Abstract
The Hierarchical Interface-enriched Finite Element Method (HIFEM) is a technique for solving problems containing discontinuities in the field gradient using finite element meshes that do not conform (match) the domain morphology. The method is suitable for analyzing problems with complex geometries or when such geometry is not known a priori. Contrary to the eXtended/Generalized Finite Element Method (X/GFEM), enrichments are placed on nodes created along interfaces, and a recursive enrichment strategy is used to resolve multiple discontinuities crossing single elements. In this manuscript we rigorously study the approximating properties and stability of HIFEM. A study on the enrichments’ polynomial order shows that the formulation does not pass the patch test as long as enrichments do not replicate the approximating properties of partition of unity shape functions. Regarding stability, we show that condition numbers of system matrices grow at the same rate as those of standard FEM—and without requiring a preconditioner. This intrinsic stability is accomplished by means of a proper construction of enrichment functions that are properly scaled as interfaces approach mesh nodes. We further demonstrate that, even without scaling, using a simple preconditioner recovers stability. The method’s stability is further demonstrated on the modeling of challenging thermal and mechanical problems with complex morphologies.
Highlights
While the finite element method (FEM) has become the standard numerical procedure for simulating a wide range of problems in solid mechanics, creating appropriate finite element (FE) meshes for problems with complex morphologies is still a laborious and time-consuming part of the modeling process
Hierarchical Interface-enriched Finite Element Method (HIFEM) yields the same accuracy and convergence rates of its predecessor [25,26], and because HIFEM is a generalization of Interface-enriched GFEM (IGFEM), the former acronym will refer to both methods
Both HIFEM and its predecessor IGFEM suffer from the same stability issues as those suffered by X/GFEM, whereby global stiffness matrices become ill-conditioned when material interfaces pass arbitrarily close to nodes of the background finite element discretization
Summary
While the finite element method (FEM) has become the standard numerical procedure for simulating a wide range of problems in solid mechanics, creating appropriate finite element (FE) meshes for problems with complex morphologies is still a laborious and time-consuming part of the modeling process. When modeling problems where several material interfaces meet at a junction (e.g., polycrystalline materials), even refining the background mesh cannot avoid the presence of multiple intersecting interfaces within an element To eliminate this restriction, Soghrati [25] introduced the Hierarchical Interface-enriched FEM (HIFEM), where a recursive algorithm is used for constructing enrichment functions in elements cut by an arbitrary number of interfaces. HIFEM yields the same accuracy and convergence rates of its predecessor [25,26], and because HIFEM is a generalization of IGFEM, the former acronym will refer to both methods Both HIFEM and its predecessor IGFEM suffer from the same stability issues as those suffered by X/GFEM, whereby global stiffness matrices become ill-conditioned when material interfaces pass arbitrarily close to nodes of the background finite element discretization. We investigate the approximation properties of HIFEM and show that the polynomial orders used in integration elements and their compatibility with those of the background mesh are crucial to the performance of HIFEM—i.e. to ensure passing simple patch tests and yielding optimal rates of convergence for problems that do not exhibit singularities
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