Conventional Iterative Methods Research Articles

Frequency domain relaxation of power system dynamics

A highly parallel frequency-domain waveform relaxation method is presented to solve transient dynamics of nonlinear systems. The method attempts to exploit parallelism offered by the frequency-domain matrix equations used to represent and solve the systems. A comparison with various other methods shows that this is a unique method in that it solves transient dynamics of nonlinear systems entirely in the frequency domain while being suitable for large-scale parallel computation. The method treats linear and nonlinear parts of the system individually. Linear blocks are easily solved in parallel in the frequency domain since superposition holds and each frequency can be handled independently of the other. Nonlinear blocks are approximated by polynomials and are solved in parallel using the describing function matrix technique. The frequency-domain-relaxation (FDR) has been used to solve for the transient response of two power system machine models. Results obtained by this method are compared to the results obtained by a conventional time-domain iteration method. It is observed that there is a reduction in the number of parallel steps required by the FDR method, which is dependent on the model and the ratio of data points required by the two methods to give solutions of comparable accuracy. >

Fully coupled solution of the equations for incompressible recirculating flows using a penalty-function finite-difference formulation

This paper describes two new solution algorithms for steady recirculating flows that use a penalty formulation to eliminate the pressure from the finite difference form of the governing equations. One algorithm uses successive substitution to linearize the equations, while the other employs the Newton-Raphson linearization. In both cases, the equations are solved in a fully coupled manner using a sparse matrix form of LU decomposition. The D'Yakonov iteration is used to avoid unnecessary factorizations of the coefficient matrix, significantly improving the computational efficiency. The Newton-Raphson linearization leads to faster convergence, but the execution times of the two methods are comparable. The algorithms converge rapidly and are robust to changes in grid size and Reynolds number. In a number of laminar two-dimensional flows, the new methods proved to be two to ten times faster than some conventional iterative methods.

A Block-Corrected Subdomain Solution Procedure for Recirculating Flow Calculations

This paper describes a robust and efficient subdomain solution procedure for two-dimensional recirculating flows. The solution domain is divided into a number of overlapping subdomains, and a direct fully coupled solution is obtained for each subdomain using a sparse matrix form of LU decomposition. An effective parabolic block correction procedure, which calculates global corrections to the tentative solution by a marching technique similar to that used for boundary layer flows, is used to accelerate the convergence of the basic procedure. The use of effective block correction is found to be essential for the success of the subdomain approach on strongly recirculating flows. In a number of laminar two-dimensional flows, the new block-corrected method performed extremely well, rivaling the best direct methods in execution time, while requiring substantially less computer storage. The new method proved to be from two to ten times faster than conventional iterative methods, while requiring only a moderate increase in storage.

Implementation of an Improved Adaptive-Implicit Method in a Thermal Compositional Simulator

Summary A multicomponent thermal simulator with an adaptive-implicit-method (AIM) formulation/inexact-adaptive-Newton (IAN) method is presented. The final coefficient matrix retains the original banded structure so that conventional iterative methods can be used. Various methods for selection of the eliminated unknowns are tested. AIM/IAN method has a lower work count per Newtonian iteration than fully implicit methods, but a wrong choice of unknowns will result in excessive Newtonian iterations. For the problems tested, the residual-error method described in the paper for selecting implicit unknowns, together with the IAN method, had an improvement of up to 28% of the CPU time over the fully implicit method.

A matrix method for partitioning the atoms of a molecule into equivalence classes

An easily vectorizable method for canonically numbering the atoms of a molecule has been devised. The method is expected to be faster than conventional iterative methods on any computer which derives significant power from parallel computations with vectors. Using the method one constructs a certain matrix for each atom of the molecule. The matrix gives a view of the molecule as seen from the particular atom. Atoms which are equivalent under automorphisms have equal matrices. The method is slower than the conventional ones on ordinary computers.

A Spectrum Enveloping Technique for Iterative Solution of Central Difference Approximations of Convection-Diffusion Equations

When a convection-diffusion equation is discretized using the central difference scheme the resulting coefficient matrix is not diagonally dominant whenever the convection terms are large. If this system of linear equations is solved using the conventional iteration methods, the iterations often fail to converge as some of the eigenvalues of the iteration matrix lie outside the unit circle $C = \{ z: | z |\leqq 1 \}$ in the complex plane. The eigenvalue spectrum of some of the iteration matrices lies inside the infinite strip $S = \{ z: | \operatorname{Real} ( z ) | < 1, | \operatorname{Imag} ( z ) | < \infty \}$. An example is that of the method of simultaneous displacements or the Jacobi method. In such cases, it is possible to enclose the eigenvalue spectrum inside an ellipse with major axis on the imaginary axis and minor axis in the real interval $( - 1,1 )$. This ellipse is used to define a convergent iteration. A practical computational algorithm is described to obtain such an iteration scheme. Numerical examples show that the spectrum enveloping technique works well when the original iterations diverge. When the original iterations converge the spectrum enveloping technique can converge even faster.

