SEARCH · Search NASA

Results for “Finite difference scheme”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 847 records · Page 47

All sources NASA NTRS DOE OSTI

NASA NTRS1987

On the validity of the modified equation approach to the stability analysis of finite-difference methods

The validity of the modified equation stability analysis introduced by Warming and Hyett was investigated. It is shown that the procedure used in the derivation of the modified equation is flawed and generally leads to invalid results. Moreover, the interpretation of the modified equation as the exact partial differential equation solved by a finite-difference method generally cannot be justified even if spatial periodicity is assumed. For a two-level scheme, due to a series of mathematical quirks, the connection between the modified equation approach and the von Neuman method established by Warming and Hyett turns out to be correct despite its questionable original derivation. However, this connection is only partially valid for a scheme involving more than two time levels. In the von Neumann analysis, the complex error multiplication factor associated with a wave number generally has (L-1) roots for an L-level scheme. It is shown that the modified equation provides information about only one of these roots.

Chang, Sin-Chung↗

NASA NTRS1988

An explicit Runge-Kutta method for 3D turbulent incompressible flows

A computer code has been developed to solve for the steady-state solution of the 3D incompressible Reynolds-averaged Navier-Stokes equations. The approach is based on the cell-center, central-difference, finite-volume formulation and an explicit one-step, multistage Runge-Kutta time-stepping scheme. The Baldwin-Lomax turbulence model is used. Techniques to accelerate the rate of convergence to a steady-state solution include the preconditioned method, the local time stepping, and the implicit residual smoothing. Improvements in computational efficiency have been demonstrated in several areas. This numerical procedure has been used to simulate the turbulent horseshoe vortex flow around an airfoil/flat-plate juncture.

Sung, Chao-Ho↗

NASA NTRS1988

Zonal techniques for flowfield simulation about aircraft

A technique for performing conservative flowfield calculations on zonal meshes is described. The underlying flow solver is an implicit, upwind finite volume scheme which can incorporate either a perfect gas or an equilibrium air equation of state. Two different approaches which yield identical results, in terms of performing a conservative flux calculation on a zonal interface, are described and compared in terms of numerical efficiency. The capability of the method to handle relatively complex geometries is demonstrated by considering the flowfield about a model SR71 aircraft.

Walters, Robert W.↗

NASA NTRS1992

Numerical simulation of conservation laws

A new numerical framework for solving conservation laws is being developed. This new approach differs substantially from the well established methods, i.e., finite difference, finite volume, finite element and spectral methods, in both concept and methodology. The key features of the current scheme include: (1) direct discretization of the integral forms of conservation laws, (2) treating space and time on the same footing, (3) flux conservation in space and time, and (4) unified treatment of the convection and diffusion fluxes. The model equation considered in the initial study is the standard one dimensional unsteady constant-coefficient convection-diffusion equation. In a stability study, it is shown that the principal and spurious amplification factors of the current scheme, respectively, are structurally similar to those of the leapfrog/DuFort-Frankel scheme. As a result, the current scheme has no numerical diffusion in the special case of pure convection and is unconditionally stable in the special case of pure diffusion. Assuming smooth initial data, it will be shown theoretically and numerically that, by using an easily determined optimal time step, the accuracy of the current scheme may reach a level which is several orders of magnitude higher than that of the MacCormack scheme, with virtually identical operation count.

Chang, Sin-Chung↗

NASA NTRS1992

Numerical study of the small scale structures in Boussinesq convection

Two-dimensional Boussinesq convection is studied numerically using two different methods: a filtered pseudospectral method and a high order accurate Essentially Nonoscillatory (ENO) scheme. The issue whether finite time singularity occurs for initially smooth flows is investigated. The numerical results suggest that the collapse of the bubble cap is unlikely to occur in resolved calculations. The strain rate corresponding to the intensification of the density gradient across the front saturates at the bubble cap. We also found that the cascade of energy to small scales is dominated by the formulation of thin and sharp fronts across which density jumps.

