Guillem Belda-Ferrín defended his PhD thesis Conformal n-dimensional Bisection for Local Refinement of Unstructured Simplicial Meshes on October 28th, 2022. The thesis was produced within the UPC doctoral program in Applied Mathematics and was supervised by Xevi Roca and Eloi Ruiz-Gironés. Currently, he is working as algorithm developer at Hexagon Manufacturing Intelligence designing path planning algorithms.
Thesis summary
In n-dimensional adaptive applications, conformal simplicial meshes must be locally modified. One systematic local modification is to bisect the prescribed simplices while surrounding simplices are bisected to ensure conformity. Although there are many conformal bisection strategies, practitioners prefer the method known as the newest vertex bisection [7–10]. This method guarantees key advantages for adaptivity whenever the mesh has a structure called reflectivity. Unfortunately, it is not known (i) how to extract a reflection structure from any unstructured conformal mesh for three or more dimensions. Fortunately, a conformal bisection method is suitable for adaptivity if it almost fulfills the newest vertex bisection advantages. These advantages are almost met by an existent multi-stage strategy in three dimensions [11]. However, it is not known (ii) how to perform multi-stage bisection for more than three dimensions.
This thesis aims to demonstrate that n-dimensional conformal bisection is possible for local refinement of unstructured conformal meshes. To this end, it proposes the following contributions. First, it proposes the first 4-dimensional two-stage method [6], showing that multi-stage bisection is possible beyond three dimensions. Second, following this possibility, the thesis proposes the first n-dimensional multi-stage method [1], and thus, it answers question (ii). Third, it guarantees the first 3-dimensional method that features the newest vertex bisection advantages [5], showing that these advantages are possible beyond two dimensions. Fourth, extending this possibility, the thesis guarantees the first n-dimensional marking method [3] that extracts a reflection structure from any unstructured conformal mesh, and thus, it answers question (i). This answer proves that local refinement with the newest vertex bisection is possible in any dimension. Fifth, this thesis shows that the proposed multi-stage method almost fulfills the advantages of the newest vertex bisection [2]. Finally, to visualize four-dimensional meshes, it proposes a simple tool to slice pentatopic meshes [4].
In conclusion, this thesis demonstrates that conformal bisection is possible for local refinement in two or more dimensions. To this end, it proposes two novel methods for unstructured conformal meshes, methods that will enable adaptive applications on n-dimensional complex geometry.
Highlighted publication: [1].
References
[1] Guillem Belda-Ferrín, Eloi Ruiz-Gironés and Xevi Roca. Conformal Marked Bisection for Local Refinement of n-Dimensional Unstructured Simplicial Meshes. Computer-Aided Design 154 (2023), 103419. pdf.
[2] Guillem Belda-Ferrín, Eloi Ruiz-Gironés and Xevi Roca. Suitability of marked bisection for local refinement of n-dimensional unstructured conformal meshes. Paper in preparation (2022).
[3] Guillem Belda-Ferrín, Eloi Ruiz-Gironés and Xevi Roca. Newest vertex bisection for unstructured n-simplicial meshes. Paper in preparation (2022).
[4] Guillem Belda-Ferrín, Eloi Ruiz-Gironés, Abel Gargallo-Peiró and Xevi Roca. Visualization of pentatopic meshes. 28th International Meshing Roundtable (2019).
[5] Guillem Belda-Ferrín, Eloi Ruiz-Gironés and Xevi Roca. Bisecting with optimal similarity bound on 3D unstructured conformal meshes. In Proceedings of the 2022 SIAM International Meshing Roundtable (2022), 86-97. pdf.
[6] Guillem Belda-Ferrín, Abel Gargallo-Peiró and Xevi Roca. Local Bisection for Conformal Refinement of Unstructured 4D Simplicial Meshes. In 27th International Meshing Roundtable (Springer, 2019), 229-247.
[7] Martin Alkämper, Fernando Gaspoz, and Robert Klöfkorn. A Weak Compatibility Condition for Newest Vertex Bisection in Any Dimension. SIAM Journal on Scientific Computing 40/6. (2018), A3853–A3872. pdf.
[8] Rob Stevenson. The Completion of Locally Refined Simplicial Partitions Created by Bisection. Mathematics of Computation 261 (2008),227-241.
[9] Christoph T. Traxler. An algorithm for adaptive mesh refinement in n dimensions. Computing 2/2 (1997),115-137.
[10] Joseph M. Maubach. Local Bisection Refinement for N-Simplicial Grids Generated by Reflection. SIAM Journal on Scientific Computing 16/1 (1995),210-227. pdf.
[11] Douglas N. Arnold, Arup Mukherjee and Luc Pouly. Locally Adapted Tetrahedral Meshes Using Bisection. SIAM Journal on Scientific Computing 22/2 (2000), 431-448.