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Results for “hyperbolic problems”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 397 records · Page 22
All sourcesNASA NTRSDOE OSTI
NASA NTRS1983

Vibrations of cantilevered doubly-curved shallow shells

Vibrational characteristics are determined for a previously unsolved class of problems, that of doubly-curved shallow shells having rectangular planforms, clamped along one edge and free on the other three. The solution procedure uses the Ritz method with algebraic polynomial trial functions. Convergence studies are made, and accurate frequencies and contour plots of mode shapes are presented for various curvature ratios, including spherical, circular cylindrical and hyperbolic paraboloidal shells. Particular emphasis is given to the effect of adding spanwise curvature to shells having chordwise curvature; numerous published references already exist for the case of zero spanwise curvature. The effects of changing aspect ratio, thickness ratio and Poisson's ratio are also studied.

Leissa, A. W.↗
NASA NTRS1990

Optimal trajectories based on linear equations

The Principal results of a recent theory of fuel optimal space trajectories for linear differential equations are presented. Both impulsive and bounded-thrust problems are treated. A new form of the Lawden Primer vector is found that is identical for both problems. For this reason, starting iteratives from the solution of the impulsive problem are highly effective in the solution of the two-point boundary-value problem associated with bounded thrust. These results were applied to the problem of fuel optimal maneuvers of a spacecraft near a satellite in circular orbit using the Clohessy-Wiltshire equations. For this case two-point boundary-value problems were solved using a microcomputer, and optimal trajectory shapes displayed. The results of this theory can also be applied if the satellite is in an arbitrary Keplerian orbit through the use of the Tschauner-Hempel equations. A new form of the solution of these equations has been found that is identical for elliptical, parabolic, and hyperbolic orbits except in the way that a certain integral is evaluated. For elliptical orbits this integral is evaluated through the use of the eccentric anomaly. An analogous evaluation is performed for hyperbolic orbits.

Carter, Thomas E.↗
NASA NTRS1993

A numerical treatment of geodynamic viscous flow problems involving the advection of material interfaces

Effective numerical treatment of multicomponent viscous flow problems involving the advection of sharp interfaces between materials of differing physical properties requires correction techniques to prevent spurious diffusion and dispersion. We develop a particular algorithm, based on modern shock-capture techniques, employing a two-step nonlinear method. The first step involves the global application of a high-order upwind scheme to a hyperbolic advection equation used to model the distribution of distinct material components in a flow field. The second step is corrective and involves the application of a global filter designed to remove dispersion errors that result from the advection of discontinuities (e.g., material interfaces) by high-order, minimally dissipative schemes. The filter introduces no additional diffusion error. Nonuniform viscosity across a material interface is allowed for by the implementation of a compositionally weighted-inverse interface viscosity scheme. The combined method approaches the optimal accuracy of modern shock-capture techniques with a minimal increase in computational time and memory. A key advantage of this method is its simplicity to incorporate into preexisting codes be they finite difference, element, or volume of two or three dimensions.

Lenardic, A.↗
NASA NTRS1991

Multi-Dimensional ENO Schemes for General Geometries

A class of ENO schemes is presented for the numerical solution of multidimensional hyperbolic systems of conservation laws in structured and unstructured grids. This is a class of shock-capturing schemes which are designed to compute cell-averages to high order accuracy. The ENO scheme is composed of a piecewise-polynomial reconstruction of the solution form its given cell-averages, approximate evolution of the resulting initial value problem, and averaging of this approximate solution over each cell. The reconstruction algorithm is based on an adaptive selection of stencil for each cell so as to avoid spurious oscillations near discontinuities while achieving high order of accuracy away from them.

