A two-dimensional fractional-order element-free Galerkin method for nonlocal elasticity and complex domain problems

TL;DR

This paper introduces a 2D meshfree fractional-order Element-Free Galerkin method that effectively solves complex nonlocal elasticity problems and irregular domain PDEs, outperforming traditional FEM in flexibility and efficiency.

Contribution

It develops the first 2D fractional-order EFG solver capable of handling complex geometries, demonstrating its application to nonlocal elasticity and validating its accuracy against benchmark results.

Findings

Successfully solves nonlocal elasticity in complex domains

Reduces computational cost compared to FEM

Accurately approximates fractional derivatives using MLS

Abstract

This study presents a meshfree two-dimensional fractional-order Element-Free Galerkin (2D f-EFG) method as a viable alternative to conventional mesh-based FEM for a numerical solution of (spatial) fractional-order differential equations (FDEs). The previously developed one-dimensional f-EFG solver offers a limited demonstration of the true efficacy of EFG formulations for FDEs, as it is restricted to simple 1D line geometries. In contrast, the 2D f-EFG solver proposed and developed here effectively demonstrates the potential of meshfree approaches for solving FDEs. The proposed solver can handle complex and irregular 2D domains that are challenging for mesh-based methods. As an example, the developed framework is employed to investigate nonlocal elasticity governed by fractional-order constitutive relations in a square and circular plate. Furthermore, the proposed approach mitigates key drawbacks of FEM, including high computational cost, mesh generation, and reduced accuracy in irregular domains. The 2D f-EFG employs 2D Moving Least Squares (MLS) approximants, which are particularly effective in approximating fractional derivatives from nodal values. The 2D f-EFG solver is employed here for the numerical solution of fractional-order linear and nonlinear partial differential equations corresponding to the nonlocal elastic response of a plate. The solver developed here is validated with the benchmark results available in the literature. While the example chosen here focuses on nonlocal elasticity, the numerical method can be extended for diverse applications of fractional-order derivatives in multiscale modeling, multiphysics coupling, anomalous diffusion, and complex material behavior.