Abstract
We aim to tackle the challenge of generating unstructured high-order meshes of complex three-dimensional bodies, which remains a significant bottleneck in the wider adoption of high-order methods. In particular we show that by adopting a variational approach to the generation process, many of the current popular high-order generation methods can be encompassed under a single unifying framework. This allows us to compare the effectiveness of these methods and to assess the quality of the meshes they produce in a systematic fashion. We present a detailed overview of the theory and formulation of the variational framework, and we highlight how such formulation can be effectively exploited to yield a highly-efficient parallel implementation. The effectiveness of this approach is examined by considering a number of two- and three-dimensional examples, where we show how the proposed approach can be used for both mesh quality optimisation and untangling of invalid high-order meshes.
Highlights
High-order methods are rapidly increasing in popularity due to their favourable numerical characteristics and ability to more effectively use modern computing hardware than traditional loworder methods
The results we present here highlight that the variational setting allows us to construct a highly efficient and robust parallel framework for high-order mesh generation, permitting the generation of very complex three-dimensional meshes in the order of minutes
This section outlines a key contribution of this work, where we show that many of the existing curvilinear mesh generation methods can be unified in a variational setting through the definition of an energy functional
Summary
High-order methods are rapidly increasing in popularity due to their favourable numerical characteristics and ability to more effectively use modern computing hardware than traditional loworder methods. In this article we significantly expand the scope of the work by investigating several additional contributions These are: the incorporation of optimisation procedures based on analytic gradients and Hessian regularisation; the implementation of an improved regularisation method used to untangle meshes and a detailed discussion of its properties; the extension of the method to permit the mesh nodes connected to the CAD geometry to slide along the curves and across the surfaces on the boundary; and, the inclusion of a wider range of examples, including hybrid prismatic–tetrahedral boundary layer meshes and very high-order quadrilateral meshes. We describe the different forms of the energy that we investigate in this article
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