Weinan, E.↗

NASA NTRS2011

Porting a Hall MHD Code to a Graphic Processing Unit

We present our experience porting a Hall MHD code to a Graphics Processing Unit (GPU). The code is a 2nd order accurate MUSCL-Hancock scheme which makes use of an HLL Riemann solver to compute numerical fluxes and second-order finite differences to compute the Hall contribution to the electric field. The divergence of the magnetic field is controlled with Dedner?s hyperbolic divergence cleaning method. Preliminary benchmark tests indicate a speedup (relative to a single Nehalem core) of 58x for a double precision calculation. We discuss scaling issues which arise when distributing work across multiple GPUs in a CPU-GPU cluster.

Dorelli, John C.↗

NASA NTRS1987

A simultaneous viscous-inviscid interaction calculation procedure for transonic turbulent flows

A new simultaneous viscous-inviscid interaction scheme has been developed for the analysis of steady turbulent transonic separated flows. The viscous and inviscid solutions are coupled through the displacement concept using a transpiration velocity. The local solutions of the full potential and boundary-layer equations are treated simultaneously using the finite-difference method. The displacement thickness is treated as an unknown and is obtained as a part of the solution through a global iteration procedure of the space marching scheme. The Cebeci-Smith and Johnson-King models are used to simulate the turbulence. The computational examples showed that the simultaneous method is more efficient and robust than the semi-inverse method for transonic flows with a strong interaction.

Lee, D.↗

NASA NTRS1995

A finite element propagation model for extracting normal incidence impedance in nonprogressive acoustic wave fields

A propagation model method for extracting the normal incidence impedance of an acoustic material installed as a finite length segment in a wall of a duct carrying a nonprogressive wave field is presented. The method recasts the determination of the unknown impedance as the minimization of the normalized wall pressure error function. A finite element propagation model is combined with a coarse/fine grid impedance plane search technique to extract the impedance of the material. Results are presented for three different materials for which the impedance is known. For each material, the input data required for the prediction scheme was computed from modal theory and then contaminated by random error. The finite element method reproduces the known impedance of each material almost exactly for random errors typical of those found in many measurement environments. Thus, the method developed here provides a means for determining the impedance of materials in a nonprogressirve wave environment such as that usually encountered in a commercial aircraft engine and most laboratory settings.

Watson, Willie R.↗

DOE OSTI2016

Well-Balanced, Conservative Finite Difference Algorithm for Atmospheric Flows

The numerical simulation of meso-, convective-, and microscale atmospheric flows requires the solution of the Euler or the Navier-Stokes equations. Nonhydrostatic weather prediction algorithms often solve the equations in terms of derived quantities such as Exner pressure and potential temperature (and are thus not conservative) and/or as perturbations to the hydrostatically balanced equilibrium state. This paper presents a well-balanced, conservative finite difference formulation for the Euler equations with a gravitational source term, where the governing equations are solved as conservation laws for mass, momentum, and energy. Preservation of the hydrostatic balance to machine precision by the discretized equations is essential because atmospheric phenomena are often small perturbations to this balance. The proposed algorithm uses the weighted essentially nonoscillatory and compact-reconstruction weighted essentially nonoscillatory schemes for spatial discretization that yields high-order accurate solutions for smooth flows and is essentially nonoscillatory across strong gradients; however, the well-balanced formulation may be used with other conservative finite difference methods. The performance of the algorithm is demonstrated on test problems as well as benchmark atmospheric flow problems, and the results are verified with those in the literature.

Ghosh, Debojyoti↗

DOE OSTI2017

A conservative nonlocal convection–diffusion model and asymptotically compatible finite difference discretization

Here in this paper, we first propose a nonlocal convection–diffusion model, in which the convection term is constructed in a special upwind manner so that mass conservation and maximum principle are maintained in any space dimension. The well-posedness of the proposed nonlocal model and its convergence to the classical local convection–diffusion model are established. A quadrature-based finite difference discretization is then developed to numerically solve the nonlocal problem and it is shown to be consistent and unconditionally stable. We further demonstrate that the numerical scheme is asymptotically compatible, that is, the approximate solutions converge to the exact solution of the corresponding local problem when δ → 0 and h → 0. Numerical experiments are also performed to complement the theoretical analysis.

