双调和方程奇异边界法的改进及在Kirchhoff板弯曲中的应用

奇异边界法(SBM)是一种基于边界离散的无网格数值方法，在很多科学计算和工程领域中得到广泛的应用．该方法在处理复杂几何区域或者多连通区域时比基本解方法(MFS)数值计算更为稳定，具有易于实施、精度高等优点．SBM数值计算的关键之处在于源强度因子的计算，特别是相对于Laplace方程更为复杂的双调和方程的边界条件下源强度因子的计算．在高阶导数边界条件下，采用反插或者“加减项”原理计算源强度因子相对繁琐．本文对双调和方程的SBM进行了改进，将其中一个插值基函数改进为非奇异基函数形式，避免计算该基函数的源强度因子，极大简化了SBM的数值计算．本文改进对MFS同样有效，可以作为对传统MFS数值算法的补充．数值算例结果表明，本文提出的改进均能得到误差很小的数值解，且算法稳定，计算效率较高．

A Modified Singular Boundary Method for Biharmonic Functions and Its Applications in Kirchhoff Plate Bending Problems

Singular boundary method (SBM) is a meshless numerical method based on boundary discretization and has been widely used in many scientific and engineering fields. Compared with the method of fundamental solutions (MFS), SBM has advantages of easy implementation and high accuracy, and can be applied in treating the complex region or the multi-connected region with high stability. The key point for SBM is the calculation of the origin intensity factor, especially under the boundary conditions of the more complex biharmonic functions compared with that of the Laplace functions. It is relatively complicated to calculate the origin intensity factors by inverse interpolation technique or "subtracting and adding-back" technique under the boundary conditions of high-order derivatives. This paper propose a modification of SBM of Biharmonic Functions by replacing one of the singular basis functions with non-singular ones to avoid the calculations for origin intensity factors and significantly simplify the computation. The modified method is also applicable and effective for MFS. It is shown through numerical examples that the modified SBM can obtain numerical solutions with minor errors, with high numerical stability and high computational efficiency.