Elastic stability and buckling loads of three dimensional nonuniform framed systems by finite element method using spatial line elements

A numerical computer method using spatial finite line elements for the determination of buckling loads of three-dimensional framed systems having complex boundary conditions is presented. A joint at end of an element experiences three rotational and three linear displacements. An element experiences axial, torsional and flexural deformations in two orthogonal planes. Buckling loads and corresponding mode vectors are determined by the solution of linear set of eigen value equations of elastic stability, where the elastic stability matrix is the product of three-dimensional sidesway flexibility matrix and the second-order sidesway stiffness matrix. Four numerical examples are included—a column, planar frame, four-legged system and three-legged triangular frame; having mixture of fixed, spherical, revolute, cylindrical and planar pairs at supports and joints.     

Helmholtz方程的楔形基区域分解法

基于楔形基函数和无网格配点法,提出了一种求解Helmholtz型方程区域分解法。该方法克服了在求解大规模问题时用一般的全域配点法所带来的配置矩阵为非对称满阵,且高度病态的问题。通过数值结果表明,该算法在求解Helmholtz型方程降低系数矩阵条件数的同时,也能够降低误差,并达到满意的收敛效果。     

A co-tree flows formulation for steady state in water distribution networks

This article describes a co-tree flows formulation for steady state simulation in water distribution networks, which reduces the original governing system of equations into a smaller set, expressed in terms of the co-tree chord flows. The formulation is derived from graph theory and matrix partitioning. The reduction in the size of the set of equations does not require any new conditions on the initial solution estimate, unlike the loop flow correction method. The proposed formulation is numerically equivalent to the link flow method. However, it requires, about 5% for the Jacobian memory storage requirements of the link flow method, thereby also drastically reducing the time of execution for solving the resulting nonlinear system, as well. Furthermore, the Jacobian matrix of the new method is symmetrical, which can reduce the memory storage by half. Thus, even for large distribution systems, there is no need for sparse matrix solvers, which trade off the storage memory with time of execution in order to manage the data requirements.     

Adaptive random sample consensus approach for segmentation of building roof in airborne laser scanning point cloud

ABSTRACT

This work proposes a three-step method for segmenting the roof planes of buildings in Airborne Laser Scanning (ALS) data. The first step aims at mainly avoiding the exhaustive search for planar roof faces throughout the ALS point cloud. Standard algorithms for processing ALS point cloud are used to isolate building regions. The second step of the proposed method consists in segmenting roof planes within building regions previously delimited. We use the RANdom SAmple Consensus (RANSAC) algorithm to detect roof plane points, taking into account two adaptive parameters for checking the consistency of ALS building points with the candidate planes: the distance between ALS building points and candidate planes; and the angle between the gradient vectors at ALS building points and the candidate planes’ normal vector. Each ALS building point is classified as consistent if computed parameters are below corresponding thresholds, which are automatically determined by thresholding histograms constructed for both parameters. As the RANSAC algorithm can generate fragmented results, in the third step, a post-processing is accomplished to merge planes that are approximately collinear and spatially close. The results show that the proposed method works properly. However, failures occur mainly in regions affected by local anomalies such as trees and antennas. Average rates around 90% and higher than 95% have been obtained for the completeness and correction quality parameters, respectively.     

Precise Hsu’s method for analyzing the stability of periodic solutions of multi-degrees-of-freedom systems with cubic nonlinearity

This paper presents a new precise Hsu’s method for investigating the stability regions of the periodic motions of an undamped two-degrees-of-freedom system with cubic nonlinearity. Firstly, the incremental harmonic balance (IHB) method is used to obtain the solution of nonlinear vibration differential equations. Hsu’s method is then adopted for computing the transition matrix at the end of one period, and the precise time integration algorithm is adjusted to improve the computational precision. The stability regions of the system obtained from the precise Hsu’s, Hsu’s and improved numerical integration methods are compared and discussed.     

