Automatic construction of quadrilateral network of curves from triangular meshes

The aim of this study is to propose a method for building quadrilateral network of curves automatically from a huge number of triangular meshes. The curve net can be served as the framework of automatic surface reconstruction. The proposed method mainly includes three stages: mesh simplification, quadrangulation and curve net generation. Mesh simplification is employed to reduce the number of meshes in accordance with a quadratic error metric for each vertex. Additional post-processing criteria are also employed to improve the shape of the reduced meshes. For quadrangulation, a front composed of a sequence of edges is introduced. An algorithm is proposed to combine each pair of triangles along the front. A new front is then formed and quadrangulation is continued until all triangles are combined or converted. For curve net generation, each edge of quadrilateral meshes is projected onto the triangular meshes to acquire a set of slicing points first. A constrained curve fitting is then employed to convert all sets of slicing points into B-spline curves, with appropriate continuity conditions across adjacent curves. Several examples have been presented to demonstrate the feasibility of the proposed method and its application in automatic surface reconstruction.     

Efficient Coding of Nontriangular Mesh Connectivity

We describe an efficient algorithm for coding the connectivity information of general polygon meshes. In contrast to most existing algorithms which are suitable only for triangular meshes, and pay a penalty for treatment of nontriangular faces, this algorithm codes the connectivity information in a direct manner. Our treatment of the special case of triangular meshes is shown to be equivalent to the Edgebreaker algorithm. Using our methods, any triangle mesh may be coded in no more than 2 bits/triangle (approximately 4 bits/vertex), a quadrilateral mesh in no more than 3.5 bits/quad (approximately 3.5 bits/vertex), and the most common case of a quad mesh with few triangles in no more than 4 bits/polygon.     

边界简化与多目标优化相结合的高质量四边形网格生成

目的 高质量四边形网格生成是计算机辅助设计、等几何分析与图形学领域中一个富有挑战性的重要问题。针对这一问题,提出一种基于边界简化与多目标优化的高质量四边形网格生成新框架。方法 首先针对亏格非零的平面区域,提出一种将多连通区域转化为单连通区域的方法,可生成高质量的插入边界;其次,提出"可简化角度"和"可简化面积比率"两个阈值概念,从顶点夹角和顶点三角形面积入手,将给定的多边形边界简化为粗糙多边形;然后对边界简化得到的粗糙多边形进行子域分解,并确定每个子域内的网格顶点连接信息;最后提出四边形网格的均匀性和正交性度量目标函数,并通过多目标非线性优化技术确定网格内部顶点的几何位置。结果 在同样的离散边界下,本文方法与现有方法所生成的四边网格相比,所生成的四边网格顶点和单元总数目较少,网格单元质量基本类似,计算时间成本大致相同,但奇异点数目可减少70% 80%,衡量网格单元质量的比例雅克比值等相关指标均有所提高。结论 本文所提出的四边形网格生成方法能够有效减少网格中的奇异点数目,并可生成具有良好光滑性、均匀性和正交性的高质量四边形网格,非常适用于工程分析和动画仿真。     

由区域生长算法实现四边形网格划分

针对当前各种三角形网格转化为四边形网格算法的缺陷，提出一种实现这种转化的算法——区域生长算法．该算法通过有选择地合并三角形来获得高质量的四边形网格，并且对两个三角形的合并增加约束条件以避免在生成四边形网格的过程中产生残余三角形；同时对生成的四边形网格进行一系列的质量改善操作。     

Fitting polynomial surfaces to triangular meshes with Voronoi squared distance minimization

This paper introduces Voronoi squared distance minimization (VSDM), an algorithm that fits a surface to an input mesh. VSDM minimizes an objective function that corresponds to a Voronoi-based approximation of the overall squared distance function between the surface and the input mesh (SDM). This objective function is a generalization of the one minimized by centroidal Voronoi tessellation, and can be minimized by a quasi-Newton solver. VSDM naturally adapts the orientation of the mesh elements to best approximate the input, without estimating any differential quantities. Therefore, it can be applied to triangle soups or surfaces with degenerate triangles, topological noise and sharp features. Applications of fitting quad meshes and polynomial surfaces to input triangular meshes are demonstrated.     

