Simulation of anisotropic diffusion processes in fluids with smoothed particle hydrodynamics

Abstract

SUMMARYAnisotropic diffusion phenomenon in fluids is simulated using smoothed particle hydrodynamics (SPH). A new SPH approximation for diffusion operator, named anisotropic SPH approximation for anisotropic diffusion (ASPHAD), is derived. Basic idea of the derivation is that anisotropic diffusion operator is first approximated by an integral in a coordinate system in which it is isotropic. The coordinate transformation is a combination of a coordinate rotation and a scaling in accordance with diffusion tensor. Then, inverse coordinate transformation and particle discretization are applied to the integral to achieve ASPHAD. Noting that weight function used in the integral approximation has anisotropic smoothing length, which becomes isotropic under the inverse transformation. ASPHAD is general and unique for both isotropic and anisotropic diffusions with either constant or variable diffusing coefficients. ASPHAD was numerically examined in some cases of isotropic and anisotropic diffusions of a contaminant in fluid, and the simulation results are very consistent with corresponding analytical solutions.

An integrative smoothed particle hydrodynamics framework for modeling cardiac function

“…Different from the previous strategies for the discretization of diffusion equation [29,30], we employ and modify the anisotropic SPH dicretization proposed by Tran-Duc et al [25]. Following Ref.…”

“…Then, we introduce a splitting reaction-by-reaction method [24] combined with quasi-steady-state (QSS) solver to capture the stiff dynamics of the transmembrane potential and the gating variables of the ionic model governed by nonlinear ordinary differential equations (ODEs). Furthermore, an anisotropic SPH discretization for diffusion equation derived by Tran-Duc et al [25] is modified by introducing a linear operator to improve the computational efficiency and using a correction kernel matrix to improve the numerical accuracy. The total Lagrangian SPH formulation is employed to predict the large deformations and the strongly anisotropic behavior of the myocardium.…”

Consistent SPH simulations of the anisotropic dispersion of a contaminant plume

“…Parallel to grid-based schemes, the method of Smoothed Particle Hydrodynamics (SPH) has also been used to simulate anisotropic dispersion [10,11,12,13,14]. SPH is a meshfree Lagrangian method based on interpolation theory.…”

“…In addition to the work of Herrera and collaborators [10,11,12], efforts to minimize and eventually remove unphysical oscillations and negative concentrations in SPH simulations of anisotropic dispersion have been made by Avesani et al [13], Tran-Duc et al [14], and more recently by Alvarado-Rodríguez et al [18]. In particular, Avesani et al [13] proposed a modified version of standard SPH based on a Moving-Least-Squares Weighted-Essentially-Non-Oscillatory (MWSPH) reconstruction technique on moving points.…”

“…They found that compared to standard SPH, the numerical solution improves even at high anisotropies of the local dispersion tensor with negative concentrations limited to about 10 −7 C 0 , where C 0 is the initial concentration. Working in the same line, Tran-Duc et al [14] reported accurate simulations of anisotropic dispersion based on a different SPH scheme, called anisotropic SPH approximation for anisotropic diffusion (ASPHAD). In ASPHAD, the diffusion operator is first approximated by an integral in a coordinate system in which it is isotropic and then by means of an inverse transformation of the integral the anisotropic character of the diffusion operator is recovered.…”