Frequency Interval Cross Gramians for Linear and Bilinear Systems

In many control engineering problems, it is desired to analyze the systems at particular frequency intervals of interest. This paper focuses on the development of frequency interval cross gramians for both linear and bilinear systems. New generalized Sylvester equations for calculating the frequency interval cross gramians are derived in order to be used to obtain information regarding controllability and observability within a single matrix. The advantage of the proposed method is that it is computationally more efficient compared to existing gramian‐based techniques since only half of the number of equations need to be solved in order to obtain information regarding the controllability and observability of a system compared to existing techniques. Numerical examples are provided to demonstrate the computational efficiency of the proposed method which uses frequency interval cross gramians relative to existing methods.     

Dynamic stability of a rectangular plate on non-homogeneous winkler foundation

The dynamic stability of a rectangular plate on non-homogeneous foundation, subjected to uniform compressive in-plane bi-axial dynamic loads and supported on completely elastically restrained boundaries is studied. The non-homogeneous foundation consists of two regions having different stiffnesses but symmetric about the centre lines of the plate. The equation governing the small amplitude motion of the system is derived by a variational method. The use of Galerkin method with reduced beam eigenfunctions transforms the system equations in matrix form. The system of coupled Mathieu-Hill equations thus obtained, are analysed by the method of multiple scales which yields the stability boundaries for different combinations of the excitation amplitude and frequency. The effects of stiffness and geometry of the foundation, boundary conditions, static load factor, in-plane load ratio and aspect ratio on the stability boundaries of the plate for first- and second-order simple and combination resonances are studied.     

Measurement of side walls of high aspect ratio microstructures

In this paper, the new method of measurements is presented for the technological control of the lithographic fabrication of high aspect ratio microstructures with complex side-wall profiles. The sampling procedure includes pouring polymer glue on a substrate with the microstructures and its hardening, sectioning the polymer matrix in the depth with planes parallel to the substrate and analysis of digital optical microscope images of the planes. The measured deviations the micro structure profiles in comparison with the designed geometry are used for the further optimization of the technological design of planar X-ray refractive lenses.     

The solution of linear,constant-coefficient,ordinary differential equations with APL

The matrix exponential technique is a method for solving linear, simultaneous constant-coefficient, differential equations. The method does not require matrix inversions, but rather the summing of a truncated exponential series. The matrix form of this method makes it particularly well-suited to the matrix-handling methods of APL. The essential portion of the APL program contains only fourteen APL statements; thus it can be readily programmed. The combination of matrix exponential method and the APL language provides a fast and accurate interactive means for solving this class of ordinary differential equations.     

Towards algebraic multigrid for elliptic problems of second order

An algebraic multigrid method is developed which can be used as a preconditioner for the solution of linear systems of equations with postitive definite matrices. The method is directed to equations which arise from the discretization of elliptic equations of second order, but only the matrix is the source for the information used by the algorithm. One has only to know whether the matrix stems from a 2-dimensional or 3-dimensional problem and whether the elliptic equations are scalar equations or belong to a system.     

大型复线性方程组预处理双共轭梯度法

当复线性方程组的规模较大或系数矩阵的条件数很大时,系数矩阵易呈现病态特性,双共轭梯度法存在不收敛和收敛速度慢的潜在问题,采用适当的预处理技术,可以改善矩阵病态特性,加快收敛速度。从实型不完全Cholesky分解预处理方法出发,构造了一种针对复线性方程组的预处理方法,结合双共轭梯度法,给出了一种预处理双共轭梯度法。数值算例表明该算法求解速度快,可靠高效,能够应用于大型复线性方程组的求解。     

Solutions of integral equations via Taylor series

The Taylor operational matrix of integration and the Taylor product operational matrix are introduced. These two operational matrices are applied to approximation of solutions of Fredholm and Volterra integral equations. The proposed method reduces solution of integral equations to the successive solution of a set of linear algebraic equations in matrix form. Owing to the simplicity of the operational matrix of integration, and the product operational matrix of the Taylor series, the algorithms derived possess considerable computational advantages over the orthogonal-polynomial approximation, provided that both input and output are analytic functions of t.     

