A higher-order variational numerical manifold method formulation and simplex integration strategy

Abstract

The numerical manifold method (NMM) is a relatively recent yet potentially powerful numerical analysis technique, integrating aspects of both traditional, meshless and Partition of Unity methods and promising to provide a single unified framework for modelling both continuum and discontinuum states. There are also similarities with the family of Hierarchical Finite Elements. There are three important characteristics that can best describe the attractiveness of NMM. First, discontinuities in the displacement field may be introduced naturally without the need for remeshing. Second, the level of approximation may be improved, either globally or locally, without the need for remeshing. Third, the integration associated with the discretization procedure may be undertaken explicitly using Simplex integration. NMM has been traditionally cast by minimization of potential energy and essential boundary conditions have been imposed by the penalty method. The original NMM has been extended by several researchers such that the level of approximation can theoretically be of any order. Whilst the foundations for higher-order NMM have been laid, only Lu (2002) has demonstrated how this may be practically implemented for any arbitrary level of approximation. This paper presents a variational higher-order NMM formulation whereby essential boundary conditions are satisfied by Lagrange multipliers, introducing an additional unknown. Furthermore, a strategy for applying Simplex integration without the need for explicit derivation of the element stiffness matrix for any order of the cover displacement functions is proposed and illustrated with a 2-D numerical example.