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NASA NTRS1991

Distributed Minimal Residual (DMR) method for acceleration of iterative algorithms

A new method for enhancing the convergence rate of iterative algorithms for the numerical integration of systems of partial differential equations was developed. It is termed the Distributed Minimal Residual (DMR) method and it is based on general Krylov subspace methods. The DMR method differs from the Krylov subspace methods by the fact that the iterative acceleration factors are different from equation to equation in the system. At the same time, the DMR method can be viewed as an incomplete Newton iteration method. The DMR method was applied to Euler equations of gas dynamics and incompressible Navier-Stokes equations. All numerical test cases were obtained using either explicit four stage Runge-Kutta or Euler implicit time integration. The formulation for the DMR method is general in nature and can be applied to explicit and implicit iterative algorithms for arbitrary systems of partial differential equations.

Lee, Seungsoo↗

NASA NTRS1990

A two-step iterative method for evolving nonlinear acoustic systems to a steady-state

A new approach for evolving two-dimensional nonlinear acoustic systems with flow to a steady state is presented. The approach is a two-step iterative method which is tested on a benchmark acoustic problem for which an exact analytical solution is available. Results are also calculated for a nonlinear acoustic problem for which an exact analytical solution is not known. Results indicate that the two-step method represents a powerful, efficient, and stable method for evolving two-dimensional acoustic systems to a steady state, and that the method is applicable to any number of spatial dimensions and to other hyperbolic systems. It is noted that for the benchmark problem only a single iteration on the method is required when the transient and steady-state field are of the same order of magnitude; however, four iterations are required when the steady-state field is several orders of magnitude smaller than the transient field. This method requires six iterations before achieving a steady state for the nonlinear test problem.

Watson, Willie R.↗

NASA NTRS1991

Comparing direct and iterative equation solvers in a large structural analysis software system

Two direct Choleski equation solvers and two iterative preconditioned conjugate gradient (PCG) equation solvers used in a large structural analysis software system are described. The two direct solvers are implementations of the Choleski method for variable-band matrix storage and sparse matrix storage. The two iterative PCG solvers include the Jacobi conjugate gradient method and an incomplete Choleski conjugate gradient method. The performance of the direct and iterative solvers is compared by solving several representative structural analysis problems. Some key factors affecting the performance of the iterative solvers relative to the direct solvers are identified.

Poole, E. L.↗

NASA NTRS1992

Upwind relaxation methods for the Navier-Stokes equations using inner iterations

A subsonic and a supersonic problem are respectively treated by an upwind line-relaxation algorithm for the Navier-Stokes equations using inner iterations to accelerate steady-state solution convergence and thereby minimize CPU time. While the ability of the inner iterative procedure to mimic the quadratic convergence of the direct solver method is attested to in both test problems, some of the nonquadratic inner iterative results are noted to have been more efficient than the quadratic. In the more successful, supersonic test case, inner iteration required only about 65 percent of the line-relaxation method-entailed CPU time.

Taylor, Arthur C., III↗

NASA NTRS1993

Helicopter trim analysis by shooting and finite element methods with optimally damped Newton iterations

Helicopter trim settings of periodic initial state and control inputs are investigated for convergence of Newton iteration in computing the settings sequentially and in parallel. The trim analysis uses a shooting method and a weak version of two temporal finite element methods with displacement formulation and with mixed formulation of displacements and momenta. These three methods broadly represent two main approaches of trim analysis: adaptation of initial-value and finite element boundary-value codes to periodic boundary conditions, particularly for unstable and marginally stable systems. In each method, both the sequential and in-parallel schemes are used, and the resulting nonlinear algebraic equations are solved by damped Newton iteration with an optimally selected damping parameter. The impact of damped Newton iteration, including earlier-observed divergence problems in trim analysis, is demonstrated by the maximum condition number of the Jacobian matrices of the iterative scheme and by virtual elimination of divergence. The advantages of the in-parallel scheme over the conventional sequential scheme are also demonstrated.

Achar, N. S.↗

NASA NTRS2001

Using Minimum-Surface Bodies for Iteration Space Partitioning

A number of known techniques for improving cache performance in scientific computations involve the reordering of the iteration space. Some of these reorderings can be considered as coverings of the iteration space with the sets having good surface-to-volume ratio. Use of such sets reduces the number of cache misses in computations of local operators having the iteration space as a domain. We study coverings of iteration spaces represented by structured and unstructured grids. For structured grids we introduce a covering based on successive minima tiles of the interference lattice of the grid. We show that the covering has good surface-to-volume ratio and present a computer experiment showing actual reduction of the cache misses achieved by using these tiles. For unstructured grids no cache efficient covering can be guaranteed. We present a triangulation of a 3-dimensional cube such that any local operator on the corresponding grid has significantly larger number of cache misses than a similar operator on a structured grid.

