Abstract
Enriched finite element methods have gained traction in recent years for modeling problems with material interfaces and cracks. By means of enrichment functions that incorporate a priori behavior about the solution, these methods decouple the finite element (FE) discretization from the geometric configuration of such discontinuities. Taking advantage of this greater flexibility, recent studies have proposed the adoption of Non-Uniform Rational B-Splines (NURBS) to preserve the interfaces' exact geometries throughout the analysis. In this article, we investigate NURBS-based geometries in the context of the Discontinuity-Enriched Finite Element Method (DE-FEM) based on linear field approximations. While optimal convergence is retained for problems with weak discontinuities without singularities, representing exact geometry via NURBS does not yield noticeable improvements when extracting stress intensity factors of cracked specimens. For low-order elements, we conclude that the benefits of exact geometry representation do not outweigh the increased complexity in formulation and implementation. The choice of linear FEs hinders the accuracy of the proposed formulation, suggesting that its full potential may only be unleashed by increasing the field representation order.
Highlights
Non-Uniform Rational B-Splines (NURBS), ubiquitous in computer-aided design (CAD),[1] offer the possibility to represent high-fidelity or even exact geometry, as opposed to the low-order polynomial geometry approximation commonly used in finite element analysis (FEA).[2]
Taking advantage of this greater flexibility, recent studies have proposed the adoption of Non-Uniform Rational B-Splines (NURBS) to preserve the interfaces’ exact geometries throughout the analysis
We investigate NURBS-based geometries in the context of the Discontinuity-Enriched Finite Element Method (DE-FEM) based on linear field approximations
Summary
Non-Uniform Rational B-Splines (NURBS), ubiquitous in computer-aided design (CAD),[1] offer the possibility to represent high-fidelity or even exact geometry, as opposed to the low-order polynomial geometry approximation commonly used in finite element analysis (FEA).[2]. Inspired by the possibility of including a priori knowledge of the solution into the FE formulation,[11] X/GFEM augments the standard FEM approximation by means of enriched terms that are able to capture jumps in the solution field (strong discontinuities) or in its gradient (weak discontinuities) This is achieved by means of suitable discontinuous enrichment functions, with corresponding enriched DOFs, which are associated with the original mesh nodes. Soghrati and Merel,[32] instead, restrict the use of NURBS to the geometric mapping alone, while the original IGFEM enrichment functions are used for the field approximation Both approaches to combining IGFEM and NURBS were used to study weak discontinuities in composite materials[31,32] and microvascular systems,[30] and in shape optimization.[33] In these works, comparisons between classical FEM or IGFEM and NURBS-enhanced methods are mostly limited to verification examples, including the integration of areas, convergence studies, and simple benchmark problems. While optimal convergence rates are still recovered for weakly discontinuous problems with no singularities, the recovery of stress intensity factors (SIFs) has no major improvement over standard DE-FEM, and the NURBS-based formulation has troubles reproducing rigid body modes and does not pass an immersed patch test
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