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At least 109 records · Page 6

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NASA NTRS1987

A practical assessment of spectral accuracy for hyperbolic problems with discontinuities

Physical and transform space filtering has been applied to the Fourier spectral collocation solution of the constant coefficient scalar wave equation with a discontinuous initial condition. High order accuracy can be extracted from the unfiltered solution. Smooth, high order Fourier space filtering gives expected polynomial order solutions away from the discontinuity. Spectral accuracy is observed with the physical space filter of Gottlieb and Tadmor.

Kopriva, David A.↗

NASA NTRS2004

Relaxation and Preconditioning for High Order Discontinuous Galerkin Methods with Applications to Aeroacoustics and High Speed Flows

This project is about the investigation of the development of the discontinuous Galerkin finite element methods, for general geometry and triangulations, for solving convection dominated problems, with applications to aeroacoustics. Other related issues in high order WENO finite difference and finite volume methods have also been investigated. methods are two classes of high order, high resolution methods suitable for convection dominated simulations with possible discontinuous or sharp gradient solutions. In [18], we first review these two classes of methods, pointing out their similarities and differences in algorithm formulation, theoretical properties, implementation issues, applicability, and relative advantages. We then present some quantitative comparisons of the third order finite volume WENO methods and discontinuous Galerkin methods for a series of test problems to assess their relative merits in accuracy and CPU timing. In [3], we review the development of the Runge-Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge-Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier-Stokes equations, and Hamilton-Jacobi-like equations.

Shu, Chi-Wang↗

NASA NTRS1984

Canonical solutions for unsteady flow fields

The initial value problem of one-dimensional gas-dynamics involving discontinuous, nonuniform initial data is discussed. Canonical solutions which are valid in a small x, t region aroung a discontinuity, and which include the first order effects of nonuniformities in the data, are derived explicitly. The theory is derived by considering a group of elementary piston problems. Solutions with a shock or with a centered expansion wave are worked out individually in order to relate initial flow properties and their gradients to the speed and acceleration of the discontinuity waves. They are then combined to represent the solution of a general initial value problem by regarding the piston path as a contact line. In addition, problems with chemical reaction are discussed in terms of elementary piston problems which involve strong detonation waves, Chapman-Jouguet detonation waves, and deflagration waves.

Liu, G. C.↗

NASA NTRS1980

The applicability of the piecewise linear current profile in the baroclinic instability problem

The applicability of the piecewise linear function in place of a similar smoothly-varying current profile is examined in the baroclinic context. Within the framework of small-perturbation linearization, the behavior of the vertical velocity and the horizontal divergence is analyzed at the discontinuity of the current shear. In the conventional geostrophic-type instability regime, the discontinuity in the horizontal divergence at the shear discontinuity is suppressed, and, therefore, the piecewise linear profile leads to a useful approximation to the true solution. In the symmetric-type instability regime, however, due to the magnified discontinuity in the horizontal divergence at the shear discontinuity, the solution thus obtained will show a major distortion, rendering the piecewise linear profile inadequate for modeling the smoothly-varying current profile. Using exemplary current profiles, numerical results are presented to demonstrate the behavior of the horizontal divergence near the discontinuity of current shear.

Hyun, J. M.↗

NASA NTRS1990

Accuracy of the ERBS definitive attitude determination system in the presence of propagation noise

Definitive attitude solutions are supposed to be the most accurate possible. For the Earth Radiation Budget Satellite (ERBS), this has been accomplished by using gyro rates to transform many nonsimultaneous observations to a common time point and then averaging to reduce the effects of observation noise. Rate quality is critical to realizing improved accuracy with this method. Gyro deterioration, which shows up as large observation residuals and discontinuities between contiguous batch solutions, now discourages using the batch approach for ERBS. To address this problem, a simple Kalman filter is tried in place of the batch estimator. The filter works well as long as the attitude is completely observable. During periods without Sun coverage, however, the extrapolated yaw may diverge and then change abruptly when the Sun returns to the sensor field of view. Causes of this behavior are discussed, and some solutions are tried that address the observability aspect of the problem.

Chu, D.↗

NASA NTRS1991

Theoretical and experimental characterization of coplanar waveguide discontinuities for filter applications

A full-wave analysis of shielded coplanar waveguide two-port discontinuities based on the solution of an appropriate surface integral equation in the space domain is presented. Frequency-dependent scattering parameters for open-end and short-end coplanar waveguide (CPW) stubs are computed using this method. The numerically derived results are compared with measurements performed in the frequency range 5-25 GHz and show very good agreement. From the scattering parameters, lumped-element equivalent circuits have been derived to model the discontinuities. The inductors and capacitors of these models have been represented by closed-form equations, as functions of the stub length, to compute the circuit element values for these discontinuities.

