Hex Mesh Research Articles

TUM.GeoFrame: automated high-order hexahedral mesh generation for shell-like structures

This paper presents a fully automated high-order hexahedral mesh generation algorithm for shell-like structures based on enhanced sweeping methods. Traditional sweeping techniques create all-hexahedral element meshes for solid structures by projecting an initial single surface mesh along a specified trajectory to a specified target surface. The work reported here enhances the traditional method for thin solids by creating conforming high-order all-hexahedral finite element meshes on an enhanced surface model with surfaces intersecting in parallel, perpendicular and skew-angled directions. The new algorithm is based on cheap projection rules separating the original surface model into a set of disjoint single surfaces and a so-called interface skeleton. The core of this process is reshaping the boundary representations of the initial surfaces, generating new sweeping templates along the intersection curves and joining the single swept hex meshes in an independently generated interface mesh.

Local transformations of hexahedral meshes of lower valence

The modification of conforming hexahedral meshes is difficult to perform since their structure does not allow easy local refinement or un-refinement such that the modification does not go through the boundary. In this paper we prove that the set of hex flipping transformations of Bern et al. (2002) [1] is the only possible local modification on a geometrical hex mesh of valence less than five i.e., with less than five edges per vertex. We propose a new basis of transformations that can generate sequences of local modifications on hex meshes of valence less than six. For quadrilateral meshes, we show the equivalence between modifying locally the number of quads on a mesh and the number of its internal vertices.

Rib fractures under anterior–posterior dynamic loads: Experimental and finite-element study

The purpose of this study was to investigate whether using a finite-element (FE) mesh composed entirely of hexahedral elements to model cortical and trabecular bone (all-hex model) would provide more accurate simulations than those with variable thickness shell elements for cortical bone and hexahedral elements for trabecular bone (hex–shell model) in the modeling human ribs. First, quasi-static non-injurious and dynamic injurious experiments were performed using the second, fourth, and tenth human thoracic ribs to record the structural behavior and fracture tolerance of individual ribs under anterior–posterior bending loads. Then, all-hex and hex–shell FE models for the three ribs were developed using an octree-based and multi-block hex meshing approach, respectively. Material properties of cortical bone were optimized using dynamic experimental data and the hex–shell model of the fourth rib and trabecular bone properties were taken from the literature. Overall, the reaction force–displacement relationship predicted by both all-hex and hex–shell models with nodes in the offset middle-cortical surfaces compared well with those measured experimentally for all the three ribs. With the exception of fracture locations, the predictions from all-hex and offset hex–shell models of the second and fourth ribs agreed better with experimental data than those from the tenth rib models in terms of reaction force at fracture (difference <15.4%), ultimate failure displacement and time (difference <7.3%), and cortical bone strains. The hex–shell models with shell nodes in outer cortical surfaces increased static reaction forces up to 16.6%, compared to offset hex–shell models. These results indicated that both all-hex and hex–shell modeling strategies were applicable for simulating rib responses and bone fractures for the loading conditions considered, but coarse hex–shell models with constant or variable shell thickness were more computationally efficient and therefore preferred.

Development of subject-specific and statistical shape models of the knee using an efficient segmentation and mesh-morphing approach

Subject-specific finite element models developed from imaging data provide functional representation of anatomical structures and have been used to evaluate healthy and pathologic knee mechanics. The creation of subject-specific models is a time-consuming process when considering manual segmentation and hexahedral (hex) meshing of the articular surfaces to ensure accurate contact assessment. Previous studies have emphasized automated mesh mapping to bone geometry from computed tomography (CT) scans, but have not considered cartilage and soft tissue structures. Statistical shape modeling has been proposed as an alternative approach to develop a population of subject models, but still requires manual segmentation and registration of a training set. Accordingly, the aim of the current study was to develop an efficient, integrated mesh-morphing-based segmentation approach to create hex meshes of subject-specific geometries from scan data, to apply the approach to natural femoral, tibial, and patellar cartilage from magnetic resonance (MR) images, and to demonstrate the creation of a statistical shape model of the knee characterizing the modes of variation using principal component analysis. The platform was demonstrated on MR scans from 10 knees and enabled hex mesh generation of the knee articular structures in approximately 1.5 h per subject. In a subset of geometries, average root mean square geometric differences were 0.54 mm for all structures and in quasi-static analyses over a range of flexion angles, differences in predicted peak contact pressures were less than 5.3% between the semi-automated and manually generated models. The integrated segmentation, mesh-morphing approach was employed in the efficient development of subject-specific models and a statistical shape model, where populations of subject-specific models have application to implant design evaluation or surgical planning.

