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At least 775 records · Page 43

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NASA NTRS1989

Calculation of steady and unsteady pressures at supersonic speeds with CAP-TSD

A finite difference technique is used to solve the transonic small disturbance flow equation making use of shock capturing to treat wave discontinuities. Thus the nonlinear effects of thickness and angle of attack are considered. Such an approach is made feasible by the development of a new code called CAP-TSD (Computational Aeroelasticity Program - Transonic Small Disturbance), and is based on a fully implicit approximate factorization (AF) finite difference method to solve the time dependent transonic small disturbance equation. The application of the CAP-TSD code to the calculation of low to moderate supersonic steady and unsteady flows is presented. In particular, comparisons with exact linear theory solutions are made for steady and unsteady cases to evaluate shock capturing and other features of the current method. In addition, steady solutions obtained from an Euler code are used to evaluate the small disturbance aspects of the code. Steady and unsteady pressure comparisons are made with measurements for an F-15 wing model and for the RAE tailplane model.

Bennett, Robert M.↗

NASA NTRS1993

An adaptive mesh-moving and refinement procedure for one-dimensional conservation laws

We examine the performance of an adaptive mesh-moving and /or local mesh refinement procedure for the finite difference solution of one-dimensional hyperbolic systems of conservation laws. Adaptive motion of a base mesh is designed to isolate spatially distinct phenomena, and recursive local refinement of the time step and cells of the stationary or moving base mesh is performed in regions where a refinement indicator exceeds a prescribed tolerance. These adaptive procedures are incorporated into a computer code that includes a MacCormack finite difference scheme wih Davis' artificial viscosity model and a discretization error estimate based on Richardson's extrapolation. Experiments are conducted on three problems in order to qualify the advantages of adaptive techniques relative to uniform mesh computations and the relative benefits of mesh moving and refinement. Key results indicate that local mesh refinement, with and without mesh moving, can provide reliable solutions at much lower computational cost than possible on uniform meshes; that mesh motion can be used to improve the results of uniform mesh solutions for a modest computational effort; that the cost of managing the tree data structure associated with refinement is small; and that a combination of mesh motion and refinement reliably produces solutions for the least cost per unit accuracy.

Biswas, Rupak↗

NASA NTRS1976

Flux-corrected transport techniques for transient calculations of strongly shocked flows

New flux-corrected transport algorithms are described for solving generalized continuity equations. These techniques were developed by requiring that the finite difference formulas used ensure positivity for an initially positive convected quantity. Thus FCT is particularly valuable for fluid-like problems with strong gradients or shocks. Repeated application of the same subroutine to mass, momentum, and energy conservation equations gives a simple solution of the coupled time-dependent equations of ideal compressible fluid dynamics without introducing an artificial viscosity. FCT algorithms span Eulerian, sliding-rezone, and Lagrangian finite difference grids in several coordinate systems. The latest FCT techniques are fully vectorized for parallel/pipeline processing.

Boris, J. P.↗

NASA NTRS1971

Transpiration and film cooling boundary layer computer program. Volume 2: Computer program and user's manual

A finite difference turbulent boundary layer computer program which allows for mass transfer wall cooling and equilibrium chemistry effects is presented. The program is capable of calculating laminar or turbulent boundary layer solutions for an arbitrary ideal gas or an equilibrium hydrogen oxygen system. Either two dimensional or axisymmetric geometric configurations may be considered. The equations are solved, in nondimension-alized physical coordinates, using the implicit Crank-Nicolson technique. The finite difference forms of the conservation of mass, momentum, total enthalpy and elements equations are linearized and uncoupled, thereby generating easily solvable tridiagonal sets of algebraic equations. A detailed description of the computer program, as well as a program user's manual is provided. Detailed descriptions of all boundary layer subroutines are included, as well as a section defining all program symbols of principal importance. Instructions are then given for preparing card input to the program and for interpreting the printed output. Finally, two sample cases are included to illustrate the use of the program.

