A high-order three-dimensional numerical manifold method enriched with derivative degrees of freedom

A three-dimensional (3D) high-order numerical manifold method (NMM) is developed based on the partition of unity method (PUM). We enrich the high-order NMM by introducing the derivative degrees of freedom associated with explicit physical significance. The global displacement in the formulation is approximated by a second-order approximation for the local displacement in conjunction with a first-order weight function. This not only helps the high-order NMM effectively avoid the problem of linear dependence that is frequently encountered in the PUM, but also renders the stress or strain at the star points continuous for the high-order NMM without the necessity of further smoothing operation. The effectiveness and robustness of the proposed new high-order NMM are demonstrated by several typical examples. Future potential developments and applications of the method are discussed.