A high order local approximation free from linear dependency with quadrilateral mesh as mathematical cover and applications to linear elastic fractures

The numerical manifold method falls into the category of the partition of unity methods. In order to enhance accuracy, high order polynomials can be specified as the local approximations. This, however, would incur rank deficiency of the stiffness matrix. In this study, a local displacement approximation is constructed over a physical patch generated from a four quadrilateral mathematical mesh. All the degrees of freedom are physically meaningful. The stresses are continuous at all nodes, suggesting that no stress polish is required. The proposed approximations have the same accuracy as the first-order polynomials, but no linear dependency inherent in the latter.