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NASA NTRS1975

High-order numerical solutions using cubic splines

The cubic spline collocation procedure for the numerical solution of partial differential equations was reformulated so that the accuracy of the second-derivative approximation is improved and parallels that previously obtained for lower derivative terms. The final result is a numerical procedure having overall third-order accuracy for a nonuniform mesh and overall fourth-order accuracy for a uniform mesh. Application of the technique was made to the Burger's equation, to the flow around a linear corner, to the potential flow over a circular cylinder, and to boundary layer problems. The results confirmed the higher-order accuracy of the spline method and suggest that accurate solutions for more practical flow problems can be obtained with relatively coarse nonuniform meshes.

Rubin, S. G.↗

NASA NTRS1998

Capacitively Stabilized Etalon Technology for Spaceborne Wind Lidar Application

Global monitoring by satellites is important for many types of environmental studies. Among these, the wind field is the single most important atmospheric state variable required for understanding atmospheric motion and predicting weather. Studies indicate that a global determination of the tropospheric wind field to an accuracy of 1-5 m/sec is critical for improved numerical weather forecasting. This measurement could be carried out with a spaceborne lidar system sensing the Doppler shift of a laser signal backscattered from the atmosphere. Over the past four years we have developed a ground-based Edge Technique lidar system and demonstrated wind measurements through the troposphere with high accuracy (1-6 m/s) and high spatial resolution. Recently, we began the design of a shuttle based wind measurement system for technology demonstration as part of the Zephyr program. In this paper, we present the characteristics of the high spectral resolution etalon filter technology for lidar wind measurement with the Edge Technique. In conclusion, a spacecraft motion compensation method has been discussed. We demonstrated experimentally that the etalon can be tuned to compensate for any satellite induced Doppler shift to an accuracy of better than 0.6 m/s over a time period of 2.5 ms.

Li, Steven X.↗

NASA NTRS1975

Structural reanalysis via a mixed method

A study is made of the approximate structural reanalysis technique based on the use of Taylor series expansion of response variables in terms of design variables in conjunction with the mixed method. In addition, comparisons are made with two reanalysis techniques based on the displacement method. These techniques are the Taylor series expansion and the modified reduced basis. It is shown that the use of the reciprocals of the sizing variables as design variables (which is the natural choice in the mixed method) can result in a substantial improvement in the accuracy of the reanalysis technique. Numerical results are presented for a space truss structure.

Noor, A. K.↗

NASA NTRS1983

Self-adjusting grid methods for one-dimensional hyperbolic conservation laws

The automatic adjustment of a grid which follows the dynamics of the numerical solution of hyperbolic conservation laws is given. The grid motion is determined by averaging the local characteristic velocities of the equations with respect to the amplitudes of the signals. The resulting algorithm is a simple extension of many currently popular Godunov-type methods. Computer codes using one of these methods can be easily modified to add the moving mesh as an option. Numerical examples are given that illustrate the improved accuracy of Godunov's and Roe's methods on a self-adjusting mesh. Previously announced in STAR as N83-15008

Harten, A.↗

NASA NTRS1990

Development of a defect stream function, law of the wall/wake method for compressible turbulent boundary layers

The method presented is designed to improve the accuracy and computational efficiency of existing numerical methods for the solution of flows with compressible turbulent boundary layers. A compressible defect stream function formulation of the governing equations assuming an arbitrary turbulence model is derived. This formulation is advantageous because it has a constrained zero-order approximation with respect to the wall shear stress and the tangential momentum equation has a first integral. Previous problems with this type of formulation near the wall are eliminated by using empirically based analytic expressions to define the flow near the wall. The van Driest law of the wall for velocity and the modified Crocco temperature-velocity relationship are used. The associated compressible law of the wake is determined and it extends the valid range of the analytical expressions beyond the logarithmic region of the boundary layer. The need for an inner-region eddy viscosity model is completely avoided. The near-wall analytic expressions are patched to numerically computed outer region solutions at a point determined during the computation. A new boundary condition on the normal derivative of the tangential velocity at the surface is presented; this condition replaces the no-slip condition and enables numerical integration to the surface with a relatively coarse grid using only an outer region turbulence model. The method was evaluated for incompressible and compressible equilibrium flows and was implemented into an existing Navier-Stokes code using the assumption of local equilibrium flow with respect to the patching. The method has proven to be accurate and efficient.

