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NASA NTRS1990

Some Aspects of Essentially Nonoscillatory (ENO) Formulations for the Euler Equations, Part 3

An essentially nonoscillatory (ENO) formulation is described for hyperbolic systems of conservation laws. ENO approaches are based on smart interpolation to avoid spurious numerical oscillations. ENO schemes are a superset of Total Variation Diminishing (TVD) schemes. In the recent past, TVD formulations were used to construct shock capturing finite difference methods. At extremum points of the solution, TVD schemes automatically reduce to being first-order accurate discretizations locally, while away from extrema they can be constructed to be of higher order accuracy. The new framework helps construct essentially non-oscillatory finite difference methods without recourse to local reductions of accuracy to first order. Thus arbitrarily high orders of accuracy can be obtained. The basic general ideas of the new approach can be specialized in several ways and one specific implementation is described based on: (1) the integral form of the conservation laws; (2) reconstruction based on the primitive functions; (3) extension to multiple dimensions in a tensor product fashion; and (4) Runge-Kutta time integration. The resulting method is fourth-order accurate in time and space and is applicable to uniform Cartesian grids. The construction of such schemes for scalar equations and systems in one and two space dimensions is described along with several examples which illustrate interesting aspects of the new approach.

Chakravarthy, Sukumar R.↗

NASA NTRS1990

GRID2D/3D: A computer program for generating grid systems in complex-shaped two- and three-dimensional spatial domains. Part 1: Theory and method

An efficient computer program, called GRID2D/3D was developed to generate single and composite grid systems within geometrically complex two- and three-dimensional (2- and 3-D) spatial domains that can deform with time. GRID2D/3D generates single grid systems by using algebraic grid generation methods based on transfinite interpolation in which the distribution of grid points within the spatial domain is controlled by stretching functions. All single grid systems generated by GRID2D/3D can have grid lines that are continuous and differentiable everywhere up to the second-order. Also, grid lines can intersect boundaries of the spatial domain orthogonally. GRID2D/3D generates composite grid systems by patching together two or more single grid systems. The patching can be discontinuous or continuous. For continuous composite grid systems, the grid lines are continuous and differentiable everywhere up to the second-order except at interfaces where different single grid systems meet. At interfaces where different single grid systems meet, the grid lines are only differentiable up to the first-order. For 2-D spatial domains, the boundary curves are described by using either cubic or tension spline interpolation. For 3-D spatial domains, the boundary surfaces are described by using either linear Coon's interpolation, bi-hyperbolic spline interpolation, or a new technique referred to as 3-D bi-directional Hermite interpolation. Since grid systems generated by algebraic methods can have grid lines that overlap one another, GRID2D/3D contains a graphics package for evaluating the grid systems generated. With the graphics package, the user can generate grid systems in an interactive manner with the grid generation part of GRID2D/3D. GRID2D/3D is written in FORTRAN 77 and can be run on any IBM PC, XT, or AT compatible computer. In order to use GRID2D/3D on workstations or mainframe computers, some minor modifications must be made in the graphics part of the program; no modifications are needed in the grid generation part of the program. This technical memorandum describes the theory and method used in GRID2D/3D.

Shih, T. I.-P.↗

NASA NTRS1989

Numerical algorithms for computations of feedback laws arising in control of flexible systems

Several continuous models will be examined, which describe flexible structures with boundary or point control/observation. Issues related to the computation of feedback laws are examined (particularly stabilizing feedbacks) with sensors and actuators located either on the boundary or at specific point locations of the structure. One of the main difficulties is due to the great sensitivity of the system (hyperbolic systems with unbounded control actions), with respect to perturbations caused either by uncertainty of the model or by the errors introduced in implementing numerical algorithms. Thus, special care must be taken in the choice of the appropriate numerical schemes which eventually lead to implementable finite dimensional solutions. Finite dimensional algorithms are constructed on a basis of a priority analysis of the properties of the original, continuous (infinite diversional) systems with the following criteria in mind: (1) convergence and stability of the algorithms and (2) robustness (reasonable insensitivity with respect to the unknown parameters of the systems). Examples with mixed finite element methods and spectral methods are provided.