Harten, Ami↗
DOE OSTI2014

A differential equation for Lerch's transcendent and associated symmetric operators in Hilbert space

The function Ψ(x,y,s)=e{sup iy}Φ(−e{sup iy},s,x), where Φ(z,s,v) is Lerch's transcendent, satisfies the following two-dimensional formally self-adjoint second-order hyperbolic differential equation, where s=1/2+iλ. The corresponding differential expression determines a densely defined symmetric operator (the minimal operator) on the Hilbert space L{sub 2}(Π), where Π=(0,1)×(0,2π). We obtain a description of the domains of definition of some symmetric extensions of the minimal operator. We show that formal solutions of the eigenvalue problem for these symmetric extensions are represented by functional series whose structure resembles that of the Fourier series of Ψ(x,y,s). We discuss sufficient conditions for these formal solutions to be eigenfunctions of the resulting symmetric differential operators. We also demonstrate a close relationship between the spectral properties of these symmetric differential operators and the distribution of the zeros of some special analytic functions analogous to the Riemann zeta function. Bibliography: 15 titles.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1976

A probability of encounter with interstellar comets and the likelihood of their existence

A theory of the probability of encounter of the sun with an interstellar comet at a distance comparable to the earth-sun distance is formulated, and a general expression is derived establishing the relationship among the influx rate of interstellar comets, the perihelion distance, the space density of the comets, the Maxwellian distribution of comet velocities in the interstellar cloud, and the cloud's systematic velocity relative to the sun. An upper limit of .0006 solar mass per cubic parsec is established for the space density of interstellar comets. The theoretical distribution of semimajor axes of interstellar comets shows that a strong hyperbolic excess must be present in the orbits of a majority of interstellar comets regardless of the dynamical characteristics of the comet cloud, except when the cloud is moving along with the sun and the distribution of individual velocities has a very low dispersion. This case, however, implies a possibility of capture by the sun and thus becomes a problem of an Oort-type cloud.

Sekanina, Z.↗
NASA NTRS1990

Robust control of systems with real parameter uncertainty and unmodelled dynamics

Two significant contributions have been made during this research period in the research 'Robust Control of Systems with Real Parameter Uncertainty and Unmodelled Dynamics' under NASA Research Grant NAG-1-1102. They are: (1) a fast algorithm for computing the optimal H(sub infinity) norm for the four-block, the two block, or the one-block optimal H(sub infinity) optimization problem; and (2) a construction of an optimal H infinity controller without numerical difficulty. In using GD (Glover and Doyle) or DGKF (Doyle, Glover, Khargonekar, and Francis) approach to solve the standard H infinity norm which required bisection search. In this research period, we developed a very fast iterative algorithm for this computation. Our algorithm was developed based on hyperbolic interpolations which is much faster than any existing algorithm. The lower bound of the parameter, gamma, in the H infinity Riccati equation for solution existence is shown to be the square root of the supremum over all frequencies of the maximum eigenvalue of a given transfer matrix which can be computed easily. The lower band of gamma such that the H infinity Riccati equation has positive semidefinite solution can be also obtained by hyperbolic interpolation search. Another significant result in this research period is the elimination of the numerical difficulties arising in the construction of an optimal H infinity controller by directly applying the Glover and Doyle's state-space formulas. With the fast iterative algorithm for the computation of the optimal H infinity norm and the reliable construction of an optimal H infinity controller, we are ready to apply these tools in the design of robust controllers for the systems with unmodelled uncertainties. These tools will be also very useful when we consider systems with structured uncertainties.

Chang, Bor-Chin↗
NASA NTRS1985

A computational study of saddle point separation and horseshoe vortex system

Incompressible flow around a cylinder-end wall junction has been simulated by solving the incompressible Navier-Stokes equations in three dimensions. The equations, cast in generalized curvilinear coordinates, are solved in time as a hyperbolic system by adding a pressure term in the continuity equation and are marched to a steady state. Various physical quantities associated with the saddle point of separation and the horseshoe vortex system are calculated. Computational and experimental results are generally consistent. The skin friction and the pressure distribution on the end wall are consistent with the physics of the problem. Secondary flows both in front of the cylinder and behind it are predicted that are in qualitative agreement with flow visualization results. The calculations also indicate a strongly nonuniform pressure loading along the length of the cylinder. A new mechanism for the existence of the recirculation bubbles behind the cylinder-end wall with relatively low ratio of cylinder height to the approaching boundary layer thickness is observed which is markedly different from its two-dimensional counterpart.