Wahls, Richard A.↗

NASA NTRS1976

Higher-order numerical solutions using cubic splines

A cubic spline collocation procedure was developed for the numerical solution of partial differential equations. This spline procedure is reformulated so that the accuracy of the second-derivative approximation is improved and parallels that previously obtained for lower derivative terms. The final result is a numerical procedure having overall third-order accuracy of a nonuniform mesh. Solutions using both spline procedures, as well as three-point finite difference methods, are presented for several model problems.

Rubin, S. G.↗

NASA NTRS1995

Testing higher-order Lagrangian perturbation theory against numerical simulations. 2: Hierarchical models

We present results showing an improvement of the accuracy of perturbation theory as applied to cosmological structure formation for a useful range of scales. The Lagrangian theory of gravitational instability of Friedmann-Lemaitre cosmogonies is compared with numerical simulations. We study the dynamics of hierarchical models as a second step. In the first step we analyzed the performance of the Lagrangian schemes for pancake models, the difference being that in the latter models the initial power spectrum is truncated. This work probed the quasi-linear and weakly non-linear regimes. We here explore whether the results found for pancake models carry over to hierarchical models which are evolved deeply into the non-linear regime. We smooth the initial data by using a variety of filter types and filter scales in order to determine the optimal performance of the analytical models, as has been done for the 'Zel'dovich-approximation' - hereafter TZA - in previous work. We find that for spectra with negative power-index the second-order scheme performs considerably better than TZA in terms of statistics which probe the dynamics, and slightly better in terms of low-order statistics like the power-spectrum. However, in contrast to the results found for pancake models, where the higher-order schemes get worse than TZA at late non-linear stages and on small scales, we here find that the second-order model is as robust as TZA, retaining the improvement at later stages and on smaller scales. In view of these results we expect that the second-order truncated Lagrangian model is especially useful for the modelling of standard dark matter models such as Hot-, Cold-, and Mixed-Dark-Matter.

Melott, A. L.↗

NASA NTRSDate not supplied

An Improved Approach to the Predictability & Reliability of the Onset of Turbulence With Shocks

The construction of numerical schemes for (a) stable and accurate simulation of turbulence with strong shocks, and for (b) obtaining correct propagation speed of discontinuities in the presence of stiff source terms share one important ingredient – minimization of numerical dissipation while maintaining numerical stability. The dual requirements to achieve both numerical stability and minimal numerical dissipation are often conflicting since existing shock capturing schemes were designed mainly to be robust for rapidly developed turbulence-free flows and for shock waves without stiff source term. For the past two decades, Yee and collaborators have focused on an improved understanding of the nonlinear behavior of different high order shock-capturing methods. It was found that even very high order methods without proper nonlinear stability and numerical dissipation control can either numerically smear the onset of turbulence due to excess numerical dissipation, or induce (onset) numerical turbulence that is not physical turbulence due to lack of proper numerical dissipation to improve nonlinear stability for long time integration. Our approach is to combine (I) and (II) below for obtaining the physically correct onset of turbulence with shocks, including problems with stiff source terms: (I) Nonlinear dynamics is utilized to complement the traditional linearized stability theory (Yee & Sweby, Yee et al., Griffiths et al., Lafon & Yee, Yee, Wang et al., Kotov et al. 1990- 2015) in order to (i) Minimize numerically induced false transition to turbulence, (ii) Minimize numerical instability due to long time integration of turbulent flows, (iii) Minimize numerically induced standing wave solutions, and (iv) Minimize wrong propagation of speed of discontinuities due to the presence of stiff source terms. (II) Our recently developed physical preserving (structural preserving) high order methods with improved nonlinear stability & accuracy that are essential in minimizing spurious numerics are used.

HECC↗

NASA NTRS2002

Impact of Quikscat Data on Numerical Weather Prediction

One of the important applications of satellite surface wind observations is to increase the accuracy of weather analyses and forecasts. Satellite surface wind data can improve numerical weather prediction (NWP) model forecasts by contributing to improved analyses of the surface wind field and air sea fluxes. Through the data assimilation process,these data can also improve atmospheric mass and motion fields in the free atmosphere above the surface. The SeaWinds scatterometer on the QuikScat satellite was launched in July 1999 and represented a dramatic departure in design from the other scatterometer instruments launched during the past decade (ERS-1,2 and NSCAT). The NASA Data Assimilation Office (DAO) was the first data assimilation center to assimilate QuikScat Seawinds data and evaluate their impact on numerical weather prediction. Following the launch of QuikScat, a detailed evaluation of the initial surface wind data sets was performed as part of a collaborative project between the Environmental Modeling Center of NCEP, NESDIS and the DAO. More recently, the impact of Quikscat data was evaluated in detailed experiments using the NCEP operational data assimilation system. As a result of the beneficial impact obtained, NCEP began operational utilization of Quikscat data. Results from these experiments as well as recent DAO assimilation experiments showing the impact of Quikscat data on stratospheric analyses and forecasts will be presented at the meeting.

