A high order numerical manifold method and its application to linear elastic continuous and fracture problems

The numerical manifold method(NMM) is a partition of unity(PU) based method. For the purpose of obtaining better accuracy with the same mesh, high order global approximation can be adopted by increasing the order of local approximations(LAs). This,however, will cause the "linear dependence"(LD) issue, where the global matrix is rank deficient even after sufficient constraints are enforced. In this paper, through quadrilateral mesh to form the mathematical cover, a high order numerical manifold method called Quad4-COLS(NMM) is developed, where the constrained and orthonormalized least-squares method(CO-LS) is used to construct the LAs. The developed Quad4-COLS(NMM) does not need extra nodes or DOFs to construct high order global approximations, while is free from the LD issue. Nine numerical tests including five tests for linear elastic continuous problems and four tests for linear elastic fracture problems are carried out to validate the accuracy and robustness of the proposed Quad4-COLS(NMM).