Preconditioned smoothed numerical manifold methods with unfitted meshes

Abstract

The numerical manifold method (NMM) has been widely utilized to solve problems involving complicated boundaries, cracks, and interfaces. Recently, strain or gradient smoothing techniques have been incorporated into NMM to improve its performance. The resulting smoothed NMMs (SNMMs) normally possess enhanced numerical properties, for example, higher accuracy, convergence, and efficiency. A challenging issue rooted in NMM and other enhanced finite element methods using unfitted meshes is the ill‐conditioning induced by extremely small cut element. In this study, we investigate the behaviors of SNMMs in the case of small cut element. It is demonstrated that the cut‐induced ill‐conditionings also exist for two types of SNMMs, namely, edge‐based SNMM and physical‐patch‐based SNMM. Furthermore, a preconditioner based on the normalization of basis functions is proposed to resolve the ill‐conditioning. Several benchmark problems demonstrate the performances of SNMMs regarding accuracy, convergence, and stability.