Atlas, Robert↗

NASA NTRS1993

Testing higher-order Lagrangian perturbation theory against numerical simulation. 1: Pancake models

We present results showing an improvement of the accuracy of perturbation theory as applied to cosmological structure formation for a useful range of quasi-linear scales. The Lagrangian theory of gravitational instability of an Einstein-de Sitter dust cosmogony investigated and solved up to the third order is compared with numerical simulations. In this paper we study the dynamics of pancake models as a first step. In previous work the accuracy of several analytical approximations for the modeling of large-scale structure in the mildly non-linear regime was analyzed in the same way, allowing for direct comparison of the accuracy of various approximations. In particular, the Zel'dovich approximation (hereafter ZA) as a subclass of the first-order Lagrangian perturbation solutions was found to provide an excellent approximation to the density field in the mildly non-linear regime (i.e. up to a linear r.m.s. density contrast of sigma is approximately 2). The performance of ZA in hierarchical clustering models can be greatly improved by truncating the initial power spectrum (smoothing the initial data). We here explore whether this approximation can be further improved with higher-order corrections in the displacement mapping from homogeneity. We study a single pancake model (truncated power-spectrum with power-spectrum with power-index n = -1) using cross-correlation statistics employed in previous work. We found that for all statistical methods used the higher-order corrections improve the results obtained for the first-order solution up to the stage when sigma (linear theory) is approximately 1. While this improvement can be seen for all spatial scales, later stages retain this feature only above a certain scale which is increasing with time. However, third-order is not much improvement over second-order at any stage. The total breakdown of the perturbation approach is observed at the stage, where sigma (linear theory) is approximately 2, which corresponds to the onset of hierarchical clustering. This success is found at a considerable higher non-linearity than is usual for perturbation theory. Whether a truncation of the initial power-spectrum in hierarchical models retains this improvement will be analyzed in a forthcoming work.

Buchert, T.↗

DOE OSTI2024

Arbitrary Order Virtual Element Methods for High‐Order Phase‐Field Modeling of Dynamic Fracture

ABSTRACT Accurate modeling of fracture nucleation and propagation in brittle and ductile materials subjected to dynamic loading is important in predicting material damage and failure under extreme conditions. Phase‐field fracture models have garnered a lot of attention in recent years due to their success in representing damage and fracture processes in a wide class of materials and under a variety of loading conditions. Second‐order phase‐field fracture models are by far the most popular among researchers (and increasingly, among practitioners), but fourth‐order models have started to gain broader acceptance since their more recent introduction. The exact solution corresponding to these high‐order phase‐field fracture models has higher regularity. Thus, numerical solutions of the model equations can achieve improved accuracy and higher spatial convergence rates. In this work, we develop a virtual element framework for the high‐order phase‐field model of dynamic fracture. The virtual element method (VEM) can be regarded as a generalization of the classical finite element method. In addition to many other desirable characteristics, the VEM allows computing on polytopal meshes. Here, we use ‐conforming virtual elements and the generalized‐ time integration method for the momentum balance equation, and adopt ‐conforming virtual elements for the high‐order phase‐field equation. We verify our virtual element framework using classical quasi‐static benchmark problems and demonstrate its capabilities with the aid of numerical simulations of dynamic fracture in brittle materials.

42 ENGINEERING↗

NASA NTRS1975

Higher-order numerical solutions using cubic splines

A cubic spline collocation procedure has recently been developed for the numerical solution of partial differential equations. In the present paper, this spline procedure is reformulated so that the accuracy of the second-derivative approximation is improved and parallels that previously obtained for lower derivative terms. The final result is a numerical procedure having overall third-order accuracy for a non-uniform mesh and overall fourth-order accuracy for a uniform mesh. Solutions using both spline procedures, as well as three-point finite difference methods, will be presented for several model problems.-

Rubin, S. G.↗

NASA NTRS2007

Investigation of a Parabolic Iterative Solver for Three-dimensional Configurations