Lasiecka, Irena↗

NASA NTRS1990

Fourier analysis of finite element preconditioned collocation schemes

The spectrum of the iteration operator of some finite element preconditioned Fourier collocation schemes is investigated. The first part of the paper analyses one-dimensional elliptic and hyperbolic model problems and the advection-diffusion equation. Analytical expressions of the eigenvalues are obtained with use of symbolic computation. The second part of the paper considers the set of one-dimensional differential equations resulting from Fourier analysis (in the tranverse direction) of the 2-D Stokes problem. All results agree with previous conclusions on the numerical efficiency of finite element preconditioning schemes.

Deville, Michel O.↗

NASA NTRS1990

Consistency and convergence for numerical radiation conditions

The problem of imposing radiation conditions at artificial boundaries for the numerical simulation of wave propagation is considered. Emphasis is on the behavior and analysis of the error which results from the restriction of the domain. The theory of error estimation is briefly outlined for boundary conditions. Use is made of the asymptotic analysis of propagating wave groups to derive and analyze boundary operators. For dissipative problems this leads to local, accurate conditions, but falls short in the hyperbolic case. A numerical experiment on the solution of the wave equation with cylindrical symmetry is described. A unified presentation of a number of conditions which have been proposed in the literature is given and the time dependence of the error which results from their use is displayed. The results are in qualitative agreement with theoretical considerations. It was found, however, that for this model problem it is particularly difficult to force the error to decay rapidly in time.

Hagstrom, Thomas↗

NASA NTRS1989

Unsteady density-current equations for highly curved terrain

New nonlinear partial differential equations containing terrain curvature and its rate of change are derived that describe the flow of an atmospheric density current. Unlike the classical hydraulic-type equations for density currents, the new equations are valid for two-dimensional, gradually varied flow over highly curved terrain, hence suitable for computing unsteady (or steady) flows over arbitrary mountain/valley profiles. The model assumes the atmosphere above the density current exerts a known arbitrary variable pressure upon the unknown interface. Later this is specialized to the varying hydrostatic pressure of the atmosphere above. The new equations yield the variable velocity distribution, the interface position, and the pressure distribution that contains a centrifugal component, often significantly larger than its hydrostatic component. These partial differential equations are hyperbolic, and the characteristic equations and characteristic directions are derived. Using these to form a characteristic mesh, a hypothetical unsteady curved-flow problem is calculated, not based upon observed data, merely as an example to illustrate the simplicity of their application to unsteady flows over mountains.

Sivakumaran, N. S.↗

NASA NTRS1990

On the Lagrangian description of unsteady boundary-layer separation. I - General theory

Although unsteady, high-Reynolds number, laminar boundary layers have conventionally been studied in terms of Eulerian coordinates, a Lagrangian approach may have significant analytical and computational advantages. In Lagrangian coordinates the classical boundary layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the motion normal to the boundary. The momentum equations, plus the energy equation if the flow is compressible, can be solved independently of the continuity equation. Unsteady separation occurs when the continuity equation becomes singular as a result of touching characteristics, the condition for which can be expressed in terms of the solution of the momentum equations. The solutions to the momentum and energy equations remain regular. Asymptotic structures for a number of unsteady 3-D separating flows follow and depend on the symmetry properties of the flow. In the absence of any symmetry, the singularity structure just prior to separation is found to be quasi 2-D with a displacement thickness in the form of a crescent shaped ridge. Physically the singularities can be understood in terms of the behavior of a fluid element inside the boundary layer which contracts in a direction parallel to the boundary and expands normal to it, thus forcing the fluid above it to be ejected from the boundary layer.

