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A Multilevel Approach for Obtaining Locally Optimal Finite Element Meshes
Rashid Mahmood and Peter K Jimack

Introduction
Automatic mesh generation is an important computational tool for the finite element analysis of a wide variety of engineering problems ranging from structural analysis through to computational fluid dynamics for example. For many of these problems the use of  unstructured grids offers many advantages over structured grids, such as permitting the straightforward triangulation of geometrically complex domains or allowing the mesh density to be adapted according to the behaviour of the solution. We are dealing mainly with the latter of these properties of unstructured grids: the natural manner in which numerous forms of mesh adaptivity are permitted.

    Broadly speaking mesh adaptivity algorithms may categorized as belonging to one of two classes. The first of these, generally referred to as h-refinement, involves adding vertices and elements to the mesh in some manner. This may be through local refinement or through more global remeshing but has the general aim of increasing the number of vertices and elements in those regions of the domain where some measure of the error is unacceptably high. The second class of approach, often referred to as r-refinement, adapts the mesh in such a way that the number of vertices and elements remains essentially unchanged. This is typically achieved through the use of node movement, where the mesh is continuously deformed so as to increase the density of vertices in those regions of the domain with the highest errors, or through the use of edge swapping, where the number and position of the vertices is held fixed but the way in which they are connected together is allowed to change.

   In this project we are considering elliptic partial differential equations which have an equivalent  variational formulation and are trying to apply different combinations of mesh adaptivity in two space dimensions. For variational problems, the fact that the exact solution minimises the energy functional provides a natural optimality criterion for the design of computational grids. Our lines of research are in the direction that given a variational problem with no prior knowledge of the behaviour of the exact solution and a finite element method, to construct an automatic procedure for finding a finite element mesh with a given  number of elements such that the energy corresponding to the finite element solution is minimum and the mechanism in finding such an optimal mesh should be efficient and robust.

    We have generalized the work of [2] to  combine different r-refinement approaches with a suitable local h-refinement strategy  to produce better locally optimal grids (in terms of energy minimisation) than using either strategy alone. The approach taken is to start with a very coarse mesh which is optimised using r-refinement. This is then refined locally to create a new mesh with a greater number of elements and vertices which can itself be optimised. By repeating this process a number of times a hierarchy of locally optimal meshes is obtained. Since the initial mesh at each level of the hierarchy is produced by local refinement of an optimal mesh at the previous level it follows that this typically provides a reliable starting point when optimising the new mesh.

Numerical Results

Example 1:
The first example we considered is the displacement field for a two dimensional linear elastic model of an overhanging cantilever beam bearing a point load, as illustrated in Figure 1. The left half of the lower boundary is fixed whilst the rest of the boundary is free.

Figure1: An illustration of the overhanging cantilever beam with a vertical point load at the end of  the cantilever.


Figure 2: A uniform mesh of 1024 elements and the corresponding locally optimised mesh using r-refinement.

    Figure 2 illustrates two meshes of 1024 elements: the first is a uniform mesh and the second is an optimised grid using r-refinement. The energies of the solutions on these meshes are  -0.306791 and  -0.329249 respectively.

Figure 3: A sequence of meshes obtained by r-refinement of an initial coarse grid and then combination of local h-refinement followed by r-refinement.

    Figure 3 is showing a sequence of meshes obtained by combining r-refinement with local h-refinement . The problem is initially solved on a uniform coarse grid containing just 64 elements. This grid is then optimised using r-refinement and the total energy reduces from -0.201352 to -0.253210, and the corresponding mesh is the first mesh in figure 3, whilst the second and third meshes contain 224 and 455 elements respectively. The total energies of the solutions on these two meshes are -0.308351 and 0.363313. This simple experiment serves to illustrate the superiority of the use of multilevel adaptivity.

Example 2:
The second example has a solution which is singular at the origin. The domain is the unit disc with a 45 degree segment removed. Dirichlet boundary conditions consistent with the exact solution are applied on the boundary. It is clear from Figure 4,5 and 6, that the multilevel approach is doing far better than the conventional approaches of adaptivity.

Figure 4: A uniform mesh of 1792 elements and the corresponding locally optimised mesh, having energies 0.438164 and 0.405547 respectively.


Figure 5: A pair of meshes of 1437 elements obtained using local one-to-four h-refinement ( top left) followed by r-refinement having energies 0.407613 and 0.398523 and a pair of meshes of 1413 elements obtained using local one-to-two h-refinement (bottom left) followed by r-refinement having energies 0.402199 and 0.398123 respectively.
 


Figure 6: A sequence of meshes obtained by r-refinement on an initial coarse grid (top left) and then combination of local h-refinement followed by r-refinement having elements 28, 107, 255 and 1275 with energies 0.549242, 0.431777, 0.402413 and 0.395183 respectively.

Ongoing Work

       ® Optimising the interaction of the r-refiniment and the h-refinement strategies.
       ® Solving three-dimensional problems.