A finite element method with mesh‐separation‐based approximation technique and its application in modeling crack propagation with adaptive mesh refinement

Abstract

This paper presents a FEM with mesh-separation-based approximation technique that separates a standard element into three geometrically independent elements. A dual mapping scheme is introduced to couple them seamlessly and to derive the element approximation. The novel technique makes it very easy for mesh generation of problems with complex or solution-dependent, varying geometry. It offers a flexible way to construct displacement approximations and provides a unified framework for the FEM to enjoy some of the key advantages of the Hansbo and Hansbo method, the meshfree methods, the semi-analytical FEMs, and the smoothed FEM. For problems with evolving discontinuities, the method enables the devising of an efficient crack-tip adaptive mesh refinement strategy to improve the accuracy of crack-tip fields. Both the discontinuities due to intra-element cracking and the incompatibility due to hanging nodes resulted from the element refinement can be treated at the elemental level. The effectiveness and robustness of the present method are benchmarked with several numerical examples. The numerical results also demonstrate that a high precision integral scheme is critical to pass the crack patch test, and it is essential to apply local adaptive mesh refinement for low fracture energy problems. due to intra-element cracking and the incompatibility caused by those hanging nodes due to local element refinement at the crack-tip at the elemental level and (ii) a unified FEM framework with some key advantages of other numerical methods such as the Hansbo and Hansbo method, the meshfree methods, the semi-analytical FEM, and the smoothed FEM (SFEM) while retaining the key feature of the standard FEM such as elemental locality.Over the last several decades, there have been rapid developments in advanced numerical methods to cope with the two major challenges mentioned earlier. To broadly categorize, major advances have been made on two dynamic fronts: (i) various meshfree (or meshless) methods and (ii) advanced FEM that can accurately and efficiently account for arbitrary discontinuities. The meshfree methods were developed with the goal of eliminating some difficulties associated with the reliance on a mesh to construct the approximation, and it can be traced back to the earlier work of Lucy [12] and Gingold and Monaghan [13]. The essence of meshfree methods is direct integration of point-wise force interactions among neighboring material points, which can effectively model the arbitrary crack formation and multiple crack interactions in solids. Numerical development along this line has been dynamic ever since the seminal work of Belytschko on the so-called elementfree Galerkin method [14] and the enriched element-free Galerkin of Fleming et al. [15]. Some of the recent fruitful advances are due to Rabczuk and colleagues (e.g., [16][17][18][19]) using extrinsically enriched methods based on the partition of unity (PoU) concept [20,21] and Duflot [22] and Barbieri et al.[23] using weight function enrichment. Recent successful ...

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