High‐quality surface remeshing using harmonic maps—Part II: Surfaces with high genus and of large aspect ratio

Abstract

This paper follows a previous one that was dealing with high-quality surface remeshing using harmonic maps (Int. J. Numer. Meth. Engng 2010; 83:403-425). In (Int. J. Numer. Meth. Engng 2010; 83:403-425), it has been demonstrated that harmonic parametrizations can be used as input for surface meshers to produce high-quality triangulations. However, two important limitations were pointed out, namely surfaces with high genus and/or of large aspect ratio. This paper addresses those two issues. We first develop a multiscale version of the harmonic parametrization of (Int. J. Numer. Meth. Engng 2010; 83:403-425) and then combine it with a multilevel partitioning algorithm to come up with an automatic remeshing algorithm that overcomes the above-mentioned limitations of harmonic maps. The overall procedure is implemented in the open-source mesh generator Gmsh (Int. E. MARCHANDISE ET AL. local checks to guarantee its bijectivity: we called that approach hybrid [1]. Non-linear algorithms are considered to be slow and will not be considered here.The concept of discrete harmonic map has been described first by Eck et al. [2]. Floater [3] introduced a concept of convex combination map that guarantees that the discrete mapping is one-to-one. Sheffer and Strurler [4] presented a constrained minimization approach, the so-called angle-based flattening (ABF), such that the variation between the set of angles of an original mesh and one of the 2D flattened version is minimized. In order to obtain a valid and flippingfree parametrization, several additional constrained algorithms are developed. More recently, they improved the performance of the ABF technique by using an advanced numerical approach and a hierarchical technique [5]. Much research has also been incorporated within the theory of differential geometry. For example, Levy et al. [6] apply the Cauchy Riemann equations to compute a least-square conformal map (LSCM). This approach is quite similar to that of Desbrun and co-workers [7] that minimize a combination of Dirichlet and distortion energy to compute a conformal map for interactive geometry remeshing. More recently, Mullen et al. [8] have also presented spectral computation of the conformal map [8].Recently, discrete parametrization methods have received some attention from non-CGspecialists and in particular, in the domain of finite element mesh generation. Here, the target application is clearly surface meshing, especially for surfaces that are defined by a triangulation only or for cross-patch meshing. In [1], we have demonstrated the use of such mappings as parametrizations allowed to generate quality finite element meshes. Yet, two important issues have not been completely addressed: reparameterization techniques fail when the surface has a large aspect ratio and/or when it has a high genus. Those issues are critical in the domain of mesh generation, maybe more than in CG. This paper aims at addressing those two issues.In [1], we have demonstrated that parametric coordinates computed using a discrete harmonic ma...

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