Polynomial solutions of certain differential equations

In this paper, the Chebyshev matrix method is applied generalisations of the Hermite, Laguerre, Legendre and Chebyshev differential equations which have polynomial solution. The method is based on taking the truncated Chebyshev series expansions of the functions in equation, and then substituting their matrix forms into the result equation. Thereby the given equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Chebyshev coefficients.     

CAVE系统中不规则多屏投影变换矩阵

常见的CAVE系统是由4个正方形投影面构成一个立体空间,将复杂的数据以可视化的方式展现,营造一个逼真的虚拟环境,带给用户一种非凡的沉浸式体验。本文在分析投影变换矩阵的基础上,通过构建CAVE投影坐标系,提出构建不规则投影变换矩阵的快速计算方法。该方法使CAVE系统不再局限于标准的立方体空间,并通过获取用户在CAVE投影坐标系下的坐标位置,实时计算投影变换矩阵,满足用户在任意空间位置下的最佳观测视角。     

Numerical Solutions of Fractional Differential Equations by Using Fractional Taylor Basis

In this paper, a new numerical method for solving fractional differential equations (FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional integration for the fractional Taylor basis is introduced. This matrix is then utilized to reduce the solution of the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of this technique.      

Nonlinear dynamic analysis of quasi-symmetric anisotropic structures

An efficient computational strategy is presented for the nonlinear dynamic analysis of quasi-symmetric anisotropic structures. A mixed formulation is used with the fundamental unknowns consisting of stress resultants, generalized displacements and velocity components. The governing semi-discrete finite element equations consist of a mixed system of algebraic and ordinary differential equations. The temporal integration of the differential equations is performed by using an explicit half-station central difference method. The three key elements of the strategy are:

• 1.(a) use of mixed finite element models with independent shape functions for the stress resultants, generalized displacements and velocity components and with the stress resultants allowed to be discontinuous at interelement boundaries
• 2.(b) operator splitting, or additive decomposition of the different arrays in the governing equations into the contributions to a symmetrized response plus correction terms
• 3.(c) application of a preconditioned conjugate gradient technique to generate the unsymmetric response of the structure, at each time step, as the sum of symmetric and antisymmetric modes, each obtained using approximately half the degrees of freedom of the original model. The preconditioning matrix is taken to be the matrix associated with the symmetrized response.The effectiveness of the proposed strategy is demonstrated by means of a numerical example and the potential of the proposed strategy for solving more complex nonlinear problems is discussed.

     

An efficient return algorithm for non-associated plasticity with linear yield criteria in principal stress space

An efficient return algorithm for stress update in numerical plasticity computations is presented. The yield criterion must be linear in principal stress space and can be composed of any number of yield planes. Each of these yield planes may have an associated or non-associated flow rule. The stress return and the formation of the constitutive matrix is carried out in principal stress space. Here the manipulations simplify and rely on geometrical arguments. The singularities arising at the intersection of yield planes are dealt with in a straightforward way also based on geometrical considerations. The method is exemplified on non-associated Mohr–Coulomb plasticity throughout the paper.     

Solution of differential equations via rationalized Haar functions

Rationalized Haar functions are developed to approximate the solution of the differential equations. Properties of rationalized Haar functions are first presented, the operational matrix of integration together with product operational matrix are utilized to reduce the computation of differential equations into algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples.     

Some symbolic computation algorithms in cosmic dynamics problems

An algorithm for computing fundamental solutions to a linear system of differential equations with a periodic matrix represented by a power series in terms of a small parameter is discussed. An algorithm based on the infinite determinant method for determining boundaries between regions of stability and instability for such a system in the parameter space is also discussed. The corresponding procedures implemented in the Mathematica system and computation results related to the elliptic restricted five-body problem are presented.     

Numerical solution of an integral equations system of the first kind by using an operational matrix with block pulse functions

This article proposes a simple efficient method for solving a Volterra integral equations system of the first kind. By using block pulse functions and their operational matrix of integration, a first kind integral equations system can be reduced to a linear system of algebraic equations. The coefficient matrix of this system is a block matrix with lower triangular blocks. Numerical examples show that the approximate solutions have a good degree of accuracy.