Dib, Nihad I.↗

NASA NTRS1999

Numerical System Solver Developed for the National Cycle Program

As part of the National Cycle Program (NCP), a powerful new numerical solver has been developed to support the simulation of aeropropulsion systems. This software uses a hierarchical object-oriented design. It can provide steady-state and time-dependent solutions to nonlinear and even discontinuous problems typically encountered when aircraft and spacecraft propulsion systems are simulated. It also can handle constrained solutions, in which one or more factors may limit the behavior of the engine system. Timedependent simulation capabilities include adaptive time-stepping and synchronization with digital control elements. The NCP solver is playing an important role in making the NCP a flexible, powerful, and reliable simulation package.

Binder, Michael P.↗

NASA NTRS1992

Unsteady, one-dimensional gas dynamics computations using a TVD type sequential solver

The efficacy of high resolution convection schemes to resolve sharp gradient in unsteady, 1D flows is examined using the TVD concept based on a sequential solution algorithm. Two unsteady flow problems are considered which include the problem involving the interaction of the various waves in a shock tube with closed reflecting ends and the problem involving the unsteady gas dynamics in a tube with closed ends subject to an initial pressure perturbation. It is concluded that high accuracy convection schemes in a sequential solution framework are capable of resolving discontinuities in unsteady flows involving complex gas dynamics. However, a sufficient amount of dissipation is required to suppress oscillations near discontinuities in the sequential approach, which leads to smearing of the solution profiles.

Thakur, Siddharth↗

DOE OSTI2024

A phase-field diffraction model for thermo-hydro-mechanical propagating fractures

This paper introduces a novel diffraction based thermo-hydraulic–mechanical (THM) model for fracture propagation using a phase-field fracture (PFF) approach. The key innovation of the THM-PFF model lies in its integrated treatment of four solution variables—displacements, phase-field, pressure, and temperature—each governed by a combination of conservation of momentum (mechanics problem), a variational inequality (constrained minimization problem), mass conservation (pressure problem), and energy conservation (temperature problem). This leads to a new formulation of a coupled variational inequality system. A major advancement is the development of an extended fixed-stress algorithm, where displacements, phase-field, pressures, and temperatures are solved in a staggered sequence. An important aspect of this work is the global coupling of pressures and temperatures across the domain using diffraction systems, with diffraction coefficients defined by material parameters weighted by the diffusive phase-field variable. To ensure robust local mass conservation, we employ enriched Galerkin finite elements (EG) for both pressure and temperature diffraction equations. By enriching the continuous Galerkin basis functions with discontinuous piecewise constants, EG accurately represents solution and parameter discontinuities while preserving local mass and energy conservation—crucial aspects for THM problems and realistic behavior. Moreover, the use of a predictor–corrector local mesh adaptivity scheme is employed, allowing the model to handle small phase-field length-scale parameters while maintaining high numerical accuracy and reasonable computational cost. Furthermore, these new model and algorithmic developments represent significant advances in the field and have been substantiated through rigorous numerical tests.

Diffraction systems↗

NASA NTRS2002

A Localized Tau Method PDE Solver

In this paper we present a new form of the collocation method that allows one to find very accurate solutions to time marching problems without the unwelcome appearance of Gibb's phenomenon oscillations. The basic method is applicable to any partial differential equation whose solution is a continuous, albeit possibly rapidly varying function. Discontinuous functions are dealt with by replacing the function in a small neighborhood of the discontinuity with a spline that smoothly connects the function segments on either side of the discontinuity. This will be demonstrated when the solution to the inviscid Burgers equation is discussed.

Cottam, Russell↗

NASA NTRS1983

Solutions of Transonic Flow in Turbomachines

Accurate approximation obtained using perturbation techniques. Pertubation procedures determine highly accurate approximations to families of nonlinear solutions either continuous or discontinuous and represent variations in some arbitrary parameter. Program written in FORTRAN IV.

Stahara, S.↗

NASA NTRS1984

Effect of hot electrons on the polar wind

A semikinetic model is used to describe the steady state collisionless flow of H(+), O(+), and electrons along diverging geomagnetic field lines in the high-latitude topside ionosphere. The effect that hot electron populations have on the polar wind is emphasized. Several such populations are considered, including the polar rain, polar showers, and polar squall. Hot electron densities and temperatures are calculated from the characteristic energy and flux measurements. The results indicate that the hot/cold electron temperature ratio varies from 10 to 10,000 and that the hot/cold electron density ratio varies from 0.001 to 0.1 at the baropause. For higher hot electron temperatures and a greater percentage of hot electrons, there is a discontinuity in the kinetic solution, which indicates the presence of a sharp transition corresponding to a contact surface between the hot and cold electrons. Along this surface, a double-layer potential barrier exists which reflects the cold ionospheric electrons and prevents their penetrations to higher altitudes.