Frumlin, Michael↗

NASA NTRS2009

Bounded-Angle Iterative Decoding of LDPC Codes

Bounded-angle iterative decoding is a modified version of conventional iterative decoding, conceived as a means of reducing undetected-error rates for short low-density parity-check (LDPC) codes. For a given code, bounded-angle iterative decoding can be implemented by means of a simple modification of the decoder algorithm, without redesigning the code. Bounded-angle iterative decoding is based on a representation of received words and code words as vectors in an n-dimensional Euclidean space (where n is an integer).

Dolinar, Samuel↗

NASA NTRSDate not supplied

Improvements in Iterative Convergence of FUN3D Solutions

This paper presents a hierarchical adaptive nonlinear iteration method (HANIM) implemented in NASA computational fluid dynamics code, FUN3D, to improve robustness and computational efficiency of FUN3Dsolutions. In contrast to the baseline iterative solver that relies on an approximate Jacobian, a simple multicolor Gauss-Seidel point-implicit iteration scheme, and linear CFL ramping, HANIM is based upon a hierarchy of modules including pre conditioner, generalized conjugate residual, realizability check, nonlinear control,and CFL adaption modules. HANIM performance is systematically compared with the performance of the baseline solver. The iterative solutions are compared for three aerodynamic benchmark cases: a subsonic separated flow around a hemisphere cylinder, a supersonic flow through a long duct, and a subsonic flow over the NASA wing-body juncture model. Two Reynolds-averaged Navier-Stokes turbulence models are used in these computations, namely, the negative variant of the linear one-equation Spalart-Allmar as model and its nonlinear extension based on quadratic constitutive relations.

Li Wang↗

NASA NTRS1973

A successive overrelaxation iterative technique for an adaptive equalizer

An adaptive strategy for the equalization of pulse-amplitude-modulated signals in the presence of intersymbol interference and additive noise is reported. The successive overrelaxation iterative technique is used as the algorithm for the iterative adjustment of the equalizer coefficents during a training period for the minimization of the mean square error. With 2-cyclic and nonnegative Jacobi matrices substantial improvement is demonstrated in the rate of convergence over the commonly used gradient techniques. The Jacobi theorems are also extended to nonpositive Jacobi matrices. Numerical examples strongly indicate that the improvements obtained for the special cases are possible for general channel characteristics. The technique is analytically demonstrated to decrease the mean square error at each iteration for a large range of parameter values for light or moderate intersymbol interference and for small intervals for general channels. Analytically, convergence of the relaxation algorithm was proven in a noisy environment and the coefficient variance was demonstrated to be bounded.

Kosovych, O. S.↗

NASA NTRS1973

Iterative methods for plasma sheath calculations: Application to spherical probe

The computer cost of a Poisson-Vlasov iteration procedure for the numerical solution of a steady-state collisionless plasma-sheath problem depends on: (1) the nature of the chosen iterative algorithm, (2) the position of the outer boundary of the grid, and (3) the nature of the boundary condition applied to simulate a condition at infinity (as in three-dimensional probe or satellite-wake problems). Two iterative algorithms, in conjunction with three types of boundary conditions, are analyzed theoretically and applied to the computation of current-voltage characteristics of a spherical electrostatic probe. The first algorithm was commonly used by physicists, and its computer costs depend primarily on the boundary conditions and are only slightly affected by the mesh interval. The second algorithm is not commonly used, and its costs depend primarily on the mesh interval and slightly on the boundary conditions.

Parker, L. W.↗

NASA NTRS1976

Convergence of an iterative procedure for large-scale static analysis of structural components

The paper proves convergence of an iterative procedure for calculating the deflections of built-up component structures which can be represented as consisting of a dominant, relatively stiff primary structure and a less stiff secondary structure, which may be composed of one or more substructures that are not connected to one another but are all connected to the primary structure. The iteration consists in estimating the deformation of the primary structure in the absence of the secondary structure on the assumption that all mechanical loads are applied directly to the primary structure. The j-th iterate primary structure deflections at the interface are imposed on the secondary structure, and the boundary loads required to produce these deflections are computed. The cycle is completed by applying the interface reaction to the primary structure and computing its updated deflections. It is shown that the mathematical condition for convergence of this procedure is that the maximum eigenvalue of the equation relating primary-structure deflection to imposed secondary-structure deflection be less than unity, which is shown to correspond with the physical requirement that the secondary structure be more flexible at the interface boundary.

