A bond-based nonlocal anisotropic diffusion model with variable matrix-valued coefficients and its asymptotically compatible meshfree discretization

Abstract

We propose a bond-based nonlocal anisotropic diffusion model with variable matrix-valued coefficients. In previous studies, the non-ordinary state-based nonlocal diffusion model [37] has been used effectively for simulating anisotropic diffusion. However, it encounters challenges such as high computational costs, complexities in implementing boundary conditions, and numerical oscillation of zero-energy mode. In this paper, we propose a novel bond-based, nonlocal anisotropic diffusion model, and the key idea is that incorporates a nonlocal operator via a kernel function, integrating matrix-valued diffusion coefficients. The influence region of our model consists of two parts: an elliptical region determined by the variable diffusion coefficient at a material point and an irregular region shaped by the coefficient at neighboring points. Furthermore, we confirm the well-posedness of the proposed model and deduce various properties, such as weak convergence and mass conservation. For computational implementation, we introduce a meshfree method that is shown to be asymptotically compatible and relies on the quadrature rule, which is compatible with the proposed nonlocal diffusion model and can effectively solve the model. To evaluate the precision and efficiency of the model, we performed comprehensive numerical experiments in both two and three dimensions. We have also confirmed the discrete maximum principle through experimental validation simultaneously.