Barakat, A. R.↗

NASA NTRS1989

Shock capturing by the spectral viscosity method

A main disadvantage of using spectral methods for nonlinear conservation laws lies in the formation of Gibbs phenomenon, once spontaneous shock discontinuities appear in the solution. The global nature of spectral methods than pollutes the unstable Gibbs oscillations overall the computational domain, and the lack of entropy dissipation prevents convergences in these cases. The Spectral Viscosity method, which is based on high frequency dependent vanishing viscosity regularization of the classical spectral methods is discussed. It is shown that this method enforces the convergence of nonlinear spectral approximations without sacrificing their overall spectral accuracy.

Tadmor, Eitan↗

NASA NTRS1995

Multiresolution Representation Using Biorthogonal Multiwavelets

We generalize Harten's multiresolution representation to biorthogonal multiwavelets. Several variants are considered. For example, a given array of discrete point values is transformed to point values and derivatives or point 'values and cell averages'. Compact Hermite interpolation is used in the decomposition and reconstruction algorithm. The resulting basis functions that are symmetric or skewsymmetric, compact, and smooth with optimal order accuracy. Harten's approach has several advantages: the multiresolution scheme is inherently discrete, non-periodic boundary conditions are easy to implement, and the representation can be extended to unstructured grids in bounded domains. We demonstrate the compression features of the new mutliwavelets by application to variable scale piecewise smooth functions with jump discontinuities typical of numerical solutions of nonlinear hyperbolic conservation laws.

Warming, Robert F.↗

NASA NTRS1997

Hypersonic Flow Computations on Unstructured Meshes

A method for computing inviscid hypersonic flow over complex configurations using unstructured meshes is presented. The unstructured grid solver uses an edge{based finite{volume formulation. Fluxes are computed using a flux vector splitting scheme that is capable of representing constant enthalpy solutions. Second{order accuracy in smooth flow regions is obtained by linearly reconstructing the solution, and stability near discontinuities is maintained by locally forcing the scheme to reduce to first-order accuracy. The implementation of the algorithm to parallel computers is described. Computations using the proposed method are presented for a sphere-cone configuration at Mach numbers of 5.25 and 10.6, and a complex hypersonic re-entry vehicle at Mach numbers of 4.5 and 9.8. Results are compared to experimental data and computations made with established structured grid methods. The use of the solver as a screening tool for rapid aerodynamic assessment of proposed vehicles is described.

Bibb, K. L.↗

NASA NTRS1972

Kinetic theory analysis of rarefied gas flow through finite length slots

An analytic study is made of the flow a rarefied monatomic gas through a two dimensional slot. The parameters of the problem are the ratios of downstream to upstream pressures, the Knudsen number at the high pressure end (based on slot half width) and the length to slot half width ratio. A moment method of solution is used by assuming a discontinuous distribution function consisting of four Maxwellians split equally in angular space. Numerical solutions are obtained for the resulting equations. The characteristics of the transition regime are portrayed. The solutions in the free molecule limit are systematically lower than the results obtained in that limit by more accurate numerical methods.

Raghuraman, P.↗

NASA NTRS1983

An investigation of the accuracy of finite difference methods in the solution of linear elasticity problems

The accuracy of the finite difference method in the solution of linear elasticity problems that involve either a stress discontinuity or a stress singularity is considered. Solutions to three elasticity problems are discussed in detail: a semi-infinite plane subjected to a uniform load over a portion of its boundary; a bimetallic plate under uniform tensile stress; and a long, midplane symmetric, fiber reinforced laminate subjected to uniform axial strain. Finite difference solutions to the three problems are compared with finite element solutions to corresponding problems. For the first problem a comparison with the exact solution is also made. The finite difference formulations for the three problems are based on second order finite difference formulas that provide for variable spacings in two perpendicular directions. Forward and backward difference formulas are used near boundaries where their use eliminates the need for fictitious grid points.

Bauld, N. R., Jr.↗

NASA NTRS1980

Evaluation of a strained-coordinate perturbation procedure - Nonlinear subsonic and transonic flows

An evaluation is made of a perturbation method devised to obtain highly accurate approximations to families of strongly nonlinear solutions which are either continuous or discontinuous, and which represent variations in some arbitrary parameter. The method first defines a unit perturbation by using two nonlinear solutions which differ from one another by a nominal change in some geometric or flow parameter, then employs this unit perturbation to predict a family of related nonlinear solutions over a range of parameter variation. Coordinate straining is incorporated into this perturbation method for determining the unit perturbation to account for the movement of discontinuities and maxima of high-gradient regions due to the perturbation. Attention is given to transonic and subsonic flows. Comparisons of the perturbation results with the corresponding 'exact' nonlinear solutions show a remarkable accuracy and range of validity of the perturbation method across the spectrum of examples considered.

Stahara, S. S.↗