Keywords: mesh generation, adaptation, tetrahedral mesh, anisotropic metric, control space.

Efficient finite element simulation of processes characterised by the presence of several coupled physical models and complex geometry, requires an application of advanced adaptive algorithms. Such methods have a substantial computational cost and their implementation as a fully automated procedure is a difficult task. The aim of the work carried out by the authors is to develop remeshing algorithms with automatic selection of the area in which the mesh is subject to modification.

In a single step of adaptation the remeshing techniques can be categorised into three groups depending on the extent of area where the modification of mesh is required: local transformation of the mesh limited to several adjacent elements (performed by insertion of additional nodes, repositioning of nodes, small topological modification such as swapping of edges and faces in neighbouring elements), remeshing of subdomain formed by a cavity created by removing a number of tetrahedra, and the generation of a new mesh for the whole domain.

This paper concerns the aspects related to the effectiveness of the retriangulation methods mentioned above. The analysis is based on the mesh generator developed by the authors. The generator applies an iterative triangulation technique based on a local non-Euclidean metric, which allows the creation of meshes of varied density and anisotropic shape of elements. Information about the metric and other parameters affecting the shape and size of the elements are stored in an additional structure called the control space (CS). The generator was also equipped with an adaptive control space, which enables the creation and utilisation of different CS structures. This feature facilitates the introduction of adaptation procedures into the mesh generator. The only requirement is to transform the error estimator dedicated to the given simulation into an appropriate metric tensor form.

In the adaptation process the information about the desired shape and size of elements is collected into a new updated control space based on the previous step of the adaptation procedure. The extent of the necessary mesh modifications can be established in two ways: by comparing the control space from two subsequent adaptation steps or by evaluating the compliance of mesh to a new updated control space. After identification of the mesh sub-areas, where the mesh needs to be modified, one of the remeshing procedures mentioned above is selected.

In the paper the methods of determining the extent of the area for retriangulation and several techniques of remeshing are discussed. The concept of the described algorithms was illustrated using two three-dimensional examples. The computational costs of such an adaptation procedure were also analysed, which showed that the technique of local retriangulation is the most time-efficient. It should be noted that the adaptation methods presented allow the mesh to be refined locally, coarsened, and also anisotropic meshes to be generated.

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