加权残数配点法解正交各向异性板的积分方程

本文推导了一般各向异性板弯曲的积分方程,运用加权残数配点法求解了正交各向异性板弯曲的积发方程,本文将部分配点取在边界上,另一部分配点取在域外,只用关于找度的基本积分方程,而不用关于转角的补充积分方程,简化了方程求解和计算程序,由于正交各向异性板没有争析形式的、实用的基本解,本文提出了两种新的近似基本解;加权双三角级数;广义各向同性板解析形式的基本解和加权双三角级数的叠加,算例表明,本文提出的解法和近似基本解适用于各类边界条件的正交各向异性板,具有简单、可靠、精度高等优点。     

变厚度板弯曲的边界元分析法

由于变厚度板弯曲问题的控制分方程复杂，直接求解其基本解推导边界积分方程建立边界元分析法较为困难，本文通过引入等效荷载，等效刚度，将此问题的控制微分方程化成与普通薄板弯曲问题基本方程相同的形式，利用求解通板弯曲问题的边界元迭代求解，建立了分析变厚度板弯曲问题的蛤法，算例表明本方法理正确，精度良好。     

薄板稳定问题的边界元法

本文基于小挠度薄板弯曲问题的基本解,建立了求解薄板稳定问题的边界积分方程,并计算了若干算例,结果表明用边界元法求解薄板的稳定问题是行之有效的.     

The BEM analysis on bending fracture problem of thin plate

In this paper, based on Betti's reciprocal theorem, a set of boundary integral equations of thin plate with a crack is introduced. These boundary integral equations can be used to solve the bending fracture problem of thin plate. In the analysis process a higher-order singular integral equation will be induced. Using the concept of the Hadamard's principal value the higher-order singular integral can be solved conveniently. By this method we have found the analytical solution of a thin plate with a straight crack under pure bending load and indicated how to use the BEM to analyze the bending and fracture problem for thin plate with a straight crack. At the end of the paper some numerical examples are given. Numerical results show the accuracy and efficiency of the algorithms.     

用杂交边界点法分析简支薄板的热弯曲问题

该文利用杂交边界点法对简支薄板的热弹性弯曲进行了分析计算.采用薄板的热弹性理论,通过薄板的修正变分原理建立了各向同性薄板的边界局部积分方程,域内变量使用基本解插值,而边界上的变量则用移动最小二乘法近似.计算时仅需边界上离散点的信息,无论变量近似还是数值积分都不需要网格,因此该方法是一种纯边界类型无网格方法.数值算例表明,杂交边界点法在分析薄板的热弯曲问题时具有效率高、精度高和收敛性好等优点.     

简支梯形底扁球壳弯曲问题的准格林函数方法

以简支梯形底扁球壳的弯曲问题为例,详细阐明了准格林函数方法的思想.即利用问题的基本解和边界方程构造一个准格林函数,这个函数满足了问题的齐次边界条件,采用格林公式将简支扁球壳弯曲问题的控制微分方程化为两个互相耦合的第二类Fredholm积分方程.边界方程有多种选择,在选定一种边界方程的基础上,可以通过建立一个新的边界方程...     

Solution to the form of Poisson equation by the boundary element methods

In this paper, the domain integral of the form of Poisson equation is translated into complete boundary integral by the fundamental solution of higher-order Laplace operator, the dimensions of the problem can be contracted into one. The numerical examples for Stokes equations show that this method is efficient.     

A NECESSARY AND SUFFICIENT FORMULATION FOR TORSION PROBLEMS OF ANISOTROPIC BODIES

Non-dimensionalized equations and boundary conditions are presentedfor the torsion problem of an anisotropic body.The error of the fundamental solutioncited in some boundary element books is pointed out after an examination of thefundamental solution.Furthermore,a necessary and sufficient boundary integralequation is given for the problem and compared with the conventional boundaryintegral equation.Numerical results show that great errors of the boundary shearstresses obtained by the conventional boundary integral equation appear with a smallerror of torsion stiffness.Meanwhile,the necessary and sufficient boundary integralequation always gives accurate results.     

Axisymmetric contact problem for an elastic half-space and a circular cover plate

The contact interaction problem for a thin circular rigid cover plate and an elastic half-space loaded at infinity by a tensile force directed in parallel to the boundary of the half-space is considered. It is assumed that the cover plate is not resistant to bending deformations. The problem can be reduced to an integral equation of the first kind whose kernel has a logarithmic singularity. The equation is solved approximately by the Multhopp-Kalandia method. The resulting approximate solution is compared with the previously obtained asymptotic solution.     

双参数地基上板弯曲问题的基本解

应用广义函数的Fourier积分变换导出了双参数地基上板弯曲问题的基本解,它可用复变量的Hankle函数表示。为便于工程应用,本文将基本解用实变量的幂级数展开,证明了该级数的一致收敛性。本文对于弹性地基板分析的解析方法和边界元法研究,将是有益的。     

面内变刚度薄板弯曲问题的挠度?弯矩耦合神经网络方法

发展了一种求解面内变刚度功能梯度薄板弯曲问题的神经网络方法. 面内变刚度薄板弯曲问题的偏微分控制方程为一复杂的4阶偏微分方程, 传统的基于强形式的神经网络解法在求解该偏微分方程时可能会遇到难以收敛、边界条件难以处理的情况. 本文基于Kirchhoff薄板弯曲理论, 提出了一种直角坐标系下任意面内变刚度薄板弯曲问题的神经网络解法. 神经网络模型包含挠度网络与弯矩网络, 分别用于预测薄板的挠度与弯矩, 从而将求解4阶偏微分方程转换为求解一系列二阶偏微分方程组, 通过对挠度、弯矩试函数的构造可使得神经网络计算结果严格满足边界条件. 在误差的反向传播中, 根据本文提出的误差函数公式计算训练误差并结合Adam优化算法更新模型的内部参数. 求解了不同边界条件、形状的面内变刚度薄板弯曲问题, 并将所得计算结果与理论解、有限元解进行对比. 研究表明, 本文模型对于求解面内变刚度薄板弯曲问题具备适应性, 虽然模型中的弯矩网络收敛较挠度网络要慢, 但本文方法在试函数的构造上更为简单、适应性更强.      