Van Dommelen, Leon L.↗

NASA NTRS1989

MHD intermediate shock discontinuities. I - Rankine-Hugoniot conditions

Recent numerical investigations have focused attention once more on the role of intermediate shocks in MHD. Four types of intermediate shock are identified using a graphical representation of the MHD Rankine-Hugoniot conditions. This same representation can be used to exhibit the close relationship of intermediate shocks to switch-on shocks and rotational discontinuities. The conditions under which intermediate discontinuities can be found are elucidated. The variations in velocity, pressure, entropy and magnetic-field jumps with upstream parameters in intermediate shocks are exhibited graphically. The evolutionary arguments traditionally advanced against intermediate shocks may fail because the equations of classical MHD are not strictly hyperbolic.

Kennel, C. F.↗

NASA NTRS1989

Recent progress on essentially non-oscillatory shock capturing schemes

An account is given of the construction of efficient implementations of 'essentially nonoscillatory' (ENO) schemes that approximate systems of hyperbolic conservation laws. ENO schemes use a local adaptive stencil to automatically obtain information from regions of smoothness when the solution develops discontinuities. Approximations employing ENOs can thereby obtain uniformly high accuracy to the very onset of discontinuities, while retaining a sharp and essentially nonoscillatory shock transition. For ease of implementation, ENO schemes applying the adaptive stencil concept to the numerical fluxes and employing a TVD Runge-Kutta-type time discretization are constructed.

Osher, Stanley↗

NASA NTRS1989

Advances in fluid dynamics

Various papers on advances in fluid dynamics are presented. Individual topics addressed include: nonunique solutions of the Euler equations; numerical solution of the biharmonic equation using a spectral multigrid method; finite-volume shape and placement for efficiency, accuracy, and physics; NND schemes and their application to numerical simulation of complex plume flow problems; symbolic analysis and computational algorithm construction; and velocity and mass flux distribution measurements of spherical glass beads in air flow in a 90-deg vertical-to-horizontal bend. Also discussed are: hyperbolic solution adaptive mesh generation; progress and future directions in CFD; numerical simulation of shock focusing; microbubble formation and splitting in a turbulent boundary layer for turbulence reduction; numerical study of spherical Taylor-Couette flow, trajectories of vortex lines beneath a free surface or above a plane; and a mathematical model for response of the coronary circulation to high sustained gravitational force fields.

Ballhaus, W. F., Jr.↗

NASA NTRS1990

Optimal finite-thrust spacecraft trajectories using collocation and nonlinear programming

A new method is described for the determination of optimal spacecraft trajectories in an inverse-square field using finite, fixed thrust. The method employs a recently developed optimization technique which uses a piecewise polynomial representation for the state and controls, and collocation, thus converting the optimal control problem into a nonlinear programming problem, which is solved numerically. This technique has been modified to provide efficient handling of those portions of the trajectory which can be determined analytically, i.e., the coast arcs. Among the problems that have been solved using this method are optimal rendezvous and transfer (including multirevolution cases) and optimal multiburn orbit insertion from hyperbolic approach.

Enright, Paul J.↗

NASA NTRS1990

Universal Lambert and Kepler algorithms for autonomous rendezvous

This paper describes Lambert and Kepler algorithms designed to be the core of an autonomous rendezvous guidance system for an onboard computer. Applications include robotic and piloted missions to the moon and planets. Flight software must be compact, fast, and totally reliable. Although high accuracy is not essential for flight, in double precision these algorithms are accurate to at least 14 places almost everywhere. Both are universal; they apply to elliptic, parabolic, hyperbolic, and even rectilinear trajectories. The algorithms are improvements to those published by Battin (1987).

Klumpp, Allan R.↗

NASA NTRS1990

High-accuracy Mars approach navigation with radio metric and optical data

The aerocapture of a space vehicle on hyperbolic approach to Mars results in tight navigation requirements at atmospheric entry. The purpose of this paper is to examine several different methods for approach navigation and to determine what accuracies are possible. The methods are broken into four groups as follows (1) navigation with only Deep Space Network (DSN) tracking of the approach vehicle, (2) navigation with the DSN plus ranging between the approach vehicle and spacecraft in orbit about Mars, (3) navigation with the DSN plus optical data involving the Martian moons, and (4) navigation with DSN range data and differenced range data involving the approach spacecraft and orbiters at Mars. If the current modeling errors that affect earth-based radio metric data, such as errors in tracking station locations, the Martian ephemeris, and differences in the quasar and planetary coordinate frames, are improved, then perhaps earth-based tracking could meet the entry error requirements imposed by aerocapture. If not, then the other three options of intervehicular range, optical data, or differenced range provide highly accurate entry knowledge at least twelve hours before entry.

