基于径向基函数物质点法的数值模拟和软件开发

摘要

物质点法的求解精度依赖于物质点和背景网格节点之间映射插值函数的选取，传统的物质点法使用的是线性插值函数，但是由于单元之间只存在着 C0连续性，传统的物质点法容易产生噪声。径向基函数保存了无网格法的优点，并具有高阶光滑连续的特点，而且径向基函数只需要局部支撑域中的节点信息来构造插值函数，这就大大简化了形函数及其导数的形成。本文将径向基函数引入物质点法以克服传统物质点法中单元数据映射产生的各种问题。同时利用 Fortran 语言开发了径向基函数物质点法数值模拟工具箱，对所开发的工具箱算例验证。%The accuracy of numerical results of the MPM depends on the selection of interpolation function in mapping between the material points and background grid nodes.In the original MPM,linear shape function is used for mapping.Due to the characteristic of C0 continuity between elements,the original MPM may suffer the numerical noise.The RBF interpolation is meshless and high degree of continuity,and the RBF interpolation function is constructed using a set of nodes within the local support domain which simplifies the formulation of shape function and its derivatives.This paper will formulate a new MPM by introducing the radial basis function interpolation to solve the problems based on mapping,and using FORTRAN language develop the R-MPM toolbox in which the radial basis function interpolation is implemented in the MPM framework,and the efficiency and accuracy of the toolbox will be proved with examples.