Size-preserving size functions and smoothing procedures for adaptive quadrilateral mesh generation

Abstract

The generation of meshes that correctly reproduce a prescribed size function is crucial for quadrilateral meshing due to two reasons. First, quadrilateral meshes are difficult to adapt to a given size field by refining or coarsening the elements without compromising the element quality. Second, after the meshing algorithm is finished, it may be necessary to apply a smoothing algorithm to improve the global quality. This smoothing step may modify the element size and the final mesh will not reproduce the prescribed element size. To solve these issues, we propose to combine the size-preserving method with a smoothing technique that takes into account both the element shape and size. The size-preserving technique allows directly generating a quadrilateral mesh that reproduces the size function, while the proposed smoother allows obtaining a high-quality mesh while maintaining the element size. In adaptive processes, the proposed approach may reduce the number of iterations to achieve convergence, since at each iteration the background mesh is properly reproduced. In addition, we detail new theoretical results that provide more insight to size-preserving size functions. Specifically, we prove that the maximum gradient of a one-dimensional size-preserving size function is bounded. Finally, several applications that show the benefits of applying the proposed techniques are presented.