A numerical framework coupling finite element and meshless methods in sequential and parallel simulations

The Finite Element Method (FEM) suffers from important drawbacks in problems involving excessive deforma-tion of elements despite being universally applied to a wide range of engineering applications. While dynamic remeshing is often offered as the ideal solution, its computational cost, numerical noise and mathematical limitations in complex geometries are impeding its widespread use. Meshless methods (MM), however, by not relying on mesh connectivity, circumvent some of these limitations, while remaining computationally more expensive than the classic FEM. These problems in MM can be improved by coupling with FEM in a FEM-MM scheme, in which MM is used within sensitive regions that undergo large deformations while retaining the more efficient FEM for other less distorted regions. Here, we present a numerical framework combining the benefits of FEM and MM to study large deformation scenarios without heavily compromising on computational efficiency. In particular, the latter is maintained through two mechanisms: (1) coupling of FEM and MM discretisation schemes within one problem, which limits MM discretisation to domains that cannot be accurately modelled in FEM, and (2) a simplified MM parallelisation approach which allows for highly efficient speed-up. The proposed approach treats the problem as a quadrature point driven problem, thus making the treatment of the constitutive models, and thus the matrix and vector assembly fully method-agnostic. The MM scheme considers the maximum entropy (max-ent) approximation, in which its weak Kronecker delta property is leveraged in parallel calculations by convexifying the subdomains, and by refining meshes at the boundary in such a way that the higher density of nodes is mainly concentrated within the bulk of the domain. The latter ensures obtaining the Kronecker delta property at the boundary of the MM domain. The results, demonstrated by means of a few applications, show an excellent scalability and a good balance between accuracy and computational cost.

Automated summary

The Finite Element Method (FEM) has limitations in problems with excessive element deformation, while dynamic remeshing is not suitable for complex geometries due to computational cost and limitations. Meshless methods (MM), although more expensive, can address these limitations. By coupling FEM and MM in a FEM-MM scheme, computational efficiency can be maintained while accurately modeling large deformation scenarios.