数值流形法的T样条局部加密算法

在岩土工程分析中求解精度控制常常是必需的，在数值流形法中可以通过控制数学覆盖网格的稀疏和覆盖位移的阶数来达到精度的要求。提出了基于等几何分析的数值流形方法，定义了相应的数学覆盖的构造形式，推导了基于二次B样条的9节点数值流形方法分析格式；针对基于Lagrange插值函数的4节点数值流形方法提出了基于T样条思想的数学覆盖网格的局部加密方法。算例计算结果表明，相对于4节点的数值流形方法，基于非均匀有理B样条的9节点数值流形方法具有更高的精度；基于T样条思想的加密网格在保持计算精度的前提下降低了自由度的数量，表明T样条加密是一种自然的局部加密算法。

T-splines local refinement algorithm for numerical manifold method

The accuracy requirements of numerical manifold method (NMM) can be reached by changing the density of mathematical covers and the order of cover functions. In this study, NMM coupled with nonuniform rational B-splines (NURBS) and T-splines in the context of isogeometric analysis is proposed. Computational formula for a 9-node NMM based on quadratic B-splines is derived. For the case of crack propagation problems where singular fields around crack tips exist, local refinement technique by the application of T-spline discretizations is incorporated into NMM, which facilitates a truly local refinement without extending the entire row of control points. A local refinement strategy for the 4-node mathematical cover mesh based on T-splines and Lagrange interpolation polynomial is proposed. The results from numerical examples show that the 9-node NMM based on NURBS has higher accuracies. The local mesh refinement using T-splines reduces the number of degrees of freedom while maintaining calculation accuracy at the same time, and T-splines refinement method is a natural truly local refinement algorithm.