A parabolic iterative solution procedure is investigated that seeks to extend the parabolic approximation used within the internal propagation module of the duct noise propagation and radiation code CDUCT-LaRC. The governing convected Helmholtz equation is split into a set of coupled equations governing propagation in the positive and negative directions. The proposed method utilizes an iterative procedure to solve the coupled equations in an attempt to account for possible reflections from internal bifurcations, impedance discontinuities, and duct terminations. A geometry consistent with the NASA Langley Curved Duct Test Rig is considered and the effects of acoustic treatment and non-anechoic termination are included. Two numerical implementations are studied and preliminary results indicate that improved accuracy in predicted amplitude and phase can be obtained for modes at a cut-off ratio of 1.7. Further predictions for modes at a cut-off ratio of 1.1 show improvement in predicted phase at the expense of increased amplitude error. Possible methods of improvement are suggested based on analytic and numerical analysis. It is hoped that coupling the parabolic iterative approach with less efficient, high fidelity finite element approaches will ultimately provide the capability to perform efficient, higher fidelity acoustic calculations within complex 3-D geometries for impedance eduction and noise propagation and radiation predictions.

Nark, Douglas M.↗

NASA NTRS1972

Numerical analysis of the asymptotic two-point boundary value solution for N-body trajectories.

Previously published asymptotic solutions for lunar and interplanetary trajectories have been modified and combined to formulate a general analytical boundary value solution applicable to a broad class of trajectory problems. In addition, the earlier first-order solutions have been extended to second-order to determine if improved accuracy is possible. Comparisons between the asymptotic solution and numerical integration for several lunar and interplanetary trajectories show that the asymptotic solution is generally quite accurate. Also, since no iterations are required, a solution to the boundary value problem is obtained in a fraction of the time required for numerically integrated solutions.

Lancaster, J. E.↗

NASA NTRS1979

Derivation of clear-column temperature profiles with high vertical resolution and accuracy

It is noted that the optimum tradeoff between accuracy and resolution for current sounders leads to an rms accuracy of 2.5 K and a vertical resolution of about 5 km, while current numerical weather prediction models require temperature profiles with an accuracy of 1-1.5 K and a vertical resolution of about 2 km in the troposphere. The present paper shows that improvements in both the vertical resolution and accuracy require improvements in the informational content of the sounding systems and cannot be achieved by numerical means alone. Results are presented from a sounding system with narrow band channels to illustrate the improvements which can be achieved with future sounders. Finally, the results of several numerical simulations are discussed which show that clear column temperature profiles can be derived in the presence of clouds with an rms accuracy of 1.5 K and vertical resolution of 2 km in the troposphere.

Chahine, M. T.↗

NASA NTRS1980

A semianalytical satellite theory for weak time-dependent perturbations

The modifications of the semianalytical satellite theory required to include these 'weak' time dependent perturbations are described. The new formulation results in additional terms in the short periodic variations but does not change the averaged equations of motion. Thus the m monthly terms are still included in the averaged equations of motion. This contrasts with the usual approach for the strongly time dependent perturbations in which the m monthly (or m daily, if tesseral harmonics are being considered) terms would be eliminated from the averaged equations of motion and included in the short periodics computation. Numerical test results for the GPS case obtained with a numerical averaging implementation of the new theory demonstrate the accuracy improvement.

Cefola, P.↗

NASA NTRS1981

Mesh generation by conformal and quasiconformal mappings

It is pointed out that many recent advances in the finite-difference solution of elliptic equations have been limited to regions whose boundary contours coincide with coordinate lines of the Cartesian coordinate system. The reason for this is related to the fact that in the case of an arbitrary curvilinear coordinate system the original equation becomes much more complex. However, there is no added complexity if an orthogonal coordinate system is generated from a conformal mapping. In the present investigation, a finite difference method developed for the construction of conformal mappings has been generalized to construct quasi-conformal mappings. It is expected that the use of more sophisticated numerical algorithms could lead to improvements in both speed and accuracy. Quasi-conformal mappings have applications not only in the solution of elliptic equations but also in other areas such as orthogonal mesh generation on surfaces and the solution of certain fluid flow problems.

Mastin, C. W.↗

NASA NTRS1992

Final results of an experiment in operational forecasting of sea breeze thunderstorms using a mesoscale numerical model

Sea breeze thunderstorms during quiescent synoptic conductions account for 40 percent of Florida rainfall, and are the dominant feature of April-October weather at the Kennedy Space Center (KSC). An effort is presently made to assess the feasibility of a mesoscale numerical model in improving the point-specific thunderstorm forecasting accuracy at the KSC, in the 2-12 hour time frame. Attention is given to the Applied Regional Atmospheric Modeling System.

Lyons, Walter A.↗