Adaptive Mesh Generation in 3 Dimensions by Means of a Delaunay Based Method. Applications to Mechanical Problems.

Abstract

Mesh adaptation is recognized as a powerful tool to compute accurate solution while minimizing the required resources (CPU time and memory space), therefore avoiding using parallel computing. It is also a way to accurately capture the physical behavior of the PDE problem in hand and, in some cases, this is the only way to access to a reasonable solution.

Mesh adaptation in two dimensions can be considered as mature and is used in various problems. Right now, in three dimensions, the question is much more tedious and only a limited number of works can be reported.

Mesh adaptation can be considered in two different ways. The first makes use of local modification of the current mesh so as to adapt it. Modification tools include well known operators such as point relocation, collapse (mesh coarsening), edge flips, point addition (mesh enrichment). Relatively easy to implement, such methods proved to give nice results in a number of cases but are not so flexible in specific when anisotropic features are desired.

The second is based on the full generation of a new (adapted) mesh based on the current one and metric data provided at the nodes of this mesh. The generation method is then a variant (widely different in various aspects) of the well know mesh generation method. The aim being not only to mesh at the best a given domain but to match the given metric, which is much more demanding.

We are concerned with this second approach and we propose a Delaunay based mesh generation method capable to complete adapted meshes. The mesh generation aspect is driven by metric data (element size and directional specification) which are defined by means of error estimates. Isotropic and anisotropic meshes can be produced.

Concrete application examples will demonstrate the flexibility of the proposed method, show the low cost of the approach which compares well with the first approach.