Feature-Sensitive Tetrahedral Mesh Generation with Guaranteed Quality

Tetrahedral meshes are being extensively used in finite element methods (FEM). This paper proposes an algorithm to generate feature-sensitive and high-quality tetrahedral meshes from an arbitrary surface mesh model. A top-down octree subdivision is conducted on the surface mesh and a set of tetrahedra are constructed using adaptive body-centered cubic (BCC) lattices. Special treatments are given to the tetrahedra near the surface such that the quality of the resulting tetrahedral mesh is provably guaranteed: the smallest dihedral angle is always greater than 5.71°. The meshes generated by our method are not only adaptive from the interior to the boundary, but also feature-sensitive on the surface with denser elements in high-curvature regions where geometric feature most likely reside. A variety of experimental results are presented to demonstrate the effectiveness and robustness of this algorithm.     

Feature-preserving adaptive mesh generation for molecular shape modeling and simulation

We describe a chain of algorithms for molecular surface and volumetric mesh generation. We take as inputs the centers and radii of all atoms of a molecule and the toolchain outputs both triangular and tetrahedral meshes that can be used for molecular shape modeling and simulation. Experiments on a number of molecules are demonstrated, showing that our methods possess several desirable properties: feature-preservation, local adaptivity, high quality, and smoothness (for surface meshes). We also demonstrate an example of molecular simulation using the finite element method and the meshes generated by our method. The approaches presented and their implementations are also applicable to other types of inputs such as 3D scalar volumes and triangular surface meshes with low quality, and hence can be used for generation/improvement of meshes in a broad range of applications.     

从表面重构的体数据实现3维网格剖分

目的 针对有限元分析中网格最优化问题,提出一种改进的生成四面体网格的自组织算法。方法 该算法首先应用几何方法将三角形表面模型重新构造成规定大小的分类体数据,同时由该表面模型建立平衡八叉树,计算用以控制网格尺寸的3维数组;然后将体数据转换成邻域内不同等值面的形态一致的边界指示数组;结合改进的自组织算法和相关3维数据的插值函数,达到生成四面体网格的目的。结果 实验结果对比表明,该方法能够生成更高比例的优质四面体,增强了对扁平面体的抑制能力,同时很好地保证了边界的一致。结论 在对封闭的3维表面网格进行有限元建模时,本文算法为其提供了一种有效、可靠的途径。     

New software developments for quality mesh generation and optimization from biomedical imaging data

In this paper we present a new software toolkit for generating and optimizing surface and volumetric meshes from three-dimensional (3D) biomedical imaging data, targeted at image-based finite element analysis of some biomedical activities in a single material domain. Our toolkit includes a series of geometric processing algorithms including surface re-meshing and quality-guaranteed tetrahedral mesh generation and optimization. All methods described have been encapsulated into a user-friendly graphical interface for easy manipulation and informative visualization of biomedical images and mesh models. Numerous examples are presented to demonstrate the effectiveness and efficiency of the described methods and toolkit.     

从表面重构的体数据实现三维网格剖分

针对有限元分析中网格最优化问题,本文提出一种改进的生成四面体网格的自组织算法。该算法首先应用几何方法将三角形表面模型重新构造成规定大小的分类体数据,同时由该表面模型建立平衡八叉树,计算用以控制网格尺寸的三维数组;然后将体数据转换成邻域内不同等值面的形态一致的边界指示数组;结合改进的自组织算法和相关三维数据的插值函数,达到生成四面体网格的目的。实验对比表明,该方法能够生成更高比例的优质四面体,同时很好地保证了边界的一致。在对封闭的三维表面网格进行有限元建模时,本文算法为其提供了一种有效、可靠的途径。     

Hexahedral mesh smoothing via local element regularization and global mesh optimization

The quality of finite element meshes is one of the key factors that affect the accuracy and reliability of finite element analysis results. In order to improve the quality of hexahedral meshes, we present a novel hexahedral mesh smoothing algorithm which combines a local regularization for each hexahedral mesh, using dual element based geometric transformation, with a global optimization operator for all hexahedral meshes. The global optimization operator is composed of three main terms, including the volumetric Laplacian operator of hexahedral meshes and the geometric constraints of surface meshes which keep the volumetric details and the surface details, and another is the transformed node displacements condition which maintains the regularity of all elements. The global optimization operator is formulated as a quadratic optimization problem, which is easily solved by solving a sparse linear system. Several experimental results are presented to demonstrate that our method obtains higher quality results than other state-of-the-art approaches.     

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.     

Feature preserving Delaunay mesh generation from 3D multi-material images

Generating realistic geometric models from 3D segmented images is an important task in many biomedical applications. Segmented 3D images impose particular challenges for meshing algorithms because they contain multi-material junctions forming features such as surface patches, edges and corners. The resulting meshes should preserve these features to ensure the visual quality and the mechanical soundness of the models. We present a feature preserving Delaunay refinement algorithm which can be used to generate high-quality tetrahedral meshes from segmented images. The idea is to explicitly sample corners and edges from the input image and to constrain the Delaunay refinement algorithm to preserve these features in addition to the surface patches. Our experimental results on segmented medical images have shown that, within a few seconds, the algorithm outputs a tetrahedral mesh in which each material is represented as a consistent submesh without gaps and overlaps. The optimization property of the Delaunay triangulation makes these meshes suitable for the purpose of realistic visualization or finite element simulations.     