Gloss, R. J.↗

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 NTRS1986

Unsteady hybrid vortex technique for transonic vortex flows and flutter applications

The well developed surface panel method is extended by adding a volume integral term allowing calculation of the full effect of compressibility. The full effect of compressibility is calculated by using mixed type finite difference scheme consistent with the mixed nature of transonic flow. The solution is obtained through successive iteration cycles for subcritical flows and for critical flows the solution is obtained through satisfying the Rankine-Hugoniot relations across the captured shock in addition to the captured shock. The method is applied to airfoils in subcritical and critical flows and the results are in good agreement with the experimental data and finite-difference solutions of the full-potential and Euler equations.

Kandil, Osama A.↗

NASA NTRS2012

Calculation of Sensitivity Derivatives in an MDAO Framework

During gradient-based optimization of a system, it is necessary to generate the derivatives of each objective and constraint with respect to each design parameter. If the system is multidisciplinary, it may consist of a set of smaller "components" with some arbitrary data interconnection and process work ow. Analytical derivatives in these components can be used to improve the speed and accuracy of the derivative calculation over a purely numerical calculation; however, a multidisciplinary system may include both components for which derivatives are available and components for which they are not. Three methods to calculate the sensitivity of a mixed multidisciplinary system are presented: the finite difference method, where the derivatives are calculated numerically; the chain rule method, where the derivatives are successively cascaded along the system's network graph; and the analytic method, where the derivatives come from the solution of a linear system of equations. Some improvements to these methods, to accommodate mixed multidisciplinary systems, are also presented; in particular, a new method is introduced to allow existing derivatives to be used inside of finite difference. All three methods are implemented and demonstrated in the open-source MDAO framework OpenMDAO. It was found that there are advantages to each of them depending on the system being solved.

Moore, Kenneth T.↗

DOE OSTI2022

Adjoint methods for quasi-symmetry of vacuum fields on a surface

Adjoint methods can speed up stellarator optimisation by providing gradient information more efficiently compared with finite-difference evaluations. Adjoint methods are herein applied to vacuum magnetic fields, with objective functions targeting quasi-symmetry and a rotational transform value on a surface. Here to measure quasi-symmetry, a novel way of evaluating approximate flux coordinates on a single flux surface without the assumption of a neighbourhood of flux surfaces is proposed. The shape gradients obtained from the adjoint formalism are evaluated numerically and verified against finite-difference evaluations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2022

A sixth order Mehrstellen scheme with an application to the Method of Local Corrections for the 3D Poisson equation

We present a sixth order finite difference scheme for Poisson’s equation when discretized with the compact 27-point stencil based on Mehrstellen corrections of the forcing function term f. Our approach results in a sixth order accurate solution error as opposed to a fourth-order error imposed by the classical Mehrstellen correction for the 19-point and 27-point stencils. The present study is a continuation of former work of Spotz and Carey (1996) on compact finite difference schemes for Poisson’s equation where sixth order convergence may be obtained under the assumption that the fourth order derivatives of f are determined analytically. Specifically, we show that sixth order convergence can still be attained when only values of f at grid points are available. The sixth order Mehrstellen scheme is further coupled with a Method of Local Corrections (MLC) 3D Poisson solver improving to sixth order accuracy the results reported in Kavouklis and Colella (2019). The MLC test case considered involves an adaptive grid that comprises 7.5 billion cells.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1981

A PANSONIC Navier-Stokes solver

A finite-difference formulation of the full Navier-Stokes equations which demonstrates a capability to economically solve two-dimensional problems has been developed. The basic algorithm was derived from the full, Reynolds-averaged, conservative, Navier-Stokes equations expressed in curvilinear coordinates. Eddy viscosity was determined by the Baldwin and Lomax algebraic turbulence model. This non-iterative, second-order accurate, implicit, numerical algorithm is based on the approximate factorization finite-difference scheme of Beam and Warming. Results indicate a facility for solving subsonic, transonic, and supersonic (hence PANSONIC) flows about arbitrary airfoils for a wide range of Reynolds numbers, Mach numbers, and angles of attack. Current computations demonstrate that vectorized implementations of this algorithm can solve steady-state, two-dimensional problems in five to ten minutes of computer time.

