An iterative method for large systems of linear structural equations

A method is described for the solution of large sets of sparse equations arising in structural analysis. This method, called partial elimination, combines the concepts of elimination and iteration in such a way that good convergence rates can be obtained using a computer storage space not much greater than that required for other iterative methods.     

An algorithm for the solution of very large banded unsymmetric linear equation systems

This paper presents an algorithm and corresponding FORTRAN program for the solution of unsymmetric banded linear equation systems. The algorithm is based on the Crout method. A special technique, called double windowing, enables the solution of very large equation systems with a total equation number reaching 30,000 and a full bandwidth in excess of 1,000. Special attention was devoted to minimization of peripheral processor time (communication with backing disc memory). An appendix lists the complete program for Cyber series computers (CDC).     

A method for determining errors in measuring systems

Translated from Izmeritel'naya Tekhnika, No. 11, pp. 27–28, November, 1992.     

New method for determining the depth of field of microscope systems

This paper presents new formulas to determine the depth of field (DOF) of optical and digital microscope systems. Unlike the conventional DOF formula, the new methods consider the interplay of geometric and diffraction optics for infinite and finite optical microscopes and for corresponding digital microscope systems. It is shown that in addition to the well understood parameters such as numerical apertures, focal length, and light wavelength, system components such as aperture stops also affect the DOF. For the same objective lens, the DOF is inversely proportional to the size of the aperture stop, and it is proportional to the focal length of the ocular lens. It is also shown that under optimal viewing and operating conditions, the visual accommodation of human observers has no meaningful impact on DOF. The new formulas reported are useful for accurately calculating the DOF of microscopes.     

Determination of eigenvalues of large structural systems in an arbitrarily specified range

For a large structural system of small bandwidth a technique combining linear interpolation on the characteristic polynomial and suppression of its previously determined roots by deflation can be used to determine its eigenvalues. The eigenvalues, however, have to be found in order, beginning from either the lowest (or the highest) to obtain monotonic convergence to the polynomial roots. A method which eliminates this restriction is presented in this paper. If eigenvalues greater than a are desired, the method, in essence, consists of modifying the deflation procedure to suppress all the eigenvalues smaller than a without actually determining them. Some consequences of the procedure developed in this paper are also presented and it is shown that the standard deflation technique is a special case of this procedure. The method has been applied to a wide range of problems with success and considerable saving in computation time.     

时滞微分方程特征值的近似求解方法

通过将时滞微分方程变换为积分方程后并进行离散,得到相应的差分方程,从理论上建立了差分方程的系统矩阵的特征值与原线性时滞微分系统的特征值之间的关系,从而得到了一种求解时滞微分方程特征值的新方法。作为算例,计算了时滞的Logistic方程的前10阶特征值。误差分析表明,提出的近似方法不但简单,并且有很高的精度。     

Interferometer method of determining the parallelism of large block gauges

Measurement Techniques -     

Wedge simulation method for calculating the reliability of linear dynamical systems

In this paper the problem of calculating the probability of failure of linear dynamical systems subjected to random excitations is considered. The failure probability can be described as a union of failure events each of which is described by a linear limit state function. While the failure probability due to a union of non-interacting limit state functions can be evaluated without difficulty, the interaction among the limit state functions makes the calculation of the failure probability a difficult and challenging task. A novel robust reliability methodology, referred to as Wedge-Simulation-Method, is proposed to calculate the probability that the response of a linear system subjected to Gaussian random excitation exceeds specified target thresholds. A numerical example is given to demonstrate the efficiency of the proposed method which is found to be enormously more efficient than Monte Carlo Simulations.     

An economical storage organization for large and small finite element systems with reference to the application of iterative approximation methods to nonlinear material behaviour

The method of data transfer from the peripheral storage unit to the central processor unit, when dealing with large solution systems, is of greater importance, as each access of peripheral stored data interrupts the active flow of arithmetic operation in the central processor core. When solving a large number of symmetrical positive-definite equations this generally leads to difficulties in channel interaction and priority, possibly even to the breakdown of the computer system or to uneconomical computing times. The problem increases when considering the iterative approximation of nonlinear problems; further still, when using the frontal solution method. It will be shown that this difficulty can be avoided by inserting two different types of buffers—a micro-buffer and a macro-buffer—into the data transfer between the central memory and an arbitrary peripheral storage unit. This method should primarily be used in those finite element (FE) programs which are based on the ‘slow’ BACKSPACE-READ-BACKSPACE commands, but even in the case of peripheral storage units such as magnetic discs (random access) or magnetic tape machines (working in one or two directions) the computing time and channel control can successfully be improved. A second achievement of this procedure is the possibility of applying a FE program economically, not only to large element systems and elements of a ‘higher order’ but also with good results for small element systems, e.g. when investigating nonlinear material behaviour with simple element types and geometrical structure. Another purpose of this text is to recommend the FORTRAN subroutine in the Appendix, whose novel features make it a useful supplement to ‘nonlinear’ FE programs.     

An iterative method for finite dimensional structural optimization problems with repeated eigenvalues

A numerical method for solving finite dimensional structural optimization problems with constraints on natural frequency and on buckling is formulated. The method treats nondifferentiable repeated eigenvalues that have been shown to systematically arise in structural optimization. Recent results on differentiability of eigenvalues are used to develop a generalized gradient projection method for structural optimization. The algorithm is shown to overcome technical difficulties associated with nondifferentiability of repeated eigenvalues. The method is used to solve buckling and vibration optimization problems in which repeated eigenvalues occur.     

An efficient simulation method for reliability analysis of linear dynamical systems using simple additive rules of probability

This paper presents a simulation technique for reliability analysis of linear dynamical systems. It is based on simple additive rules of probability (in contrast to other probabilistic approaches such as importance sampling). It is shown that the proposed appoach is identical to a newly developed approach, Importance Sampling using Elementary Events (ISEE) [Au SK, Beck JL. First excursion probabilities for linear sytems by very efficient importance sampling. Probabl Eng Mech 2001;16(3):193–208]. A simple formula for the coefficient of variation of the estimator of the failure probability using the samples is also given. A 10-story building model with nonstationary excitation is utilized to demonstrate the accuracy and efficiency of the proposed method.     

A subspace iteration method for the eigensolution of large undamped gyroscopic systems

This paper presents an efficient and numerically stable algorithm for computing accurately the eigenvalues of smallest moduli and the associated eigenvectors of the quadratic eigenproblem which arises in the analysis of spinning systems. Derived from the well-known standard subspace iteration method, this solution procedure takes full advantage of the banded configuration of the structural matrices, and of the specific nature of gyroscopic systems. A numerical example is presented to show the efficiency and accuracy of the algorithm.