Keywords: mesh generation, mesh adaptation, metric tensor, anisotropic metric, metric source, control space, surface curvature, model geometry.

Mesh generators have to fulfil a number of conditions in order to construct proper discretizations of models. An important requirement is the ability to precisely adjust the size, shape and quality of elements in the selected areas of the modelled domain. Depending on application, the anisotropic elements may also be created.

In the proposed approach, prescription of size and shape of the generated elements is introduced through the use of appropriate metrics, subjected to local changes in various sub-domains of the three-dimensional object [1]. Throughout the process of mesh construction and adaptation, different representations of the metrics are applied and stored in an auxiliary discrete structure of control space.

Another issue is the selection of metric sources for generation of a proper volume mesh. The goal of the work is an automated recognition of the properties of the discretized domain in order to create meshes with elements of adequate quality.

For each surface patch defining the model boundary, the metric from the curvature of the surface and its contours is determined and included in the control space structure [2]. In case of parametric patches the surface curvature can be determined directly, using the second fundamental form. In some cases the quality of the parameterization of the surface patch may be insufficient and it is necessary to use another method of local curvature approximation: either directly from the discrete points on the surface or using a reparameterization based on the local surface approximation. A similar procedure is necessary if the volume mesh boundary is defined in the discrete form of a surface mesh.

Other metric data based on the geometry of the model may be also gathered from the information sources, such as short features or proximity of edges and faces in the model. Depending on the application, the control space may also include metric sources resulting from the simulation or prescribed by the user.

The construction of a three-dimensional mesh proceeds hierarchically; starting from a surface mesh (generated or given directly), and then creating a volume mesh on the basis of the surface discretization. At each of the intermediate steps additional metric information is collected resulting from the geometry of the model and its surface. Some data may be gathered during the initial model inspection at the beginning of the discretization process. Such sources are used directly to form the initial control space. Other sources may become available only after completion of some intermediate meshing steps. In such cases the additional metric data are employed to update the control space, which is then used to properly adjust the current mesh and also for supervising the subsequent meshing steps.

In the paper the techniques of retrieving the necessary sizing information from geometric description of the discretized model and its introduction into the control space structure are emphasized.

1

T. Jurczyk, "Efficient Algorithms of Automatic Discretization of Non-Trivial Three-Dimensional Geometries and its Object-Oriented Implementation", PhD thesis, AGH University of Science and Technology, Kraków, Poland, November 2007.

2

T. Jurczyk, B. Glut, "Adaptive Control Space Structure for Anisotropic Mesh Generation", in Proc. of ECCOMAS CFD 2006 European Conference on Computational Fluid Dynamics, Egmond aan Zee, The Netherlands, 2006.

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