A linearly independent high-order numerical manifold method with physically meaningful degrees of freedom

Numerical manifold method (NMM) is established based on the two cover systems (including mathematical cover and physical cover) and the contact loop, and can be used to solve the continuous and discontinuous deformation problems in the geotechnical engineering in a unified way. Similar to other numerical methods based on the partition of unity (PU) theory, the NMM can also improve the computational accuracy by increasing the orders of local displacement functions freely without mesh refinement, though this may cause the global stiffness matrix singular, leading to the linear dependence issue. In this study, a new local displacement function of high-order polynomials is proposed. The new function is applied to solve the general elastic problems. The results show that the linear dependence is eliminated. Compared with the traditional NMM based on the first order polynomials, higher precision is reached. Stresses at nodes are continuous. All the degrees of freedom defined on a physical patch are physically meaningful, with the third to fifth simply being the strain components at the interpolation point of the patch. As a result, the stresses at the interpolation points can be directly obtained. The proposed procedure can be easily extended to other PU-based methods.