Elliptic Solver Research Articles

Two Vectorized Algorithms for the Effective Calculation of Mass-Consistent Flow Fields

Abstract The purpose of this paper is the calculation of mass-consistent wind velocity fields over complex orography on the basis of existing measurements. Measured data are used to generate an initial wind velocity field that in general does not satisfy continuity. For the adjustment of this velocity field a three-dimensional elliptic differential equation is solved. A transformation of this equation to a terrain-following coordinate system ensures the proper consideration of the orography. Two numerical algorithms for the solution of the transformed equation are presented. One algorithm makes use of a fast direct elliptic solver based on Fourier analysis, the other utilizes the red-black SOR method. Both algorithms achieve full vectorization on computers like the CYBER 205. A test problem is defined to compare the two algorithms with regard to the computing time: In the case of small terrain roughness, the algorithm using the fast direct elliptic solver is recommended, in the opposite case the red-black...

A Comparison of Elliptic Solvers for General Two-Dimensional Regions

We present results of some numerical experiments performed to compare the relative efficiency and storage requirements of some elliptic solvers on general two-dimensional regions. The techniques employed by the solvers included: capacitance matrix techniques, sparse matrix techniques, multi-grid techniques and preconditioned conjugate gradient techniques. Special emphasis is placed on the many right-hand side case, as motivated by applications to time simulation studies.

Application of a two-dimensional grid solver for three-dimensional problems

A two-dimensional elliptic grid solver is presented and its application to various three-dimensional configurations, both internal and external, is demonstrated. The method uses proper forcing terms to cluster grid points near boundaries with a specified grid spacing and allows grid lines to intersect the boundaries at a specified angle. By segmenting the region, grid results are generated for sharp leading edged configurations and wing-vertical tail combinations.

Fast elliptic solvers—an overview

The purpose of this paper is to present a brief survey of fast direct methods for solving elliptic boundary-value problems. The methods reviewed are based on Fourier analysis, block reduction techniques, and marching algorithms. First, the Poisson equation with Dirichlet and mixed boundary conditions are considered. Then we go to more general elliptic problems and irregular regions.

Plasma equilibrium calculations by line successive over relaxation

Line successive over relaxation (LSOR) is an iterative method for solving elliptic differential equations. LSOR takes advantage of the CRAY vector capabilities as compared to the point successive over relaxation (SOR) method, which does not vectorize. The substantial advantages of LSOR on a vectorizing machine are not well-known, except in the field of aerodynamics. By minor modification of the traditional SOR elliptic equation solver, we find that in certain coordinates an increase of a factor of two or greater in convergence time can be realized. As a model problem for comparison of SOR and LSOR, the numerical solution of Poisson's equation will be reviewed in Sec. II. In Sec. III, we discuss the decreased computation time on the National Fusion Energy Computer Center (NMFECC) CRAY computers found with LSOR applied to the iterative solution of plasma equilibria. In Sec. IV, the conditions for which LSOR is most useful are summarized.

Finite difference solution to the flow between two rotating spheres

Abstract This paper describes a numerical method for calculating incompressible viscous flows between two concentric rotating spheres. The dependent variables describing the axisymmetric flow field are the azimuthal components of the vorticity, of the velocity vector potential and of the velocity. The coupled set of governing partial differential equations is written as a system of strictly second-order equations by introducing vorticity conditions of an integral character in a meridional plane. Such conditions generalize the one-dimensional integral conditions employed by Dennis and Singh to calculate steady-state solutions of the same problem using Gegenbauer polynomials and finite differences. The basic equations are discretized in space and in time by means of the finite-difference method. A fourth-order accurate centred-difference approximation of the advection terms is employed and a nonlinearly implicit scheme for the discrete time integration is here considered. A general finite-difference algorithm for steady-state and time-dependent problems is obtained which has no relaxation parameter and makes extensive use of fast elliptic solvers. The numerical results obtained by the present method are found to be in good agreement with the literature and confirm the nonuniqueness of the steady-state solution in a narrow spherical gap at certain regimes.

