Smoothed particle hydrodynamics of anisotropic diffusions

Diffusion problems with anisotropic features arise in various scientific and engineering disciplines. However, accurately describing these processes in numerical simulations poses challenges due to the erroneous approximation and the appearance of spurious oscillations and negative concentrations. As a Lagrangian mesh-less method, Smoothed Particle Hydrodynamics (SPH) offers unified framework for addressing multi-physics problems involving diffusion process. Despite these benefits, its application to anisotropic diffusion remains limited, due to the lack of accurate and stable particle approximation. In this paper, we developed a complete Hessian matrix formulation for anisotropic diffusion to achieve 2nd-order accuracy and numerical stability. Specifically, the 2nd-order reproducing approximation of the complete Hessian matrix is used for the coordinate transformation to obtain the anisotropic diffusion operator. The formulation is further applied for simulating diffusion processes in thin structures, where anisotropic resolution is handled with the adaptive SPH method. To validate the proposed formulation, firstly, anisotropic contaminant diffusion in a fluid is simulated, showing strong consistency with analytical solutions and producing smooth, oscillation-free results even in the presence of discontinuities. Secondly, the diffusion of a physical field with a predefined initial distribution is modeled using anisotropic resolution and the adaptive SPH method. Across various anisotropic ratios, the accuracy is examined in relation to the truncation error. Finally, this formulation is applied to modeling practical applications, including fluid-structure interaction in a thin porous membrane and anisotropic transmembrane potential transport in the left ventricle. These results demonstrate that the proposed formulation can accurately and stably solve complex anisotropic diffusion problems and diffusion problem using anisotropic kernel and spatial resolutions in diverse physical settings.