Direct solution of large systems of linear equations

A very efficient computer subroutine for the direct solution of large numbers of simultaneous linear equations is presented. Basically the program uses Gauss elimination on positive-definite symmetrical systems. The specific features are that systems of very large size and bandwidth can be solved and that all operations on zero elements are eliminated. Also, the program is very simple and can be incorporated into existing programs with a minimum of effort. The amount of backup storage available on the computer used will govern the maximum size of the system which can be solved. A FORTRAN IV listing of the subroutine is given.     

Analysis for P−Δ effects in seismic response of buildings

A matrix formulation to account for P?Δ effects in computer seismic analysis of multistory buildings is presented. The method employs a linear solution approach requiring no iteration and can be used for static or dynamic elastic analyses. Amplified P?Δ effects resulting from inelastic displacement levels which may occur during a major earthquake can be accounted for in an approximate manner. The method has been implemented in a computer program and sample seismic analyses of a 31-story building are presented. Observations on the appropriate use and interpretation of analyses are made.     

A natural solution to the modal equations

The direct solution for all of the mode shapes of undamped, linear elastic systems in free vibration is accomplished by means of a converging iteration scheme applied to the modal equations. Previous versions of the digital computer program used in this solution find successive mode shapes with a decreasing accuracy. The present version employs a natural solution, so that every mode shape is found with the same accuracy.     

Solving large elliptic difference equations on CYBER 205

Preconditioned conjugate gradient methods are considered for the solution of large elliptic difference equations on the CYBER 205 computer. An approximate polynomial preconditioning procedure is introduced for systems of equations derived from high-order numerical approximations to elliptic partial differential equations. Computational results for high-order versus low-order methods are presented, and the comparison of the conjugate gradient method used in conjunction with various preconditioning shows the new preconditioning strategy to be very effective for solving algebraic equations that arise from high-order discretization techniques.     

The analysis of nonlinear cable net systems and their supporting structures

The analysis of large cable net systems is feasible only with the aid of high speed digital computers. However, even computerized design of cable nets requires careful formulation and the selection of efficient solution techniques. This paper describes the solution of a general integrated structural system which includes three-dimensional beam members and cable elements that is useful to the design engineer. This solution is used to analyze an example prestressed cable structure, and results are presented to demonstrate the efficiency and accuracy of different solution techniques and to illustrate the effect of variation of important parameters.NET 3 is the general solution program. Input includes: (1) cable sizes, weights, and stress-strain characteristics; (2) initial coordinate points and cable forces due to prestressing; (3) support conditions for the cable net; (4) applied loads; and (5) data for supporting structure. The initial shape and cable forces are computed with the aid of an auxilliary program, SHAPE, given the coordinates of the cable end points, the initial prestressing forces, and an initial interior coordinate. The cable stress-strain characteristics may be linear or nonlinear. The supporting structure may be replaced with equivalent spring stiffnesses or actual three-dimensional elements.The assumptions used in the solution are: (1) cable elements are straight between nodal points and have no flexural stiffnesses; (2) the roofing and decking provide no stiffness; (3) all loads are applied at the nodal points; and (4) all supporting structural elements are elastic. It should be noted that no displacements of the cable nodal points are neglected. Even at boundaries, the elastic stiffness can be considered. Temperature changes can be analyzed to determine their effect upon the structure.Cable net structures usually have many elements and require relatively large amounts of computer time. In addition many loading and geometric conditions must be considered. To be useful as a practical design tool, it was essential that the program be as efficient as possible and require a minimum of storage. The solution utilizes a variation of the Newton-Raphson method. Initially the tangent-modulus stiffness is used. At each iteration the change in the element force is compared with the change in element force for the previous iteration to determine if the convergence rate is within a prescribed value. If it is acceptable, the stiffness for the previous iteration is used; if unacceptable for any element, tangent stiffnesses for all elements are recalculated. This technique attempts to combine the best features of the Newton-Raphson and the modified Newton-Raphson methods by increasing convergence probability and decreasing solution time.To minimize storage requirements, NET 3 has the capability of utilizing tapes in the problem solution. The storage requirements are determined internally before the solution is initiated. If the required storage exceeds that alloted for the system of equations, the solution is effected through the use of tapes.Output for the cables includes nodal point displacements, nodal point forces, and element forces; for beam elements, forces and moments are given. The solution has been verified by comparison with results by other investigators.An elliptical shaped structure, 220 by 240 ft, has been analyzed with the program to investigate the influence of several parameters. The short direction cables were prestressed, and a uniform load of 40 psf was assumed. The variables investigated include: (1) initial prestress force; (2) degree of stiffness of supporting structure; (3) cable sizes; (4) sag-span ratio; and (5) temperature change.     

