Unstructured and semi-structured hexahedral mesh generation methods

Discretization techniques such as the finite element method, the finite volume method or the discontinuous Galerkin method are the most used simulation techniques in ap- plied sciences and technology. These methods rely on a spatial discretization adapted to the geometry and to the prescribed distribution of element size. Several fast and robust algorithms have been developed to generate triangular and tetrahedral meshes. In these methods local connectivity modifications are a crucial step. Nevertheless, in hexahedral meshes the connectivity modifications propagate through the mesh. In this sense, hexahedral meshes are more constrained and therefore, more difficult to gener- ate. However, in many applications such as boundary layers in computational fluid dy- namics or composite material in structural analysis hexahedral meshes are preferred. In this work we present a survey of developed methods for generating structured and unstructured hexahedral meshes.Peer ReviewedPostprint (published version

Learning topological operations on meshes with application to block decomposition of polygons

We present a learning based framework for mesh quality improvement on unstructured triangular and quadrilateral meshes. Our model learns to improve mesh quality according to a prescribed objective function purely via self-play reinforcement learning with no prior heuristics. The actions performed on the mesh are standard local and global element operations. The goal is to minimize the deviation of the node degrees from their ideal values, which in the case of interior vertices leads to a minimization of irregular nodes.Comment: Submitted to Computer-Aided Design Journal. Presented at 17th US National Conference on Computational Mechanics, Albuquerque, N

Large-scale Geometric Data Decomposition, Processing and Structured Mesh Generation

Mesh generation is a fundamental and critical problem in geometric data modeling and processing. In most scientific and engineering tasks that involve numerical computations and simulations on 2D/3D regions or on curved geometric objects, discretizing or approximating the geometric data using a polygonal or polyhedral meshes is always the first step of the procedure. The quality of this tessellation often dictates the subsequent computation accuracy, efficiency, and numerical stability. When compared with unstructured meshes, the structured meshes are favored in many scientific/engineering tasks due to their good properties. However, generating high-quality structured mesh remains challenging, especially for complex or large-scale geometric data. In industrial Computer-aided Design/Engineering (CAD/CAE) pipelines, the geometry processing to create a desirable structural mesh of the complex model is the most costly step. This step is semi-manual, and often takes up to several weeks to finish. Several technical challenges remains unsolved in existing structured mesh generation techniques. This dissertation studies the effective generation of structural mesh on large and complex geometric data. We study a general geometric computation paradigm to solve this problem via model partitioning and divide-and-conquer. To apply effective divide-and-conquer, we study two key technical components: the shape decomposition in the divide stage, and the structured meshing in the conquer stage. We test our algorithm on vairous data set, the results demonstrate the efficiency and effectiveness of our framework. The comparisons also show our algorithm outperforms existing partitioning methods in final meshing quality. We also show our pipeline scales up efficiently on HPC environment

Workshop on the Integration of Finite Element Modeling with Geometric Modeling

The workshop on the Integration of Finite Element Modeling with Geometric Modeling was held on 12 May 1987. It was held to discuss the geometric modeling requirements of the finite element modeling process and to better understand the technical aspects of the integration of these two areas. The 11 papers are presented except for one for which only the abstract is given

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

CAD-Based Porous Scaffold Design of Intervertebral Discs in Tissue Engineering

With the development and maturity of three-dimensional (3D) printing technology over the past decade, 3D printing has been widely investigated and applied in the field of tissue engineering to repair damaged tissues or organs, such as muscles, skin, and bones, Although a number of automated fabrication methods have been developed to create superior bio-scaffolds with specific surface properties and porosity, the major challenges still focus on how to fabricate 3D natural biodegradable scaffolds that have tailor properties such as intricate architecture, porosity, and interconnectivity in order to provide the needed structural integrity, strength, transport, and ideal microenvironment for cell- and tissue-growth. In this dissertation, a robust pipeline of fabricating bio-functional porous scaffolds of intervertebral discs based on different innovative porous design methodologies is illustrated. Firstly, a triply periodic minimal surface (TPMS) based parameterization method, which has overcome the integrity problem of traditional TPMS method, is presented in Chapter 3. Then, an implicit surface modeling (ISM) approach using tetrahedral implicit surface (TIS) is demonstrated and compared with the TPMS method in Chapter 4. In Chapter 5, we present an advanced porous design method with higher flexibility using anisotropic radial basis function (ARBF) and volumetric meshes. Based on all these advanced porous design methods, the 3D model of a bio-functional porous intervertebral disc scaffold can be easily designed and its physical model can also be manufactured through 3D printing. However, due to the unique shape of each intervertebral disc and the intricate topological relationship between the intervertebral discs and the spine, the accurate localization and segmentation of dysfunctional discs are regarded as another obstacle to fabricating porous 3D disc models. To that end, we discuss in Chapter 6 a segmentation technique of intervertebral discs from CT-scanned medical images by using deep convolutional neural networks. Additionally, some examples of applying different porous designs on the segmented intervertebral disc models are demonstrated in Chapter 6

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