基于有限变形理论的数值流形方法研究

原有数值流形方法通过积累每一时步的小变形而得到结构最终的大变形，然而，当结构发生大变形、大转动时往往产生较大计算误差. 针对该问题，从动量守恒方程以及应力边界条件的积分弱形式出发，引入流形方法的插值函数，建立了基于有限变形理论的数值流形方法. 通过对比改进前后流形方法的计算迭代格式，指出了原有流形方法计算大变形问题时的误差来源. 最后，通过大变形悬臂梁和旋转块体算例对有限变形流形方法进行了验证. 数值结果表明，改进后的流形方法能够很好地处理大变形大转动问题，消除了转动所带来的计算误差，其计算结果与解析解及ABAQUS 软件求得的数值解相吻合. 

STUDY ON NUMERICAL MANIFOLD METHOD BASED ON FINITE DEFORMATION THEORY

The large deformation of structure calculated by original numerical manifold method (NMM) is cumulated by small deformation calculated in each time step. However, when the structure undergoes large deformation and large rotation, the calculation strategy used in original NMM will lead to calculation error. In order to solve this problem, in this study, combining the interpolation function of NMM, the Numerical Manifold Method based on finite deformation theory is deduced from integral weak form of the momentum conservation equation and the stress boundary conditions. The comparison between the iteration schemes of original NMM and improved NMM points out the sources of error calculated by original NMM for the large deformation problem. Finally, examples of large deformation cantilever and rotation block are employed to exam the improved NMM. The numerical result shows that the improved NMM handles the problem involving in large deformation and large rotation very well. The result calculated by the improved NMM eliminates the errors caused by rotation of structure and is coincide with the analytical solution and Abaqus numerical solution very well.