We extend the Discontinuity-Enriched Finite Element Method (DE-FEM) to simulate intersecting discontinuities, such as those encountered in polycrystalline materials, multi-material wedge problems, and branched cracks. The proposed hierarchical enrichment functions capture weak and strong discontinuities at junctions within a single formulation. Several numerical applications to branched cracks and polycrystalline microstructures under both thermal and mechanical loads are presented to demonstrate the proposed method. Results indicate that DE-FEM can accurately capture complex discontinuous primal and gradient fields and attain convergence rates comparable to those of standard FEM using fitted meshes. The main advantages of DE-FEM equipped with the proposed junction enrichment functions lie in the method's ability to model intersecting discontinuities using meshes that are completely decoupled from them and its robustness in reproducing correct displacement and strain jumps across them, as demonstrated by a patch test. This work thus highlights the potential of DE-FEM for applications to problems characterized by the presence of multiple intersecting discontinuities, posing a valid alternative to traditional FEM and eXtended/Generalized Finite Element (X/GFEM) Methods.

In this work, an object-oriented geometric engine is proposed to solve problems with discontinuities, for instance, material interfaces and cracks, by means of unfitted, immersed, or enriched finite element methods (FEMs). Both explicit and implicit representations, such as geometric entities and level sets, are introduced to describe configurations of discontinuities. The geometric engine is designed in an object-oriented way and consists of several modules. For efficiency, a (Formula presented.) -d tree data structure that partitions the background mesh is constructed for detecting cut elements whose neighbors are found by means of a dual graph structure. Moreover, the implementation for creating enriched nodes, integration elements, and physical groups is described in detail, and the corresponding pseudo-code is also provided. The complexity and efficiency of the geometric engine are investigated by solving 2-D and 3-D discontinuous models. The capability of the geometric engine is demonstrated on several numerical examples. Topology optimization and problems with intersecting discontinuities are handled with enriched FEMs, where enriched discretizations obtained from the geometric engine are used for the analysis. Furthermore, polycrystalline structures that overlap with an unfitted mesh are considered, where integration elements are created so they align with grain boundaries. Another example shows that the Stanford bunny, which is discretized by a surface mesh with triangular elements, can be fully immersed into a 3-D background mesh. Finally, we share a list of main findings and conclude that the proposed geometric engine is general, robust, and efficient.

We propose an enriched finite element formulation to address the computational modeling of contact problems and the coupling of non-conforming discretizations in the small deformation setting. The displacement field is augmented by enriched terms that are associated with generalized degrees of freedom collocated along non-conforming interfaces or contact surfaces. The enrichment strategy effectively produces an enriched node-to-node discretization that can be used with any constraint enforcement criterion; this is demonstrated with both multi-point constraints and Lagrange multipliers, the latter in a generalized Newton implementation where both primal and Lagrange multiplier fields are updated simultaneously. We show that the node-to-node enrichment ensures continuity of the displacement field—without locking—in mesh coupling problems, and that tractions are transferred accurately at contact interfaces without the need for stabilization. We also show the formulation is stable with respect to the condition number of the stiffness matrix by using a simple Jacobi-like diagonal preconditioner.

The application of fiber-reinforced polymer (FRP) composites is gaining increasing popu-larity in impact-resistant devices, automotives, biomedical devices and aircraft structures due to their high strength-to-weight ratios and their potential for impact energy absorption. Impact-induced high loading rates can result in significant changes of mechanical properties (e.g., elastic modulus and strength) before strain softening occurs and failure characteristics inside the strain localization zone (e.g., failure mechanisms and fracture energy) for fiber-reinforced polymer composites. In general, these phenomena are called the strain rate effects. The underlying mechanisms of the observed rate-dependent deformation and failure of composites take place among multiple length and time scales. The contributing mechanisms can be roughly classified as: the viscosity of composite constituents (polymer, fiber and interfaces), the rate-dependency of the fracture mechanisms, the inertia effects, the thermomechanical dissipation and the characteristic fracture time. Numerical models, including the viscosity type of constitutive models, rate-dependent cohesive zone models, enriched equation of motion and thermomechanical numerical models, are useful for a better understanding of these contributing factors of strain rate effects of FRP composites.