A stable one-point quadrature rule for three-dimensional numerical manifold method

We present a numerically stable one-point quadrature rule for the stiffness matrix and mass matrix of the three-dimensional numerical manifold method (3D NMM). The rule simplifies the integration over irregularly shaped manifold elements and overcomes locking issues, and it does not cause spurious modes in modal analysis. The essential idea is to transfer the integral over a manifold element to a few moments to the element center, thereby deriving a one-point integration rule by the moments and making modifications to avoid locking issues. For the stiffness matrix, after the virtual work is decomposed into moments, higher-order moments are modified to overcome locking issues in nearly incompressible and bending-dominated conditions. For the mass matrix, the consistent and lumped types are derived by moments. In particular, the lumped type has the clear advantage of simplicity. The proposed method is naturally suitable for 3D NMM meshes automatically generated from a regular grid. Numerical tests justify the accuracy improvements and the stability of the proposed procedure.