NURBS enhanced HIFEM: A fully mesh-independent method with zero geometric discretization error

Abstract

This manuscript introduces a NURBS-enhanced Hierarchical Interface-enriched Finite Element Method (NeHIFEM) for the mesh-independent treatment of multiphase problems with complex morphologies. The NeHIFEM implements a non-isoparametric mapping similar to that introduced in the NURBS Enhanced Finite Element Method (NEFEM) for the exact geometrical modeling of problems with curved materials interfaces. In this work, we propose a new generalized NURBS-enhanced mapping for triangular elements with three curvilinear edges, together with an isoparametric approach for evaluating basis functions in these elements to facilitate the construction of enrichment functions in the NeHIFEM. This unique capability allows the implementation of simple structured meshes for creating the computational model, while fully eliminating the geometric discretization error without introducing additional degrees of freedom. A new approach for the construction of higher-order enrichment functions in the NeHIFEM is also introduced to accurately capture gradient discontinuities along materials interfaces. Several example problems are presented to shed light on the accuracy and convergence rate of the NeHIFEM. We also show the application of this method for simulating the thermal and structural responses of heterogeneous materials with complex microstructures.