摘要
无网格方法现在已成为一类有效的数值算法,能够处理许多基于网格的计算方法中存在的问题,如对复杂几何形状划分网格较困难、在分析大变形问题时会出现网格扭曲和对移动边界问题需要进行网格重构等。本文首先综述了无网格方法的发展历史和研究现状,同时对不同数值算法在板壳问题中的应用情况进行了介绍。
杂交边界点法是近几年发展起来的一种新的边界类型无网格方法,在形函数插值和数值积分时均不需要划分网格,且继承了边界元法降维和计算精度高的优点。但杂交边界点法目前求解的问题的控制方程都是二阶偏微分方程。而对于另一类物理问题,例如板弯曲问题,其控制方程为四阶偏微分方程。为此,本文提出了四阶偏微分方程边值问题的杂交边界点法,进一步完善和发展了杂交边界点法的理论和应用。
杂交边界点法的基本思想是利用修正变分原理将原问题的解转化为边界节点的局部积分方程,其关键在构造合适的修正泛函。对于双调和方程边值问题,修正泛函中有四个相互独立的场函数,即域内势函数、边界上的势函数及其法向导数和二阶导数；对于薄板弯曲问题,修正泛函中有五个相互独立的场函数,即域内的挠度、边界上的挠度和法向转角、边界上单位长度法向弯矩和等效剪力。利用调和方程和双调和方程的基本解的线性组合对域内场函数进行插值,边界上的场函数则采用移动最小二乘法进行近似。这样获得了四阶偏微分方程边值问题的杂交边界点法理论公式。
在求解薄板弯曲问题时,引入双互易理论,利用径向基函数对薄板上作用的横向载荷和弹性地基产生的反力进行插值。将问题的解分为通解和特解两部分。非齐次方程的特解采用径向基函数插值得到,在杂交边界点法中应用修正的边界条件对通解进行求解。本文计算了各种不同几何形状、不同边界支撑和不同横向载荷作用的薄板弯曲问题,并对该方法中的自由参数进行了敏感性分析。
将多互易理论和杂交边界点法结合,提出了多互易杂交边界点法,对非齐次势问题进行了分析。对于控制方程中的非齐次项,采用多互易法的思想对其进行不断微分,利用高阶基本解对问题的特解进行插值。其特点是在计算时不需要域内插值点,仅需边界节点进行离散。利用该方法求解了弹性扭转问题和带热源的热传导问题。
本文的研究表明：杂交边界点法不仅能够分析二阶偏微分方程的边值问题,而且在求解四阶偏微分方程边值问题时也很有效。该方法具有计算精度高、收敛速度快等特点,适合求解各种复杂形状、复杂边界条件和复杂横向载荷作用的薄板弯曲问题,可以用其解决各种工程实际问题。
Meshless methods have emerged as a class of effective numerical methods which are capable of avoiding the difficulties encountered in the computational meshed base methods. For example, these include the problems for the complex geometries meshing, mesh distortion due to large deformation and remeshing in the moving boundary problems. At first, the original literature and recent developments of the meshless methods are briefly reviewed, and the application of some numerical methods to plate and shell problems is introduced.
The hybrid boundary node method (HBNM) is a boundary type meshless method in the recent years, which requires no cell either for the interpolation of the solution variables or for the numerical integration, and inherits the reduced dimensionality and high accuracy advantages of the BEM. Until now, the governing equation of the problems which have been solved successfully is the second order partial differential equation. For the other physical problems, such as the plate bending, it is need to solve the fourth order partial differential equation. Therefore, the hybrid boundary node method for the fourth order partial differential equation is developed. The present study further improves and develops the theory and application of the HBNM.
In the present method, the modified variational principle is employed, and the solution of the original problem is transformed into the local integration equations of the boundary nodes. The key of this method is how to construct the suitable modified functional. For the boundary value problems of the biharmonic equation, the modified functional involves four types of the independent variables, i.e., the potentials inside the domain, the potentials, the normal fluxes and the vorticities on the boundary. For the thin plate bending problems, the modified functional involves five types of the independent variables, i.e., the deflections inside the domain, the deflections and normal slopes on the boundary, the normal bending moments and effective shear forces per unit length on the boundary. The domain variables are interpolated by a linear combination of the fundamental solutions of both the biharmonic equation and Laplace's equation, whereas the boundary variables are approximated using the moving least squares. Based on the above analysis, the formulas of the hybrid boundary node method for the boundary value problems of the fourth order partial differential equation are obtained.
For the thin plate bending problems, the dual reciprocity method is introduced to deal with the transverse distributed load and the subgrade reaction. The solution in this method is divided into two parts, i.e., the complementary and particular solutions. The particular one is obtained by the radial basis function interpolation, while the modified boundary conditions are applied in the hybrid boundary node method to solve the complementary one. The thin plates with various shapes and boundary conditions under various transverse loads are analyzed. At the same time, the influences of some computational parameters on the performance of the present method are investigated.
Combining the multiple reciprocity method and the hybrid boundary node method, the multiple reciprocity hybrid boundary node method is proposed, and is used to solve the inhomogeneous potential problems. Based on the multiple reciprocity method, the inhomogeneous term of the governing equation is successively operated by differentiation operator. The particular solution is interpolated by the high-order fundamental solutions. No internal points are required in this method. The multiple reciprocity hybrid boundary node method is implemented for solving the elastic torsion problems and heat conductivity problems with heat source.
The study shows that not only the boundary value problems of the second order partial differential equation, but also the boundary value problems of the fourth order partial differential equation can be effectively solved by the hybrid boundary node method. This method possesses the high accuracy, the good performance of convergence, and is suitable for analyzing all kinds of plates with arbitrary shapes and complicated boundary conditions and transverse distributed load. Therefore, the hybrid boundary node method is an attractive and prospective method in solving the practical engineering problems.
引文
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