An efficient analysis for thin plates of general quadrilateral shape subject to bending stresses

Isotropic quadrilateral plates subject to bending are analysed by the Rayleigh-Ritz method. It is shown that, by defining the plate in a warped coordinate system, suitable global functions for the plate displacements which fit most boundary conditions of practical significance can be readily derived. The performance of the method is assessed by comparing the results with those obtained from different theoretical solutions and/or experiment. Good correlation is obtained between all sets of results, whilst those due to the proposed method indicate considerable advantages over the alternative solutions, both in terms of computational efficiency and the ease with which the requirements for an acceptably accurate solution can be pre-determined.     

Application of isogeometric method to free vibration of Reissner–Mindlin plates with non-conforming multi-patch

The isogeometric method is used to study the free vibration of thick plates based on Mindlin theory. The Non-uniform Rational B-Spline (NURBS) basis functions are employed to build the thick plate’s geometry models and serve as the shape functions for solution field approximation in finite element analysis. The Reissner–Mindlin plates built with multiple NURBS patches are investigated, in which several patches of the model have multi-interface and different patches may share a common point. In order to solve the non-conforming interface problems, Nitsche method is employed to glue different NURBS patches and only refers to the coupling conditions in this work. Various plate shapes, different boundary conditions and several kinds of thickness-span ratios are considered to verify the validity of the presented method. The dimensionless frequencies for different cases are obtained by solving the eigenvalue equation problems and compared with the existing reference solutions or the results calculated by ABAQUS software. Several numerical examples exhibit the effectiveness of the isogeometric approach. It shows that the natural frequencies of the Reissner–Mindlin plate can be successfully predicted by the combination of isogeometric analysis and Nitsche method.     

Exact solutions in the reinforcement of a circular plate under concentrated loads

A thin elastic plate, supported along the boundary, is loaded either by a single concentrated force at a given internal point, or by two equal and opposite concentrated forces at given internal diametrically opposite points. In each problem, the compliance, defined as the work done by the load, can be reduced on stiffening the plate by the addition to the boundary of an elastic strip whose cross-section is variable but whose total volume is prescribed. The cross-sectional area that minimises the compliance provides the optimal reinforcement for both problems. The respective compliances are compared with those obtained when the plate’s boundary is reinforced by a strip of uniform cross-section and of the given volume.     

Geometrically nonlinear analysis of a Reissner type plate by the boundary element method

The boundary element method is used in the geometrically nonlinear analysis of laterally loaded isotropic plates taking into account the effects of transverse shear deformation. This paper presents the general equations for finite deformation of a Reissner type plate, and also gives an integral formulation of the Von Kármán type nonlinear governing equations which involve coupling between in-plane and out-of-plane deformation. The boundary and the domain of the plate are discretized to solve nonlinear plate bending problems. All unknown variables are at the boundary. An iterative procedure is applied to achieve linearization of the nonlinear equations. Some numerical results of the computation are compared with the analytical solutions and other numerical techniques, and good agreement is obtained.     

Application of boundary-domain element method to the free vibration problem of plate structures

The boundary-domain element method is applied to the free vibration problem of thin-walled plate structures. The static fundamental solutions are used for the derivation of the integral equations for both in-plane and out-of-plane motions. All the integral equations to be implemented are regularized up to an integrable order and then discretized by means of the boundary-domain element method. The entire system of equations for the plate structures composed of thin elastic plates is obtained by assembling the equations for each plate component satisfying the equilibrium and compatibility conditions on the connected edge as well as the boundary conditions. The algebraic eigenvalue equation is derived from this system of equations and is able to be solved by using the standard solver to obtain eigenfrequencies and eigenmodes. Numerical analysis is carried out for a few example problems and the computational aspects are discussed.     

Solution of Polynomial Systems Derived from Differential Equations

Nonlinear two-point boundary value problems arise in numerous areas of application. The existence and number of solutions for various cases has been studied from a theoretical standpoint. These results generally rely upon growth conditions of the nonlinearity. However, in general, one cannot forecast how many solutions a boundary value problem may possess or even determine the existence of a solution. In recent years numerical continuation methods have been developed which permit the numerical approximation of all complex solutions of systems of polynomial equations. In this paper, numerical continuation methods are adapted to numerically calculate the solutions of finite difference discretizations of nonlinear two-point boundary value problems. The approach taken here is to perform a homotopy deformation to successively refine discretizations. In this way additional new solutions on finer meshes are obtained from solutions on coarser meshes. The complicating issue which the complex polynomial system setting introduces is that the number of solutions grows with the number of mesh points of the discretization. To counter this, the use of filters to limit the number of paths to be followed at each stage is considered.     

