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NASA NTRS1993

The piecewise-linear predictor-corrector code - A Lagrangian-remap method for astrophysical flows

We describe a time-explicit finite-difference algorithm for solving the nonlinear fluid equations. The method is similar to existing Eulerian schemes in its use of operator-splitting and artificial viscosity, except that we solve the Lagrangian equations of motion with a predictor-corrector and then remap onto a fixed Eulerian grid. The remap is formulated to eliminate errors associated with coordinate singularities, with a general prescription for remaps of arbitrary order. We perform a comprehensive series of tests on standard problems. Self-convergence tests show that the code has a second-order rate of convergence in smooth, two-dimensional flow, with pressure forces, gravity, and curvilinear geometry included. While not as accurate on idealized problems as high-order Riemann-solving schemes, the predictor-corrector Lagrangian-remap code has great flexibility for application to a variety of astrophysical problems.

Lufkin, Eric A.↗

NASA NTRS1999

Sensitivity Analysis for Coupled Aero-structural Systems

A novel method has been developed for calculating gradients of aerodynamic force and moment coefficients for an aeroelastic aircraft model. This method uses the Global Sensitivity Equations (GSE) to account for the aero-structural coupling, and a reduced-order modal analysis approach to condense the coupling bandwidth between the aerodynamic and structural models. Parallel computing is applied to reduce the computational expense of the numerous high fidelity aerodynamic analyses needed for the coupled aero-structural system. Good agreement is obtained between aerodynamic force and moment gradients computed with the GSE/modal analysis approach and the same quantities computed using brute-force, computationally expensive, finite difference approximations. A comparison between the computational expense of the GSE/modal analysis method and a pure finite difference approach is presented. These results show that the GSE/modal analysis approach is the more computationally efficient technique if sensitivity analysis is to be performed for two or more aircraft design parameters.

Giunta, Anthony A.↗

NASA NTRS1984

Application of numerical optimization to rotor aerodynamic design

Based on initial results obtained from the performance optimization code, a number of observations can be made regarding the utility of optimization codes in supporting design of rotors for improved performance. (1) The primary objective of improving the productivity and responsiveness of current design methods can be met. (2) The use of optimization allows the designer to consider a wider range of design variables in a greatly compressed time period. (3) Optimization requires the user to carefully define his problem to avoid unproductive use of computer resources. (4) Optimization will increase the burden on the analyst to validate designs and to improve the accuracy of analysis methods. (5) Direct calculation of finite difference derivatives by the optimizer was not prohibitive for this application but was expensive. Approximate analysis in some form would be considered to improve program response time. (6) Program developement is not complete and will continue to evolve to integrate new analysis methods, design problems, and alternate optimizer options.

Pleasants, W. A., III↗

NASA NTRS1989

Nonlinear filters for efficient shock computation

A new type of methods for the numerical approximation of hyperbolic conservation laws with discontinuous solution is introduced. The methods are based on standard finite difference schemes. The difference solution is processed with a nonlinear conservation form filter at every time level to eliminate spurious oscillations near shocks. It is proved that the filter can control the total variation of the solution and also produce sharp discrete shocks. The method is simpler and faster than many other high resolution schemes for shock calculations. Numerical examples in one and two space dimensions are presented.

Engquist, Bjorn↗

NASA NTRS1984

Spectral methods for the Euler equations: Fourier methods and shock-capturing

Spectral methods for compressible flows are introduced in relation to finite difference and finite element techniques within the framework of the method of weighted residuals. Current spectral collocation methods are put in historical context. The basic concepts of Fourier spectral collocation methods are provided. Filtering strategies for shock-capturing approaches are also presented. Fourier shock capturing techniques are evaluated using a one dimensional, periodic astrophysical ""nozzle'' problem.

Hussaini, M. Y.↗

NASA NTRS1985

Spectral methods for the Euler equations. I - Fourier methods and shock capturing

Spectral methods for compressible flows are introduced in relation to finite difference and finite element techniques within the framework of the method of weighted residuals. Current spectral collociation methods are put into historical context. The basic concepts of Fourier spectral collocation methods are provided. Filtering strategies for shock-capturing approaches are also presented. Fourier shock-capturing techniques are evaluated using a one-dimensional, periodic astrophysical 'nozzle' problem.

