A single point integration rule for numerical manifold method without locking and hourglass issues

Due to the salient feature of cutting operation, the numerical manifold method (NMM) can deal with an any-shaped problem domain by the simplest regular grid. However, this usually creates many irregularly shaped lower-order manifold elements. As a result, the NMM not only needs lots of integration points, but also encounters severe locking issues on nearly incompressible or bending-dominated conditions. This study shows a robust single-point integration rule to handle the above issue in the two-dimensional NMM. The essential idea is to separate the virtual work of an element in terms of moments to the center, so that a zero-order main term and higher-order stabilizing terms are obtained. Further, the volumetric locking and the shearing locking are avoided by modifications to the spherical part and shearing part of the stabilizing terms, and hourglass deformation is overcome since stabilizing terms are always non-zero. Consequently, in addition to fewer integration points, the rule improves accuracy since it is free from locking or hourglass issues. Numerical examples verify the robustness and accuracy improvement of the new rule.