The optimisation of finite element meshes

• Main Author: Kelly, Alan
• Format: Dissertation
• Language: English
• Published: University of Glasgow 2014
• Subjects: T Technology (General)

Among the several numerical methods which are available for solving complex problems in many areas of engineering and science such as structural analysis, fluid flow and bio-mechanics, the Finite Element Method (FEM) is the most prominent. In the context of these methods, high quality meshes can be crucial to obtaining accurate results. Finite Element meshes are composed of elements and the quality of an element can be described as a numerical measure which estimates the effect that the size/shape of an element will have on the accuracy of an analysis. In this thesis, the strong link between mesh geometry and the accuracy and efficiency of a simulation is explored and it is shown that poor quality elements cause both interpolation errors and poor conditioning of the global stiffness matrix. Numerical optimisation is the process of maximising or minimising an objective func- tion, subject to constraints on the solution. When this is applied to a finite element mesh it is referred to as mesh optimisation, where the quality of the mesh is the objec- tive function and the constraints include, for example, the domain geometry, maximum element size, etc. A mesh optimisation strategy is developed with a particular focus on optimising the quality of the worst elements in a mesh. Using both two and three dimensional examples, the most efficient and effective combination of element quality measure and objective function is found. Many of the problems under consideration are characterised by very complex geometries. The nodes lying on the surfaces of such meshes are typically treated as unmovable by most mesh optimisation software. Techniques exist for moving such nodes as part of the mesh optimisation process, however, the resulting mesh geometry and area/volume is often not conserved. This means that the optimised mesh is no longer an accurate discretisation of the original domain. Therefore, a method is developed and demonstrated which optimises the positions of surface nodes while respecting the geometry and area/volume of a domain. At the heart of many of the problems being considered is the Arbitrary Lagrangian Eulerian (ALE) formulation where the need to ensure mesh quality in an evolving mesh is very important. In such a formulation, a method of determining the updated nodal positions is required. Such a method is developed using mesh optimisation techniques as part of the FE solution process and this is demonstrated using a two-dimensional, axisymmetric simulation of a micro-fluid droplet subject to external excitation. While better quality meshes were observed using this method, the time step collapsed resulting in simulations requiring significantly more time to complete. The extension of this method to incorporate adaptive re-meshing is also discussed.