Hussaini, M. Y.↗

DOE OSTI2021

Local Explosion Detection and Infrasound Localization by Reverse Time Migration Using 3-D Finite-Difference Wave Propagation

Infrasound data are routinely used to detect and locate volcanic and other explosions, using both arrays and single sensor networks. However, at local distances (< km) topography often complicates acoustic propagation, resulting in inaccurate acoustic travel times leading to biased source locations when assuming straight-line propagation. Here we present a new method, termed Reverse Time Migration-Finite-Difference Time Domain (RTM-FDTD), that integrates numerical modeling into the standard RTM back-projection process. Travel time information is computed across the entire potential source grid via FDTD modeling to incorporate the effects of topography. The waveforms are then back-projected and stacked at each grid point, with the stack maximum corresponding to the likely source. We apply our method to three volcanoes with different network configurations, source-receiver distances, and topography. At Yasur Volcano, Vanuatu, RTM-FDTD locates explosions within ~20 m of the source and differentiates between multiple vents. RTM-FDTD produces a more accurate location for the two Yasur subcraters than standard RTM and doubles the number of detected events. At Sakurajima Volcano, Japan, RTM-FDTD locates the source within 50 m of the active vent despite notable topographic blocking. The RTM-FDTD location is similar to that from the Time Reversal Mirror method, but is more computationally efficient. Lastly, at Shishaldin Volcano, Alaska, RTM and RTM-FDTD both produce realistic source locations (<50 m) for ground-coupled airwaves recorded on a four-station seismic network. We show that RTM is an effective method to detect and locate infrasonic sources across a variety of scenarios, and by integrating numerical modeling, RTM-FDTD produces more accurate source locations and increases the detection capability.

58 GEOSCIENCES↗

NASA NTRS1988

Three-dimensional zonal grids about arbitrary shapes by Poisson's equation

A method for generating 3-D finite difference grids about or within arbitrary shapes is presented. The 3-D Poisson equations are solved numerically, with values for the inhomogeneous terms found automatically by the algorithm. Those inhomogeneous terms have the effect near boundaries of reducing cell skewness and imposing arbitrary cell height. The method allows the region of interest to be divided into zones (blocks), allowing the method to be applicable to almost any physical domain. A FORTRAN program called 3DGRAPE has been written to implement the algorithm. Lastly, a method for redistributing grid points along lines normal to boundaries will be described.

Sorenson, Reese L.↗

NASA NTRS1988

Three-dimensional zonal grids about arbitrary shapes by Poisson's equation

Sorenson, Reese L.↗

NASA NTRS1993

Shape optimization for aerodynamic efficiency and low observability

Field methods based on the finite-difference approximations of the time-domain Maxwell's equations and the potential-flow equation have been developed to solve the multidisciplinary problem of airfoil shaping for aerodynamic efficiency and low radar cross section (RCS). A parametric study and an optimization study employing the two analysis methods are presented to illustrate their combined capabilities. The parametric study shows that for frontal radar illumination, the RCS of an airfoil is independent of the chordwise location of maximum thickness but depends strongly on the maximum thickness, leading-edge radius, and leadingedge shape. In addition, this study shows that the RCS of an airfoil can be reduced without significant effects on its transonic aerodynamic efficiency by reducing the leading-edge radius and/or modifying the shape of the leading edge. The optimization study involves the minimization of wave drag for a non-lifting, symmetrical airfoil with constraints on the airfoil maximum thickness and monostatic RCS. This optimization study shows that the two analysis methods can be used effectively to design aerodynamically efficient airfoils with certain desired RCS characteristics.

Vinh, Hoang↗

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.↗

DOE OSTI2022

High-order dimensionally-split Cartesian embedded boundary method for non-dissipative schemes

Centered finite-difference schemes are commonly used for high-fidelity turbulent flow simulations in canonical configurations because of their non-dissipative property and computational efficiency. However, their use in flow simulations over complex geometries is limited by the requirements of a structured grid and a stable boundary treatment in the absence of artificial (numerical) dissipation. Cartesian embedded boundary (EB) approaches provide an efficient structured-grid framework to apply difference schemes over complex domains. However, they are often restricted to low orders of accuracy because of numerical instabilities at the embedded boundaries and the issues of small-cell problem that are difficult to address with high-order accuracy. The present work discusses a systematic approach to obtain high-order EB methods with non-dissipative centered schemes in the interior. This approach, based on satisfying the primary and secondary conservation conditions, is employed to derive EB schemes that are up to sixth-order accurate in the interior and fourth-order accurate globally for hyperbolic, parabolic as well as incompletely parabolic problems. The proposed finite-difference discretization is, by construction, dimensionally split and addresses the small-cell problem without any cell/geometry transformations, thus, highly simplifying implementation in a flow solver. Various linear and non-linear numerical tests are performed to evaluate the stability and the accuracy of the proposed EB schemes.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1977