角点悬空弯曲厚矩形板的边界积分法

在边界积分法中引用了拟基本系统矩形板,在该拟基本系统与实际系统之间应用功的互等定理,得到一挠曲面方程的积分表达式,只要对此表达式进行极简单的积分便可得到该挠曲面方程,这比直接求解Reissner挠度控制方程要简单,边界积分法的求解过程概念清晰,计算伊始便给出了挠曲面方程的总体表达式.以Reissner厚板理论为基础,应用边界积分法研究了角点悬空厚矩形板的弯曲问题,给出了在集中荷载作用下两邻边固定另两邻边自由且角点悬空弯曲厚矩形板的封闭解析解,并给出了相应的数据和图表以供工程上的应用和参考.     

各向异性柱体扭转的充分必要的边界积分方程

本文列出各向异性柱体扭转问题无量纲化后的有关方程的边界条件推导和验证了基本解，并指出一些书中基本解列式有误＾［９－１１］，列出了充分必要的边界积分方 程，进行了数据计算，并与习用的边界积分方程所得结果进行了比较，表明在退化值附近，习用的边界积分方程所得的边界剪应力会出现巨大的误差，扭转刚度的误差则要小得多，而充要的边界积分方程计算的结果则如终保持良好的精度，再次显示了它的优点。     

Amended influence matrix method for removal of rigid motion in the interior BVP for plane elasticity

A conventional complex variable boundary integral equation(CVBIE) in plane elasticity is provided. After using the Somigliana identity between a particular fundamental stress field and a physical stress field, an additional integral equality is obtained. By adding both sides of this integral equality to both sides of the conventional CVBIE, the amended boundary integral equation(BIE) is obtained. The method based on the discretization of the amended BIE is called the amended influence matrix method.With this method, for the Neumann boundary value problem(BVP) of an interior region,a unique solution for the displacement can be obtained. Several numerical examples are provided to prove the efficiency of the suggested method.     

薄板弯曲问题的虚边界元-最小二乘法

本文提出求解任意形状的薄板弯曲问题的虚边界元-最小二乘法。本法首先利用薄板弯曲平衡方程的格林函数和离开实际边界上分布的未知的横向荷载和法向弯矩函数建立满足实际边界条件的积分方程;然后采用最小二乘法和沿虚边界分段离散化的待定的分布横向荷载和法向弯矩函数得到求上述积分方程离散化数值解的线性代数方程组。导出了一系列的数值积分的公式,并求解了许多例题,数值结果说明本法完全避免了奇异积分及其复杂的处理方法和耗时的运算,而且在边界及其附近区域解的精度比普通边界元(以后简称边界元)法大大地提高了。     

一种处理弹性动力边界元法中奇异积分的方法

利用边界元法求解瞬态弹性动力学问题时,时域基本解函数的分段连续性和奇异性为该问题的求解带来很大的困难。为了解决时域基本解中的奇异性问题,本文依据柯西主值的定义,对经过时间解析积分之后的时域基本解进行奇异值分解,将其分成奇异和正则积分两部分;其中正则部分可通过采用常规高斯积分方法来计算,而奇异部分具有简单的形式,可以利用解析积分计算。经过上述操作之后,就可以达到直接消除时域基本解中奇异积分的目的。和传统方法相比,本文方法并不依赖静力学基本解来消除奇异性,是一种直接求解方法。最后给定两个数值算例来验证本文提出方法的正确性和可行性,结果表明使用本文算法可以解决弹性动力学边界积分方程中的奇异性问题。     

GREEN QUASIFUNCTION METHOD FOR FREE VIBRATION OF CLAMPED THIN PLATES

The Green quasifunction method is employed to solve the free vibration problem of clamped thin plates.A Green quasifunction is established by using the fundamental solution and boundary equation of the...     

Axisymmetric bending for thick laminated circular plate under a concentrated load

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Boundary integral equations for the bending problem of plates on two-parameter foundation

By means of Fourier integral transformation of generalized fimction, the fimdamental solution for the bending problem of plales on two-parameter foundation is derived in this paper, and the fundamental solution is expanded into a uniformly convergent series.On the basis of the above work, two boundary integral eguations which are suitable to arhitrary shapes and arbitrary boundary conditions are established by means of the Rayleigh-Green identity. The content of the paper provides the powerful theories for the application of BEM in this problem.     

ANALYSIS OF SYMMETRIC LAMINATED RECTANGULAR PLATES IN PLANE STRESS

Symmetric laminated plates used usually are anisotropic plates. Based on the fundamental equation for anisotropic rectangular plates in plane stress problem, a general analytical solution is established accurately by method of stress function. Therefore the general formula of stress and displacement in plane is given. The integral constants in general formula can be determined by boundary conditions. This general solution is composed of solutions made by trigonometric function and hyperbolic function, which can satisfy the problem of arbitrary boundary conditions along four edges, and the algebraic polynomial solutions which can satisfy the problem of boundary conditions at four corners. Consequently this general solution can be used to solve the plane stress problem with arbitrary boundary conditions. For example, a symmetric laminated square plate acted with uniform normal load, tangential load and nonuniform normal load on four edges is calculated and analyzed.