Flipping Cubical Meshes

We define and examine flip operations for quadrilateral and hexahedral meshes, similar to the flipping transformations previously used in triangular and tetrahedral mesh generation.Comment: 20 pages, 24 figures. Expanded journal version of paper from 10th International Meshing Roundtable. This version removes some unwanted paragraph breaks from the previous version; the text is unchange

A case study in hexahedral mesh generation: Simulation of the human mandible

We provide a case study for the generation of pure hexahedral meshes for the numerical simulation of physiological stress scenarios of the human mandible. Due to its complex and very detailed free-form geometry, the mandible model is very demanding. This test case is used as a running example to demonstrate the applicability of a combinatorial approach for the generation of hexahedral meshes by means of successive dual cycle eliminations, which has been proposed by the second author in previous work. We report on the progress and recent advances of the cycle elimination scheme. The given input data, a surface triangulation obtained from computed tomography data, requires a substantial mesh reduction and a suitable conversion into a quadrilateral surface mesh as a first step, for which we use mesh clustering and b-matching techniques. Several strategies for improved cycle elimination orders are proposed. They lead to a significant reduction in the mesh size and a better structural quality. Based on the resulting combinatorial meshes, gradient-based optimized smoothing with the condition number of the Jacobian matrix as objective together with mesh untangling techniques yielded embeddings of a satisfactory quality. To test our hexahedral meshes for the mandible model within an FEM simulation we used the scenario of a bite on a ‘hard nut.’ Our simulation results are in good agreement with observations from biomechanical experiments

Finding Hexahedrizations for Small Quadrangulations of the Sphere

This paper tackles the challenging problem of constrained hexahedral meshing. An algorithm is introduced to build combinatorial hexahedral meshes whose boundary facets exactly match a given quadrangulation of the topological sphere. This algorithm is the first practical solution to the problem. It is able to compute small hexahedral meshes of quadrangulations for which the previously known best solutions could only be built by hand or contained thousands of hexahedra. These challenging quadrangulations include the boundaries of transition templates that are critical for the success of general hexahedral meshing algorithms. The algorithm proposed in this paper is dedicated to building combinatorial hexahedral meshes of small quadrangulations and ignores the geometrical problem. The key idea of the method is to exploit the equivalence between quad flips in the boundary and the insertion of hexahedra glued to this boundary. The tree of all sequences of flipping operations is explored, searching for a path that transforms the input quadrangulation Q into a new quadrangulation for which a hexahedral mesh is known. When a small hexahedral mesh exists, a sequence transforming Q into the boundary of a cube is found; otherwise, a set of pre-computed hexahedral meshes is used. A novel approach to deal with the large number of problem symmetries is proposed. Combined with an efficient backtracking search, it allows small shellable hexahedral meshes to be found for all even quadrangulations with up to 20 quadrangles. All 54,943 such quadrangulations were meshed using no more than 72 hexahedra. This algorithm is also used to find a construction to fill arbitrary domains, thereby proving that any ball-shaped domain bounded by n quadrangles can be meshed with no more than 78 n hexahedra. This very significantly lowers the previous upper bound of 5396 n.Comment: Accepted for SIGGRAPH 201

7th International Meshing Roundtable '98

The goal of the 7th International Meshing Roundtable is to bring together researchers and developers from industry, academia, and government labs in a stimulating, open environment for the exchange of technical information related to the meshing process. In the past, the Roundtable has enjoyed significant participation from each of these groups from a wide variety of countries