Konopliv, Alex↗

NASA NTRS1991

Faster Algorithm For Computation Of Incompressible Flow

Improved algorithm yields faster numerical solutions of Navier-Stokes equations of steady or unsteady three-dimensional flow of incompressible fluid. In artificial-compressibility method, unsteady flow treated as incompressible in advancing from one time step to next, but at each time step (or in steady state), fluid treated as having variable compressibility enabling propagation of flow field, and subiterations performed in increments of pseudotime until effects of compressibility subside. Directly couples pressure and velocity fields at same time step and converts elliptic incompressible Navier-Stokes equations to hyperbolic form more amenable to numerical integration.

Rogers, S. E.↗

NASA NTRS1991

Overview Of Methods For Computation Of Shocks

NASA technical memorandum provides systematic overview of class of conservative finite-difference shock-capturing numerical-integration methods for solution of hyperbolic conservation laws. Unified and generalized formulation for one class of methods presented.

Yee, H. C.↗

NASA NTRS1990

Spectral methods for time dependent problems

Spectral approximations are reviewed for time dependent problems. Some basic ingredients from the spectral Fourier and Chebyshev approximations theory are discussed. A brief survey was made of hyperbolic and parabolic time dependent problems which are dealt with by both the energy method and the related Fourier analysis. The ideas presented above are combined in the study of accuracy stability and convergence of the spectral Fourier approximation to time dependent problems.

Tadmor, Eitan↗

NASA NTRS1990

Spurious frequencies as a result of numerical boundary treatments

The stability theory for finite difference Initial Boundary-Value approximations to systems of hyperbolic partial differential equations states that the exclusion of eigenvalues and generalized eigenvalues is a sufficient condition for stability. The theory, however, does not discuss the nature of numerical approximations in the presence of such eigenvalues. In fact, as was shown previously, for the problem of vortex shedding by a 2-D cylinder in subsonic flow, stating boundary conditions in terms of the primitive (non-characteristic) variables may lead to such eigenvalues, causing perturbations that decay slowly in space and remain periodic time. Characteristic formulation of the boundary conditions avoided this problem. A more systematic study of the behavior of the (linearized) one-dimensional gas dynamic equations under various sets of oscillation-inducing legal boundary conditions is reported.

Abarbanel, Saul↗

NASA NTRS1990

An investigation of design optimization using a 2-D viscous flow code with multigrid

Computational fluid dynamics (CFD) codes have advanced to the point where they are effective analytical tools for solving flow fields around complex geometries. There is also a need for their use as a design tool to find optimum aerodynamic shapes. In the area of design, however, a difficulty arises due to the large amount of computer resources required by these codes. It is desired to streamline the design process so that a large number of design options and constraints can be investigated without overloading the system. There are several techniques which have been proposed to help streamline the design process. The feasibility of one of these techniques is investigated. The technique under consideration is the interaction of the geometry change with the flow calculation. The problem of finding the value of camber which maximizes the ratio of lift over drag for a particular airfoil is considered. In order to test out this technique, a particular optimization problem was tried. A NACA 0012 airfoil was considered at free stream Mach number of 0.5 with a zero angle of attack. Camber was added to the mean line of the airfoil. The goal was to find the value of camber for which the ratio of lift over drag is a maximum. The flow code used was FLOMGE which is a two dimensional viscous flow solver which uses multigrid to speed up convergence. A hyperbolic grid generation program was used to construct the grid for each value of camber.

Doria, Michael L.↗