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.

翻译：各向异性扩散问题广泛存在于科学与工程领域的诸多方面。作为一种拉格朗日无网格方法，SPH（光滑粒子流体动力学）因其在处理对流项方面的优势，在求解扩散问题中具有独特优势。然而，由于需要稳健且通用的SPH公式来精确逼近二阶导数，其在各向异性扩散求解中的应用仍受限。本文基于SPH公式改进了一种二阶导数模型，获得了包含拉普拉斯算子元素的完整Hessian矩阵。为验证所提出的SPH格式，首先采用各向异性核函数耦合各向异性分辨率的方法，模拟了薄壁结构内按预设函数分布的标量扩散过程。在不同各向异性比率下，所得结果均与理论解高度吻合。随后，模拟了流体中污染物的各向异性扩散。模拟结果与相应解析解高度一致，表明该算法能在具有间断性的污染物输运问题中获得光滑解且无伪振荡，并达到二阶精度。最后，我们运用新开发的SPH公式处理了流体通过多孔薄膜的扩散问题以及左心室内跨膜电位的各向异性输运问题，证明了所提出的SPH框架在求解复杂各向异性问题方面的能力。