Hexahedral mesh generation constraints

For finite element analyses within highly elastic and plastic structural domains, hexahedral meshes have historically offered some benefits over tetrahedral finite element meshes in terms of reduced error, smaller element counts, and improved reliability. However, hexahedral finite element mesh generation continues to be difficult to perform and automate, with hexahedral mesh generation taking several orders of magnitude longer than current tetrahedral mesh generators to complete. Thus, developing a better understanding of the underlying constraints that make hexahedral meshing difficult could result in dramatic reductions in the amount of time necessary to prepare a hexahedral finite element model for analysis. In this paper, we present a survey of constraints associated with hexahedral meshes (i.e., the conditions that must be satisfied to produce a hexahedral mesh). In presenting our formulation of these constraints, we will utilize the dual of a hexahedral mesh. We also discuss how incorporation of these constraints into existing hexahedral mesh generation algorithms could be utilized to extend the class of geometries to which these algorithms apply. We also describe a list of open problems in hexahedral mesh generation and give some context for future efforts in addressing these problems.     

The geometric element transformation method for mixed mesh smoothing

The geometric element transformation method (GETMe) is a geometry-based smoothing method for mixed and non-mixed meshes. It is based on a simple geometric transformation applicable to elements bounded by polygons with an arbitrary number of nodes. The transformation, if applied iteratively, leads to a regularization of the polygons. Global mesh smoothing is accomplished by averaging the new node positions obtained by local element transformations. Thereby, the choice of transformation parameters as well as averaging weights can be based on the element quality which leads to high quality results. In this paper, a concept of an enhanced transformation approach is presented and a proof for the regularizing effect of the transformation based on eigenpolygons is given. Numerical examples confirm that the GETMe approach leads to superior mesh quality if compared to other geometry-based methods. In terms of quality it can even compete with optimization-based techniques, despite being conceptually significantly simpler.     

An approach to automated decomposition of volumetric mesh

Mesh decomposition is critical for analyzing, understanding, editing and reusing of mesh models. Although there are many methods for mesh decomposition, most utilize only triangular meshes. In this paper, we present an automated method for decomposing a volumetric mesh into semantic components. Our method consists of three parts. First, the outer surface mesh of the volumetric mesh is decomposed into semantic features by applying existing surface mesh segmentation and feature recognition techniques. Then, for each recognized feature, its outer boundary lines are identified, and the corresponding splitter element groups are setup accordingly. The inner volumetric elements of the feature are then obtained based on the established splitter element groups. Finally, each splitter element group is decomposed into two parts using the graph cut algorithm; each group completely belongs to one feature adjacent to the splitter element group. In our graph cut algorithm, the weights of the edges in the dual graph are calculated based on the electric field, which is generated using the vertices of the boundary lines of the features. Experiments on both tetrahedral and hexahedral meshes demonstrate the effectiveness of our method.     

Simultaneous untangling and smoothing of quadrilateral and hexahedral meshes using an object-oriented framework

In this work, we present a simultaneous untangling and smoothing technique for quadrilateral and hexahedral meshes. The algorithm iteratively improves a quadrilateral or hexahedral mesh by minimizing an objective function defined in terms of a regularized algebraic distortion measure of the elements. We propose several techniques to improve the robustness and the computational efficiency of the optimization algorithm. In addition, we have adopted an object-oriented paradigm to create a common framework to smooth meshes composed by any type of elements, and using different minimization techniques. Finally, we present several examples to show that the proposed technique obtains valid meshes composed by high-quality quadrilaterals and hexahedra, even when the initial meshes contain a large number of tangled elements.     

Non-iterative approach for global mesh optimization

This paper presents a global optimization operator for arbitrary meshes. The global optimization operator is composed of two main terms, one part is the global Laplacian operator of the mesh which keeps the fairness and another is the constraint condition which reserves the fidelity to the mesh. The global optimization operator is formulated as a quadratic optimization problem, which is easily solved by solving a sparse linear system. Our global mesh optimization approach can be effectively used in at least three applications: smoothing the noisy mesh, improving the simplified mesh, and geometric modeling with subdivision-connectivity. Many experimental results are presented to show the applicability and flexibility of the approach.     

