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。

Virtual Boundary Element-Least Square Method for Solving Bending Problems of Thin Plate

In this paper a virtual boundary element-least square method for solving bending problems of thin plate with arbitrary shape under the action of arbitrary transverse loads is presented. At first, the Green function of thin plate goverment equation and the unknown to be determining transverse loads and the normal bending moments distributed on the virtual boundary placed apart some distance from the real boundary are used to formulate a system of integral equation satisfing the real boundary conditions by superposition method. Secondly, a system of linear algebraic equation is obtained by numerically solving the above system of integral equation using the least square method and the piece-wise discrete distributing transverse loads and the normal bending moments acted along the virtual boundary. The results of some examples show that the singular integral and it' s complex treatment and calculation consuming much time are all avoided in this method. The accuracy of the solution near the boundary (including the boundary) is improved obviously.