The finite element response matrix method for the solution of the neutron transport equation

The finite element response matrix method has been applied to the solution of the neutron transport equation. This method has previously been applied to the neutron diffusion equation for coarse mesh reactor analysis with excellent results. As with the diffusion equation implementation, the transport method employs a local-global projection technique to allow treatment of internal heterogeneities that would normally not be resolved by the coarse global mesh that is needed for computational efficiency. However, since the transport equation includes the angular domain, the local-global projection technique has been extended to angle as well as space. Since the response matrices do not depend on the multiplication factor, a conventional fission source iteration method is utilized for criticality problems. The method has been applied to the one-dimensional and two-dimensional neutron transport equations. For one-dimensional geometries, the local-global projection method yields excellent results, indicating the potential of this approach as a viable coarse mesh transport method. Numerical results are presented for several one-dimensional configurations that were analyzed with varying choices of local and global meshes in the spatial domain. Preliminary results with two-dimensional applications indicate that computational times may be an order of magnitude faster than with the conventional finite element solution of the two-dimensional transport equation.

Critical points on a perfectly 8- or 6-connected thin binary line

A set of “logical pixel logic” based procedures is described and illustrated for detecting the near-minimal number of pixels on a perfectly 8- or 6-connected binary thin line which have to be retained to adequately approximate the original pixel string with a connected sequence of straight-line segments. The found critical points or pixels on the line are for practical purposes the same as those chosen by a person, were he asked to perform the same task. The method is fast and in practically all cases the approximating connected sequence of straight-line segments does not deviate by more than one pixel distance from the original pixel string. Experiments on Chinese characters have given very good results. The method could also be used as a preprocessing step to conventional iterative methods based on minimizing a distance criterion between the approximating straight-line segments and the pixels on the original binary line. It will be very useful in any field in which computational speed is one of the major concerns, such as industrial vision systems.

A non-iterative method for fitting two-component decay curves

A method for fitting a two-component decay curve without iterative procedure is presented. The method utilizes a linearization technique of the sum of exponential functions. Artifical decay data have been used to test the present method and a comparison with the conventional iterative methods has been made.

A new non-iterative method for fitting Lorentzian to Mössbauer spectra

A new method for fitting a Lorentzian function without an iterative procedure is presented. The method is quicker and simpler than the previously proposed method of non-iterative fitting. Comparison with the previous method and with the conventional iterative method has been made. It is shown that the present method gives satisfactory results.

An efficient method for deriving mass-consistent flow fields from wind observations in rough terrain

Mass-consistent wind fields can be generated for observed wind data. The usual process involves the minimal adjustment of a trial wind field by relaxation techniques to achieve nondivergent flow. This paper describes how linear combinations of a limited number of solutions obtained by the conventional iterative relaxation methods can be used to achieve the same results with greatly reduced computational requirements. The solutions must only be obtained for linearly independent data sets. The application described uses eigenvectors of the covariance matrix of the input wind component data as the linearly-independent data sets. The model with which this technique is applied uses a relaxation scheme to achieve mass-conservative flow with minimum difference between the initial trial wind field obtained by interpolation and the resulting mass-conserving flow.

AC-HVDC solution and security assessment using a diakoptical method

The paper describes a direct solution of an HVDC link within an AC loadflow solution and a direct security analysis using diakoptical techniques. The direct solution is faster than the more conventional boundary iterative method and therefore is more economical when used for regular calculation such as security assessment and transient stability analysis. It can easily be incorporated in existing AC loadflow algorithms.

Suboptimal control of neutral systems

Abstract The usual approach for obtaining the optimal control for a neutral system with quadratic cost consists of first deriving a set of necessary conditions for optimality and then using conventional iterative numerical methods to find a control satisfying those conditions. The burden of computation in this approach is enormous. The suboptimum iterative scheme proposed in this paper does not proceed along those lines. Instead, the neutral system is first converted into a non-linear time-delay system. The non-linear delay term is then treated like an extra perturbing input. A linear non-delay system is optimized at each stage and the resulting sequence of control functions converges rapidly to a suboptimal control for the original problem. One numerical example illustrates the technique

A simple iterative method for sub-optimal control of linear time delay systems with quadratic cost†

Abstract The usual approach for obtaining the optimal control For a linear time-delay system with a quadratic cost consists of first deriving a set of necessary conditions for optimality and then using conventional iterative numerical methods to find a control satisfying those conditions. The burden of computation in this approach is enormous. The iterative scheme proposed in this paper does not proceed along these lines. Instead, the delay term is treated liko an extra perturbing input. A linear non-delay system is optimized at each stage, and the resulting sequence of control functions converges rapidly to the sub-optimal control for the original problem, as illustrated by two numerical examples.

Optimization of coupled non-linear systems†

A coupling perturbation method is developed for near optimum design of non-linear systems. A scalar parameter ϵ is introduced in an nth-order system such that for ϵ ϵ = the system decouples into two (or more) independent low-order sub-systems. By the same procedure ϵ is introduced into the performance index. The near optimal control u¯ is defined as a truncated power series in ϵ. It is shown that each term of the above series can be sequentially obtained by separate sub-system calculations. Thus a considerable saving in computational effort is achieved Two examples are presented and the results compared with those obtained by conventional iterative methods. Applicability of the method to problems which cannot be solved by conventional methods is also discussed.

GENERAL COMPUTER METHOD OF ANALYSIS OF CONDUCTION AND DIFFUSION IN BIOLOGICAL SYSTEMS WITH DISTRIBUTIVE SOURCES.

A method of analysis by use of the computer has been developed which permits the physical laws governing conduction, or diffusion, to be expressed in first-order finite form. Solutions are obtained in the one-dimensional case for rectangular, cylindrical, and spherical geometries without the use of transcendental functions or other conventional mathematical iterative methods. The case of the cylindrical calorimeter with concentric insulating and conducting layers of varying chemical properties and with distributive heat sources arising from biochemical reactions at the center is treated in the transient and steady-state case.