Algorithms and data structures for adaptive multigrid elliptic solvers

With the advent of multigrid iteration, the large linear systems arising in numerical treatment of elliptic boundary value problems can be solved quickly and reliably. This frees the researcher to focus on the other issues involved in numerical solution of elliptic problems: adaptive refinement, error estimation and control, and grid generation. Progress is being made on each of these issues and the technology now seems almost at hand to put together general purpose elliptic software having reliability and efficiency comparable to that of library software for ordinary differential equations. This paper looks at the components required in such general elliptic solvers and suggests new approaches to some of the issues involved. One of these issues is adaptive refinement and the complicated data structures required to support it. These data structures must be carefully tuned, especially in three dimensions where the time and storage requirements of algorithms are crucial. Another major issue is grid generation. The options available seem to be curvilinear fitted grids, constructed on interactive graphics systems, and unfitted Cartesian grids, which can be constructed automatically. On several grounds, including storage requirements, the second option seems preferrable for the well behaved scalar elliptic problems considered here. A variety of techniques for treatment of boundary conditions on such grids have been described previously and are reviewed here. A new approach, which may overcome some of the difficulties encountered with previous approaches, is also presented.

Direct solution of the discretized poisson—Neumann problem on a domain composed of rectangles

Finite-difference approximations for the two-dimensional Poisson-Neumann problem ( 1 λ) div λ grad p = q , λ > 0on R λ n · grad p = g R on ∂R result in a large system of linear equations Lp = q. The n × n matrix L is singular with rank( L) = n − 1. A solution exists if q satisfies the consistency condition v T q = 0, where L T V = 0, v ≠ 0. A direct-solution scheme is described for the case where R can be decomposed into a set of rectangular domains each having a possibly different but constant material coefficient λ. We assume that this problem has to be solved repeatedly for many vectors q. The method is efficient in that the number of operations is of order n log n. This efficiency is gained by using fast elliptic solvers for each single rectangle and a proper variant of the socalled influence- or capacitance-matrix technique. Here, on each rectangle a separate Poisson-Neumann problem has to be solved. The essential point is the manner in which the consistency condition is enforced for each rectangle. The new method has been successfully applied in a code FLUX to analyze fluid-structure interactions.

Generation of three-dimensional boundary-fitted curvilinear coordinate systems for wing/wing-tip geometries using the elliptic solver method

A three-dimensional elliptic solver technique is utilized to generate surface-fitted coordinates about wing/wing-tip configurations. The method is applicable to wings of arbitrary section profile and camber, leading-edge sweep, taper ratio, and spanwise thickness variation. The basic theory of three-dimensional elliptic mappings is developed along with a method to compute interior coordinate control functions. Examples of grids generated about several wing/wing-tip geometries are given. A 49 × 33 × 17 grid requires about 3 minutes of CPU time on a CYBER 203 computer.

Two-Dimensional Unsteady Euler-Equation Solver for Arbitrarily Shaped Flow Regions

A new technique is described for solving supersonic fluid dynamics problems containing multiple regions of continuous flow, each bounded by a permeable or impermeable surface. Region boundaries are, in general, arbitrarily shaped and time dependent. Discretization of such a region for solution by conventional finite difference procedures is accomplished using an elliptic solver which alleviates the dependence on a particular base coordinate system. Multiple regions are coupled together through the boundary conditions. The technique has been applied to a variety of problems including a shock diffraction problem and supersonic flow over a pointed ogive.

ITERATIVE NONLINEAR THERMAL ANALYSIS OF DIRECTIONAL SOLIDIFICATION USING A FAST ELLIPTIC SOLVER

The material characteristics of directionally solidified superalloy crystals require carefully controlled solidification rates, temperature field distributions, and maximum temperature gradients. Experimentally, these can be determined by vacuum casting of cylindrical rods. Corresponding analytical studies yield nonlinear equation systems due to radiative boundary conditions and latent heat effects, which in the past required lengthy finite-difference or finite-element solutions. For the quasi-steady freezing encountered in long cylindrical alloy bars, we employed an efficient iteration scheme based on a fast elliptic equation solver previously reported in the literature. This paper discusses the principal aspects of the computational scheme and gives results showing the two-dimensional temperature fields encountered in solidification processes subject to various input conditions.

