Parallel Unstructured Mesh Adaptation and Applications

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Abstract

In this thesis we develop 2D parallel unstructured mesh adaptationmethods for the solution of partial differential equations (PDEs) bythe finite element method (FEM). Additionally, we develop a novelblock preconditioner for the iterative solution of the linear systemsarising from the finite element discretisation of the Föppl-von Kàrmànequations.Two of the problems arising in the numerical solution of PDEs by FEMare the memory constraints that limit the solution of large problems,and the inefficiency of solving the associated linear systems bydirect or iterative solvers.We initially focus on mesh adaptation, which is a memory demandingtask of the FEM. The size of the problem increases by adding moreelements and nodes to the mesh during mesh refinement. In problemsinvolving a large number of elements, the problem size is limited bythe memory available on a single processor.In order to be able to work with large problems, we use a domaindecomposition approach to distribute the problem over multipleprocessors. One of the main objectives of this thesis is thedevelopment of 2D parallel unstructured mesh adaptation methods forthe solution of PDEs by the FEM in a variety of problems; includingdomains with curved boundaries, holes and internal boundaries. Ournewly developed methods are implemented in the software libraryoomph-lib, an open-source object oriented multi-physicssoftware library implementing the FEM. We validate and demonstratetheir utility in a set of increasingly complex problems ranging fromscalar PDEs to fully coupled multi-physics problems.Having implemented and validated the infrastructure which facilitatesthe finite-element-based discretisation of PDEs in a distributedenvironment, we shift our focus to the second problem concerning thisthesis and one of the major challenges in the computational solutionof PDEs: the solution of the large linear systems arising from theirdiscretisation.For sufficiently large problems, the solution of their associatedlinear system by direct solvers becomes impossible or inefficient,typically because of memory and time constraints. We therefore focuson preconditioned Krylov subspace methods whose efficiency dependscrucially on the provision of a good preconditioner. Thesepreconditioners are invariably problem dependent. We illustrate theirapplication and development in the solution of two elasticity problemswhich give rise to relatively large problems.First we consider the solution of a linear elasticity problem andcompute the stress distribution near a crack tip where strong localmesh refinement is required. We then consider the deformation of thinplates which are described by the nonlinear Föppl-von Kàrmànequations. A key contribution of this work is the development of anovel block preconditioner for the iterative solution of theseequations, we present the development of the preconditioner anddemonstrate its practical performance.

Keyword(s)

Elasticity; Föppl-von Kàrmàn; Parallel unstructured mesh adaptation; Preconditioner; oomph-lib

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