An Operator Splitting Algorithm For The Three Dimensional Advection Diffusion Equation PDF Download

The goal of this work was to develop a numerical algorithm for simulation of 3-D unsteady advection-dominated transport problems using unstructured grids, and to apply the algorithm to simulate the mass transport of some physiologically-significant species in several arterial geometries. A 3-D Characteristic Galerkin scheme for solving the hyperbolic advection equation was developed and tested. The scheme projects information from the Lagrangian (departure) grid onto the Eulerian grid at Gauss quadrature points of the Eulerian elements. This Gaussian quadrature based projection makes the scheme easily and efficiently extendible to three dimensions. A variable time step technique was devised and implemented to treat the Dirichlet inlet boundary conditions. Two efficient algorithms were devised for searching for feet of the characteristics within the finite element mesh. The properties of the Characteristic Galerkin scheme were examined by several 2-D numerical tests. The scheme exhibited good phase, Accuracy and stability behaviour. An algorithm based on operator splitting was developed for solving unsteady, advection-dominated transport problems. The algorithm incorporates the 3-D Characteristic Galerkin scheme to treat advection terms, and a standard Galerkin treatment of the diffusion terms. Up to third-order operator-splitting was implemented and validated against several analytical solutions. The algorithm showed consistent error performances and demonstrated good performance in modeling advection-dominated transport problems. Capturing the thin boundary layers formed near the walls or around objects in advection-dominated problems can be a major difficulty. It is necessary from an efficiency viewpoint to use high aspect ratio elements that are thin perpendicular to the boundary/object wall and wide along it. A general strategy was devised and implemented for generating stretched tetrahedral elements needed to efficiently capture the thin boundary layers. The algorithms described above were employed to calculate mass transfer patterns in two symmetric and asymmetric stenotic arterial models with equal cross-sectional area reduction. The complex flow field due to the stenosis in the asymmetric case resulted in mass transfer patterns which were substantially different than those exhibited by the axisymmetric stenosis. This implies that accurate representation of the arterial geometries is essential in mass transfer studies. The algorithms were also applied to calculate mass transfer in a right coronary artery model with a realistic geometry. The results suggest that anatomically correct, patient-specific arterial models are needed before any conclusion is drawn towards the existence of a link between the mass transport variations and atherosclerosis.