三维实体仿真建模的网格自动生成方法

有限元网格模型的生成与几何拓扑特征和力学特性有直接关系。建立网格模型时，为了更真实地反映原几何形体的特征，在小特征尺寸或曲率较大等局部区域网格应加密剖分；为提高有限元分析精度和效率，在待分析的开口、裂纹、几何突变、外载、约束等具有应力集中力学特性的局部区域，网格应加密剖分。为此，该文提出了基于几何特征和物理特性相结合的网格自动生成方法。该方法既能有效地描述几何形体，又能实现应力集中区域的网格局部加密及粗细网格的均匀过渡。实例表明本方法实用性强、效果良好。     

An automatic 3D mesh generation method for domains with multiple materials

This paper describes an automatic and efficient approach to construct unstructured tetrahedral and hexahedral meshes for a composite domain made up of heterogeneous materials. The boundaries of these material regions form non-manifold surfaces. In earlier papers, we developed an octree-based isocontouring method to construct unstructured 3D meshes for a single material (homogeneous) domain with manifold boundary. In this paper, we introduce the notion of a material change edge and use it to identify the interface between two or several different materials. A novel method to calculate the minimizer point for a cell shared by more than two materials is provided, which forms a non-manifold node on the boundary. We then mesh all the material regions simultaneously and automatically while conforming to their boundaries directly from volumetric data. Both material change edges and interior edges are analyzed to construct tetrahedral meshes, and interior grid points are analyzed for proper hexahedral mesh construction. Finally, edge-contraction and smoothing methods are used to improve the quality of tetrahedral meshes, and a combination of pillowing, geometric flow and optimization techniques is used for hexahedral mesh quality improvement. The shrink set of pillowing schemes is defined automatically as the boundary of each material region. Several application results of our multi-material mesh generation method are also provided.     

基于医学体数据生成四面体网格的方法

为了从医学体数据直接构造四面体网格,提出一种基于栅格的网格生成算法.该算法的主要思想是从背景栅格中提取并填充代表区域边界的等值面.首先,对医学体数据进行预处理与采样,构建一个背景栅格.其次,用对偶方法从栅格提取三角表面网格,用于分段线性逼近等值面.然后,对栅格中所有位于等值面之内或与等值面相交的立方体,用预定义的模板分解成四面体单元.最后,用Laplacian平滑技术优化四面体网格.在均匀网格的基础上,研究了自适应网格生成算法,在保持网格几何精度的同时精简单元数量,以提高有限元计算效率.给出了从CT数据生成人体股骨远端四面体网格的实例,该网格模型被用于虚拟膝关节镜手术.     

Mesh Quality: A Function of Geometry, Error Estimates or Both?

The issue of mesh quality for unstructured triangular and tetrahedral meshes is considered. The theoretical background to finite element methods is used to understand the basis of present-day geometrical mesh quality indicators. A survey of more recent research in the development of finite element methods describes work on anisotropic meshing algorithms, and on providing good error estimates that reveal the relationship between the error and both the mesh and the solution gradients. The reality of solving complex three-dimensional problems is that such indicators are presently not available for many problems of interest. A simple tetrahedral mesh quality measure using both geometrical and solution information is described. Some of the issues in mesh quality for unstructured tetrahedral meshes are illustrated by means of two simple examples.     

A CPU–GPU framework for optimizing the quality of large meshes

The automatic generation of 3D finite element meshes (FEM) is still a bottleneck for the simulation of large fluid dynamic problems. Although today there are several algorithms that can generate good meshes without user intervention, in cases where the geometry changes during the calculation and thousands of meshes must be constructed, the computational cost of this process can exceed the cost of the FEM. There has been a lot of work in FEM parallelization and the algorithms work well in different parallel architectures, but at present there has not been much success in the parallelization of mesh generation methods. This paper will present a massive parallelization scheme for re-meshing with tetrahedral elements using the local modification algorithm. This method is frequently used to improve the quality of elements once the mesh has been generated, but we will show it can also be applied as a regeneration process, starting with the distorted and invalid mesh of the previous step. The parallelization is carried out using OpenCL and OpenMP in order to test the method in a multiple CPU architecture and also in Graphics Processing Units (GPUs). Finally we present the speedup and quality results obtained in meshes with hundreds of thousands of elements and different parallel APIs.     

