We propose an extrinsic, continuous-Galerkin (CG), extended finite element method (XFEM) that generalizes the work of Hansbo and Hansbo to allow multiple Heaviside enrichments within a single element in a hierarchical manner. This approach enables complex, evolving XFEM surfaces in 3D that cannot be captured using existing CG-XFEM approaches. We describe an implementation of the method for 3D static elasticity with linearized strain for modeling open cracks as a salient step towards modeling progressive fracture. The implementation includes a description of the finite element model, hybrid implicit/explicit representation of enrichments, numerical integration method, and novel degree-of-freedom (DoF) enumeration algorithm. This algorithm supports an arbitrary number of enrichments within an element, while simultaneously maintaining a CG solution across elements. Additionally, our approach easily allows an implementation suitable for distributed computing systems. Enabled by the DoF enumeration algorithm, the proposed method lays the groundwork for a computational tool that efficiently models progressive fracture. To facilitate a discussion of the complex enrichment hierarchies, we develop enrichment diagrams to succinctly describe and visualize the relationships between the enrichments (and the fields they create) within an element. This also provides a unified language for discussing extrinsic XFEM methods in the literature. We compare several methods, relying on the enrichment diagrams to highlight their nuanced differences.
翻译:我们建议采用外源、连续-伽勒金(CG),即扩大的有限元素法(XFEM),将汉斯博和汉斯博的工作概括化,以便允许在一个单一元素中以等级方式进行多重高振场浓缩。这一方法使3D中复杂的、不断演进的XFEM表面能够使用现有的CG-XFEM方法进行3D静态弹性,无法用现有的CG-XFEM方法加以捕捉。我们描述了3D静态弹性和线性弹性方法的实施情况,用于模拟开阔裂的模型,这是走向逐步断裂的模型的显著步骤。实施的方法包括描述有限元素模型、浓缩差异、数字集成法和新颖的自由程度查算法的混合隐含/清晰度表示。这种算法支持一个元素中任意的浓缩数量,同时维护各元素之间的一个CG解决方案。此外,我们的方法很容易使分布式计算系统适合实施。根据 DoF 算法,拟议的方法为一种高效模拟逐渐断裂的计算工具奠定了基础。为了便利讨论复杂的浓缩结构之间的讨论,我们还开发了X的浓缩结构,从而在分析其直观地讨论了其浓缩结构中的浓缩结构,我们还分析了了其直图。