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NASA NTRS1999

Comparison of AGE and Spectral Methods for the Simulation of Far-Wakes

Turbulent flow simulation methods based on finite differences are attractive for their simplicity, flexibility and efficiency, but not always for accuracy or stability. This report demonstrates that a good compromise is possible with the Advected Grid Explicit (AGE) method. AGE has proven to be both efficient and accurate for simulating turbulent free-shear flows, including planar mixing layers and planar jets. Its efficiency results from its localized fully explicit finite difference formulation (Bisset 1998a,b) that is very straightforward to compute, outweighing the need for a fairly small timestep. Also, most of the successful simulations were slightly under-resolved, and therefore they were, in effect, large-eddy simulations (LES) without a sub-grid-scale (SGS) model, rather than direct numerical simulations (DNS). The principle is that the role of the smallest scales of turbulent motion (when the Reynolds number is not too low) is to dissipate turbulent energy, and therefore they do not have to be simulated when the numerical method is inherently dissipative at its resolution limits. Such simulations are termed 'auto-LES' (LES with automatic SGS modeling) in this report.

Bisset, D. K.↗

NASA NTRS2002

A Multi-domain Spectral Method for Supersonic Reactive Flows

This paper has a dual purpose: it presents a multidomain Chebyshev method for the solution of the two-dimensional reactive compressible Navier-Stokes equations, and it reports the results of the application of this code to the numerical simulations of high Mach number reactive flows in recessed cavity. The computational method utilizes newly derived interface boundary conditions as well as an adaptive filtering technique to stabilize the computations. The results of the simulations are relevant to recessed cavity flameholders.

Don, Wai-Sun↗

NASA NTRS2011

Challenges at Petascale for Pseudo-Spectral Methods on Spheres (A Last Hurrah?)

Conclusions: a) Proper software abstractions should enable rapid-exploration of platform-specific optimizations/ tradeoffs. b) Pseudo-spectra! methods are marginally viable for at least some classes of petascaie problems. i.e., GPU based machine with good bisection would be best. c) Scalability at exascale is possible, but the necessary resolution will make algorithm prohibitively expensive. Efficient implementations of realistic global transposes are mtricate and tedious in MPI. PS at petascaie requires exploration of a variety of strategies for spreading local and remote communic3tions. PGAS allows far simpler implementation and thus rapid exploration of variants.

Clune, Thomas↗

NASA NTRS1992

Adaptive mesh strategies for the spectral element method

An adaptive spectral method was developed for the efficient solution of time dependent partial differential equations. Adaptive mesh strategies that include resolution refinement and coarsening by three different methods are illustrated on solutions to the 1-D viscous Burger equation and the 2-D Navier-Stokes equations for driven flow in a cavity. Sharp gradients, singularities, and regions of poor resolution are resolved optimally as they develop in time using error estimators which indicate the choice of refinement to be used. The adaptive formulation presents significant increases in efficiency, flexibility, and general capabilities for high order spectral methods.

Mavriplis, Catherine↗

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 NTRS2001

A High-Order Finite Spectral Volume Method for Conservation Laws on Unstructured Grids

A time accurate, high-order, conservative, yet efficient method named Finite Spectral Volume (FSV) is developed for conservation laws on unstructured grids. The concept of a 'spectral volume' is introduced to achieve high-order accuracy in an efficient manner similar to spectral element and multi-domain spectral methods. In addition, each spectral volume is further sub-divided into control volumes (CVs), and cell-averaged data from these control volumes is used to reconstruct a high-order approximation in the spectral volume. Riemann solvers are used to compute the fluxes at spectral volume boundaries. Then cell-averaged state variables in the control volumes are updated independently. Furthermore, TVD (Total Variation Diminishing) and TVB (Total Variation Bounded) limiters are introduced in the FSV method to remove/reduce spurious oscillations near discontinuities. A very desirable feature of the FSV method is that the reconstruction is carried out only once, and analytically, and is the same for all cells of the same type, and that the reconstruction stencil is always non-singular, in contrast to the memory and CPU-intensive reconstruction in a high-order finite volume (FV) method. Discussions are made concerning why the FSV method is significantly more efficient than high-order finite volume and the Discontinuous Galerkin (DG) methods. Fundamental properties of the FSV method are studied and high-order accuracy is demonstrated for several model problems with and without discontinuities.