A local cell quality metric and variational grid smoothing algorithm

A local cell quality metric is introduced and used to construct a variational functional for a grid smoothing algorithm. A maximum principle is proved and the properties of the local quality measure, which combines element shape and size control metrics, are investigated. Level set contours are displayed to indicate the effect of cell distortion. The approach is demonstrated for meshes of triangles and quadrilaterals in 2D and a test case with hexahedral cells in 3D. Issues such as the use of a penalty for folded meshes and the effect of valence change in the mesh patches are considered.     

Hexahedral mesh development of free-formed geometry: The human femur exemplified

The objective of the present research is to develop a standard hexahedral Finite Element (FE) model of the human femur accounting for the material characteristics of cortical bone, cancellous bone and bone marrow. The anatomical data were acquired from the Visible Human Project. A detailed outline of the steps necessary in developing hexahedral FE meshes from computed tomography (CT) data is provided, along with a section on modelling strategies providing comprehensive suggestions on how to overcome meshing difficulties due to geometrical non-linearities. The stress and deformation results are discussed.     

A comparison of two optimization methods for mesh quality improvement

We compare inexact Newton and block coordinate descent optimization methods for improving the quality of a mesh by repositioning the vertices, where the overall quality is measured by the harmonic mean of the mean-ratio metric. The effects of problem size, element size heterogeneity, and various vertex displacement schemes on the performance of these algorithms are assessed for a series of tetrahedral meshes.

Automated Conformal Hexahedral Meshing Constraints, Challenges and Opportunities

Automated hexahedral element meshing has been the ‘Holy Grail’ of mesh generation research for years. The rigid connectivity and shape constraints of these meshes provide the challenge. The ever-present economic pressure for automating meshing of complex geometries, difficult transitions and large mesh sizes establishes the opportunity. This paper will systematically review the requirements of the hexahedral meshing challenge, the various approaches to the solution of the problem (along with their respective attributes), and some musings about future research opportunities.     

Hexahedral and Tetrahedral Mesh Untangling

We investigate a well-motivated mesh untangling objective function whose optimization automatically produces non-inverted elements when possible. Examples show the procedure is highly effective on tetrahedral meshes and on many hexahedral meshes constructed via mapping or sweeping algorithms.     

Robust mesh smoothing

This paper proposes a vertex-estimation-based, feature-preserving smoothing technique for meshes. A robust mesh smoothing operator called mean value coordinates flow is introduced to modify mean curvature flow and make it more stable. Also the paper proposes a three-pass vertex estimation based on bilateral filtering of local neighbors which is transferred from image processing settings and a Quasi-Laplacian operation, derived from the standard Laplacian operator, is performed to increase the smoothness order of the mesh rapidly whilst denoising meshes efficiently, preventing volume shrinkage as well as preserving sharp features of the mesh. Compared with previous algorithms, the result shows it is simple, efficient and robust.     

SUSAN structure preserving filtering for mesh denoising

Motivated by the impressive effect of the SUSAN operator for low level image processing and its usage simplicity, we extend it to denoise the 3D mesh. We use the angle between the normals on the surface to determine the SUSAN area; each point has associated itself with the SUSAN area that is has a similar continuity feature to the point. The SUSAN area avoids the feature to be taken as noise effectively, so the SUSAN operator gives the maximal number of suitable neighbors with which to take an average, whilst no neighbors from unrelated regions are involved. Thus, the entire structure can be preserved. We also extend the SUSAN operator to two-ring neighbors by a squared umbrella-operator to improve the surface smoothness with little loss of detailed features. Details of the SUSAN structure preserving noise reduction algorithm are discussed along with the test results in this paper.     