Kaul, U. K.↗
NASA NTRS1989

Local error estimates for discontinuous solutions of nonlinear hyperbolic equations

Let u(x,t) be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data. Suppose u sub epsilon(x,t) is the solution of an approximate viscosity regularization, where epsilon greater than 0 is the small viscosity amplitude. It is shown that by post-processing the small viscosity approximation u sub epsilon, pointwise values of u and its derivatives can be recovered with an error as close to epsilon as desired. The analysis relies on the adjoint problem of the forward error equation, which in this case amounts to a backward linear transport with discontinuous coefficients. The novelty of this approach is to use a (generalized) E-condition of the forward problem in order to deduce a W(exp 1,infinity) energy estimate for the discontinuous backward transport equation; this, in turn, leads one to an epsilon-uniform estimate on moments of the error u(sub epsilon) - u. This approach does not follow the characteristics and, therefore, applies mutatis mutandis to other approximate solutions such as E-difference schemes.

Tadmor, Eitan↗
DOE OSTI2016

Semi-implicit time integration of atmospheric flows with characteristic-based flux partitioning

Here, this paper presents a characteristic-based flux partitioning for the semi-implicit time integration of atmospheric flows. Nonhydrostatic models require the solution of the compressible Euler equations. The acoustic time scale is significantly faster than the advective scale, yet it is typically not relevant to atmospheric and weather phenomena. The acoustic and advective components of the hyperbolic flux are separated in the characteristic space. High-order, conservative additive Runge-Kutta methods are applied to the partitioned equations so that the acoustic component is integrated in time implicitly with an unconditionally stable method, while the advective component is integrated explicitly. The time step of the overall algorithm is thus determined by the advective scale. Benchmark flow problems are used to demonstrate the accuracy, stability, and convergence of the proposed algorithm. The computational cost of the partitioned semi-implicit approach is compared with that of explicit time integration.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1979

Recent developments in finite element analysis for transonic airfoils

The prediction of aerodynamic forces in the transonic regime generally requires a flow field calculation to solve the governing non-linear mixed elliptic-hyperbolic partial differential equations. Finite difference techniques were developed to the point that design and analysis application are routine, and continual improvements are being made by various research groups. The principal limitation in extending finite difference methods to complex three-dimensional geometries is the construction of a suitable mesh system. Finite element techniques are attractive since their application to other problems have permitted irregular mesh elements to be employed. The purpose of this paper is to review the recent developments in the application of finite element methods to transonic flow problems and to report some recent results.

Hafez, M. M.↗
NASA NTRS1986

Uniformly high order accurate essentially non-oscillatory schemes 3

In this paper (a third in a series) the construction and the analysis of essentially non-oscillatory shock capturing methods for the approximation of hyperbolic conservation laws are presented. Also presented is a hierarchy of high order accurate schemes which generalizes Godunov's scheme and its second order accurate MUSCL extension to arbitrary order of accuracy. The design involves an essentially non-oscillatory piecewise polynomial reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell. The reconstruction algorithm is derived from a new interpolation technique that when applied to piecewise smooth data gives high-order accuracy whenever the function is smooth but avoids a Gibbs phenomenon at discontinuities. Unlike standard finite difference methods this procedure uses an adaptive stencil of grid points and consequently the resulting schemes are highly nonlinear.

