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All sourcesNASA NTRSDOE OSTI
NASA NTRS1983

The Application of the Implicit Alternating-direction Numerical Technique to Thermal Analysis Involving Conduction and Convection

A computerized model was developed for analyzing the temperature distribution of a two dimensional body which is located at or near the soil surface and is partially exposed to solar radiation. The body may have one or more interior cavities containing air or another fluid. The methodology which evolved is also applicable to a general class of thermal analyses involving a body surrounded by a semi-infinite medium exposed to surface radiation energy. The theoretical analysis, numerical procedure, and a sample case are discussed.

Hughes, R. D.↗
DOE OSTI2021

A unified formulation of splitting-based implicit time integration schemes

Splitting-based time integration approaches such as fractional step, alternating direction implicit, operator splitting, and locally one dimensional methods partition the system of interest into components, and solve individual components implicitly in a cost-effective way. Here this work proposes a unified formulation of splitting time integration schemes in the framework of general-structure additive Runge–Kutta (GARK) methods. Specifically, we develop implicit-implicit (IMIM) GARK schemes, provide the order conditions for this class, and explain their application to partitioned systems of ordinary differential equations. We show that classical splitting methods belong to the IMIM GARK family, and therefore can be studied in this unified framework. New IMIM-GARK splitting methods are developed and tested using parabolic systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1997

Improving Discrete-Sensitivity-Based Approach for Practical Design Optimization

In developing the automated methodologies for simulation-based optimal shape designs, their accuracy, efficiency and practicality are the defining factors to their success. To that end, four recent improvements to the building blocks of such a methodology, intended for more practical design optimization, have been reported. First, in addition to a polynomial-based parameterization, a partial differential equation (PDE) based parameterization was shown to be a practical tool for a number of reasons. Second, an alternative has been incorporated to one of the tedious phases of developing such a methodology, namely, the automatic differentiation of the computer code for the flow analysis in order to generate the sensitivities. Third, by extending the methodology for the thin-layer Navier-Stokes (TLNS) based flow simulations, the more accurate flow physics was made available. However, the computer storage requirement for a shape optimization of a practical configuration with the -fidelity simulations (TLNS and dense-grid based simulations), required substantial computational resources. Therefore, the final improvement reported herein responded to this point by including the alternating-direct-implicit (ADI) based system solver as an alternative to the preconditioned biconjugate (PbCG) and other direct solvers.

Baysal, Oktay↗
NASA NTRS1997

Adventures with Implicit Methods

In this lecture we trace the historical developments of alternating direction implicit methods. In particular, we emphasize contributions originating in the Computational Fluid Dynamics Branch at Ames Research Center in the 1970's and early 1980's. Joe Steger played a seminal role in demonstrating the practicality of using an efficient, vectorized, implicit code for solving the compressible Navier-Stokes equations. Numerous discussions with Joe had a significant impact on our own research and it is a pleasure to dedicate this lecture to honor his memory.

Warming, Robert F.↗
NASA NTRS1977

An implicit shock-fitting scheme for unsteady transonic flow computations

The alternating-direction implicit scheme developed by NASA Ames for unsteady transonic flows has been modified to include a shock-fitting algorithm as well as an analytically stretched coordinate system. The shock-fitting procedure treats shock waves as discontinuities normal to the free stream. Improvements in shock position and the unsteady pressure distributions are obtained by this modification. The various types of shock motion observed experimentally by Tijdeman are well simulated in calculations using the modified computational scheme. The method of detecting shock wave formation and the procedure for fitting a moving shock wave are illustrated. Results for a pulsating parabolic arc airfoil and for an NACA 64A006 airfoil with oscillating quarter-chord flap are presented and discussed.

Yu, N. J.↗
NASA NTRS1987

A diagonal implicit multigrid algorithm for the Euler equations

A multigrid implementation of the Alternating Direction Implicit algorithm has been developed to solve the Euler equations of inviscid, compressible flow. The equations are approximated using a finite-volume spatial approximation with added dissipation provided by an adaptive blend of second and fourth differences. For computational efficiency, the equations are diagonalized by a local similariity transformation so that only a decoupled system of scalar pentadiagonal systems need be solved along each line. Results are computed for transonic flows past airfoils and include pressure distributions to verify the accuracy of the basic scheme and convergence histories to demonstrate the efficiency of the method.