An approach to surface retouching and mesh smoothing

In this paper, we discuss a novel, fast, practical algorithm for surface modification of geometric objects. A space-mapping technique is used to transform a given or damaged part of a surface into a different shape in a continuous manner. The proposed approach is used for surface-retouching and mesh-smoothing problems. The technique, in fact, is based on a local processing of polygonal data that can be applied to the fairing of 3D meshes. We consider shape transformation as a general type of operation for surface modification and attempt to approach the problem from a single point of view, namely, that of the space-mapping technique based on the implementation of radial-basis functions. Experimental results are included to demonstrate the functionality of our mesh-modeling tool.     

Context-aware mesh smoothing for biomedical applications

Smoothing algorithms allow to reduce artifacts from mesh generation, but often degrade accuracy. Thus, we present a method that identifies staircase artifacts which result from image inhomogeneities and binary segmentation in medical image data for subsequent removal by adaptive mesh smoothing. This paper makes the following specific contributions: caps, which are flat regions, resulting from segmentation or clipping at the endings of anatomical structures are detected and modified by smoothing; the effects of the adaptive smoothing method involving context information are quantitatively analyzed with respect to accuracy and their influence on blood flow simulations; the image stack orientation, which is relevant for this context-aware smoothing approach, is estimated automatically from the surface models. Thus, context-aware smoothing enables to adaptively smooth artifact areas, while non-artifact features can be preserved. The approach has been applied to CT neck datasets, as well as phantom data and the results are evaluated regarding smoothness and model accuracy. The accuracy of model orientation estimation and cap detection has been evaluated for clinical and phantom data. Finally, context-aware smoothing has been applied to CT angiography data for the simulation of blood flow. The simulation results are presented and prove the general suitability of context-aware smoothing.     

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. Du 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.     

Shape optimization of quad mesh elements

We study the problem of optimizing the face elements of a quad mesh surface, that is, re-sampling a given quad mesh to make it possess, as much as possible, face elements of some desired aspect ratio and size. Unlike previous quad mesh optimization/improvement methods based on local operations on a small group of elements, we adopt a global approach that does not introduce extra singularities and therefore preserves the original quad structure of the input mesh. Starting from a collection of quad patches extracted from an input quad mesh, two global operations, i.e. re-sampling and re-distribution, are performed to optimize the number and spacings of grid lines in each patch. Both operations are formulated as simple optimization problems with linear constraints.     

As-rigid-as-possible mesh deformation and its application in hexahedral mesh generation

This paper presents an efficient and stable as-rigid-as-possible mesh deformation algorithm for planar shape deformation and hexahedral mesh generation. The deformation algorithm aims to preserve two local geometric properties: scale-invariant intrinsic variables and elastic deformation energy, which are together represented in a quadric energy function. To preserve these properties, the position of each vertex is further adjusted by iteratively minimizing this quadric energy function to meet the position constraint of the controlling points. Experimental results show that the deformation algorithm is efficient, and can obtain physically plausible results, which have the same topology structure with the original mesh. Such a mesh deformation method is useful to project the source surface mesh onto the target surfaces in hexahedral mesh generation based on sweep method, and application results show that the proposed method is feasible to mesh projection not only between similar surface contours but also dissimilar surface contours.     

Automatic mesh generation and modification techniques for mixed quadrilateral and hexahedral element meshes of non-manifold models

A typical geometric model usually consists of both solid sections and thin-walled sections. Through using a suitable dimensional reduction algorithm, the model can be reduced to a non-manifold model consisting of solid portions and two-dimensional portions which represent the mid-surfaces of the thin-walled sections. It is desirable to mesh the solid entities using three-dimensional elements and the surface entities using two-dimensional elements. This paper proposes a robust scheme to automatically generate such a mesh of mixed two-dimensional and three-dimensional elements. It also ensures that the mesh is conforming at the interface of the non-manifold geometries. Different classes of problems are identified and their corresponding solutions are presented.