A target construction methodology for mesh quality improvement

In the target matrix optimization paradigm (TMOP), it has long been understood that one must create a set of target matrices before the mesh can be optimized. But there is still no general method to create correct, effective targets in response to a specific mesh quality improvement goal. The TMOP literature describes how certain sets of target matrices can be used to control the shape or size of mesh elements, but those examples address only a fraction of the problems that can occur in mesh quality improvement and were not derived from a general framework for target matrix construction. In this work, a general method of target construction is introduced based on an independent set of geometric parameters that are intrinsic to the Jacobian matrices upon which TMOP is based. The parameters enable a systematic approach to target definition and construction. The approach entails two parts. The first part defines correspondences between available primary data (stuff about the mesh and/or the physical solution) and secondary data (e.g., a field of error estimates). Once the correspondences are established, the primary data are processed into intermediate field data existing on mesh sample points. The second part creates a model that represents the values of the geometric target parameters as functions of the secondary data. The model is then tested numerically to establish model constants and effectiveness. This systematic approach to target construction is illustrated in a set of examples to show how it can be applied to common problems in mesh optimization such as equalization of geometric properties, preservation of existing good quality, and adaptation of the mesh to the physical solution. The result is a systematic method of target construction for TMOP that can be applied to a wide variety of planar and volume mesh quality improvement tasks.