一种基于SMP的并行逐次超松弛迭代法

逐次超松弛迭代方法被广泛应用于油藏数值模拟中压力方程的求解.其并行实现是提高模拟速度的重要途径.传统并行方案大都只是在一次迭代内进行数据划分,而没有进一步将数据划分与迭代空间划分相结合,故针对SOR算法和SMP（symmetric multi-processors）系统的特点,以OpenMP为并行化实现工具,提出了基于SMP的并行逐次超松弛迭代方法（parallelSOR）.方法通过改变不同迭代步内数据点的更新次序,使不同区域内的数据点可以并行执行多次迭代.总结出针对三维油藏区域在数据空间划分和迭代空间合并上相对较优的策略,分析了迭代过程中网格块的生长形状.与传统的并行策略相比,该方法具有可减小同步开销、改进数据局部性、cache命中率高等优点.实验结果表明,该方法具有较高的加速比和效率.     

Substructuring and equation system solutions in finite element analysis

Dealing with large structural systems it is convenient and often almost necessary to employ a substructure technique in order not to exceed the available computer capacity. Efficient solution of systems of equations is very essential in finite element analysis. In this paper, first, a procedure is described as to how to systematically reduce the number of assembled equations on different substructure levels. The size of a system of equations to be solved at a time is reduced by condensation of different types of unknowns at the element level. Second, a generalized Gaussian elimination scheme is presented for the solution of systems of equations in substructuring. The same scheme may be very useful in solutions of mixed finite element models, equation systems with Lagrangian multipliers and moderately ill-conditioned problems. The system properties and the initially neatly banded forms of system equations are preserved during the solution process. Lastly, three examples illustrate solutions of some complex structural problems. The method presented is applicable to a wide range of physical problems and can achieve a substantial computer economy.     

Theoretical and experimental study of the U-Pu-Zn system

A method for analyzing binary phase diagrams developed by L. Brewer for a desk top calculator is applied to determining lattice stability and excess free energy parameters for the U-Pu, Pu-Zn, and U-Zn binary systems. These data were used to calculate a ternary phase diagram. Several experimental U-Pu-Zn melts were prepared. The results of the computer generated phase diagrams prepared with the calculator generated parameters and with regular solution theory constants are compared with each other and with the experimental data. Although the regular solution theory constants gave better agreement, we believe that the desk top calculator program provides a valuable method for analysis of binary phase systems. The computer generated phase diagram was very valuable in guiding experimentation and in interpreting the results.     

Solving large linear algebraic systems in the context of integrable non-abelian Laurent ODEs

The paper reports on a computer algebra program LSSS (Linear Selective Systems Solver) for solving linear algebraic systems with rational coefficients. The program is especially efficient for very large sparse systems that have a solution in which many variables take the value zero. The program is applied to the symmetry investigation of a non-abelian Laurent ODE introduced recently by M. Kontsevich. The computed symmetries confirmed that a Lax pair found for this system earlier generates all first integrals of degree at least up to 14.     

病态线性系统的控制模型设计及其迭代算法

建立一种解决病态线性系统的控制模型，得到求解病态线性系统近似解的一种新算法．该算法为病态线性系统利用计算机迭代求解提供了直接的算法依据，同时也适合于常态线性系统近似解的计算．最后将新算法与传统的Gauss—Seidel算法进行比较．     

An algorithm for nonlinear system parameter identification

An algorithm for system parameter identification will be presented in this paper. The method is applicable to linear and nonlinear systems with known structures. It is applied in this paper to systems in which both system and measurement noises can be neglected. The algorithm requires less time per iteration and less computer storage than the quasilinearization method. A shaft position control system with multiple nonlinearities will be used to illustrate the method.     

Parallel analysis of rotationally periodic structures

A parallel finite element solution algorithm for analysing large rotationally periodic structures on MIMD parallel computer systems is described. For a rotationally periodic structure, the global stiffness matrix under the corresponding symmetric coordinate system is periodic, i.e. possesses isomorphic properties, so that the global equation system can be transformed into a number of smaller equation systems which are fully decoupled. These decoupled equation systems then can be solved simultaneously on a multiprocessor parallel computer. The algorithm also generates the decoupled equation systems in parallel, without explicitly assembling the global stiffness matrix of the structure. A prototype implementation of the algorithm on an array of transputers is presented, and the efficiency of the program is also studied in this paper. Finally, a numerical example is given to demonstrate the speedup of the program.     

Object-oriented programming of distributed iterative equation solvers

An object-oriented approach is used to develop classes and frameworks for the implementation of distributed iterative equation solution. The software is implemented using the .NET framework, and builds upon previous work by the author. Development of the framework for iterative solution makes good use of interfaces to isolate sources of complexity. The framework is used for three different solution scenarios (i) conjugate gradient iteration on a single matrix; (ii) conjugate gradient iteration when domain decomposition is used; and (iii) using the Schur complement approach. Moreover, the framework is used for both local and remote objects. The .NET framework makes it very straightforward to program distributed applications, and the object-oriented approach greatly facilitates the software development. The framework was used in a finite element program and the speed-up results are shown.     