Cooper, G. K.↗

DOE OSTI2022

Spectral quadrature for the first principles study of crystal defects: Application to magnesium

In this work, we present an accurate and efficient finite-difference formulation and parallel implementation of Kohn-Sham Density (Operator) Functional Theory (DFT) for non periodic systems embedded in a bulk environment. Specifically, employing non-local pseudopotentials, local reformulation of electrostatics, and truncation of the spatial Kohn-Sham Hamiltonian, and the Linear Scaling Spectral Quadrature method to solve for the pointwise electronic fields in real-space and the non-local component of the atomic force, we develop a parallel finite difference framework suitable for distributed memory computing architectures to simulate non-periodic systems embedded in a bulk environment. Choosing examples from magnesium-aluminum alloys, we first demonstrate the convergence of energies and forces with respect to spectral quadrature polynomial order, and the width of the spatially truncated Hamiltonian. Next, we demonstrate the parallel scaling of our framework, and show that the computation time and memory scale linearly with respect to the number of atoms. Next, we use the developed framework to simulate isolated point defects and their interactions in magnesium-aluminum alloys. Our findings conclude that the binding energies of divacancies, Al solute-vacancy and two Al solute atoms are anisotropic and are dependent on cell size. Furthermore, the binding is favorable in all three cases.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1977

Assessment of existing analytic methods for prediction of high angle-of-attack loads on delta wings at supersonic speeds

An assessment of the applicability of four loading prediction methods to high angle-of-attack conditions for simplified wing-body configurations is provided. The methods are: The tangent wedge approximation, the linear theory methods of Middleton and Woodward, and a shock-fitting finite-difference technique. Estimates obtained by these methods were compared with experimental pressure data on delta wings to examine the effects of Mach number, camber, sweep angle, and angle of attack. Results indicate that all of the methods provided reasonable estimates at moderate angles of attack. At these moderate angles of attack, the methods of Middleton and Woodward provided good estimates at Mach numbers higher than those usually associated with linear theory. Only the finite-difference method provided reasonable load estimates at high angles of attack.

Landrum, E. J.↗

NASA NTRS1979

Middle atmosphere project. A semi-spectral numerical model for the large-scale stratospheric circulation

The complete model is a semispectral model in which the longitudinal dependence is represented by expansion in zonal harmonics while the latitude and height dependencies are represented by a finite difference grid. The model is based on the primitive equations in the log pressure coordinate system. The lower boundary of the model domain is set at the 100 mb level (i.e., near the tropopause) and the effects of tropospheric forcing are included in the lower boundary condition. The upper boundary is at approximately 96 km, and the latitudinal extent is either global or hemispheric. The basic differential equations and boundary conditions are outlined. The finite difference equations are described. The initial conditions are discussed and a sample calculation is presented. The FORTRAN code is given in the appendix.

Holton, J. R.↗

NASA NTRS1984

Variational objective analyses for cyclone studies

The basic analysis equations, i.e., the two horizontal momentum equations, the hydrostatic equation, and the integrated continuity equation were derived for the nonlinear vertical coordinate, nondimensionalized, and expressed in finite differences on a staggered grid. Special care was taken to transform the hydrostatic equation and the pressure gradient terms of the horizontal momentum equations to nearly eliminate truncation error over steeply sloping terrain. This formulation also eliminated explicit reference to orographically induced variations in the thermodynamic variables so that the variational adjustments are on the scale of the meteorological perturbations. The analysis equations were subjected to the Euler-Lagrange operations as expressed for finite differences and an additional set of five partial differential equations was derived, bringing to nine the number of equations in Model I. Higher order terms, terms containing observed quantities, and terms containing none of the variables to be adjusted were grouped into forcing functions and the equations were solved for the zero order terms. Zero order variables were eliminated between these equations and there resulted two diagnostic equations which take the form of general linear second order partial differential equations with nonconstant coefficients.

