This thesis addresses the solution of the steady and unsteady incompressible Navier-Stokes (INS) equations employing finite element discretizations based on Galerkin variational formulations. The goal is the simulation of blood flow (hemodynamics) in realistic geometries reconstructed in-vivo. Advection-dominated incompressible flows, occurring at physiologic conditions in large arteries, constitute a challenging class of problems both in terms of numerical stability and computational cost. In the context of unsteady incompressible fluid flow simulations a new formulation based on the pressure-correction algorithm featuring discontinuous velocity and continuous pressure a is proposed and validated with numerical experiments. In this configuration we are able exploit the ability of dG to deal with convection-dominated flows maintaining a less expensive Galerkin discretization for the pressure projection step and, therefore, obtaining an effective solution process. The ability to simulate the blood flow behaviour in complex geometries reconstructed in-vivo employing high-order finite element discretizations has been demonstrated. In this context, discontinuous Galerkin (dG) methods offer many advantages: LBB stable equal-order discretizations can be devised, the extension to arbitrary unstructured and nonconforming grids is straightforward, and the resulting discretization displays an increased stability in the high Reynolds regimes. Flexibility, however, comes at a price. In particular, memory requirements as well as the increased computational cost have discouraged wide adoption of these methods up to now. In this thesis we consider effective strategies to overcome these limitations and make available to CFD practioners the flexibility and the good properties previously described. Coupled variables dG discretization of the INS equations are combined with efficient pseudo-transient continuation methods in order to exploit fully implicit discretizations of the non-linear term in the achievement of steady state solutions. Trivial implementation of hp-adaptivity on non-conforming meshes is one of the most attractive and documented dG formulations pros. The benefits of future based adaptive mesh refinement are investigated on simple two-dimensional test cases.
(2010). Galerkin methods for incompressible fluid flow simulations: application to hemodynamics [doctoral thesis - tesi di dottorato]. Retrieved from http://hdl.handle.net/10446/610
Galerkin methods for incompressible fluid flow simulations: application to hemodynamics
BOTTI, Lorenzo Alessio
2010-04-09
Abstract
This thesis addresses the solution of the steady and unsteady incompressible Navier-Stokes (INS) equations employing finite element discretizations based on Galerkin variational formulations. The goal is the simulation of blood flow (hemodynamics) in realistic geometries reconstructed in-vivo. Advection-dominated incompressible flows, occurring at physiologic conditions in large arteries, constitute a challenging class of problems both in terms of numerical stability and computational cost. In the context of unsteady incompressible fluid flow simulations a new formulation based on the pressure-correction algorithm featuring discontinuous velocity and continuous pressure a is proposed and validated with numerical experiments. In this configuration we are able exploit the ability of dG to deal with convection-dominated flows maintaining a less expensive Galerkin discretization for the pressure projection step and, therefore, obtaining an effective solution process. The ability to simulate the blood flow behaviour in complex geometries reconstructed in-vivo employing high-order finite element discretizations has been demonstrated. In this context, discontinuous Galerkin (dG) methods offer many advantages: LBB stable equal-order discretizations can be devised, the extension to arbitrary unstructured and nonconforming grids is straightforward, and the resulting discretization displays an increased stability in the high Reynolds regimes. Flexibility, however, comes at a price. In particular, memory requirements as well as the increased computational cost have discouraged wide adoption of these methods up to now. In this thesis we consider effective strategies to overcome these limitations and make available to CFD practioners the flexibility and the good properties previously described. Coupled variables dG discretization of the INS equations are combined with efficient pseudo-transient continuation methods in order to exploit fully implicit discretizations of the non-linear term in the achievement of steady state solutions. Trivial implementation of hp-adaptivity on non-conforming meshes is one of the most attractive and documented dG formulations pros. The benefits of future based adaptive mesh refinement are investigated on simple two-dimensional test cases.File | Dimensione del file | Formato | |
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