Two-Time Scale Controlled Markov Chains: A Decomposition and Parallel Processing Approach

This correspondence deals with a class of ergodic control problems for systems described by Markov chains with strong and weak interactions. These systems are composed of a set of subchains that are weakly coupled. Using results already available in the literature one formulates a limit control problem the solution of which can be obtained via an associated nondifferentiable convex programming (NDCP) problem. The technique used to solve the NDCP problem is the Analytic Center Cutting Plane Method (ACCPM) which implements a dialogue between, on one hand, a master program computing the analytical center of a localization set containing the solution and, on the other hand, an oracle proposing cutting planes that reduce the size of the localization set at each main iteration. The interesting aspect of this implementation comes from two characteristics: (i) the oracle proposes cutting planes by solving reduced sized Markov Decision Problems (MDP) via a linear program (LP) or a policy iteration method; (ii) several cutting planes can be proposed simultaneously through a parallel implementation on processors. The correspondence concentrates on these two aspects and shows, on a large scale MDP obtained from the numerical approximation ldquoa la Kushner-Dupuisrdquo of a singularly perturbed hybrid stochastic control problem, the important computational speed-up obtained.     

关于线性时变离散系统的稳定性

本文研究线性时变离散系统的稳定性，采用一种解的估计技巧，简化了［１］用Ｇａｕｓｓ－Ｓｅｉｄｅｌ迭代法建立的稳定性判据的证明，并获得一些新的代数判据。     

An algorithm for solution of large eigenvalue problems

A method for calculating natural frequencies and mode shapes of large structural systems using conjugate gradient (CG) algorithm with element-by-element (EBE) preconditioning and the basic subspace iteration is proposed. In this method, the entire analysis can be carried out elementwise without the need to construct and triangularize the structural stiffness matrix. Thus very large systems can be solved with a minimum amount of incore memory and the task of running a separate processor (program) for bandwidth minimization becomes unnecessary. Potential for further improvement on the proposed scheme by using computers with parallel processing capability is also discussed. Finally, examples are given to show the improved performance of EBE preconditioning as compared to the traditional CG algorithm.     

遗传算法在可靠性约束的光网络带宽分配中的应用

光网络中广泛使用OSPF作为路由协议,这使得路由选择智能化.然而,如何根据既定的网络拓扑和业务量规划各链路带宽及其可靠性成为一个难题,在综合考虑网络成本和可靠性的基础上,给出了两种链路带宽算法,即迭代法和遗传算法.在设计带宽遗传算法时,提出将带宽分解为两个指标(相对尺度和绝对尺度),从而解决了编码经各种遗传运算仍为可行解的问题.从实验结果看出,带宽遗传算法能很好地逼近最优解,结果令人满意.     

Transient analysis of flexible multi-body systems. Part II: Application to aircraft landing

This investigation presents a method for transient analysis of a large-scale multi-body aircraft consisting of interconnected rigid and flexible bodies that undergo large angular rotations. Elastic components of the aircraft are discretized using the finite element method. The system equations of motion and nonlinear algebraic constraint equations describing joints between different components are written in the Lagrangian formulation using a finite set of coupled reference and modal coordinates. The system differential equations of motion and algebraic constraint equations are computer-generated and integrated forward in time using an explicit-implicit direct numerical integration algorithm coupled with a Newton-Raphson type iteration in order to check on constraint violations. Impact and intermittent motion events are accounted for by using a generalized momentum balance that predict jump discontinuities in the generalized velocities as well as jump discontinuities in the system reaction forces. The formulation presented and the computer program developed are used to simulate the impact between the landing gear and the runway. The method is also used to predict the dynamic behavior of the aircraft during the traverse of an abrupt elevation change in the runway.     

Solution of finite element systems on concurrent processing computers

A new computer program architecture for the solution of finite element systems using concurrent processing is presented. The basic approach involves the automatic creation of substructures. A host provides control over a set of processors, each of which is assigned initially to one substructure, then dynamically reassigned to the common interface for the solution of the complete system of substructures. Algorithm details are presented fo each phase of the analysis.Results of analysis of large plate bending problems on a hypercube multicomputer are reported. For a system with 2,000 equations, an efficiency of 80 percent of the maximum theoretical value was obtained using 16 processors.     

An automated,energy-based,finite-difference procedure for the elastic collapse of rectangular plates and panels

A FORTRAN IV, large capacity, computer program has been developed to determine collapse loads and bifurcation loads for linear and nonlinear prebuckling behavior for fiber-reinforced, laminated, rectangular plates and panels under general loading systems and boundary conditions. The program is based on the principle of total potential energy and uses finite-differences in the discretization process. Whole-station spacing has been used to calculate the strain energy associated with an area-element and an orthogonal finite-difference grid that provides for variable spacings in perpendicular directions is incorporated.Numerical results are presented that compare favorably with results obtained via the general computer program STAGS. Other numerical results are presented that illustrate the types of boundary conditions, applied loads, cut-outs and initial geometric imperfections that can be handled by the present program. A brief study of the effect of panel construction and initial geometric imperfections on the buckling behavior of fiber-reinforced panels is presented.