具有边梁加固的板弯曲问题的变分-差分方法研究

On the Finite Difference Method Based on the Principle of Variation of the Bending Problem of Plate with Beam-strengthened Edges

具有边梁加固的板的弯曲问题,其平衡方程模型为四阶椭圆型偏微分方程的边值问题,其中的自然边界条件涉及到了沿板边的切线和法线方向的高阶导数,对于非均匀、变厚度的板,该问题还具有"变系数"的特点.由问题的变分模型入手,应用变分-差分方法构造了该边值问题的一个差分格式.由于该方法能够结合平衡方程模型中的边界条件以消除沿板边的高阶导数项,因而,所得差分算子仅仅依赖于板面网格结点,并且保持了差分算子的对称、正定性质.同时,将已得算法在计算机上进行了数值模拟,并与现有文献进行了对比计算.结果显示本文所给出的算法具有较高的精确度,该算法将可用于定量地揭示板与边梁之间相互作用的规律,为工程设计提供参考依据.

The balance equation model of the bending problem of plate with beam-strengthened edges, where the natural boundary conditions are related to the higher order derivatives of the boundary tangent and normal, is the boundary value problem of the fourth- order elliptic partial differential equations. For the plate that is non--homogenous and has variable thickness, the problem has the characteristics of variable coefficients. In this thesis, starting from the variation model, the finite difference scheme of the boundary problem was constructed with FDM based on the principle of variation. The method incorporated the boundary condition of the balance equation to resolve the higher order derivative items in the boundary, so the difference operator that only relied on the network nodes of the plate. Moreover, the symmetry and positive definiteness of the difference operator were kept. At the same time, according to the difference equations, the Matlab program is programmed and developed to do numerical simulation. In comparison with the available document, comparing calculation was processed . The result showed the algorithm has satisfying accuracy. This algorithm can show quantitatively the international law of the plate and the boundary beam. At the same time, the algorithm can provide reference for engineering design.