Caughey, David A.↗
NASA NTRS1986

Linearized form of implicit TVD schemes for the multidimensional Euler and Navier-Stokes equations

Linearized alternating direction implicit (ADI) forms of a class of total variation diminishing (TVD) schemes for the Euler and Navier-Stokes equations have been developed. These schemes are based on the second-order-accurate TVD schemes for hyperbolic conservation laws developed by Harten (1983, 1984). They have the property of not generating spurious oscillations across shocks and contact discontinuities. In general, shocks can be captured within 1-2 grid points. These schemes are relatively simple to understand and easy to implement into a new or existing computer code. One can modify a standard three-point central-difference code by simply changing the conventional numerical dissipation term into the one designed for the TVD scheme. For steady-state applications, the only difference in computation is that the current schemes require a more elaborate dissipation term for the explicit operator; no extra computation is required for the implicit operator. Numerical experiments with the proposed algorithms on a variety of steady-state airfoil problems illustrate the versatility of the schemes.

Yee, H. C.↗
NASA NTRS1991

A Fourier-Chebyshev pseudospectral method for solving steady 3-D Navier-Stokes and heat equations in cylindrical cavities

A Fourier-Chebyshev pseudospectral method for solving steady 3D Navier-Stokes equations in cylindrical cavities is presented and discussed. The general method is pseudo-unsteady and uses a semi-implicit finite difference scheme for the time discretization. The generalized ADI (Alternating Direction Implicit) procedure is then applied to reduce the problem to a successive solution of one-dimensional problems. The spatial approximation uses a Fourier-Galerkin approximation in the periodic direction and a Chebyshev-collocation approximation in the other directions. Difficulties related to the pressure are surmounted by using the artifical compressibility method. A suitable variable change was chosen to avoid the problem of singularity at the axis generated by cylindrical coordinates. The method is first tested on an advection-diffusion equation and then on the Navier-Stokes equations. Finally, the method is illustrated by a convection problem of a differentially heated fluid.

Pulicani, J. P.↗
NASA NTRS1993

Implicit multigrid techniques for compressible flows

Recent advances in the development of the diagonalized alternating direction implicit multigrid method for compressible aerodynamic problems are reviewed. These include the extension of the method originally developed for the Euler equations to include viscous effects, the computation of turbulent flows and the implementation on parallel computers of the scheme on multiblock rids.

Caughey, David A.↗
NASA NTRS2008

An Initial Investigation of the Effects of Turbulence Models on the Convergence of the RK/Implicit Scheme

A three-stage Runge-Kutta (RK) scheme with multigrid and an implicit preconditioner has been shown to be an effective solver for the fluid dynamic equations. This scheme has been applied to both the compressible and essentially incompressible Reynolds-averaged Navier-Stokes (RANS) equations using the algebraic turbulence model of Baldwin and Lomax (BL). In this paper we focus on the convergence of the RK/implicit scheme when the effects of turbulence are represented by either the Spalart-Allmaras model or the Wilcox k-! model, which are frequently used models in practical fluid dynamic applications. Convergence behavior of the scheme with these turbulence models and the BL model are directly compared. For this initial investigation we solve the flow equations and the partial differential equations of the turbulence models indirectly coupled. With this approach we examine the convergence behavior of each system. Both point and line symmetric Gauss-Seidel are considered for approximating the inverse of the implicit operator of the flow solver. To solve the turbulence equations we use a diagonally dominant alternating direction implicit (DDADI) scheme. Computational results are presented for three airfoil flow cases and comparisons are made with experimental data. We demonstrate that the two-dimensional RANS equations and transport-type equations for turbulence modeling can be efficiently solved with an indirectly coupled algorithm that uses the RK/implicit scheme for the flow equations.

Swanson, R. C.↗
NASA NTRS1984

An implict LU scheme for the Euler equations applied to arbitrary cascades

An implicit scheme for solving the Euler equations is derived and demonstrated. The alternating-direction implicit (ADI) technique is modified, using two implicit-operator factors corresponding to lower-block-diagonal (L) or upper-block-diagonal (U) algebraic systems which can be easily inverted. The resulting LU scheme is implemented in finite-volume mode and applied to 2D subsonic and transonic cascade flows with differing degrees of geometric complexity. The results are presented graphically and found to be in good agreement with those of other numerical and analytical approaches. The LU method is also 2.0-3.4 times faster than ADI, suggesting its value in calculating 3D problems.

Buratynski, E. K.↗
NASA NTRS2001

An Implicit LU/AF FDTD Method

There has been some recent work to develop two and three-dimensional alternating direction implicit (ADI) FDTD schemes. These ADI schemes are based upon the original ADI concept developed by Peaceman and Rachford and Douglas and Gunn, which is a popular solution method in Computational Fluid Dynamics (CFD). These ADI schemes work well and they require solution of a tridiagonal system of equations. A new approach proposed in this paper applies a LU/AF approximate factorization technique from CFD to Maxwell s equations in flux conservative form for one space dimension. The result is a scheme that will retain its unconditional stability in three space dimensions, but does not require the solution of tridiagonal systems. The theory for this new algorithm is outlined in a one-dimensional context for clarity. An extension to two and threedimensional cases is discussed. Results of Fourier analysis are discussed for both stability and dispersion/damping properties of the algorithm. Results are presented for a one-dimensional model problem, and the explicit FDTD algorithm is chosen as a convenient reference for comparison.

