Narrow Results

This paper presents a review on the numerical manifold method (NMM), which covers the basic theories of the NMM, such as NMM components, NMM displacement approximation, formulations of the discrete system of equations, integration scheme, imposition of the boundary conditions, treatment of contact problems involved in the NMM, and also the recent developments and applications of the NMM. Modeling the strong discontinuities within the framework of the NMM is specially emphasized. Several examples demonstrating the capability of the NMM in modeling discrete block system, strong discontinuities, as well as weak discontinuities are given. The similarities and distinctions of the NMM with various other numerical methods such as the finite element method (FEM), the extended finite element method (XFEM), the generalized finite element method (GFEM), the discontinuous deformation analysis (DDA), and the distinct element method (DEM) are investigated. Further developments on the NMM are suggested.

Interface problems exist widely in various engineering problems and their high-precision simulation is of great importance. A new computational approach for dealing with interface problems is proposed based on the recently developed integral-generalized finite difference (IGFD) scheme. In this method, the research domain is divided into several subdomains by interfaces, and discretization schemes are established independently in each subdomain. A new cross-subdomain integration scheme is introduced to connect these subdomains. Several two-dimensional elasticity models containing material interfaces are studied to test the effectiveness of the proposed method. The results show that the recently proposed approach without the help of discontinuous functions or auxiliary equations that are commonly used in other numerical methods (e.g., extended finite element method and boundary element method) enables obtaining high accuracy and efficiency in interface problems. The proposed method has great potential in the application of material interface problems in solid mechanics and, furthermore, weak discontinuity problems in various fields.