Harten, A.↗
DOE OSTI2018

Numerical solution of linear and nonlinear hyperbolic telegraph type equations with variable coefficients using shifted Jacobi collocation method

The telegraph equation is one of the important equations of mathematical physics. In this work, a spectral collocation scheme is proposed for the numerical solutions of one- and two-dimensional linear telegraph equations and telegraph equations with nonlinear forcing term. The homogeneous initial and boundary conditions are satisfied exactly by expanding the unknown variable using polynomial bases of functions which are built upon the Jacobi polynomials. The suggested scheme is successfully developed for the aforementioned problem with nonhomogeneous data. Extensive numerical experiments are presented to verify the efficiency of the proposed scheme.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

A model and numerical method for compressible flows with capillary effects

A new model for interface problems with capillary effects in compressible fluids is presented together with a specific numerical method to treat capillary flows and pressure waves propagation. This new multiphase model is in agreement with physical principles of conservation and respects the second law of thermodynamics. A new numerical method is also proposed where the global system of equations is split into several submodels. Each submodel is hyperbolic or weakly hyperbolic and can be solved with an adequate numerical method. This method is tested and validated thanks to comparisons with analytical solutions (Laplace law) and with experimental results on droplet breakup induced by a shock wave.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2017

Simulations of single- and two-phase shock tubes and gravity-driven wave problems with the RELAP-7 nuclear reactor system analysis code

Propagation of pressure waves in single and two-phase flows is of key interest for nuclear safety analyses, reactor design, construction and life extension of nuclear reactors. In this work, we assess the capabilities of RELAP-7, the next-generation nuclear reactor system code, to solve single- and two-phase flows developing pressure waves. RELAP-7 is based on a 7-equation, two-phase flow model, with distinct phasic pressures. It is well established that these hyperbolic conservation laws can develop shocks and discontinuities and thus, require a stabilization numerical method. The all-Mach flow Entropy Viscosity Method (EVM) is now employed in RELAP-7 as a viscous stabilization approach. In this paper, we briefly describe the implementation of the EVM and use open-literature test cases that exhibit pressure waves to show that the entropy viscosity method performs adequately for these single- and two-phase cases.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2026

Mathematical modelling of the concave front in the adjacent high explosive detonation problem

This study presents an analysis of the transition-zone in adjacent high explosive (HE) detonation problems which uses a $D, 𝜅, \dot{D}$ relationship, where $D$ is the detonation front-normal velocity, 𝜅 is the detonation front curvature and $\dot{D}$ is the time derivative of detonation front-normal velocity. Our approach extends the traditional $(D, 𝜅)$ model to accurately predict the behaviour of both diverging and converging detonation shock fronts. Our findings affirm that a hyperbolic type of front evolution equation, enhanced with wave acceleration, provides a robust framework for modelling complex shock front dynamics in HE materials. This approach not only captures the natural effects of straightness and boundary slope jumps in the transition-zone but also bridges the gap between mathematical predictions and experimental observations, offering insights into the behaviour of both diverging and converging detonation propagations in a homogeneous HE.

acceleration↗
NASA NTRS1986

A spectral multidomain method for the solution of hyperbolic systems

A multidomain Chebyshev spectral collocation method for solving hyperbolic partial differential equations were developed. Though spectral methods are global methods, an attractive idea is to break a computational domain into several domains, and a way to handle the interfaces is described. The multidomain approach offers advantages over the use of a single Chebyshev grid. It allows complex geometries to be covered, and local refinement can be used to resolve important features. For steady state problems it reduces the stiffness associated with the use of explicit time integration as a relaxation scheme. Furthermore, the proposed method remains spectrally accurate. Results showing performance of the method on one dimensional linear models and one and two dimensional nonlinear gas dynamics problems are presented.

Kopriva, D.↗
NASA NTRS1988

Non-oscillatory central differencing for hyperbolic conservation laws

Many of the recently developed high resolution schemes for hyperbolic conservation laws are based on upwind differencing. The building block for these schemes is the averaging of an appropriate Godunov solver; its time consuming part involves the field-by-field decomposition which is required in order to identify the direction of the wind. Instead, the use of the more robust Lax-Friedrichs (LxF) solver is proposed. The main advantage is simplicity: no Riemann problems are solved and hence field-by-field decompositions are avoided. The main disadvantage is the excessive numerical viscosity typical to the LxF solver. This is compensated for by using high-resolution MUSCL-type interpolants. Numerical experiments show that the quality of results obtained by such convenient central differencing is comparable with those of the upwind schemes.

Nessyahu, Haim↗