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The finite element method (FEM) is commonly used for modeling continuum media, while particle simulation methods like the so-called discrete element method (DEM) are used for discrete systems. Coupling the discrete (DEM) and continuum (FEM) methods is conventionally achieved through a direct mapping between discrete particles and finite elements. Coarse-graining (CG) is a micro–macro transition (discrete-to-continuum) method that maps discrete particle data onto smooth, differentiable fields that satisfy the continuum equations. By choosing an appropriate length scale (the coarse-graining width c), the coarse-grained fields are then homogenized and projected onto a FEM spatial discretization. This concept is utilized here to reformulate FEM-DEM coupling methods, both surface and volume, where in the limiting case of c→0, the classical coupling is recovered. For surface coupling, the discrete particle–surface contact forces are first mapped onto a continuous surface traction field (using CG) which is then coupled to the continuum FEM model. For volume coupling (also known as the Arlequin framework), the homogenization operators are enriched with CG functions, offering a non-local coupling approach between discrete particles, their continuum fields, and the finite elements. The CG enrichment represents a new strategy that consists of (1) a discrete-to-continuum mapping and (2) a continuum-to-continuum coupling based on “CG-enriched homogenization” (CGH). It is shown for surface coupling that the CG-enriched formulation not only leads to more accurate results, conserving symmetry, but also reduces energies generated by the coupling. For volume coupling, there is consistently less numerical dissipation with than without CG-enrichment, especially when the dynamic load contains high-frequency content. Finally, the optimal CG widths are identified for very simple test cases, with which the surface or volume coupling performs best. CGH can be potentially extended beyond the present examples, by considering other continuum fields (e.g., higher-order) and equations (e.g., multi-physics), and used to formulate other multi-scale modeling methods. |
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