Direct extraction of surface meshes from implicitly represented heterogeneous volumes

This paper describes a novel algorithm to extract surface meshes directly from implicitly represented heterogeneous models made of different constituent materials. Our approach can directly convert implicitly represented heterogeneous objects into a surface model separating homogeneous material regions, where every homogeneous region in a heterogeneous structure is enclosed by a set of two-manifold surface meshes. Unlike other discretization techniques of implicitly represented heterogeneous objects, the intermediate surfaces between two constituent materials can be directly extracted by our algorithm. Therefore, it is more convenient to adopt the surface meshes from our approach in the boundary element method (BEM) or as a starting model to generate volumetric meshes preserving intermediate surfaces for the finite element method (FEM). The algorithm consists of three major steps: firstly, a set of assembled two-manifold surface patches coarsely approximating the interfaces between homogeneous regions are extracted and segmented; secondly, signed distance fields are constructed such that each field expresses the Euclidean distance from points to the surface of one homogeneous material region; and finally, coarse patches generated in the first step are dynamically optimized to give adaptive and high-quality surface meshes. The manifold topology is preserved on each surface patch.     

基于LEPP递变的四面体网格自适应剖分算法

为了更合理地进行四面体网格剖分,提出了一种根据待剖分对象形态不同进行网格密度自适应调整的四面体网格剖分方法。该方法首先采用BCC(body-centered cubic)网格初始化网格空间,并根据表面曲率的大小以及距离物体表面的远近,采用LEPP(longest edge propagation path)算法由外至内对初始化后的网格空间进行不同尺度的细分;然后对横跨表面的网格进行调整,以形成对象的表面形态;最后采用以质量函数引导的拉普拉斯平滑与棱边收缩(edge collapse)的方法对网格的质量进行优化来最终得到待剖分对象的四面体网格。结果表明,该方法所生成的网格不仅具有自适应的网格密度,而且网格质量比常用的Advancing Front算法也有所提高。对于基于3维断层图像或表面模型进行有限元建模,该方法不失为一种行之有效的好方法。     

Tetrahedral finite element mesh generation from NURBS solid models

In this paper, the development and the implementation of a tetrahedral meshing algorithm for generation of finite element meshes from NURBS solid models is presented. The meshing algorithm is based on a Delaunay technique, and makes use of some spatial data structures. The algorithm is capable of generating both uniform and varying size four-node and ten-node tetrahedral meshes. The algorithm has been implemented in a building block approach as part of a software library. It has been used as a practical tool in engineering design processes. Several representative test cases illustrate the effectiveness of the automatic solid mesh generator.     

Transformation of Hexaedral Finite Element Meshes into Tetrahedral Meshes According to Quality Criteria

The paper is concerned with algorithms for transforming hexahedral finite element meshes into tetrahedral meshes without introducing new nodes. Known algorithms use only the topological structure of the hexahedral mesh but no geometry information. The paper provides another algorithm which is then extented such that quality criteria for the splitting of faces are respected.     

三角形条带网格模型几何压缩方法研究

三角形条带为三角形网格提供了一种紧凑的表示方法，使快速的绘制和传输三角形网格成为可能，因此对由三角形条带构成的网格压缩进行研究具有重要的意义．本文使用Triangle Fixer方法对三角形条带构成的三维模型拓扑信息进行了压缩，并采用3阶自适应算术编码进一步提高压缩率；同时结合量化、平行四边形顶点坐标预测以及算术编码来实现三角形网格几何信息的压缩，在几何模型质量基本没有损失的情况下，获得了很好的压缩性能．     

基于边长的三维形状插值

形状插值在计算机图形学和几何处理中是一个极其重要而基础的问题,在计算机动画等领域有 着广泛应用。注意到在平面三角网格和三维四面体网格插值问题中,对边长平方插值等价于对回拉度量进行插 值,因此具有等距扭曲和共形扭曲同时有界的良好性质。通过将其推广至曲面三角网格,提出了一种完全基于 边长的曲面三角网格插值算法。给定边长,在重建网格阶段,使用牛顿法对边长误差能量进行优化。并且给出 了其海森矩阵的解析正定化形式,从而避免了高代价的特征值分解步骤。注意到四面体网格的边长平方插值结 果具有极低曲率,意味着只需少许修改即可将其压平从而嵌入三维空间。因此提出先将曲面三角网格四面体化, 再从四面体网格的插值结果提取表面。然后将这表面作为初始化用于边长误差能量的牛顿迭代,从而使得收敛 结果更加接近全局最优。在一系列三角网格上进行了实验,结果说明了本文方法比之前方法的边长误差更小, 且得到的结果还是有界扭曲的。     

面向有限元网格生成的单元尺寸场光滑化算法

针对单元尺寸场的合适与否会直接影响到后续有限元网格质量的问题,提出一种尺寸修正算法来优化单元尺寸场。在Borouchaki等提出的H变化量（BOROUCHAKI H, HECHT F, FREY P J. Mesh gradation control. International Journal for Numerical Methods in Engineering, 1998,43(6):1143-1165）的基础上,引入尺寸梯度概念,进行一系列公式推导,得到二维的单元尺寸场的合理过渡要求,从而以定义在非结构背景网格的单元尺寸场为例,改进Borouchaki修正算法,提出了一种最少量地重置尺寸场中节点单元尺寸值,最大化地全局光滑单元尺寸场的新算法。最后给出若干实例的网格生成效果图,证明算法能帮助工程应用的模型生成更高质量的网格,跟其他修正算法相比,网格尺寸过渡明显更均匀。