HexBox: Interactive Box Modeling of Hexahedral Meshes

We introduce HexBox, an intuitive modeling method and interactive tool for creating and editing hexahedral meshes. Hexbox brings the major and widely validated surface modeling paradigm of surface box modeling into the world of hex meshing. The main idea is to allow the user to box-model a volumetric mesh by primarily modifying its surface through a set of topological and geometric operations. We support, in particular, local and global subdivision, various instantiations of extrusion, removal, and cloning of elements, the creation of non-conformal or conformal grids, as well as shape modifications through vertex positioning, including manual editing, automatic smoothing, or, eventually, projection on an externally-provided target surface. At the core of the efficient implementation of the method is the coherent maintenance, at all steps, of two parallel data structures: a hexahedral mesh representing the topology and geometry of the currently modeled shape, and a directed acyclic graph that connects operation nodes to the affected mesh hexahedra. Operations are realized by exploiting recent advancements in grid- based meshing, such as mixing of 3-refinement, 2-refinement, and face-refinement, and using templated topological bridges to enforce on-the-fly mesh conformity across pairs of adjacent elements. A direct manipulation user interface lets users control all operations. The effectiveness of our tool, released as open source to the community, is demonstrated by modeling several complex shapes hard to realize with competing tools and techniques

Optimal Dual Schemes for Adaptive Grid Based Hexmeshing

Hexahedral meshes are an ubiquitous domain for the numerical resolution of partial differential equations. Computing a pure hexahedral mesh from an adaptively refined grid is a prominent approach to automatic hexmeshing, and requires the ability to restore the all hex property around the hanging nodes that arise at the interface between cells having different size. The most advanced tools to accomplish this task are based on mesh dualization. These approaches use topological schemes to regularize the valence of inner vertices and edges, such that dualizing the grid yields a pure hexahedral mesh. In this paper we study in detail the dual approach, and propose four main contributions to it: (i) we enumerate all the possible transitions that dual methods must be able to handle, showing that prior schemes do not natively cover all of them; (ii) we show that schemes are internally asymmetric, therefore not only their implementation is ambiguous, but different implementation choices lead to hexahedral meshes with different singular structure; (iii) we explore the combinatorial space of dual schemes, selecting the minimum set that covers all the possible configurations and also yields the simplest singular structure in the output hexmesh; (iv) we enlarge the class of adaptive grids that can be transformed into pure hexahedral meshes, relaxing one of the tight requirements imposed by previous approaches, and ultimately permitting to obtain much coarser meshes for same geometric accuracy. Last but not least, for the first time we make grid-based hexmeshing truly reproducible, releasing our code and also revealing a conspicuous amount of technical details that were always overlooked in previous literature, creating an entry barrier that was hard to overcome for practitioners in the field

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

At-Most-Hexa Meshes

AbstractVolumetric polyhedral meshes are required in many applications, especially for solving partial differential equations on finite element simulations. Still, their construction bears several additional challenges compared to boundary‐based representations. Tetrahedral meshes and (pure) hex‐meshes are two popular formats in scenarios like CAD applications, offering opposite advantages and disadvantages. Hex‐meshes are more intricate to construct due to the global structure of the meshing, but feature much better regularity, alignment, are more expressive, and offer the same simulation accuracy with fewer elements. Hex‐dominant meshes, where most but not all cell elements have a hexahedral structure, constitute an attractive compromise, potentially unlocking benefits from both structures, but their generality makes their employment in downstream applications difficult. In this work, we introduce a strict subset of general hex‐dominant meshes, which we term 'at‐most‐hexa meshes', in which most cells are still hexahedral, but no cell has more than six boundary faces, and no face has more than four sides. We exemplify the ease of construction of at‐most‐hexa meshes by proposing a frugal and straightforward method to generate high‐quality meshes of this kind, starting directly from hulls or point clouds, for example, from a 3D scan. In contrast to existing methods for (pure) hexahedral meshing, ours does not require an intermediate parameterization of other costly pre‐computations and can start directly from surfaces or samples. We leverage a Lloyd relaxation process to exploit the synergistic effects of aligning an orientation field in a modified 3D Voronoi diagram using the norm for cubical cells. The extracted geometry incorporates regularity as well as feature alignment, following sharp edges and curved boundary surfaces. We introduce specialized operations on the three‐dimensional graph structure to enforce consistency during the relaxation. The resulting algorithm allows for an efficient evaluation with parallel algorithms on GPU hardware and completes even large reconstructions within minutes

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

Selective Padding for Polycube‐Based Hexahedral Meshing

Hexahedral meshes generated from polycube mapping often exhibit a low number of singularities but also poor quality elements located near the surface. It is thus necessary to improve the overall mesh quality, in terms of the minimum Scaled Jacobian (MSJ) or average Scaled Jacobian (ASJ). Improving the quality may be obtained via global padding (or pillowing), which pushes the singularities inside by adding an extra layer of hexahedra on the entire domain boundary. Such a global padding operation suffers from a large increase of complexity, with unnecessary hexahedra added. In addition, the quality of elements near the boundary may decrease. We propose a novel optimization method which inserts sheets of hexahedra so as to perform selective padding, where it is most needed for improving the mesh quality. A sheet can pad part of the domain boundary, traverse the domain and form singularities. Our global formulation, based on solving a binary problem, enables us to control the balance between quality improvement, increase of complexity and number of singularities. We show in a series of experiments that our approach increases the MSJ value and preserves (or even improves) the ASJ, while adding fewer hexahedra than global padding