of the form: {dx/dt = f(x, u)} and (2) the discrete lattice dynamics involved of cellular automata. Most of the CA literature involves a discretization of a partial differential equation system of the type encountered in physics problems (e.g. fluid and gas problems). Both of these directions requires much thinking and new development of mathematical fundamentals before they may be utilized in the ATC work. Rather than consider CA as 'discretization' of PDE systems, I believe that the ATC applications will require a completely different and new mathematical methodology, a sort of discrete analogue of jet bundles and/or the sheaf-theoretic techniques to topologists. Here too, I have begun work on virtually 'virgin' mathematical ground (at least from an 'applied' point of view) which will require considerable preliminary work.

Hermann, Robert↗
NASA NTRS1974

A theory for predicting composite laminate warpage resulting from fabrication

Linear laminate theory is used with the moment-curvature relationship to derive equations for predicting end deflections due to warpage without solving the coupled fourth-order partial differential equations of the plate. Composite micro- and macrohyphenmechanics are used with laminate theory to assess the contribution of factors such as ply misorientation, fiber migration, and fiber and/or void volume ratio nonuniformity on the laminate warpage. Using these equations, it was found that a 1 deg error in the orientation angle of one ply was sufficient to produce warpage end deflection equal to two laminate thicknesses in a 10 inch by 10 inch laminate made from 8 ply Mod-I/epoxy. Using a sensitivity analysis on the governing parameters, it was found that a 3 deg fiber migration or a void volume ratio of three percent in some plies is sufficient to produce laminate warpage corner deflection equal to several laminate thicknesses. Tabular and graphical data are presented which can be used to identify possible errors contributing to laminate warpage and/or to obtain an a priori assessment when unavoidable errors during fabrication are anticipated.

Chamis, C. C.↗
NASA NTRS1975

A theory for predicting composite laminate warpage resulting from fabrication

Linear laminate theory is used in conjunction with the moment-curvature relationship to derive equations for predicting end deflections due to warpage without solving the coupled fourth-order partial differential equations of the plate. Using these equations, it is found that a 1 deg error in the orientation angle of one ply is sufficient to produce warpage end deflection equal to two laminate thicknesses in a 10 inch by 10 inch laminate made from 8-ply Mod-I/epoxy. From a sensitivity analysis on the governing parameters, it is found that a 3 deg fiber migration or a void volume ratio of three percent in some plies is sufficient to produce laminate warpage corner deflection equal to several laminate thicknesses. Tabular and graphical data are presented which can be used to identify possible errors contributing to laminate warpage and/or to obtain an a priori assessment when unavoidable errors during fabrication are anticipated.

Chamis, C. C.↗
NASA NTRS1975

Gravitational collapse of a turbulent vortex with application to star formation

The gravitational collapse of a rotating cloud or vortex is analyzed by expanding the dependent variables in the equations of motion in two-dimensional Taylor series in the space variables. It is shown that the gravitation and rotation terms in the equations are of first order in the space variables, the pressure gradient terms are of second order, and the turbulent viscosity term is of third order. The presence of a turbulent viscosity insures that the initial rotation is solid-body-like near the origin. The effect of pressure on the collapse process is found to depend on the shape of the initial density disturbance at the origin. Dimensionless collapse times, as well as the evolution of density and velocity, are calculated by solving numerically the system of nonlinear ordinary differential equations resulting from the series expansions. The axial inflow plays an important role and allows collapse to occur even when the rotation is large. An approximate solution of the governing partial differential equations is also given; the equations are used to study the spacial distributions of the density and velocity.

Deissler, R. G.↗
NASA NTRS1975

Gravitational collapse of a turbulent vortex with application to star formation

The gravitational collapse of a rotating cloud or vortex is analyzed by expanding the dependent variables in the equations of motion in two-dimensional Taylor series in the space variables. It is shown that the gravitation and rotation terms in the equations are of first order in the space variables, the pressure gradient terms are of second order, and the turbulent viscosity term is of third order. The presence of a turbulent viscosity insures that the initial rotation is solid-body-like near the origin. The effect of pressure on the collapse process is found to depend on the shape of the intial density disturbance at the origin. Dimensionless collapse times, as well as the evolution of density and velocity, are calculated by solving numerically the system of nonlinear ordinary differential equations resulting from the series expansions. The axial inflow plays an important role and allows collapse to occur even when the rotation is large. An approximate solution of the governing partial differential equations is also given, in order to study the spacial distributions of the density and velocity.

Deissler, R. G.↗
NASA NTRS1976

Gravitational collapse of a turbulent vortex with application to star formation

The gravitational collapse of a rotating cloud or vortex is analyzed by expanding the dependent variables in the equations of motion in two-dimensional Taylor series in the space variables. It is shown that the gravitational and rotational terms in the equations are of first order in the space variables, the pressure-gradient terms are of second order, and the turbulent-viscosity term is of third order. The presence of turbulent viscosity ensures that the initial rotation is solid-body-like near the origin. The effect of pressure on the collapse process is found to depend on the shape of the initial density disturbance at the origin. Dimensionless collapse times, as well as the evolution of density and velocity, are calculated by solving numerically the system of nonlinear ordinary differential equations resulting from the series expansions. The axial flow is always inward and allows collapse to occur (axially) even when the rotation is large. An approximate solution of the governing partial differential equations is also given in order to study the spatial distributions of the density and velocity.

Deissler, R. G.↗
NASA NTRS1986

Optimal control of aeroassisted plane change maneuver using feedback expansions

A guidance law for an aeroassisted plane change maneuver is developed by an asymptotic expansion technique using a small parameter which essentially represents the ratio of the inertial forces to the atmospheric forces. This guidance law minimizes the energy loss while meeting terminal constraints on the altitude, flight path angle, and heading angle. By neglecting the inertial forces, the resulting optimization problem is integrable and can be determined in closed form. This zeroth-order solution is the first term in an asymptotic series solution of the Hamilton-Jacobi-Bellman equation. The remaining terms are determined from the solution of a first-order, linear partial differential equation whose solution requires only quadrature integration. Our initial results in using this guidance scheme are encouraging.

Mishne, D.↗
NASA NTRS1989

Numerical computation of shock wave-turbulent boundary layer interaction in transonic flow over an axisymmetric curved hill

A control-volume based finite difference computation of a turbulent transonic flow over an axisymmetric curved hill is presented. The numerical method is based on the SIMPLE algorithm, and hence the conservation of mass equation is replaced by a pressure correction equation for compressible flows. The turbulence is described by a k-epsilon turbulence model supplemented by a near-wall turbulence model. In the method, the dissipation rate in the region very close to the wall is obtained from an algebraic equation and that for the rest of the flow domain is obtained by solving a partial differential equation for the dissipation rate. The other flow equations are integrated up to the wall. It is shown that the present turbulence model yields the correct location of the compression shock. The other computational results are also in good agreement with experimental data.

Kim, S.-W.↗