基于3DS模型的四边形网格生成算法研究

针对文献[1]中提出的四边形网格形变因子的不足,提出了曲面中改进的四边形形变因子。根据此约束条件,研究并实现了一种将3DS模型表面三角形网格转化为四边形网格的合并算法,并对极少的残余三角形进行拆分处理,实现彻底转化。经过对已有模型的实际转化实验,表明了本算法的有效性。     

Parallel quadrilateral subdomain generation for finite element analysis

In this article the Sub-domain Generation Method (SGM) previously developed for finite element meshes composed of triangular elements is generalised to account for meshes composed of arbitrary quadrilateral elements. A neural network of enhanced accuracy was developed to predict the number of quadrilaterals generated within each coarse quadrilateral element after refinement using an adaptive re-meshing procedure. Partitioning of the meshes was undertaken on the coarse mesh using a genetic algorithm to optimise the partitions considering both load balancing and interprocessor communication of the subsequent finite element analysis. A series of examples of increasing refinement were decomposed by considering a single coarse mesh of a fixed number of elements.     

Automatic generation of finite element meshes

This paper describes a new algorithm for the automatic generation of finite element meshes of arbitrary multiply connected domains. The strategy is based upon the construction of a mapping from the generated mesh into a regular one. The scheme is designed for maximum flexibility and is able to generate meshes of triangular or quadrilateral curved elements. Several examples are presented to illustrate the applicability of the algorithm.     

基于棱边约束的曲面体矩形片划分

对于带有特征的网格给出了一种将它们划分为一系列矩形片的算法．首先基于给定的一些特征，在网格上建立一个优化的三角剖分，在得到一系列的三角片或多边形片后，利用固定角保留的方法进行预处理；然后对这些三角片及多边形片之间的连接关系进行分类；最后针对不同的连接关系，定义相应的规则来提取矩形片．     

An improved technique for eliminating duplicate nodes in a two-dimensional mesh generator

A simple technique is presented to eliminate duplicate nodes and nodal parameters in a twodimensional mesh generator which was described previously in this journal. The improved technique suggested here is simpler and faster than other techniques. It requires only slight modifications on the original mesh generation program and can avoid duplicate nodes for both triangular and quadrilateral elements.     

Surface remeshing with robust high-order reconstruction

Remeshing is an important problem in variety of applications, such as finite element methods and geometry processing. Surface remeshing poses some unique challenges, as it must deliver not only good mesh quality but also good geometric accuracy. For applications such as finite elements with high-order elements (quadratic or cubic elements), the geometry must be preserved to high-order (third-order or higher) accuracy, since low-order accuracy may undermine the convergence of numerical computations. The problem is particularly challenging if the CAD model is not available for the underlying geometry, and is even more so if the surface meshes contain some inverted elements. We describe remeshing strategies that can simultaneously produce high-quality triangular meshes, untangling mildly folded triangles and preserve the geometry to high-order of accuracy. Our approach extends our earlier works on high-order surface reconstruction and mesh optimization by enhancing its robustness with a geometric limiter for under-resolved geometries. We also integrate high-order surface reconstruction with surface mesh adaptation techniques, which alter the number of triangles and nodes. We demonstrate the utilization of our method to meshes for high-order finite elements, biomedical image-based surface meshes, and complex interface meshes in fluid simulations.     

Lloyd’s energy minimization in the L p norm for quadrilateral surface mesh generation

Indirect methods recombine the elements of triangular meshes to produce quadrilaterals. The resulting quadrilaterals are usually randomly oriented, which is not desirable. However, by aligning the vertices of the initial triangular mesh, precisely oriented quads can be produced. Levy’s algorithm is a non-linear optimization procedure that can align points according to a locally defined orientation matrix. It minimizes an energy functional based on the L _p distance. The triangulation of a set of vertices smoothed with Levy’s algorithm is mainly composed of right-angled triangles, which is ideal for quad recombination. An implementation of Levy’s algorithm based on numerical integration is presented. The implementation has the advantage of not modifying the edge meshes. It also features automatic calculation of the orientation angle. When used in combination with an indirect recombination algorithm, it can create quads of varying size and orientation. It has been tested on two-dimensional surfaces as well as globally parametrized three-dimensional surfaces. The results demonstrate an increase in the number of nodes having four neighbors and an improvement of the quads quality. The development took place in the framework of the Gmsh free software.     

Winslow Smoothing on Two-Dimensional Unstructured Meshes

The Winslow equations from structured elliptic grid generation are adapted to smoothing of two-dimensional unstructured meshes using a finite difference approach. We use a local mapping from a uniform N-valent logical mesh to a local physical subdomain. Taylor Series expansions are then applied to compute the derivatives which appear in the Winslow equations. The resulting algorithm for Winslow smoothing on unstructured triangular and quadrilateral meshes gives generally superior qualilty than traditional Laplacian smoothing, while retaining the resistance to mesh folding on structured quadrilateral meshes.     