Wang, Z. J.↗

NASA NTRS1972

Effect of radiometric errors on accuracy of temperature-profile measurement by the spectral-scanning method.

The spectral-scanning method may be used to determine the temperature profile of a jet- or rocket-engine exhaust stream by measurements of gas radiation and transmittance at two or more wavelengths. A single, fixed line of sight is used, using immobile radiators and radiometers outside of the gas stream, and there is no interference with the flow. A given radiometric error causes an error in computed temperatures. The ratio between temperature error and radiometric error depends on profile shape, path length, temperature level, the strength of line absorption, and the absorption coefficient and its temperature coefficient. These influence the choice of wavelengths for any given gas. Conditions for minimum temperature error are derived. Numerical results are presented for a two wavelength measurement on a family of profiles that may be expected in a practical case of H2 - O2 combustion. Under favorable conditions, the fractional error in temperature approximates the fractional error in radiant-flux measurement.

Buchele, D. R.↗

NASA NTRS1988

Spectral element methods: Algorithms and architectures

Spectral element methods are high-order weighted residual techniques for partial differential equations that combine the geometric flexibility of finite element methods with the rapid convergence of spectral techniques. Spectral element methods are described for the simulation of incompressible fluid flows, with special emphasis on implementation of spectral element techniques on medium-grained parallel processors. Two parallel architectures are considered: the first, a commercially available message-passing hypercube system; the second, a developmental reconfigurable architecture based on Geometry-Defining Processors. High parallel efficiency is obtained in hypercube spectral element computations, indicating that load balancing and communication issues can be successfully addressed by a high-order technique/medium-grained processor algorithm-architecture coupling.

Fischer, Paul↗

NASA NTRS1988

Spectral element methods - Algorithms and architectures

Spectral element methods are high-order weighted residual techniques for partial differential equations that combine the geometric flexibility of finite element methods with the rapid convergence of spectral techniques. Spectral element methods are described for the simulation of incompressible fluid flows, with special emphasis on implementation of spectral element techniques on medium-grained parallel processors. Two parallel architectures are considered; the first, a commercially available message-passing hypercube system; the second, a developmental reconfigurable architecture based on Geometry-Defining Processors. High parallel efficiency is obtained in hypercube spectral element computations, indicating that load balancing and communication issues can be successfully addressed by a high-order technique/medium-grained processor algorithm-architecture coupling.

Fischer, Paul↗

NASA NTRS1996

Chebyshev polynomials in the spectral Tau method and applications to Eigenvalue problems

Chebyshev Spectral methods have received much attention recently as a technique for the rapid solution of ordinary differential equations. This technique also works well for solving linear eigenvalue problems. Specific detail is given to the properties and algebra of chebyshev polynomials; the use of chebyshev polynomials in spectral methods; and the recurrence relationships that are developed. These formula and equations are then applied to several examples which are worked out in detail. The appendix contains an example FORTRAN program used in solving an eigenvalue problem.

Johnson, Duane↗

NASA NTRSDate not supplied

Spectral Gap Filling Method Using the Spectral Information from CLARREO-Pathfinder Measured Earth-Reflected Solar Spectra to Support CPF-CERES Intercalibration

The high-accuracy Earth-reflected solar spectral radiances from CLARREO Pathfinder (CPF) will serve as in-orbit benchmark measurements for intercalibrating other satellite imagers. The CPF team is developing a state-of-the-art intercalibration method to calibrate the shortwave channel (300-5000 nm) of the CERES instrument against the CPF reference with an uncertainty of 0.3%. Because the CPF instrument only covers spectral range from 350 nm to 2300 nm, the team has developed a spectral extension method to fill in the gaps between the spectral domains of CPF and the CERES shortwave channel. The method solely relies on using the hyperspectral information from the CPF wavelengths to predict the radiances below 350 nm and above 2300 nm. To quantify the relationship between the CPF spectrum and the non-overlapping portions of the spectrum between CPF and CERES, the team has simulated a large ensemble of spectra that cover the spectral range for both CPF and CERES. A hyperspectral bidirectional reflectance model for land surface based on thousands of measured land surface reflectance has been developed. This land surface model as well as the Cox-Munk model for ocean surface are used in the spectral gap filling. Various water/ice clouds, aerosols, atmosphere profiles and surfaces were used to simulate Earth-reflected radiances from 200 nm to 15 um under different solar and sensor geometries. The targeted uncertainty for the CPF-extended broadband CERES radiances is less than 0.1%. The previously derived training spectra were updated to replace the Kurucz solar irradiance model with the TSIS-1 Hybrid Solar Reference Spectrum. The influence of the scalar assumption in radiative transfer model equations on the uncertainty in estimating the broadband radiances was also investigated.