Frame Fields for Hexahedral Mesh Generation

As a discretized representation of the volumetric domain, hexahedral meshes have been a popular choice in computational engineering science and serve as one of the main mesh types in leading industrial software of relevance. The generation of high quality hexahedral meshes is extremely challenging because it is essentially an optimization problem involving multiple (conflicting) objectives, such as fidelity, element quality, and structural regularity. Various hexahedral meshing methods have been proposed in past decades, attempting to solve the problem from different perspectives. Unfortunately, algorithmic hexahedral meshing with guarantees of robustness and quality remains unsolved. The frame field based hexahedral meshing method is the most promising approach that is capable of automatically generating hexahedral meshes of high quality, but unfortunately, it suffers from several robustness issues. Field based hexahedral meshing follows the idea of integer-grid maps, which pull back the Cartesian hexahedral grid formed by integer isoplanes from a parametric domain to a surface-conforming hexahedral mesh of the input object. Since directly optimizing for a high quality integer-grid map is mathematically challenging, the construction is usually split into two steps: (1) generation of a feature-aligned frame field and (2) generation of an integer-grid map that best aligns with the frame field. The main robustness issue stems from the fact that smooth frame fields frequently exhibit singularity graphs that are inappropriate for hexahedral meshing and induce heavily degenerate integer-grid maps. The thesis aims at analyzing the gap between the topologies of frame fields and hexahedral meshes and developing algorithms to realize a more robust field based hexahedral mesh generation. The first contribution of this work is an enumeration of all local configurations that exist in hexahedral meshes with bounded edge valence and a generalization of the Hopf-Poincaré formula to octahedral (orthonormal frame) fields, leading to necessary local and global conditions for the hex-meshability of an octahedral field in terms of its singularity graph. The second contribution is a novel algorithm to generate octahedral fields with prescribed hex-meshable singularity graphs, which requires the solution of a large non-linear mixed-integer algebraic system. This algorithm is an important step toward robust automatic hexahedral meshing since it enables the generation of a hex-meshable octahedral field. In the collaboration work with colleagues [BRK+22], the dataset HexMe consisting of practically relevant models with feature tags is set up, allowing a fair evaluation for practical hexahedral mesh generation algorithms. The extendable and mutable dataset remains valuable as hexahedral meshing algorithms develop. The results of the standard field based hexahedral meshing algorithms on the HexMesh dataset expose the fragility of the automatic pipeline. The major contribution of this thesis improves the robustness of the automatic field based hexahedral meshing by guaranteeing local meshability of general feature aligned smooth frame fields. We derive conditions on the meshability of frame fields when feature constraints are considered, and describe an algorithm to automatically turn a given non-meshable frame field into a similar but locally meshable one. Despite the fact that local meshability is only a necessary but not sufficient condition for the stronger requirement of meshability, our algorithm increases the 2% success rate of generating valid integer-grid maps with state-of-the-art methods to 57%, when compared on the challenging HexMe dataset

Parallel paving: An algorithm for generating distributed, adaptive, all-quadrilateral meshes on parallel computers

Paving is an automated mesh generation algorithm which produces all-quadrilateral elements. It can additionally generate these elements in varying sizes such that the resulting mesh adapts to a function distribution, such as an error function. While powerful, conventional paving is a very serial algorithm in its operation. Parallel paving is the extension of serial paving into parallel environments to perform the same meshing functions as conventional paving only on distributed, discretized models. This extension allows large, adaptive, parallel finite element simulations to take advantage of paving`s meshing capabilities for h-remap remeshing. A significantly modified version of the CUBIT mesh generation code has been developed to host the parallel paving algorithm and demonstrate its capabilities on both two dimensional and three dimensional surface geometries and compare the resulting parallel produced meshes to conventionally paved meshes for mesh quality and algorithm performance. Sandia`s {open_quotes}tiling{close_quotes} dynamic load balancing code has also been extended to work with the paving algorithm to retain parallel efficiency as subdomains undergo iterative mesh refinement

ICASE/LaRC Workshop on Adaptive Grid Methods

Solution-adaptive grid techniques are essential to the attainment of practical, user friendly, computational fluid dynamics (CFD) applications. In this three-day workshop, experts gathered together to describe state-of-the-art methods in solution-adaptive grid refinement, analysis, and implementation; to assess the current practice; and to discuss future needs and directions for research. This was accomplished through a series of invited and contributed papers. The workshop focused on a set of two-dimensional test cases designed by the organizers to aid in assessing the current state of development of adaptive grid technology. In addition, a panel of experts from universities, industry, and government research laboratories discussed their views of needs and future directions in this field