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

TL;DR

This paper presents a meshfree two-dimensional f-EFG method for solving spatial fractional-order PDEs in nonlocal elasticity on complex domains, extending the previous 1D solver by employing 2D MLS approximants to evaluate fractional derivatives of order $\alpha \in (0,1]$ with horizons $l_A$ and $l_B$. The framework is demonstrated on both rectangular and circular Kirchhoff plates under linear and geometrically nonlinear regimes, with validation against established benchmarks and robustness to nonuniform node distributions. Convergence studies show accurate results with modest node counts, and a comprehensive cost analysis indicates competitive efficiency relative to mesh-based FEM for FDEs. The method's flexibility and demonstrated accuracy suggest strong potential for tackling irregular geometries and multiphysics problems in nonlocal elasticity and beyond.

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.

Figures (5)

• Figure 1: Illustration of isotropic Kirchhoff plates with respective choices for Cartesian coordinates. Plates are subject to transverse loads on top surface.
• Figure 2: Transverse displacement for the linear (infinitesimal) response of nonlocal rectangular plate compared for different node distributions.
• Figure 3: A schematic representation of the non-uniform grid in a rectangular plate.
• Figure 4: Normalized transverse displacement along the length of rectangular plate at $y=b/2$ under a UDTL. The linear elastic response is compared for both uniform and non-uniform node distributions.
• Figure 5: Comparison of the transverse displacement profiles for circular plate subject to UDTL considering local elastic (filled color) and nonlocal elastic response (partial opacity).