Number Of Simultaneous Linear Equations Research Articles

New class ôN of statistical models: Transfer matrix eigenstates, chain Hamiltonians, factorizable S matrix

Statistical models corresponding to a new class of braid matrices (ôN;N⩾3) presented in a previous paper are studied. Indices labeling states spanning the Nr dimensional base space of T(r)(θ), the rth order transfer matrix are so chosen that the operators W (the sum of the state labels) and (CP) (the circular permutation of state labels) commute with T(r)(θ). This drastically simplifies the construction of eigenstates, reducing it to solutions of relatively small number of simultaneous linear equations. Roots of unity play a crucial role. Thus for diagonalizing the 81 dimensional space for N=3, r=4, one has to solve a maximal set of five linear equations. A supplementary symmetry relates invariant subspaces pairwise [W=(r,Nr) and so on] so that only one of each pair needs study. The case N=3 is studied fully for r=(1,2,3,4). Basic aspects for all (N,r) are discussed. Full exploitation of such symmetries lead to a formalism quite different from, possibly generalized, algebraic Bethe ansatz. Chain Hamiltonians are studied. The specific types of spin flips they induce and propagate are pointed out. The inverse Cayley transform of the YB matrix giving the potential leading to factorizable S matrix is constructed explicitly for N=3 as also the full set of R̂tt relations. Perspectives are discussed in a final section.

Design of composite helicopter rotor blades to meet given cross-sectional properties

AbstractThis paper examines the design of a composite helicopter rotor blade to meet given cross-sectional properties. As with many real-world problems, the choice of objective and design variables can lead to a problem with a non-linear and/or non-convex objective function, which would require the use of stochastic optimisation methods to find an optimum. Since the objective function is evaluated from the results of a finite element analysis of the cross section, the computational expense of using stochastic methods would be prohibitive. It is shown that by choosing appropriate simplified design variables, the problem becomes convex with respect to those design variables. This allows deterministic optimisation methods to be used, which is considerably more computationally efficient than stochastic methods. It is also shown that the design variables can be chosen such that the response of each individual cross-sectional property can be closely modelled by a linear approximation, even though the response of a single objective function to many design parameters is non-linear. The design problem may therefore be reformulated into a number of simultaneous linear equations that are easily solved by matrix methods, thus allowing an optimum to be located with the minimum number of computationally expensive finite element analyses.

A computationally efficient technique for designing frequency sampling filters

In a recent paper, a technique for designing linear phase frequency sampling filters was proposed that approximates a desired frequency response by minimizing the mean square error over the stopbands subject to constraints on the filters amplitude response. This technique results in a large number of simultaneous linear equations the solution of which determines the filter's impulse response. The filter's frequency samples which are used to implement the filter are then determined by computing the discrete Fourier transform of this impulse response. In this brief, a modification of this technique is developed. This modified technique also approximates a desired frequency response by minimizing the mean square error over the stopbands subject to constraints on the filter's amplitude response. Additionally, however, it allows passbands to be approximated by a weighted mean square error. This modified technique results in a set of simultaneous linear equations, the solution of which directly determines the filter's nonzero frequency samples. Because the number of nonzero frequency samples is typically much less than the number of impulse response elements, this technique requires a significantly smaller number of simultaneous linear equations than the other technique.

Finding all solutions of piecewise-linear resistive circuits using simple sign tests

This paper presents an efficient algorithm for finding all solutions of piecewise-linear resistive circuits. The algorithm uses two types of sign tests; one is a new test that is proposed in this paper, and the other is the test proposed by Yamamura and Ochiai (1992). The computational complexity of the new test is much smaller than that of Yamamura and Ochiai's test. These tests eliminate many linear regions that do not contain a solution. Therefore, the number of simultaneous linear equations to be solved is substantially reduced. The proposed algorithm is very simple and efficient. >

Accurate and rapid thermal regenerator calculations

Means are described whereby the speed of calculation of the performance of thermal regenerators can be considerably increased using the method of Hill and Willmott ( Int. J. Heat Mass Transfer 30, 241–249 (1987) which was developed from the method of Razelos ( Wärme- und Stoffübertr. 12, 59–71 (1979). In particular the number of simultaneous linear equations which must be solved for the non-symmetric case is halved. Advantage is taken of Richardson's extrapolation method.