Beggs, John H.↗
NASA NTRS1980

Implicit computations of unsteady transonic flow governed by the full-potential equation in conservation form

An alternating-direction implicit algorithm is presented for solving the conservative, full-potential equation for unsteady, transonic flow. A new development is the time-linearization of the density function. This linearization reduces the solution process from one of solving a system of two equations at each mesh point to one of solving a single equation. Two sample cases are computed. First, a one-dimensional traveling shock wave is computed and compared with the analytic solution. Second, a two-dimensional case is computed of a flow field that results from a thickening and subsequently thinning airfoil. The resulting flow field, which includes a traveling shock wave, is compared to the flow field obtained from the low-frequency, small-disturbance, transonic equation.

Goorjian, P. M.↗
NASA NTRS1975

A cubic spline approximation for problems in fluid mechanics

A cubic spline approximation is presented which is suited for many fluid-mechanics problems. This procedure provides a high degree of accuracy, even with a nonuniform mesh, and leads to an accurate treatment of derivative boundary conditions. The truncation errors and stability limitations of several implicit and explicit integration schemes are presented. For two-dimensional flows, a spline-alternating-direction-implicit method is evaluated. The spline procedure is assessed, and results are presented for the one-dimensional nonlinear Burgers' equation, as well as the two-dimensional diffusion equation and the vorticity-stream function system describing the viscous flow in a driven cavity. Comparisons are made with analytic solutions for the first two problems and with finite-difference calculations for the cavity flow.

Rubin, S. G.↗
NASA NTRS1987

Chemistry-split techniques for viscous reactive blunt body flow computations

The weak-coupling structure between the fluid and species equations has been exploited and resulted in three, closely related, time-iterative implicit techniques. While the primitive variables are solved in two separated groups and each by an Alternating Direction Implicit (ADI) factorization scheme, the rate-species Jacobian can be treated in either full or diagonal matrix form, or simply ignored. The latter two versions render the split technique to solving for species as scalar rather than vector variables. The solution is completed at the end of each iteration after determining temperature and pressure from the flow density, energy and species concentrations. Numerical experimentation has shown that the split scalar technique, using partial rate Jacobian, yields the best overall stability and consistency. Satisfactory viscous solutions were obtained for an ellipsoidal body of axis ratio 3:1 at Mach 35 and an angle of attack of 20 degrees.

Li, C. P.↗
NASA NTRS1997

Gradient-Based Aerodynamic Shape Optimization Using ADI Method for Large-Scale Problems

A gradient-based shape optimization methodology, that is intended for practical three-dimensional aerodynamic applications, has been developed. It is based on the quasi-analytical sensitivities. The flow analysis is rendered by a fully implicit, finite volume formulation of the Euler equations.The aerodynamic sensitivity equation is solved using the alternating-direction-implicit (ADI) algorithm for memory efficiency. A flexible wing geometry model, that is based on surface parameterization and platform schedules, is utilized. The present methodology and its components have been tested via several comparisons. Initially, the flow analysis for for a wing is compared with those obtained using an unfactored, preconditioned conjugate gradient approach (PCG), and an extensively validated CFD code. Then, the sensitivities computed with the present method have been compared with those obtained using the finite-difference and the PCG approaches. Effects of grid refinement and convergence tolerance on the analysis and shape optimization have been explored. Finally the new procedure has been demonstrated in the design of a cranked arrow wing at Mach 2.4. Despite the expected increase in the computational time, the results indicate that shape optimization, which require large numbers of grid points can be resolved with a gradient-based approach.

Pandya, Mohagna J.↗
NASA NTRS1975

An investigation of several numerical procedures for time-asymptotic compressible Navier-Stokes solutions

The status of an investigation of four numerical techniques for the time-dependent compressible Navier-Stokes equations is presented. Results for free shear layer calculations in the Reynolds number range from 1000 to 81000 indicate that a sequential alternating-direction implicit (ADI) finite-difference procedure requires longer computing times to reach steady state than a low-storage hopscotch finite-difference procedure. A finite-element method with cubic approximating functions was found to require excessive computer storage and computation times. A fourth method, an alternating-direction cubic spline technique which is still being tested, is also described.

Rudy, D. H.↗