Achtemeier, G. L.↗

NASA NTRS1991

Computational aspects of sensitivity calculations in linear transient structural analysis

The calculation of sensitivities of displacements, velocities, accelerations, and stresses in linear, structural, and transient response problems is studied. Several existing sensitivity calculation methods and two new methods are compared for three example problems. Approximation vectors such as vibration mode shapes are used to reduce the dimensionality of the finite model. To be computationally practical in large-order problems, the overall finite difference methods must use the approximation vectors from the original design in the analyses of the perturbed models. This was found to result in poor convergence of stress sensitivities in several cases. Two semianalytical techniques are developed to overcome this poor convergence. Both new methods result in very good convergence of the stress sensitivities; the computational cost is much less than would result if the vibration modes were recalculated and then used in an overall finite difference method.

Greene, W. H.↗

NASA NTRS1974

A study of the diffusional behavior of a two-phase metal matrix composite exposed to a high temperature environment

The progress of diffusion-controlled filament-matrix interaction in a metal matrix composite where the filaments and matrix comprise a two-phase binary alloy system was studied by mathematically modeling compositional changes resulting from prolonged elevated temperature exposure. The analysis treats a finite, diffusion-controlled, two-phase moving-interface problem by means of a variable-grid finite-difference technique. The Ni-W system was selected as an example system. Modeling was carried out for the 1000 to 1200 C temperature range for unidirectional composites containing from 6 to 40 volume percent tungsten filaments in a Ni matrix. The results are displayed to show both the change in filament diameter and matrix composition as a function of exposure time. Compositional profiles produced between first and second nearest neighbor filaments were calculated by superposition of finite-difference solutions of the diffusion equations.

Tenney, D. R.↗

NASA NTRS1985

A hybrid numerical technique for predicting the aerodynamic and acoustic fields of advanced turboprops

A hybrid numerical procedure is presented for the prediction of the aerodynamic and acoustic performance of advanced turboprops. A hybrid scheme is proposed which in principle leads to a consistent simultaneous prediction of both fields. In the inner flow a finite difference method, the Approximate-Factorization Alternating-Direction-Implicit (ADI) scheme, is used to solve the nonlinear Euler equations. In the outer flow the linearized acoustic equations are solved via a Boundary-Integral Equation (BIE) method. The two solutions are iteratively matched across a fictitious interface in the flow so as to maintain continuity. At convergence the resulting aerodynamic load prediction will automatically satisfy the appropriate free-field boundary conditions at the edge of the finite difference grid, while the acoustic predictions will reflect the back-reaction of the radiated field on the magnitude of the loading source terms, as well as refractive effects in the inner flow. The equations and logic needed to match the two solutions are developed and the computer program implementing the procedure is described. Unfortunately, no converged solutions were obtained, due to unexpectedly large running times. The reasons for this are discussed and several means to alleviate the situation are suggested.

Homicz, G. F.↗

NASA NTRS1973

Supersonic turbulent boundary-layer flows with tangential slot injection.

An implicit finite-difference method has been developed for the solution of the compressible boundary-layer equations. This method is applied to tangential slot injection into supersonic turbulent boundary-layer flows. In addition, the effects induced by the interaction between the boundary-layer displacement thickness and the external pressure field are considered. Three different eddy viscosity models have been used to specify the turbulent momentum exchange. One model depends on the species concentration profile, and the species conservation equation has been included in the system of governing partial differential equations. For air injected into air, the freestream and injected gases are treated as separate species which have common fluid properties. Calculations were made and results were compared with experimental data at freestream Mach numbers of 2.4 and 6.0 and with results of another finite-difference method. Good agreement was obtained for the reduction of wall skin friction with slot injection.

Miner, E. W.↗