Automated refinement of conformal quadrilateral and hexahedral meshes

Abstract

SUMMARYConformal refinement using a shrink and connect strategy, known as pillowing or buffer insertion, contracts and reconnects contiguous elements of an all-quadrilateral or an all-hexahedral mesh in order to locally increase vertex density without introducing hanging nodes or non-cubical elements. Using layers as shrink sets, the present method automates the anisotropic refinement of such meshes according to a prescribed size map expressed as a Riemannian metric field. An anisotropic smoother further enhances vertex clustering to capture the features of the metric. Both two-and three-dimensional test cases with analytic control metrics confirm the feasibility of the present approach and explore strategies to minimize the trade-off between element shape quality and size conformity. Additional examples using discrete metric maps illustrate possible practical applications. Although local vertex removal and reconnection capabilities have yet to be developed, the present refinement method is a step towards an automated tool for conformal adaptation of all-quadrilateral and all-hexahedral meshes.

A close look at valences in hexahedral element meshes

“…In agreement with the published literature, for hexahedral meshes, it is true that node valences which deviate significantly from perfect will yield element topology which will only admit poor element quality [3,5,13,[19][20][21]. However, the inverse is not true.…”

“…Tchon et al [5] also use node valence to measure topology in hexahedral meshes indicating that a valence of six is perfect. They note that their refinement procedures degrade mesh quality in the same regions that node valence is modified from six.…”

“…There are many published hexahedral mesh modification techniques such as grafting [6], mesh cutting [7], refinement [5,10], mesh matching [8], and coarsening [9]. Each of these techniques must make decisions as to where and what modifications will be made to effect the desired changes to the mesh.…”

“…In spite of the difficultly with hexahedral elemental topology optimization, there are many published hexahedral mesh modification techniques [5][6][7][8][9][10], such as adaptive refinement and coarsening, which take an input hexahedral mesh and modify it in some way to meet the analysts' objectives for the analysis. Most of these methods follow prescribed patterns or templates for mesh modification with little if any real-time decision making between potential topology changes.…”

Quad-Dominant Mesh Adaptation Using Specialized Simplicial Optimization

“…However, although it can improve both vertex density and element shape, smoothing is limited by a fixed mesh topology. Mesh refinement and coarsening, on the other hand, involve connectivity modifications [9,10,11,12]. The major issue with conformal refinement-coarsening is, however, that element shape quality cannot be maintained without a powerful local reconnection method and the coveted cubical flip operator is still only theoretical in three dimensions [13].…”