Hyperspectral↗

NASA NTRS2016

Modified Spectral Fatigue Methods for S-N Curves With MIL-HDBK-5J Coefficients

The rainflow method is used for counting fatigue cycles from a stress response time history, where the fatigue cycles are stress-reversals. The rainflow method allows the application of Palmgren-Miner's rule in order to assess the fatigue life of a structure subject to complex loading. The fatigue damage may also be calculated from a stress response power spectral density (PSD) using the semi-empirical Dirlik, Single Moment, Zhao-Baker and other spectral methods. These methods effectively assume that the PSD has a corresponding time history which is stationary with a normal distribution. This paper shows how the probability density function for rainflow stress cycles can be extracted from each of the spectral methods. This extraction allows for the application of the MIL-HDBK-5J fatigue coefficients in the cumulative damage summation. A numerical example is given in this paper for the stress response of a beam undergoing random base excitation, where the excitation is applied separately by a time history and by its corresponding PSD. The fatigue calculation is performed in the time domain, as well as in the frequency domain via the modified spectral methods. The result comparison shows that the modified spectral methods give comparable results to the time domain rainflow counting method.

Irvine, Tom↗

NASA NTRS1992

A spectral overlay method for dissimilar materials

The spectral overlay method is applied to examine the stress distribution along the interface between the two different materials. The essential feature of the spectral overlay method is that the high resolution of localized steep gradients can be achieved by overlaying a spectral interpolant on a standard finite element mesh. In order to evaluate this method and compare it with conventional finite element method, a test problem with high gradients in Poisson equation has been devised. The results show that this method is very powerful in capturing the shape of the gradient field and its peak value. Solutions for two problems with interfaces between dissimilar materials are then given to illustrate the effectiveness of this method.

Belytschko, T.↗

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 NTRS1983

A spectral numerical method for the Navier-Stokes equations with applications to Taylor-Couette flow

A new spectral method for solving the incompressible Navier-Stokes equations in a plane channel and between concentric cylinders is presented. The method uses spectral expansions which inherently satisfy the boundary conditions and the continuity equation and yield banded matrices which are efficiently solved at each time step. In addition, the number of dependent variables is reduced, resulting in a reduction in computer memory requirements. Several test problems have been computed for the channel flow and for flow between concentric cylinders, including Taylor-Couette flow with axisymmetric Taylor vortices and wavy vortices. In all cases, agreement with available experimental and theoretical results is very good.

Moser, R. D.↗

NASA NTRS1995

Solution of acoustic workshop problems by a spectral multidomain method

We use a new staggered grid Chebyshev spectral multidomain method to solve three of the Workshop benchmark problems. The method defines solution unknowns at the nodes of the Chebyshev Gauss quadrature, and the fluxes at the nodes of the Chebyshev Gauss-Lobatto quadrature. The Chebyshev spectral method gives exponentially convergent phase and dissipation errors. The multidomain approximation gives the method flexibility. Using the method, we solve problems in Categories 1 and 5 of the benchmark problems.

Kopriva, Davis A.↗

NASA NTRS1983

Spectral multigrid methods for elliptic equations 2

A detailed description of spectral multigrid methods is provided. This includes the interpolation and coarse-grid operators for both periodic and Dirichlet problems. The spectral methods for periodic problems use Fourier series and those for Dirichlet problems are based upon Chebyshev polynomials. An improved preconditioning for Dirichlet problems is given. Numerical examples and practical advice are included.

Zang, T. A.↗

NASA NTRS1984

Spectral multigrid methods for elliptic equations II

Zang, T. A.↗