基于仿射速度矩阵改进物质点法的大位移滑坡研究

Large-displacement landslides based on affine velocity matrix-improved material point method

为克服物质点法的数值噪声问题,将仿射质点网格法的动量映射格式应用于经典物质点法,即在动量映射过程中额外维护一个仿射速度矩阵,进而构建出由仿射速度矩阵和物质点的速度矢量共同表示的速度场。通过理论推导证明了仿射速度矩阵是速度梯度的一阶近似,进而利用仿射速度矩阵极大地简化了旋率、变形率的求解过程和单时间步内的计算流程,使计算效率成倍提高。采用可GPU并行计算的Taichi编程语言进行算法编程,使求解效率提高数百倍。通过模拟沙柱自然成堆和铝棒堆积体模型试验分别验证了改进算法克服数值噪声的有效性、数值准确性和程序的计算效率;基于典型形状边坡的滑坡模拟,验证了算法和程序在滑坡分析中的适用性及优势。

In order to overcome the numerical noise problem of the material point method, the momentum mapping scheme of the affine particle-in-cell method is applied to the classical material point method, that is, in the process of momentum mapping,an additional affine velocity matrix is maintained, and then the velocity field represented by the affine velocity matrix and velocity vector of material point is constructed. It is proved that the affine velocity matrix is the first-order approximation of velocity gradient. The affine velocity matrix greatly simplifies the solving process of spin rate and rate of deformation as well as the computation process in a single time step, so that the computational efficiency is doubled. The Taichi programming language, which can be used for GPU parallel computing, is used for algorithm programming, which improves the solution efficiency hundreds of times. The effectiveness of the improved algorithm to overcome numerical noise, the numerical accuracy and the computational efficiency of the program are verified by simulating the natural stacking of sand columns and the stacking of aluminum rods. By simulating the landslide of a typical shape slope, the applicability and advantages of the algorithm and program in landslide analysis are verified.