Simulating anisotropic diffusion processes with smoothed particle hydrodynamics

TL;DR

This work tackles the challenge of accurately simulating anisotropic diffusion within a mesh-free SPH framework by developing a full Hessian-based second-derivative operator and incorporating an adaptive anisotropic kernel via a transformation tensor $\mathbf{G}$. By transforming the diffusion operator to an isotropic form in a transformed coordinate system and then mapping back, the method achieves robust, second-order accuracy for diffusion tensors $\mathbf{D}$ and accounts for cross-derivative effects. The approach is validated across 2D and 3D diffusion problems, fluid-structure interactions in thin membranes, and cardiac electrophysiology, demonstrating stability, reduced spurious oscillations, and significant computational savings when using nonisotropic resolution. The resulting ASPH framework enables efficient and accurate simulation of complex anisotropic diffusion problems in porous media, thin structures, and cardiac tissue, with potential for broad scientific and engineering impact.

Abstract

Diffusion problems with anisotropic features arise in the various areas of science and engineering fields. As a Lagrangian mesh-less method, SPH has a special advantage in addressing the diffusion problems due to the the benefit of dealing with the advection term. But its application to solving anisotropic diffusion is still limited since a robust and general SPH formulation is required to obtain accurate approximations of second derivatives. In this paper, we modify a second derivatives model based on the SPH formulation to obtain a full version of Hessian matrix consisting of the Laplacian operator elements. To verify the proposed SPH scheme, firstly, the diffusion of a scalar which distributes following a pre-function within a thin structure is performed by using anisotropic resolution coupling anisotropic kernel. With various anisotropic ratios, excellent agreements with the theoretical solution are achieved. Then, the anisotropic diffusion of a contaminant in fluid is simulated. The simulation results are very consistent with corresponding analytical solutions, showing that the present algorithm can obtain smooth solution without the spurious oscillations for contaminant transport problems with discontinuities, and achieve second-order accuracy. Subsequently, we utilize this newly developed SPH formulation to tackle the problem of the fluid diffusion through a thin porous membrane and the anisotropic transport of transmembrane potential within the left ventricle, demonstrating the capabilities of the proposed SPH framework in solving the complex anisotropic problems.

Figures (22)

• Figure 2.1: Partially saturated porous medium.
• Figure 2.2: Finite deformation process on a body $\mathcal{B}$.
• Figure 4.1: (a) Isotropic with spherical smoothing kernel and (b) anisotropic resolutions with elliptical smoothing kernel.
• Figure 6.1: 2D nonoisotropic diffusion: comparisons of the $\frac{d\phi}{dt}$ contour obtained from (a) the analytical solution and (b)-(f) the present ASPH method with different anisotropic ratios.
• Figure 6.2: Diffusion in a rectangle: the time evolution of $\phi$ contour at different time instants using ASPH with anisotropic ratio r = 4.0, $N_y$ = 20.