Two-dimensional Hermitian numerical manifold method

Abstract

Numerous approaches have been proposed to enhance the accuracy and convergence of the numerical manifold method (NMM) in recent years, but most, if not all, of these approaches cannot ensure C1 continuity. Hermitian interpolation is an effective approach for obtaining high-order approximations. However, the requirement of rectangular meshes hinders the application of this approach in the finite element method. Taking advantage of the freedom in meshing in NMM, Hermitian interpolation is incorporated into NMM to obtain the C1 approximation. In contrast to the common high-order NMM, the Hermitian NMM (HNMM) improves the accuracy and convergence without causing the linear dependence problem. Moreover, the degrees of freedom (DOFs) of the mathematical nodes inside the physical domain have physical meanings, and the strains at nodes can be obtained directly without the need for extra postprocessing. The proposed HNMM is verified by solving numerous benchmark linear elastic problems, and the results are compared against those of linear and cubic Lagrangian NMMs. The numerical solutions for these examples confirm the remarkable superiority of the HNMM over the Lagrangian NMMs in terms of accuracy, convergence and efficiency.