六面体覆盖的高阶数值流形方法的探讨

采用高阶近似位移覆盖函数,基于六面体数学覆盖网格建立了三维数值流形方法分析格式,给出了相应的子矩阵。利用MATLAB编制了与之对应的计算程序,对简单的地下洞室模型进行了计算,并将计算结果与其他数值分析方法结果进行了比较,证明了分析格式及相应程序的正确性和有效性。结果表明,当数值流形方法的覆盖函数推广到高阶情况时,其求解精度会有相应的提高。最后,对该方法在隧道及地下工程的应用前景作了展望。

Discussion on high-order numerical manifold method based on hexahedron element

By introducing two covers, namely, the mathematical cover and physical cover, the numerical manifold method can solve continuous and discontinuous problems in geotechnical engineering in unified way. At present, the application of NMM is mainly limited to two-dimensional analysis which is not realistic for many underground engineering. In this paper, three-dimensional numerical manifold method based on hexahedron element is proposed; and two kinds of high-order displacement functions are used as the cover function. The stiffness matrix, loading matrix, displacement resistance matrix are given according to the cover function; and the methods of handling rank deficiency induced by high degree polynomials as local approximation is discussed. Corresponding program is made by Matlab. One simple example is calculated by the program and the numerical results, being compared with the result of other methods, proves its correctness and effectiveness. It shows that the solution accuracy of numerical manifold method will be improved, when high-order displacement function is used. An example of underground chamber is also presented. Finally, the prospects of the method in tunnel and underground engineering are given.