Computer aspects of three-dimensional finite element analysis of stresses and strains in the intact heart

Three main computing aspects encountered in the stress and strain analysis of the intact heart using the finite element technique, namely, the automatic partitioning of the myocardium of the heart into finite elements, the assemblage of the structural stiffness matrix of the myocardium, and the storage and retrieval of its nonzero coefficients are described. The influence of boundary conditions and different relaxation factors on the speed of convergence to the final solution are also described. Programming for the solution of the large number of simultaneous linear equations generated by the finite element method on computers with capacity not exceeding 32K words of memory required special attention to very compact storage of the stiffness matrix and the retrieval of its coefficients. Computation time for the three-dimensional stress and strain analysis of a heart represented by approximately 7000 finite elements was less than 20–25 min on a CDC 3500 computer operated in the multibatch MASTER mode.

Steady-state transmission through networks containing periodically operated switches

THE GENERAL problem of transmitting signals through linear systems in which one or more parameters vary periodically with time has an extensive literature. One widely used method is based on Fourier series representation of the varying parameters. This leads to an infinite number of simultaneous linear equations expressing the relations between the coefficients in the corresponding Fourier series representation of the steady-state response. The solution of the equations can be expressed in terms of determinants of infinite order which in turn can be evaluated by various approximation techniques. In practical cases it is often permissible to neglect all but a few dominant components; the number of equations is thereby made finite and reasonably small.

ON THE EXTENDED USE OF KLEINLOGEL'S RAHMENFORMELN

This paper contains but a single idea. Its purpose is to show how, by the use of Kleinlogel's Rahmenformeln, the number of simultaneous linear equations that occur in the analysis of indeterminate structures can be reduced materially by using a primary structure, itself made up of indeterminate parts, provided the bending moment diagrams can be drawn for them. These are obtained from the Rahmenformeln. In order to recall to the reader's mind the well known moment-area method a brief summary of that process is included. A practical numerical example of the analysis of a frame indeterminate to the seventh degree is worked out in detail. Two arrangements of the indeterminate-components are chosen in the illustrative example. The final bending moment diagram is shown for a simple loading case and the influence line for the bending moment at one point is discussed briefly.

A Method for the Stress Analysis of Helicopter Blades

Abstract : The method of analysis presented herein is applicable to both fixed and hinged rotors. Although the equations developed apply directly to rectangular blades, the method may be extended to cover blades having other shapes. Up to this time, an exact general solution has been found for even the simplest form of the differential equation governing rotor bending moments, although solutions can be determined for certain combinations of constants (these solutions are not general). Because of the lack of a general exact solution, various approximate methods such as series, perturbation methods, etc., were tried, but convergence was poor in most cases, necessitating the use of an unwieldy number of terms. The solution presented in this paper is based on the method of finite differences and depends on teh evaluation of a number of simultaneous linear equations. The method leads to a rapid determination of the bending moment at any point on the blade.

On the Solution of the Numerical Simultaneous Equations Arising in the Analysis of Redundant Structures

The application of strain energy or slope-deflection methods in the analysis of redundant structures leads to a number of simultaneous linear equations with numerical coefficients; the equations may be obtained in such order that each successive equation contains one new unknown, until all the unknowns are so included. This is the only condition essential for the method to be described in the present paper, but the labour is much reduced in slope-deflection and strain energy applications by the fact that most (or all) of the equations contain very few of the unknowns. The method to be given reduces the solving of these equations to a column of successive evaluations, followed by the solution, by algebraic methods, of a small number of simultaneous equations; and a final column of evaluations. In the remaining paragraphs a number of problems are examined to show how the equations may be obtained in suitable sequence for the method to apply. Following an application to the determination of secondary stresses, the operations involved in the moment-distribution method and in this method are compared. A numerical example is worked out in the simple case of §2, and it is shown how any order of mathematical accuracy in the roots may be ensured, provided that sufficient figures have been retained to permit that accuracy.