Application of the extended Kantorovich method to the bending of variable thickness plates

The governing equations of the classical plate theory for a uniform or a unidirectional variable thickness rectangular plate under transverse applied loading are solved by means of the extended Kantorovich method. The plate may be either simply supported or clamped along the edges. The solution procedure is iterative and must be carried out numerically. This necessitates the calculation of the two missing pieces of boundary data along the edges of the plate. The missing boundary data are determined utilizing the method of adjoints of the shooting method. The numerical values of the deflection and bending moments for uniform and variable thickness plates are compared with those from the exact solutions and finite element analysis, respectively.     

The optimal solution for the flow of a fourth-grade fluid with partial slip

The steady flow of a non-Newtonian fluid when slippage between the plate and the fluid occurs is considered. The constitutive equations of the fluid are modeled for a fourth-grade non-Newtonian fluid with partial slip; they give rise to nonlinear boundary value problems. Analytical solutions are obtained using powerful analytic techniques for solving nonlinear problems, homotopy perturbation and optimal homotopy asymptotic methods. The results obtained are compared with the numerical results and it is shown that solutions exist for all values of the non-Newtonian parameters. The solutions valid for the no-slip condition for all values of the non-Newtonian parameters can be derived as special cases of the present analysis. Finally the solutions are discussed using a graphical approach.     

A boundary element analysis of symmetric laminated composite shallow shells

This paper presents a boundary element formulation for the analysis of symmetric laminated composite shallow shells where only the boundary is discretized. Classical plate bending and plane elasticity formulations are coupled and effects of curvature are treated as body forces. Fundamental solutions for elastostatic formulations are used and body forces are written as a sum of approximation functions multiplied by unknown coefficients. Two approximation functions are used. Domain integrals which arise in the formulation are transformed into boundary integrals by the radial integration method. Results for the approximation functions are compared and the accuracy of the proposed formulation is assessed by results from literature. It was shown that results obtained with the approximation function called augmented thin plate spline present very good agreement with literature even for shells that are not so shallow.     

Dynamic response of plate to moving loads: Structural impedance method

An algorithm based on a structural impedance approach has been developed to study the transient response of plates with arbitrary boundary conditions and subjected to moving loads. Thin plate theory is assumed for the plate model and the algorithm places no restrictions on the loading conditions. The algorithm accounts for the complete dynamic interactions between the moving loads and the plate. Therefore, the method can be applied to the general moving mass problems and to the simplified moving force and static problems. The accuracy of the algorithm is verified by comparing the results obtained from the structural impedance method (SIM), in the case of moving force solutions, with the available exact solutions and with other numerical results. The dynamic deflections obtained from the moving mass solutions are compared with available experimental results. Parametric analyses over a wide spectrum of velocities and mass ratios indicate that the inertial effect on dynamic deflections is pronounced when the vehicle is traveling at high speed.     

Axisymmetric dynamic stability of a bimodulus thick circular plate

The object of the present paper is to investigate the dynamic stability of bimodulus thick circular and annular plates subjected to a combination of a pure dynamic bending and a uniform dynamic extensional stress in the plane of the plate. The Mindlin plate finite element model is established to solve the dynamic stability problems of an axisymmetric circular plate. A ring type element is chosen to approach the axisymmetric problem. The obtained results of the dynamic stability region are shown to be very good when compared with the closed form solutions for ordinary plates. The influences of various parameters on the dynamic stability boundary are studied.     

Optimization of boundary control at one end of a string in the presence of a model nonlocal condition

The paper is devoted to problems of optimal boundary control of string vibrations under nonlocal conditions that relate values of string displacement or its derivative at the boundary point with their values at the interior point. The solutions for all problems of boundary control are obtained in explicit analytic form.     

Shape-free polygonal hybrid displacement-function element method for analyses of Mindlin–Reissner plates

A high-performance shape-free polygonal hybrid displacement-function finite-element method is proposed for analyses of Mindlin–Reissner plates. The analytical solutions of displacement functions are employed to construct element resultant fields, and the three-node Timoshenko’s beam formulae are adopted to simulate the boundary displacements. Then, the element stiffness matrix is obtained by the modified principle of minimum complementary energy. With a simple division, the integration of all the necessary matrices can be performed within polygonal element region. Five new polygonal plate elements containing a mid-side node on each element edge are developed, in which element HDF-PE is for general case, while the other four, HDF-PE-SS1, HDF-PE-Free, IHDF-PE-SS1, and IHDF-PE-Free, are for the edge effects at different boundary types. Furthermore, the shapes of these new elements are quite free, i.e., there is almost no limitation on the element shape and the number of element sides. Numerical examples show that the new elements are insensitive to mesh distortions, possess excellent and much better performance and flexibility in dealing with challenging problems with edge effects, complicated loading, and material distributions.

