A Comparative Study Of Three Mesh-free Methods

This thesis aims to comparatively study three mesh-free methods from the theoretical and numerical points of view,Smoothed Particle Hydrodynamics(SPH),Dissipative Particle Dynamics(DPD)and Smoothed Dissipative Particle Dynamics(SDPD).Theoretically,SPH,DPD and SDPD have the similar formulation,which includes the conservative force(describing the fluid compressibility)and the dissipative force(describing the fluid viscosity).SPH is a macroscopic method,but DPD and SDPD are mesoscopic,such that the latter two have a random force(describing the system temperature).DPD has no rigorous theoretical foundation,because it is based on the assumption that DPD has a fixed formulation structure.SPH and SDPD are derived from discretizing the Navier-Stokes equations.As a result,DPD has no clear physical meaning,and its parameters are determined by an empirical formula.SPH and DPD have clear physical meaning,and their parameters are set by the real physical parameters.In addition,all of them have no correspondence with the molecules.This is reasonable for SPH,because it is is a macroscopic method.However,it is a challenging issue for DPD and SDPD,which has attracted a lot of attention recently.SDPD is not only an improvement of DPD to address its issue of no clear physical meaning,but also an extension of SPH in the mesoscale.It has the advantages of both SPH and DPD,and is promising in various mesoscopic simulations,such as polymer,biomedical fields,etc.Numerically,we carry out four benchmark cases: self-diffusion,Couette flow,Poiseuille flow and cavity flow,and further comparatively study three mesh-free methods.Moreover,we also simulate blood flow in a microvessel by SDPD due to its superiority,to show the application of SDPD.In the self-diffusion problem,SPH has no self-diffusion because it has no random force.Both DPD and SDPD give the similar self-diffusion coefficients with the theoretical results,and SDPD is closer than DPD to the theoretical values.In the Couette,Poiseuille and cavity flow,all of the three methods can generate the similar velocity profiles with the exact solutions.SPH shows more smooth results due to no random force,while DPD and SDPD present a slight oscillation around the exact solutions.SPH is not involved with the system temperature,whereas DPD and SDPD can keep an almost system temperature.More importantly,DPD simulation is very complex,because there are a lot of parameters required setting,and these parameters have no correspondences with the real physical quantities.In DPD and SDPD,however,these parameters are given directly by the physical quantities.It can be concluded that SDPD has obvious advantages.In view of this,we use SDPD to simulate the movement,deformation and aggregation of red blood cells in microvessel,to show an application of SDPD in cell simulations.