摘要
无网格法因为不需要划分网格,可以避免网格畸变问题,使得其广泛应用于大变形和一些复杂问题.径向基函数配点法是一种典型的强形式无网格法,这种方法具有完全不需要任何网格、求解过程简单、精度高、收敛性好以及易于扩展到高维空间等优点,但是由于其采用全域的形函数,在求解高梯度问题时存在精度较低和无法很好地反应局部特性的缺点.针对这个问题,本文引入分区径向基函数配点法来求解局部存在高梯度的大变形问题.基于完全拉格朗日格式,采用牛顿迭代法建立了分区径向基函数配点法在大变形分析中的增量求解模式.这种方法将求解域根据其几何特点划分成若干个子域,在子域内构建径向基函数插值,在界面上施加所有的界面连续条件,构建分块稀疏矩阵统一求解.该方法仍然保持超收敛性,且将原来的满阵转化成了稀疏矩阵,降低了存储空间,提高了计算效率.相比较于传统的径向基函数配点法和有限元法,这种方法能够更好地反应局部特性和求解高梯度问题.数值分析表明该方法能够有效求解局部存在高梯度的大变形问题.
The meshfree methods can avoid grid distortion problems because it does not need to be meshed, which make them widely used in large deformations and other complicated problems. Radial basis collocation method(RBCM) is a typical strong form meshfree method. This method has the advantages of no need for any mesh, simple solution process,high precision, good convergence and easy expansion to high-dimensional problems. Since the global shape function is used, this method has the disadvantages of low precision and poor representation to local characteristics when solving high gradient problems. To resolve this issue, this paper introduces finite subdomain radial basis collocation method to solve the large deformation problem with local high gradients. Based on the total Lagrangian formulation, the Newton iteration method is used to establish the incremental solution scheme of the FSRBCM in large deformation analysis. This method partitions the solution domain into several subdomains according to its geometric characteristics, then constructs radial basis function interpolation in the subdomains, and imposes all the interface continuous conditions on the interfaces, which results in a block sparse matrix for the numerical solution. The proposed method has super convergence and transforms the original full matrix into a sparse matrix, which reduces the storage space and improves the computational efficiency.Compared to the traditional RBCM and finite element method(FEM), this method can better reflect local characteristics and solve high gradient problems. Numerical simulations show that the method can effectively solve the large deformation problems with local high gradients.
引文
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