几类薄板弯曲问题边界元迭代求解的统一格式

弹性地基板、具有初曲率板及变厚度板弯曲问题的控制微分方程较复杂,直接求解问题基本解建立边界积分方程较为困难。本文通过引入等效荷载,将此类问题的控制策分方程化为与普通板弯曲基本方程相同的形式,利用求解一般板弯曲问题的边界元法迭代求解,建立了分析这几类薄板弯曲问题的统一边界元方法。     

应力边界元法解平面热弹性问题

本文提出了求解平面热弹性问题的应力边界元法。利用应力法由平面热弹性问题的基本方程出发,简要地叙述了边界积分方程的建立,给出了位移单值条件。这种方法适用于应力边界值问题。作为数值计算例,计算了圆形区域和具有偏心圆孔的圆形区域的热应力,得到了满意的结果。应力边界元法也可应用于平板弯曲问题。     

Kirchhoff型裂纹板的边界元法

本文用的复变函数理论,导出了含裂板弯曲问题的基本解。该基本解满足自由裂纹的边界条件。将其引入直接或间接积分方程中,只要对板的外边界进行离散,就可计算有限尺寸裂纹板的弯曲问题。算例表明,本文所得到的基本解用以求解裂纹板弯曲问题划分的单元较少,精度较高。本文的方法还可用以求解含有形状比较复杂的裂纹或孔洞板弯曲问题的基本解。     

板弯曲问题三维虚边界元分析

本文抛弃以往解板弯曲问题的假设,直接从三维弹性力学微分方程出发,依据三维弹性力学问题的Kelvin解,应用最小二乘法建立了三维虚边界元法解板弯曲问题的一般方法。文中给出了具有各种约束的矩形板的数值算例,以作为本方法的应用。本文方法与边界元直接法相比,优点在于无需处理奇异积分,且系数阵是对称的：再者,本文方法思想简单,且程序实现容易,易于被工程界接受。     

板弯曲问题三维边界元分析

本文抛弃以往解板弯曲问题的假设,直接从三维弹性力学微分方程出发,依据三给弹性力学问题的Ｋｅｌｖｉｎ解,应用最小二乘法建立了三维虚边界元法解板弯曲问题的一般方法,文中给出了具有各种约束的矩形板的数值算例,以作为本方法的作用,本文方法与边界元直接法相比,优点在于需处理奇积分,且系数阵是对称的;再者,本文方法是思想简单,且程序实现容易,易于被工程界接受。     

无奇点边界元法分析板的稳定

根据板的稳定问题控制微分方程，利用无奇点边界元法（域外奇点法）离散比，导出稳定特征方程，从而求出临界荷载因子。经编程计算例题，效果良好。     

BEM—FEM分析弹性地基加反

用无奇点边界元法处理弹性基上的薄板；用有限元法处理加肋板上的肋梁弹性地基上的薄板与格栅之间的平衡与协调关系，将两种方法建立的方程进行耦联导出一组基本方程；求解板上选点的挠度和有关参数，进而求得板和肋梁的内力。本方法适于任意形状，多种边界条件以及不均匀地基上的加肋板。     

与权函数对应的积分方程及其联立解法

边界元法一般采用控制方程的基本解作为权函数,这往往能在控制方程为齐次时可避免域积分。但当问题复杂,基本解不能求得时,此法便产生了困难。虽有人也曾偶尔用非基本解函数作为权函数,但本文将系统地探讨函娄与边界积分方程的关系,所涉及的各类定解问题都用Laplace基本解和kelvin基本解作为权函数,并提出边界点公式和内点公式联立求解的方法。这不但避免了求基本解的困难,同时也为编制能求解多种问题的多功能电算程序提供了方便,使程序的长度缩短了,编程和调试的难度也降低了。     

波浪与弹性板作用的数值模拟

该文采用高阶有限元和边界元联合的方法求解波浪与弹性板的相互作用。其中流场采用边界元法求解,结构弹性响应方程采用基于Mindlin板理论的有限元方法求解,通过模态叠加技术实现了弹性板变形与流场相互作用的解耦。通过对一矩形板的计算,验证了该文方法与他人试验结果和数值模拟结果都吻合良好。利用这一模型进一步分析了波浪与弹性圆形板的作用问题,并对圆形板运动响应的收敛性进行了分析。     