Iterative Nonlinear Thermal Analysis of Directional Solidification Using a Fast Elliptic Solver

Iterative Nonlinear Thermal Analysis of Directional Solidification Using a Fast Elliptic Solver

Fast elliptic solvers and three-dimensional fluid-structure interactions in a pressurized water reactor

A numerical method is described for analysis of the response of internal structures in the vessel of a pressurized water reactor to a sudden depressurization (“blowdown”). The three-dimensional geometry and fluid-structure interactions are taken into account. Potential flow is assumed for a compressible fluid with constant speed of sound which is appropriate for subcooled water. Any linear-elastic dynamic model in terms of a finite number of degrees of freedom can be employed for the internal structure, in particular the core barrel; here the model “CYLDY2” is implemented. An implicit and stable time-differencing scheme is used. Spatially, finite differences or spectral approximations are introduced. The resultant set of linear equations is solved efficiently by means of fast elliptic solvers and the capacitance, or influence, matrix technique. The resultant code, FLUX2, is applied to predict the dynamics in case of the planned HDR experiment. A sensitivity analysis shows that truncation errors can be kept sufficiently small. By comparison of results with and without fluid-structure interactions it is found that the coupling has an essential effect on the resultant maximum stress.

Numerical Solution of the Two-Dimensional Premixed Laminar Flame Equations

The present paper presents a numerical solution method for the two-dimensional elliptic partial differential equations modeling a premixed laminar flame freely propagating at constant pressure between parallel plates having a constant, cool wall temperature. The technique developed is an efficient, semidirect (semi-iterative) method termed multistep damped Picard iteration. Picard linearization of the nonlinear terms together with variable damping as the solution progressed and a sparse matrix elliptic solver allowed rapid solution to be obtained as a limit of a convergent sequence of solutions. The flame-quenching limit was determined by iterating the solution with respect to the distance separating the plates and approaching the l imit state to within a specific criterion. Global kinetics and average transport properties were used for a simulated propane-air mixture; radiation loss was neglected throughout the analysis. The method was used to study the effect of the ignition temperature on the flame parameters as well as to reveal the two-dimensional structure of the flame at the quenching limit.

A Fast Semidirect Method for Computing Transonic Aerodynamic Flows

A fast, semidirect, iterative computational method, previously introduced for finite-difference solution of subsonic and slightly supercritical flow over airfoils, is extended both to apply to strongly supercritical conditions and to include full second-order accuracy in computing inviscid flows over airfoils. The nonlinear small-disturbance equations are solved iteratively by a direct, linear, elliptic solver. General, fully conservative, type-dependent difference equations are formulated, including parabolic- and shock-point transition operators that provide consistency with the integral conservation laws. These equations specialize to either first-order or to fully second-order-accurate equations. Various free parameters are evaluated for rapid convergence of the first-order scheme. Resulting pressure distributions and computing times are compared with the improved Murman-Cole line-relaxation method.

A generalized-capacity-matrix technique for computing aerodynamic flows

Abstract A numerical generalized-capacity-matrix technique is developed for application to aerodynamic flow computations. This technique allows the very fast direct (noniterative) numerical elliptic solvers to be used in poblems with arbitrary internal boundaries and with a wide class of boundary conditions, including numerical application of the Kutta condition on an airfoil without iteration. Accuracy, speed, and usefulness of the technique are demonstrated with linear problems for potential flows over airfoil shapes. The method's main advantages, however, can be exploited within iterative procedures for a variety of complex flow problems governed by systems of equations not necessarily elliptic or linear

Direct numerical solution of three‐dimensional equations containing elliptic operators

AbstractA direct three‐dimensional elliptic solver is presented for application in a wide class of numerical methods for solving partial differential equations in physics and engineering. The derived algorithm and FORTRAN code implement Buzbee, Golub and Nielson's proposed extension of Buneman's Cyclic‐Reduction Poisson solver to three dimensions. Both a ‘most direct’ cyclic reduction and a revised method (to eliminate roundoff error difficulties) are derived. Tests on an IBM 360/67 computer, using various optional combinations of subroutines, showed significant differences in accuracy and computing time, with the optimum subroutine combination depending on mesh size.