Local topological modification of hexahedral meshes. Part I: A set of dual-based operations

For hexahedral meshes, it is difficult to make topological modifications which preserve conformal property and which are local to the modified elements. This is most easily understood in how changes affect the dual surfaces (sheets) and lines (chords), which have non­local extent in hexahedral meshes. A set of three operations is proposed which represent distinct local changes to the hex mesh and its dual. These operations are shown to compose the larger set of flipping operations described by Bern et. al. The relation between these dual­based operations and the insertion of a Boy surface in the dual, described by Bern, is also discussed.

Automatic nested refinement: a technique for the generation of high quality multi-block structured grids for multi-scale problems using GridPro

A technique for the generation of conformal adaptive refinement of hex meshes is presented. Automatic nested refinement is a technique for generating recursively nested topology automatically. It can be applied inside GridPro’s topological paradigm to generate block structured grids, which can resolve tiny features in the problem while providing for a smooth and consistent way to transition to a larger scale. The selection of self-similar, ‘fractal-like’ topological templates makes the selection of number of levels easy, while making the technique feasible for infinite levels of adaptive refinement. The technique has been programmed and integrated into the GUI of GridPro, making it very accessible and easy to use. This method illustrates a way of generating structured grids in an unstructured way, made possible because of the topological paradigm of GridPro. Because the technique uses the topological paradigm, it inherits all the advantages the paradigm offers, including the ability to handle very complex geometries, parametric variation of surfaces, and the ease of use, speed and quality of GridPro. The technique has been illustrated using a variety of applications. This method has proved to be a fast, efficient, automatic and reliable means to perform physical simulations that have a disparity in scale.

Construction Techniques for Cubical Complexes, Odd Cubical 4-Polytopes, and Prescribed Dual Manifolds

We provide a number of new construction techniques for cubical complexes and cubical polytopes, and thus for cubifications (hexahedral mesh generation). As an application we obtain an instance of a cubical 4-polytope that has a nonorientable dual manifold (a Klein bottle). This confirms an existence conjecture of Hetyei (1995). More systematically, we prove that every normal crossing codimension one immersion of a compact 2-manifold into R 3 is PL-equivalent to a dual manifold immersion of a cubical 4-polytope. As an instance we obtain a cubical 4-polytope with a cubification of Boy's surface as a dual manifold immersion, and with an odd number of facets. Our explicit example has 17,718 vertices and 16,533 facets. Thus we get a parity-changing operation for three-dimensional cubical complexes (hex meshes); this solves problems of Eppstein, Thurston, and others.

Automatic hexahedral mesh generation for FEM using shape recognition technique and tree method [EM field analysis

In this paper, a method for automatic hexahedral (Hex) mesh generation for electromagnetic field analysis is presented. In this method, the shape recognition technique and the tree method are used. Utilizing the shape recognition technique, a mesh is generated in a complicated analysis domain, and a high-quality Hex mesh is obtained. By using the tree method, control of the mesh density is easily enabled. In this proposed system, a modest amount of input data is necessary, such as object parameters and the mesh density data.

The generation of hexahedral meshes for assembly geometry: survey and progress

AbstractThe finite element method is being used today to model component assemblies in a wide variety of application areas, including structural mechanics, fluid simulations, and others. Generating hexahedral meshes for these assemblies usually requires the use of geometry decomposition, with different meshing algorithms applied to different regions. While the primary motivation for this approach remains the lack of an automatic, reliable all‐hexahedral meshing algorithm, requirements in mesh quality and mesh configuration for typical analyses are also factors. For these reasons, this approach is also sometimes required when producing other types of unstructured meshes. This paper will review progress to date in automating many parts of the hex meshing process, which has halved the time to produce all‐hex meshes for large assemblies. Particular issues which have been exposed due to this progress will also be discussed, along with their applicability to the general unstructured meshing problem. Published in 2001 by John Wiley &amp; Sons, Ltd.

