Advances and perspectives in numerical manifold method and its applications

Numerical manifoldmethod(NMM) is a new and generalized numerical method based onfinite cover technique. It is similar to finite element method,meshless method and partition of unit method on the approximationprinciple of field functions, but has its own characteristics andadvantages on mesh generation, cover system and approximationfunction. The important advances on numerical manifold method andits applications in recent years were reviewed in this paper.Performance assessment for manifold elements with differentphysical cover showed that manifold elements have highercomputational accuracy than finite element, and the computationalaccuracy can be improved by employing high order cover functions.Theoretical studies on manifold method have also been extend tothree dimension high order manifold method from two dimension loworder manifold method, to nonlinear manifold method from linearmanifold method, to Galerkin weighted residual manifold methodfrom based on energy principle. Incompatible manifold method andmeshless manifold method have been investigated as well. Moreover,the researches on some relative theories of NMM, such as automaticgeneralization of cover system, the form of cover functions andtreatments of boundary conditions, have been developed. Inapplications of manifold method, discontinuous deformationproblems on rock damage and crack propagation have been analyzedmore deeply than before, and it has been extend to apply invarious fields of engineering gradually, such as metal forming,porous media mechanics, temperature field etc. Based on the recentwork, some perspectives in research topics for NMM and itsapplications in computational fluid dynamics (CFD), metal forming,multiphysical field and other domains are presented.