In this research, the finite element method (FEM) was used to solve the nonlinear, incompressible, transient, three dimensional Navier-Stokes equations in their non-conservative form. Linear tetrahedron elements were employed with the elegant, equal order interpolation for both pressure and velocity. The characteristic based split scheme was formulated in a fully implicit manner to circumvent the time step restrictions of the classical explicit formulations. The monolithic (single step, fully coupled solution procedure for pressures and velocity) form of the CBS scheme was also derived and its suitability was positively demonstrated. Casting the CBS scheme in a monolithic framework, results in the generation of a pressure stabilization term in the mass conservation equation, thereby circumventing the LBB restriction by the elimination of the zero pressure block. An account of all the steps involved in discretizing the Navier-Stokes equations (both in split and monolithic frameworks) was presented in meticulous detail, which included the derivation of the convective and pressure stabilization terms, linearization of the non-linear terms and the consequent derivation of the highly efficient analytical jacobian matrix, along with the temporal and spatial discretizations of the corresponding terms. The monolithic and the split version of the CBS scheme were integrated into a parallel, scalable and extensible Fortran90 software called IFENs. The development of IFENs started during the course of this research and all of its components have been designed and implemented by the author of this thesis. Multi processor parallelism was achieved using the Intel implementation of the most widely used/preferred, Message Passing Interface (MPI) standard. The parallel support needed for the use of a variety of parallel, linear, iterative solvers belonging to the Krylov subspace family (e.g. GMRES and its variants, CG, BiCG, BiCG- stab, etc.), parallel non linear solvers belonging to the Newton-Krylov family (line search newton, trust region newton, nonlinear GMRES, etc.) and parallel preconditioners (incomplete LU, Additive Shwarz Method - ASM, algebraic multigrid, etc.), was provided by the incorporation of PETSc into IFENs. PETSc is a state of the art, non-trivial toolkit, which represents a collection of several parallel libraries useful in high performance scientific computing. Keeping in mind the specific requirements of IFENs, a custom mesh partitioner was implemented. It operated on meshes that were renumbered using bandwidth reducing algorithms like Revere Cuthill Mckee. The possibility of using established domain decomposition libraries like ParMETIS was explored and demonstrated to be counter productive for the demands of this research. After the preliminary testing and validation of the procedures adopted before and during the execution of IFENs, large, high definition domains representative of human arteries (specifically, carotid bifurcations, found in the neck) were considered and the complete incompressible set of Navier-Stokes equations were solved for pressure and velocity fields. During the tenure of this research more than 1000 recorded parallel test cases were executed to test various components of IFENs, as well as various simulations representative of a wide variety of problems. IFENs can easily handle meshes with tens of millions of elements. The largest mesh used for the purpose of this research contained 14.58 million tetrahedrons and 2.489 million nodes, which on average required just 7 minutes per timestep, while executing the classical split framework of the CBS scheme. Results from the simulation of 9 carotid meshes, representative of 4 carotid geometries were presented and found to be in good agreement with the available ultrasound data. The flow fields were analysed and post processed using different techniques for each case. The haemodynamic wall parameters like time averaged wall shear stress and oscillatory shear index were calculated and mapped onto the corresponding boundary nodes. The region in the carotid bifurcation susceptible to the deposition of plaques and consequent stenosis were pointed out and other anomalies were highlighted.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:678367 |
| Date | January 2014 |
| Creators | Kapoor, Amarpal Singh |
| Publisher | Swansea University |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://cronfa.swan.ac.uk/Record/cronfa42216 |
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