Feature based hex meshing methodology: feature recognition and volume decomposition

Considerable progress has been made on automatic hexahedral mesh generation in recent years. A few automated meshing algorithms (e.g. mapping, submapping, sweeping) have proven to be very reliable on certain classes of geometry. While it is always worth pursuing general algorithms viable on arbitrary geometry, a combination of the well-established algorithms is ready to take on classes of complicated geometry. By partitioning the entire geometry into meshable pieces matched with appropriate meshing algorithms, the original geometry becomes meshable and may achieve better mesh quality. Each meshable portion is recognized as a meshing feature. This paper, which is a part of the feature based meshing methodology, presents the work on shape recognition and volume decomposition to automatically decompose a CAD model into hex meshable volumes. There are four phases in this approach: Feature Determination to extract decomposition features; Cutting Surfaces Generation to form the cutting surfaces; Body Decomposition to get the imprinted volumes; and Meshing Algorithm Assignment to match volumes decomposed with appropriate meshing algorithms. This paper focuses on describing feature determination and volume decomposition; the last part has been described in another paper. The feature determination procedure is based on the CLoop feature recognition algorithm that is extended to be more general. Some decomposition and meshing results are demonstrated in the final section.

OCTREE-BASED HEXAHEDRAL MESH GENERATION

An octree-based algorithm for the generation of hexahedral element meshes is presented. The algorithm works in three steps: (i) The geometry to be meshed is approximated by an octree structure. (ii) An unstructured hexahedral element mesh is derived from the octree. (iii) The mesh is adapted to the boundary of the geometry. We focus on step (ii) and describe an algorithm that constructs a hex mesh for a given octree structure.

H-Morph: an indirect approach to advancing front hex meshing

H-Morph is a new automatic algorithm for the generation of a hexahedral-dominant finite element mesh for arbitrary volumes. The H-Morph method starts with an initial tetrahedral mesh and systematically transforms and combines tetrahedral into hexahedra. It uses an advancing front technique where the initial front consists of a set of prescribed quadrilateral surface facets. Fronts are individually processed by recovering each of the six quadrilateral faces of a hexahedron from the tetrahedral mesh. Recovery techniques similar to those used in boundary constrained Delaunay mesh generation are used. Tetrahedral internal to the six hexahedral faces are then removed and a hexahedron is formed. At any time during the H-Morph procedure a valid mixed hexahedral-tetrahedral mesh is in existence within the volume. The procedure continues until no tetrahedral remain within the volume, or tetrahedral remain which cannot be transformed or combined into valid hexahedral elements. Any remaining tetrahedral are typically towards the interior of the volume, generally a less critical region for analysis. Transition from tetrahedral to hexahedra in the final mesh is accomplished through pyramid shaped elements. Advantages of the proposed method include its ability to conform to an existing quadrilateral surface mesh, its ability to mesh without the need to decompose or recognize special classes of geometry, and its characteristic well-aligned layers of elements parallel to the boundary. Example test cases are presented on a variety of models.

An automated process for 3D hexahedral mesh regeneration in metal forming

This paper describes the method for automatic remeshing of the distorted hexahedral finite elements and its application in metal forming processes. It first generates a regular mesh inside the bounding box that circumscribes the given volume to be meshed. Secondly extra elements of the regular mesh that lie outside the volume are removed from the regular mesh. Then, the geometric features of the old mesh (i.e. sharp edges and corners nodes) are determined by established criteria and corresponding features of the new mesh are traced down to guarantee geometric consistency between the two meshes. After that, a Boundary Element Layer (BEL) is constructed to form new hex mesh. By classification of all surface nodes of the regular mesh into face nodes, edge nodes and corner nodes and by the corresponding treatment to different types of surface nodes, the resulted new mesh gives much-improved hex elements that best suit local geometry. Finally a smoothing process is done to reposition the edge nodes, face nodes and internal nodes sequentially before the new mesh is exported for use in continued calculation. The density of the new mesh can be controlled to be adaptive to strain gradient during smoothing.