42 ENGINEERING↗

DOE OSTI2017

Modeling of wave processes in blocky media with porous and fluid-saturated interlayers

The wave processes in blocky media are analyzed by applying different mathematical models, wherein the elastic blocks interact with each other via pliant interlayers with the complex mechanical properties. Four versions of constitutive equations are considered. In the first version, an elastic interaction between the blocks is simulated within the framework of linear elasticity theory, and the model of elastic–plastic interlayers is constructed to take into account the appearance of irreversible deformation of interlayers at short time intervals. In the second one, the effects of viscoelastic shear in the interblock interlayers are taken into the consideration using the Poynting–Thomson rheological scheme. In the third option, the model of an elastic porous material is used in the interlayers, where the pores collapse if an abrupt compressive stress is applied. In the fourth case, the model of a fluid-saturated material with open pores is examined based on Biot's equations. The collapse of pores is modeled by the generalized rheological approach, wherein the mechanical properties of a material are simulated using four rheological elements. Three of them are the traditional elastic, viscous and plastic elements, the fourth element is the so-called rigid contact, which is used to describe the behavior of materials with the different resistance to tension and compression. It was shown that the thermodynamically consistent model is provided, which means that the energy balance equation is fulfilled for an entire blocky structure, where the kinetic and potential energy of the system is the sum of the kinetic and potential energies of the blocks and interlayers. Under numerical implementation of the interlayers models, the dissipationless finite difference Ivanov's method was used. The splitting method by spatial variables in the combination with the Godunov gap decay scheme was applied in the blocks. As a result, robust and stable computational algorithms are built and tested. Using MPI technology, the parallel software was designed for the modeling of wave processes in 2D setting. The numerical results are presented, discussed and future studies are outlined. - Highlights: • We model the propagation of waves in a blocky medium with compliant interlayers. • Models of porous and fluid saturated interlayers are proposed using Biot's approach. • A parallel computational algorithm is developed for supercomputers of cluster type. • Problems of instant rotation of central and boundary blocks are solved numerically. • Wave motions are compared in the cases of elastic and fluid-saturated interlayers.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1989

An Eigenvalue Analysis of finite-difference approximations for hyperbolic IBVPs

The eigenvalue spectrum associated with a linear finite-difference approximation plays a crucial role in the stability analysis and in the actual computational performance of the discrete approximation. The eigenvalue spectrum associated with the Lax-Wendroff scheme applied to a model hyperbolic equation was investigated. For an initial-boundary-value problem (IBVP) on a finite domain, the eigenvalue or normal mode analysis is analytically intractable. A study of auxiliary problems (Dirichlet and quarter-plane) leads to asymptotic estimates of the eigenvalue spectrum and to an identification of individual modes as either benign or unstable. The asymptotic analysis establishes an intuitive as well as quantitative connection between the algebraic tests in the theory of Gustafsson, Kreiss, and Sundstrom and Lax-Richtmyer L(sub 2) stability on a finite domain.

Warming, Robert F.↗

NASA NTRS1990

An eigenvalue analysis of finite-difference approximations for hyperbolic IBVPs

Warming, Robert F.↗

DOE OSTI2019

A FFT-based numerical implementation of mesoscale field dislocation mechanics: Application to two-phase laminates