动力弹塑性刚体界面元法

本文探讨了用刚体界面元法求解二维动力弹塑性问题的理论．首先，从连续介质的质点运动微分方程出发，利用其等效积分方程，通过引入刚体模式的位移场函数建立了刚体界面元的基本动力方程．为了以渐进解法求解弹塑性问题，将界面元简化的应力矢量引入塑性流动理论，从而给出了界面元的弹塑性本构关系．     

A semianalytic solution for radially supported curved plates in bending

By substituting the basic function in the circumferential direction satisfying the boundary conditions of the radial edges in the plate bending equation of the curved plate and performing a suitable transformation, an ordinary differential equation is resulted. The resulting equation is solved by finite difference technique using a small number of discrete variables. The method takes into account the orthotropy of the plate as well as the variation of its thickness. Examples have been presented for a variety of radially supported curved plates of different boundary conditions under uniform load and point loads. Excellent accuracy has been obtained wherever comparisons have been possible.     

Exact solutions for extensible circular curved timoshenko beams with nonhomogeneous elastic boundary conditions

Summary A generalized Green function ofnth-order ordinary differential equation with forcing function composed of the delta function and its derivatives is obtained. The generalized Green function can be easily and effectively applied to both the boundary value problems and the initial value problems. The generalized Green function is expressed in terms ofn linearly independent normalized homogeneous solutions. It is the generalization of those given by Pan and Hohenstein, and Kanwal. Accordingly, the exact solution for static analysis of an extensible circular curved Timoshenko beam with general nonhomogeneous elastic boundary conditions, subjected to any transverse, tangential and moment loads is obtained. The three coupled governing differential equations are uncoupled into one complete sixth-order ordinary differential characteristic equation in the tangential displacement. The explicit relations between the angle of rotation due to bending, the transverse displacement and the tangential displacement are obtained. The deflection curves due to a unit generalized displacement at nodal coordinate, and the exact element stiffness matrix are derived based on the solution for the general system. A finite element method can be developed based on the results for the dynamic analysis. Meanwhile, the stiffness locking phenomena accompanied in some other curved beam element methods does not exist in the proposed method.     

A fast multipole boundary element method for solving the thin plate bending problem

A fast multipole boundary element method (BEM) for solving large-scale thin plate bending problems is presented in this paper. The method is based on the Kirchhoff thin plate bending theory and the biharmonic equation governing the deflection of the plate. First, the direct boundary integral equations and the conventional BEM for thin plate bending problems are reviewed. Second, the complex notation of the kernel functions, expansions and translations in the fast multipole BEM are presented. Finally, a few numerical examples are presented to show the accuracy and efficiency of the fast multipole BEM in solving thin plate bending problems. The bending rigidity of a perforated plate is evaluated using the developed code. It is shown that the fast multipole BEM can be applied to solve plate bending problems with good accuracy. Possible improvements in the efficiency of the method are discussed.     

The boundary element method for thick plates on a Winkler foundation

In this paper the application of the boundary element method to thick plates resting on a Winkler foundation is presented. The Reissner plate bending theory is used to model the plate behaviour. The Winkler foundation model is represented by continuous springs which are directly incorporated into the governing differential equation. The fundamental solutions are constructed using operator decoupling technique. These fundamental solutions represent three different cases depending on the problem constants. The explicit forms of the boundary and internal point kernels are given in all cases. Quadratic isoparametric boundary elements are used to model the plate boundary. Several examples are presented to demonstrate the accuracy of the present formulation. © 1998 John Wiley & Sons, Ltd.     

A unified 3D‐based technique for plate bending analysis using scaled boundary finite element method

This paper presents a unified technique for solving the plate bending problems by extending the scaled boundary finite element method. The formulation is based on the three‐dimensional governing equation without enforcing the kinematics of plate theory. Only the in‐plane dimensions are discretised into finite elements. Any two‐dimensional displacement‐based elements can be employed. The solution along the thickness is expressed analytically by using a matrix function. The proposed technique is consistent with the three‐dimensional theory and applicable to both thick and thin plates without exhibiting the numerical locking phenomenon. Moreover, the use of higher order spectral elements allows the proposed technique to better represent curved boundaries and to achieve high accuracy and fast convergence. Numerical examples of various plate structures with different thickness‐to‐length ratios demonstrate the applicability and accuracy of the proposed technique. Copyright © 2012 John Wiley & Sons, Ltd.     

