基于径向基函数配点法的梁板弯曲问题分析

采用径向基函数配点法分析考虑剪切效应的梁板弯曲问题，该方法利用径向基函数作为近似函数，基于配点法离散方程，通过最小二乘法求解。径向基函数配点法在离散和计算过程中不需要任何形式的网格划分，是一种真正的无网格法；径向基函数可以用一元函数来描述多元函数，存在明显的储存和运算简单的特点；而基于配点法求解不需要积分，提高了计算效率。分析考虑剪切效应的薄梁板问题时，传统的有限元法或无网格法求解均会存在剪切锁闭问题，而径向基函数在全域内存在无限连续性，能够准确地满足Kirchhoff约束条件，因此径向基函数配点法能够消除剪切锁闭现象，而且不会出现应力波动。该方法的优势在于，其不仅易于离散、精度高，而且具有指数收敛率，计算效率高。数值算例验证了上述结论和该方法的稳定性。

RADIAL BASIS COLLOCATION METHOD FOR BENDING PROBLEMS OF BEAM AND PLATE

Radial basis collocation method is introduced to analyze the bending problems of Timoshenko beam and Reissner-Mindlin plate.The radial basis functions are employed for approximation,the collocation method is utilized for discretization,and the least squares approach is adopted to solve the discretized equations.No mesh will be required in the discretization and resolution and so radial basis collocation method is a truly meshfree method.1-D radial basis functions can represent all the 2-D or 3-D radial functions which greatly reduce the memory space. No integration will be used in collocation method which improves the computational efficiency.For resolving the problems of thin Timoshenko beam and Reissner-Mindlin plate,analysis demonstrates that radial basis collocation method is free of locking since the shape functions with infinite continuity can satisfy the Kirchhoff constraint conditions,and no stress oscillation is observed,while conventional finite element method and conventional meshfree methods suffer locking problems.The advantages of this approach include easy discretization and implementation,and possessing exponential convergence and high efficiency.Numerical examples validate the conclusions and the stability of this proposed method.