In this study, we present an enhanced crystal plasticity elasto-viscoplastic fast Fourier transform (EVPFFT) formulation coupled with a phenomenological Mesoscale Field Dislocation Mechanics (MFDM) theory here named MFDM-EVPFFT formulation. In contrast with classic CP-EVPFFT, the model is able to tackle plastic flow and hardening due to polar dislocation density distributions or geometrically necessary dislocations (GNDs) in addition to statistically stored dislocations (SSDs). The model also considers GND mobility through a GND density evolution law numerically solved with a recently developed filtered spectral approach, which is here coupled with stress equilibrium. The discrete Fourier transform method combined with finite differences is applied to solve both lattice incompatibility and Lippmann-Schwinger equations in an augmented Lagrangian numerical scheme. Numerical results are presented for two-phase laminate composites with plastic channels and elastic second phase. It is shown that both GND densities and slip constraint at phase boundaries influence the overall and local hardening behavior. In contrast with the CP-EVPFFT formulation, a channel size effect is predicted on the shear flow stress with the present MFDM-EVPFFT formulation. The size effect originates from the progressive formation of continuous screw GND pile-ups from phase boundaries to the channel center. The effect of GND mean free path on local and global responses is also examined for the two-phase composite.

36 MATERIALS SCIENCE↗

DOE OSTI2017

Orbital Advection with Magnetohydrodynamics and Vector Potential

Orbital advection is a significant bottleneck in disk simulations, and a particularly tricky one when used in connection with magnetohydrodynamics. We have developed an orbital advection algorithm suitable for the induction equation with magnetic potential. The electromotive force is split into advection and shear terms, and we find that we do not need an advective gauge since solving the orbital advection implicitly precludes the shear term from canceling the advection term. We prove and demonstrate the third order in time accuracy of the scheme. The algorithm is also suited to non-magnetic problems. Benchmarked results of (hydrodynamical) planet–disk interaction and of the magnetorotational instability are reproduced. We include detailed descriptions of the construction and selection of stabilizing dissipations (or high-frequency filters) needed to generate practical results. The scheme is self-consistent, accurate, and elegant in its simplicity, making it particularly efficient for straightforward finite-difference methods. As a result of the work, the algorithm is incorporated in the public version of the Pencil Code, where it can be used by the community.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1973

The transonic aerofoil problem with embedded shocks.

The integral equation approach to the mixed flow problem of infinite wings at high subsonic speeds is adopted for non-circulatory and circulatory (lifting) flows. The solutions are determined from a system of non-linear algebraic equations and, to ensure always unique solutions, the method of differentiation with respect to a parameter has been applied. The resulting Cauchy problem is then solved with the linearised flow solution as the initial value vector. For the case of embedded shocks in the flow field, the method of steepest descent has been added to the calculation scheme. Results for subcritical and supercritical flows past aerofoils are given and compared with solutions obtained by finite-difference techniques.

Norstrud, H.↗

NASA NTRS1974

Solutions for laminar boundary layers with separation and reattachment

Numerical solutions of the laminar, incompressible boundary-layer equations are presented for flows involving separation and reattachment. Regular solutions are obtained with an inverse approach in which either the displacement thickness or the skin friction is specified, and the pressure is deduced from the solution. A stream function/vorticity formulation of the boundary-layer equations is used to eliminate the unknown pressure. Solutions of the resulting finite-difference equations, in which the flow direction is taken into account, are obtained by a successive column iteration scheme. Results are compared with Klineberg and Steger's (1974) separated boundary-layer calculations, and with Briley's (1971) solution of the Navier-Stokes equations for a separated region.

Carter, J. E.↗

NASA NTRS1987

A two-dimensional finite difference program for thermal analysis of rocket thrust chambers

A two-dimensional finite difference computer model for thermal analysis of rocket thrust chambers has been developed. The model uses an iterative scheme for calculating the temperature distribution within the chamber wall and implements a successive overrelaxation formula for a quick convergence. The inputs of the models are the dimensions of the thrust chamber wall, types of materials used, heat transfer coefficients and temperatures of the hot gas and the coolant. The resulting output of the program consists of the nodal temperature distribution, heat transfer to the coolant and heat transfer from the hot gas.

Naraghi, Mohammad H.↗