Calculation of unsteady two-dimensional laminar and turbulent boundary layers with fluctuations in external velocity

A numerical method is proposed for calculating unsteady two-dimensional laminar and turbulent boundary layers with allowance for fluctuations in external flow velocity. An eddy-viscosity formulation is used to model the Reynolds shear stress term appropriate to turbulent flow, and the governing boundary layer equations are solved by a two-point finite-difference scheme. For illustration, the method is applied to the calculation of the phase angles between the shear stress at the wall and an oscillating external laminar boundary layer on a flat plate. The effectiveness of the method (for engineering purposes) is demonstrated by comparing calculations for turbulent flows with experimental data.

Cebeci, T.↗

NASA NTRS1996

Multigrid Techniques for Highly Indefinite Equations

A multigrid method for the solution of finite difference approximations of elliptic PDE's is introduced. A parallelizable version of it, suitable for two and multi level analysis, is also defined, and serves as a theoretical tool for deriving a suitable implementation for the main version. For indefinite Helmholtz equations, this analysis provides a suitable mesh size for the coarsest grid used. Numerical experiments show that the method is applicable to diffusion equations with discontinuous coefficients and highly indefinite Helmholtz equations.

Shapira, Yair↗

NASA NTRS1992

An initial investigation into methods of computing transonic aerodynamic sensitivity coefficients

Research conducted during the period from July 1991 through December 1992 is covered. A method based upon the quasi-analytical approach was developed for computing the aerodynamic sensitivity coefficients of three dimensional wings in transonic and subsonic flow. In addition, the method computes for comparison purposes the aerodynamic sensitivity coefficients using the finite difference approach. The accuracy and validity of the methods are currently under investigation.

Carlson, Leland A.↗

NASA NTRS1972

Theoretical study of stress concentrations at circular holes and inclusions in strain hardening materials.

Nonlinear boundary value problems of an infinite elastic-plastic plate with a circular hole subjected to pure tension and pure shear at infinity are solved by a method involving Fourier series and finite difference. On the basis of these solutions, the validity of Neuber's relationship between the stress and strain concentration factors for the plane stress problems is examined and a generalized Stowell formula for the stress concentration factor is proposed for problems in which the applied loading may be pure shear as well as pure tension and, furthermore, other stress states. By the same method of solution, the stress distributions around a rigid circular cylindrical inclusion embedded in an infinite rigid-plastic matrix subjected to uniform transverse pure shear and tension are obtained.

Huang, W. C.↗

NASA NTRS1983

Solution of Poisson equations for 3-dimensional grid generations

A method for generating three dimensional, finite difference grids about complicated geometries by using Poisson equations is developed. The inhomogenous terms are automatically chosen such that orthogonality and spacing restrictions at the body surface are satisfied. Spherical variables are used to avoid the axis singularity, and an alternating-direction-implicit (ADI) solution scheme is used to accelerate the computations. Computed results are presented that show the capability of the method. Since most of the results presented have been used as grids for flow-field computations, this is indicative that the method is a useful tool for generating three-dimensional grids about complicated geometries.

Fujii, K.↗

NASA NTRS1977

Calculation of transonic potential flowfields about complex, three-dimensional configurations

Methods for extending iterative, finite-difference calculations of transonic potential flowfields to complex three-dimensional configurations are discussed. One particularly attractive approach is to use relatively simple conformal mappings in combination with shearing transformations to generate computational domains that are nearly-conformally mapped from the physical space in one family of coordinate surfaces, and which map the complex boundaries to grid surfaces. The application of such a method to a general wing-body combination or to a multi-bladed fan is discussed. A transformation to map the wing-fuselage or fan-hub combination to a convenient computational domain is proposed. The transformation is useful in its own right for treating the two-dimensional problems of flow past a profile in a wind tunnel or through a cascade. Some results of preliminary calculations are presented.

Caughey, D. A.↗