In this thesis we develop 2D parallel unstructured mesh adaptation methods for the solution of partial differential equations (PDEs) by the finite element method (FEM). Additionally, we develop a novel block preconditioner for the iterative solution of the linear systems arising from the finite element discretisation of the Föppl-von Kàrmàn equations. Two of the problems arising in the numerical solution of PDEs by FEM are the memory constraints that limit the solution of large problems, and the inefficiency of solving the associated linear systems by direct or iterative solvers. We initially focus on mesh adaptation, which is a memory demanding task of the FEM. The size of the problem increases by adding more elements and nodes to the mesh during mesh refinement. In problems involving a large number of elements, the problem size is limited by the memory available on a single processor. In order to be able to work with large problems, we use a domain decomposition approach to distribute the problem over multiple processors. One of the main objectives of this thesis is the development of 2D parallel unstructured mesh adaptation methods for the solution of PDEs by the FEM in a variety of problems; including domains with curved boundaries, holes and internal boundaries. Our newly developed methods are implemented in the software library oomph-lib, an open-source object oriented multi-physics software library implementing the FEM. We validate and demonstrate their utility in a set of increasingly complex problems ranging from scalar PDEs to fully coupled multi-physics problems. Having implemented and validated the infrastructure which facilitates the finite-element-based discretisation of PDEs in a distributed environment, we shift our focus to the second problem concerning this thesis and one of the major challenges in the computational solution of PDEs: the solution of the large linear systems arising from their discretisation. For sufficiently large problems, the solution of their associated linear system by direct solvers becomes impossible or inefficient, typically because of memory and time constraints. We therefore focus on preconditioned Krylov subspace methods whose efficiency depends crucially on the provision of a good preconditioner. These preconditioners are invariably problem dependent. We illustrate their application and development in the solution of two elasticity problems which give rise to relatively large problems. First we consider the solution of a linear elasticity problem and compute the stress distribution near a crack tip where strong local mesh refinement is required. We then consider the deformation of thin plates which are described by the nonlinear Föppl-von Kàrmàn equations. A key contribution of this work is the development of a novel block preconditioner for the iterative solution of these equations, we present the development of the preconditioner and demonstrate its practical performance.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:727916 |
| Date | January 2016 |
| Creators | Perez Sansalvador, Julio |
| Publisher | University of Manchester |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://www.research.manchester.ac.uk/portal/en/theses/parallel-unstructured-mesh-adaptation-and-applications(26248d4d-48a6-4101-a687-004218e39cb4).html |
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