结合边折叠和局部优化的网格简化算法

针对目前网格简化算法在将三维模型简化到较低分辨率时,网格模型的细节特征丢失、网格质量不佳的问题,提出一种保持特征的高质量网格简化算法。引入顶点近似曲率的概念,并将其与边折叠的误差矩阵结合,使得简化模型的细节特征在最大限度上得到保持。同时分析简化后三角网格的质量,对三角网格作局部优化处理,减少狭长三角形的数量,提高简化模型的网格质量。使用Apple模型和Horse模型进行实验,并与一种经典的基于边折叠的网格简化算法以及其改进算法之一进行对比。实验结果显示,两种对比算法三角网格分布过于均匀,局部细节模糊不清,而所提算法的三角网格在曲率大的区域稠密,在平坦处稀疏,细节特征清晰可辨;简化模型的几何误差的数量值与两种对比算法处于同一数量级;所提算法的简化网格的平均质量远高于两种对比算法。实验结果表明,在不扩大几何误差的情况下,所提算法不仅具有较强的细节特征保持能力,而且简化模型的网格质量较高,视觉效果较好。     

Generation of bi-monotone patches from quadrilateral mesh for reverse engineering

Thanks to recent improvements, computational methods can now be used to convert triangular meshes into quadrilateral meshes so that the quadrilateral elements capture well the principal curvature directional fields of surfaces and intrinsically have surface parametric values. In this study, a quadrilateral mesh generated using the mixed integer quadrangulation technique of Bommes et al. is used for input. We first segment a quadrilateral mesh into four-sided patches. The feature curves inside these patches are then detected and are constrained to act as the patch boundaries. Finally, the patch configuration is improved to generate large patches. The proposed method produces bi-monotone patches, which are appropriate for use in reverse engineering to capture the surface details of an object. A shape control parameter that can be adjusted by the user during the patch generation process is also provided to support the creation of patches with good bi-monotone shapes. This study mainly targets shape models of mechanical parts consisting of major smooth surfaces with feature curves between them.     

HP90: A general and flexible Fortran 90 hp-FE code

A general 2D-hp-adaptive Finite Element (FE) implementation in Fortran 90 is described. The implementation is based on an abstract data structure, which allows to incorporate the full hp-adaptivity of triangular and quadrilateral finite elements. The h-refinement strategies are based on h2-refinement of quadrilaterals and h4-refinement of triangles. For p-refinement we allow the approximation order to vary within any element. The mesh refinement algorithms are restricted to 1-irregular meshes. Anisotropic and geometric refinement of quadrilateral meshes is made possible by additionally allowing double constrained nodes in rectangles. The capabilities of this hp-adaptive FE package are demonstrated on various test problems. Received: 18 December 1997 / Accepted: 17 April 1998     

Face coloring in unstructured CFD codes

We propose an algorithm for preventing race conditions in the evaluation of the surface integral contributions in edge-based CFD solvers by coloring the faces (or edges) of the computational mesh. We use a partitioning algorithm that separates the edges of triangular elements into three groups, the faces of quadrilateral and tetrahedral elements into four groups, and the faces of hexahedral elements into six groups. Our method is also applicable to hybrid meshes. We then extend this partitioning to adaptively refined, nonconforming meshes. We use the coloring to reduce code memory requirements by eliminating buffering. The coloring is also used to renumber and reorder elemental data to optimize reading and writing to memory, thus reducing access latencies and accelerating computations.     

一种用于自动生成二维全四边形有限元网格的改进的铺设算法

本文所述改进算法删除了原有算法的诸多繁琐之处,改进了原有算法的许多不足,并加入了一些新的算法规则。实例测试结果表明,用本文所述改进算法生产的四边形网格具有网格分布均匀,四格四边形接近于规则四边等特点。     

极大熵条件下的动态搜索优化图象重建方法

提出一种满足多种准则的动态搜索优化图象重建方法。从图象场的本质出发，设立了３个准则函数。熵函数的导数是非线性的，本文将其变换为近似的线性公式以获得迭代公路。动态搜索则尽可能避免很多算法在迭代过程中对图象校正过量或不足的问题。     

四边形网格间接生成方法

研究了基于背景三角网格的四边形网格间接生成算法,并针对三角形合并过程中容易残留三角形的缺陷提出了确定侧边的详细算法,该算法主要是依据背景三角网格中边的位置和前沿边的情形,通过背景三角网格中已存在的边、边交换或边分割确定侧边,以避免在三角形合并过程中残留三角形单元。最后给出实例验证了算法的有效性。