Quad mesh generation for k-sided faces and hex mesh generation for trivalent polyhedra

Two mesh generation methods are presented in this paper: one for quadrilateral meshing of faces with convex vertices but an arbitrary number of sides, the other hexahedral meshing of polyhedra with convex edges, vertices connected to three edges and an arbitrary number of faces each of which has at least three sides. Efficient mapping methods for determining nodal positions are developed based on an extension to transfinite mapping and it is shown that these are identical to those obtained by iterative isoparametric smoothing.

THE WHISKER WEAVING ALGORITHM: A CONNECTIVITY-BASED METHOD FOR CONSTRUCTING ALL-HEXAHEDRAL FINITE ELEMENT MESHES

This paper introduces a new algorithm called whisker weaving for constructing unstructured, all-hexahedral finite element meshes. Whisker weaving is based on the Spatial Twist Continuum (STC), a global interpretation of the geometric dual of an all-hexahedral mesh. Whisker weaving begins with a closed, all-quadrilateral surface mesh bounding a solid geometry, then constructs hexahedral element connectivity advancing into the solid. The result of the whisker weaving algorithm is a complete representation of hex mesh connectivity only: Actual mesh node locations are determined afterwards. The basic step of whisker weaving is to form a hexahedral element by crossing or intersecting dual entities. This operation, combined with seaming or joining operations in dual space, is sufficient to mesh simple block problems. When meshing more complex geometries, certain other dual entities appear such as blind chords, merged sheets, and self-intersecting chords. Occasionally specific types of invalid connectivity arise. These are detected by a general method based on repeated STC edges. This leads into a strategy for resolving some cases of invalidities immediately. The whisker weaving implementation has so far been successful at generating meshes for simple block-type geometries and for some non-block geometries. Mesh sizes are currently limited to a few hundred elements. While the size and complexity of meshes generated by whisker weaving are currently limited, the algorithm shows promise for extension to much more general problems.

THE WHISKER WEAVING ALGORITHM: A CONNECTIVITY‐BASED METHOD FOR CONSTRUCTING ALL‐HEXAHEDRAL FINITE ELEMENT MESHES

This paper introduces a new algorithm called whisker weaving for constructing unstructured, all-hexahedral finite element meshes. Whisker weaving is based on the Spatial Twist Continuum (STC), a global interpretation of the geometric dual of an all-hexahedral mesh. Whisker weaving begins with a closed, all-quadrilateral surface mesh bounding a solid geometry, then constructs hexahedral element connectivity advancing into the solid. The result of the whisker weaving algorithm is a complete representation of hex mesh connectivity only: Actual mesh node locations are determined afterwards. The basic step of whisker weaving is to form a hexahedral element by crossing or intersecting dual entities. This operation, combined with seaming or joining operations in dual space, is sufficient to mesh simple block problems. When meshing more complex geometries, certain other dual entities appear such as blind chords, merged sheets, and self-intersecting chords. Occasionally specific types of invalid connectivity arise. These are detected by a general method based on repeated STC edges. This leads into a strategy for resolving some cases of invalidities immediately. The whisker weaving implementation has so far been successful at generating meshes for simple block-type geometries and for some non-block geometries. Mesh sizes are currently limited to a few hundred elements. While the size and complexity of meshes generated by whisker weaving are currently limited, the algorithm shows promise for extension to much more general problems.

Case report

Sheet inflation is an enhanced and more general version of the classic pillowing procedure [1] used to modify hexahedral meshes. The flexibility of sheet inflation makes it a valuable tool for hex mesh generation, modification and topology optimization. However, it is still difficult to generate self-intersecting sheet within a local region while assuring the mesh quality. This paper proposes an approach to achieving optimized complex sheet inflation under various constraints. The approach can generate complex sheets that intersect themselves more than once and maximize the quality of the resultant mesh. We successfully apply this approach to mesh matching and mesh boundary optimizing.