Austin, F.↗

NASA NTRS1976

An iterative decoupling solution method for large scale Lyapunov equations

A great deal of attention has been given to the numerical solution of the Lyapunov equation. A useful classification of the variety of solution techniques are the groupings of direct, transformation, and iterative methods. The paper summarizes those methods that are at least partly favorable numerically, giving special attention to two criteria: exploitation of a general sparse system matrix structure and efficiency in resolving the governing linear matrix equation for different matrices. An iterative decoupling solution method is proposed as a promising approach for solving large-scale Lyapunov equation when the system matrix exhibits a general sparse structure. A Fortran computer program that realizes the iterative decoupling algorithm is also discussed.

Athay, T. M.↗

NASA NTRS1977

Comparison of results obtained by solving the radiative transfer equation with an iterative method and a spherical harmonics method

Fluxes and intensities of light scattered by a model atmosphere are computed by a spherical harmonics approximation and by an iterative method of solving the radiative transfer equation and are compared. The large differences in the net fluxes and intensities reported by Dave and Armstrong (1974) for the two methods are reduced here by making a few changes in the iterative routine. Decreasing the polar angle increment from 2 to 1 deg in the iterative method of computing the source function does not improve the results as suggested by Dave and Armstrong.

Bahethi, O. P.↗

NASA NTRS1977

A block iterative LU solver for weakly coupled linear systems

A hybrid technique, called the block iterative LU solver, is proposed for solving the linear equations resulting from a finite element numerical analysis of certain fluid dynamics problems where the equations are weakly coupled between distinct sets of variables. Either the block Jacobi iterative method or the block Gauss-Seidel iterative solver is combined with LU decomposition.

Cooke, C. H.↗

NASA NTRS1981

A multigrid algorithm for steady transonic potential flows around aerofoils using Newton iteration

The application of multigrid relaxation to transonic potential flow calculation was investigated. Conservative potential flows around aerofoils were taken as test problems. The solution algorithm was based on Newton iteration. It was found that the iteration to the circulation has to be kept outside the multigrid algorithm. To obtain meaningful norms of residuals, difference formulas with asymptotic scaling were introduced. Nonlinear instability problems were solved by upwind differencing using mass flux vector splitting instead of artificial viscosity or artificial density. It is suggested that the algorithms efficiency be increased by improving the iteration on the shock positions even though this is a highly nonlinear process.

Boestoel, J. W.↗

NASA NTRS1981

An iterative finite element-integral technique for predicting sound radiation from turbofan inlets

A new iterative solution technique for predicting the sound field radiated from a turbofan inlet is presented. The sound field is divided into two regions; the sound field within the inlet which is computed using the finite element method and the radiation field outside the inlet which is calculated using an integral solution technique. A 'unified' solution is obtained by matching the finite element and integral solutions at the interface between the interior and exterior regions. The applicability of the iterative technique is demonstrated by considering several simple cases for which exact or 'classical' solutions for the sound field are available. These examples show that the proper solution is obtained within five iterations. The overall accuracy of the method is demonstrated by comparison with experimental data.

Horowitz, S. J.↗

NASA NTRS1982

An iterative finite element-integral technique for predicting sound radiation from turbofan inlets in steady flight

A new iterative solution technique for predicting the sound field radiated from a turbofan inlet in steady flight is presented. The sound field is divided into two regions: the sound field within and near the inlet which is computed using the finite element method and the radiation field beyond the inlet which is calculated using an integral solution technique. A continuous solution is obtained by matching the finite element and integral solutions at the interface between the two regions. The applicability of the iterative technique is demonstrated by comparison of experimental results with the theoretical results for several different inlet configurations with and without flow. These examples show that good agreement between experiment and theory is obtained within five iterations.

Horowitz, S. J.↗

NASA NTRS1983

Accelerating an iterative process by explicit annihilation

A slowly convergent stationary iterative process can be accelerated by explicitly annihilating (i.e., eliminating) the dominant eigenvector component of the error. The dominant eigenvalue or complex pair of eigenvalues can be estimated from the solution during the iteration. The corresponding eigenvector or complex pair of eigenvectors can then be annihilated by applying an explicit Richardson process over the basic iterative method. This can be done entirely in real arithmetic by analytically combining the complex conjugate annihilation steps. The technique is applied to an implicit algorithm for the calculation of two dimensional steady transonic flow over a circular cylinder using the equations of compressible inviscid gas dynamics. This demonstrates the use of explicit annihilation on a nonlinear problem.

Jespersen, D. C.↗