Bending analysis of thick laminated plates using extended Kantorovich method

This study is concerned with bending of moderately thick rectangular laminated plates with clamped edges. The governing equations, based on Reissner first-order shear deformation plate theory; in terms of deflection and rotations of the plate include a system of three second-order, partial differential equations (PDEs). Application of extended Kantorovich method (EKM) to the system of partial differential equations reduces the governing equations to a double set of three second-order ordinary differential equations in the variables x and y. These sets of equations were then solved in an iterative manner until convergence was achieved. Normally three to four iterations are enough to get the final results with desired accuracy. It is demonstrated that, unlike other weighted residual methods, in the extended Kantorovich method initial guesses to start iterations are arbitrary and not even necessary to satisfy the boundary conditions. Results of this study also reveal that the convergence of the EKM is rapid and the method is an efficient way to solve system of PDEs of the same type. To compare the results of this study, the problem was also analyzed using commercial finite element software, ANSYS. Results show reasonably good agreement with the finite element analysis.     

Analysis of functionally graded plates by meshless method: A purely analytical formulation

Functionally graded plates under static and dynamic loads are investigated by the local integral equation method (LIEM) in this paper. Plate bending problem is described by the Reissner moderate thick plate theory. The governing equations for the functionally graded material with respect to the neutral plane are presented in the Laplace transform domain and therefore the in-plane and bending problems are uncoupled. Both isotropic and orthotropic material properties are considered. The local integral equation method is developed with the locally supported radial basis function (RBF) interpolation. As the closed forms of the local boundary integrals are obtained, there are no domain or boundary integrals to be calculated numerically in this approach. The solutions of the nodal values for the entire plate are obtained by solving a set of linear algebraic equation system with certain boundary conditions. Details of numerical procedures are presented and the accuracy and convergence characteristics of the method are examined. Several examples are presented for the functionally graded plates under static and dynamic loads and the accuracy for proposed method has been observed compared with 3D analytical solutions.     

Boundary element analysis of reinforced shear deformable shells

In this paper, stiffened shear‐deformable shells are analysed using the boundary element method. Coupled boundary integral equations are presented for describing curved shells under general loading conditions. The equations are based on boundary integral equations for plane stress and plate bending, with coupling terms arising from the curvature of the shell. Domain integrals are transformed into boundary integrals using the dual reciprocity technique. Stiffeners are modelled as curved beams, continuously attached to the shell. Numerical solutions calculated using the present method are compared with finite element results in two examples. Copyright © 2002 John Wiley & Sons, Ltd.     

Hybrid-Trefftz stress elements for plate bending

The hybrid-Trefftz stress element is used to emulate conventional finite elements for analysis of Kirchhoff and Mindlin-Reissner plate bending problems. The element is hybrid because it is based on the independent approximation of the stress-resultant and boundary displacement fields. The Trefftz variant is consequent on the use of the formal solutions of the governing Lagrange equation to approximate the stress-resultant field. In order to emulate conventional elements, nodal functions are used to approximate the displacements on the boundary of the element. Duality is used to set up the element solving system. The associated variational statements and conditions for existence and uniqueness of solutions are recovered. Triangular and quadrilateral elements are tested and characterized in terms of convergence, sensitivity to shear-locking, and shape distortion. Their relative performance is assessed using assumed strain Mixed Interpolation of Tensorial Components (MITC) elements and recently proposed Trefftz-based elements. This relative assessment is extended to a hypersingular problem to illustrate the effect of enriching the domain and boundary approximation bases.     

A meshless method for Kirchhoff plate bending problems

In this work a meshless method for the analysis of bending of thin homogeneous plates is presented. This meshless method is based on the use of radial basis functions to build an approximation of the general solution of the partial differential equations governing the Kirchhoff plate bending problem. In order to obtain a symmetric and non‐singular linear equation system the Hermite collocation method is used. To assess the formulation a series of plates with different boundary conditions are analysed. Comparisons are made with other results available in the literature. Copyright © 2001 John Wiley & Sons, Ltd.