     

A finite element method for the free vibration of plates allowing for transverse shear deformation

An isoparametric quadrilateral plate bending element is introduced and its use for the free vibration analysis of both thick and thin plates is examined. Plates of rectangular planform and of orthotropic materials are analysed and excellent results are obtained. The element performance is assessed by comparison with well established analytical and numerical solutions based on Mindlin's thick plate theory, three dimensional elasticity solutions and solutions based on thin plate theory. The ease with which the element may be implemented is stressed. The use of an eigenvalue economiser which produces considerable economy in the computer solution is demonstrated. Various mass lumping schemes and numerical integration rules used in the construction of the element mass matrix are also examined.     

Boundary element formulation for two-dimensional creep problems using isoparametric quadratic elements

A boundary element formulation for creep and time-dependent material behaviour problems based on an initial strain approach is presented. The details of numerical algorithm are shown where isoparametric quadratic elements are used both for the boundary elements and the quadrilateral domain cells. The Euler method with automatic time-step control scheme is implemented for time integration. Two creep power laws, time-hardening and strain-hardening, are employed to analyse a number of problems, including a square plate, a plate with a circular hole and a plate with a semi-circular notch subjected to a uniaxial load. The results are compared with analytical solutions where available and the corresponding finite element solutions.     

一种基于边界约束的流形展开方法

在流形学习的谱方法中, 流形展开被表述为优化问题. 这些优化问题的解是退化的, 即所有的样本将被嵌入到同一个点. 为了避免退化解, 谱方法对嵌入坐标人为地强加了一个单位协方差矩阵约束. 然而, 该约束往往导致流形展开的失真非常明显. 本文提出一种新的流形学习方法, 彻底抛弃了人为的单位协方差矩阵约束. 主要思路是先对流形边界进行嵌入, 然后再求流形内部的嵌入; 流形边界的嵌入位置被确定后, 流形内部样本的嵌入位置将被边界拉开, 使得它们不会都收缩到一个点上, 从而避免了退化解的出现. 将流形边界的嵌入位置作为边界条件, 求解一个线性方程组来得到内部样本的嵌入; 该线性方程组反映了尽量保持邻近样本间距离不变的要求. 流形边界的嵌入由简化流形的嵌入求出; 为此, 本文还设计了一种流形边界检测算法以及一种流形简化算法. 与目前代表性的几种流形学习方法进行了比较实验, 结果表明了本文方法的有效性, 其展开失真比谱方法明显要小.     

哈密顿体系下矩形薄板自由振动的一般解

根据弹性薄板自由振动问题的基本方程，把问题引入到哈密顿对偶体系中．x方向模拟为时间，选取弯矩，等效剪力，转角和挠度为对偶向量，得到了在不同边界条件时关于x轴对称和反对称时的解析解．算例研究了四边固支薄板的自由振动情形，从而推广了哈密顿体系的应用范围，验证了哈密顿体系求解方法在自由振动问题中的有效性．     

结构光在集装箱顶角加强板焊缝定位中的应用

针对集装箱顶角加强板的焊接特点,设计了一种基于结构光的自动焊缝定位方法。该方法主要有两个特点：一是将摄像机安装在机器人上,使光轴垂直工作平面,通过比较面积大小来获得深度信息,从而将三维信息的获取问题转化为二维信息的提取,并利用仿射变换获得空间信息;二是利用激光器与摄像机相配合的方式进行定位,由于激光束在焊接环境中的特征明显,可降低图像处理的难度。具体实现过程是利用两条线激光打在加强板的一个角上,在加强板和顶板之间的焊缝处出现转折,通过对激光条纹的有效处理,获得4对转折点的图像坐标,然后拟合直线求出加强板的焊接初始位置及方向,再结合事先制作的图像模板将加强板的其他特征点逐一求出。该系统有效地利用了激光器的优势,可靠地解决了加强板焊缝的自动定位问题。     

Slow periodic motions with internal sliding modes in variable structure systems

Singularly perturbed relay systems (SPRS) in which the reduced systems have the stable periodic motions with internal sliding modes are studied. The slow motion integral manifold of such systems consists of the parts which correspond to the different values of relay control and the solutions may contain the jumps from one part of the slow manifold to another. For such systems a theorem about existence and stability of the periodic solutions is proved. An algorithm of asymptotic representation for this periodic solutions using boundary layer method is presented. It is proved that in the neighbourhood of the break away point the asymptotic representation starts with the first order boundary layer function.