Multiscale multimesh finite element method | $\text{M}^2$-FEM: Hierarchical mesh-decoupling for integral structural theories

This study presents a generalized multiscale multimesh finite element method ($\text{M}^2$-FEM) that addresses several long-standing challenges in the numerical simulation of integral structural theories, often used to model multiscale and nonlocal effects. The major challenges in the numerical simulation of integral boundary value problems are primarily rooted in the coupling of the spatial discretization of the global (parent) and integral (child) domains which severely restricts the computational efficiency of existing algorithms by imposing an implicit trade-off in the accuracy achieved by the child domain and in the resources dedicated to the simulation of the overall parent domain. One of the most defining contributions of this study consists in the development of a mesh-decoupling technique that generates isolated sets of meshes such that the parent and child domains can be discretized and approximated independently. This mesh-decoupling has a multi-fold impact on the simulation of integral theories such that, when compared to existing state-of-the-art techniques, the proposed algorithm achieves simultaneously better numerical accuracy and efficiency (hence allowing a greater flexibility in both mesh size and computational cost trade-off decisions), greater ability to adopt generalized integral kernel functions, and the ability to handle non-regular (non-rectangular) domains via unstructured meshing. In this study, we choose a benchmark problem based on an extended version of the Eringen's nonlocal elasticity theory (implicitly, a multiscale theory) that leverages the use of generalized attenuation kernels and non-constant horizons of nonlocality. Nonetheless, the proposed $\text{M}^2$-FEM algorithm is very general and it can be applied to a variety of integral theories, even beyond structural elasticity.