基于黎曼度量参数曲面四边形网格生成方法及UG实现

论文给出了基于黎曼度量的参数曲面网格生成的改进铺砖算法。阐述了曲面自身的黎曼度量,并且运用黎曼度量计算二维参数域上单元节点的位置,从而使映射到三维物理空间的四边形网格形状良好。文中对原有铺砖法相交处理进行了改进,在运用铺砖法的同时调用UG-NX强大的二次开发库函数获取相应的信息,直接在UG-NX模型的表面生成四边形网格。算例表明,该法能在曲面上生成质量好的网格。     

Automatic metric 3D surface mesh generation using subdivision surface geometrical model. Part 2: Mesh generation algorithm and examples

In this paper, a new metric advancing front surface mesh generation scheme is suggested. This new surface mesh generator is based on a new geometrical model employing the interpolating subdivision surface concept. The target surfaces to be meshed are represented implicitly by interpolating subdivision surfaces which allow the presence of various sharp and discontinuous features in the underlying geometrical model. While the main generation steps of the new generator are based on a robust metric surface triangulation kernel developed previously, a number of specially designed algorithms are developed in order to combine the existing metric advancing front algorithm with the new geometrical model. As a result, the application areas of the new mesh generator are largely extended and can be used to handle problems involving extensive changes in domain geometry. Numerical experience indicates that, by using the proposed mesh generation scheme, high quality surface meshes with rapid varying element size and anisotropic characteristics can be generated in a short time by using a low‐end PC. Finally, by using the pseudo‐curvature element‐size controlling metric to impose the curvature element‐size requirement in an implicit manner, the new mesh generation procedure can also generate finite element meshes with high fidelity to approximate the target surfaces accurately. Copyright © 2003 John Wiley & Sons, Ltd.     

An algebraic method for smoothing surface triangulations on a local parametric space

This paper presents a new procedure to improve the quality of triangular meshes defined on surfaces. The improvement is obtained by an iterative process in which each node of the mesh is moved to a new position that minimizes a certain objective function. This objective function is derived from algebraic quality measures of the local mesh (the set of triangles connected to the adjustable or free node). If we allow the free node to move on the surface without imposing any restriction, only guided by the improvement of the quality, the optimization procedure can construct a high‐quality local mesh, but with this node in an unacceptable position. To avoid this problem the optimization is done in the parametric mesh, where the presence of barriers in the objective function maintains the free node inside the feasible region. In this way, the original problem on the surface is transformed into a two‐dimensional one on the parametric space. In our case, the parametric space is a plane, chosen in terms of the local mesh, in such a way that this mesh can be optimally projected performing a valid mesh, that is, without inverted elements. Several examples and applications presented in this work show how this technique is capable of improving the quality of triangular surface meshes. Copyright © 2005 John Wiley & Sons, Ltd.     

Geometry independence for a meshing engine for 2D manifolds

The ‘meshing engine’ of the title is a software component that generates unstructured triangular meshes of two‐dimensional triangles for a variety of contexts. The mesh generation is based on the well‐known technique of iterative Delaunay refinement, for which the Euclidean metric is intrinsic. The meshing engine is to be connected to applications specific host programs which can use a geometry that is different from the intrinsic geometry of the mesh, i.e. locally, the Euclidean plane. An application may require a surface mesh for embedding in a three‐dimensional geometry, or it might use a Riemannian metric to specify a required anisotropy in the mesh, or both. We focus on how the meshing engine can be designed to be independent of the embedding geometry of a host program but conveniently linked to it. A crucial tool for these goals is the use of an appropriate local co‐ordinate system for the triangles as seen by the meshing engine. We refer to it as the longest edge co‐ordinate system. Our reference to ‘linking’ the meshing engine and host system is both general and technical in the sense that the example meshes provided in the paper have all been generated by the same object code of a prototype of such a meshing engine linked to host programs defining different embedding geometries. Copyright © 2004 John Wiley & Sons, Ltd.     

A new finite point generation scheme using metric specifications

A new approach to generate finite point meshes on 2D flat surface and any bi‐variate parametric surfaces is suggested. It can be used to generate boundary‐conforming anisotropic point meshes with node spacing compatible with the metric specifications defined in a background point mesh. In contrast to many automatic mesh generation schemes, the advancing front concept is abandoned in the present method. A few simple basic operations including boundary offsetting, node insertion and node deletion are used instead. The point mesh generation schemeis initialized by a boundary offsetting procedure. The point mesh quality is then improved by node insertion and deletion such that optimally spaced nodes will fill up the entire problem domain. In addition to the point mesh generation scheme, a new way to define the connectivity of a point mesh is also suggested. Furthermore, based on the connectivity information, a new scheme to perform smoothing for a point mesh is proposed toimprove the node spacing quality of the mesh. Timing shows thatdue to the simple node insertion and deletion operations, the generation speed of the new scheme is nearly 10 times faster than a similar advancing front mesh generator. Copyright © 2000 John Wiley & Sons, Ltd.     

On curvature element‐size control in metric surface mesh generation

A new procedure is suggested for controlling the element‐size distribution of surface meshes during automatic adaptive surface mesh generation. In order to ensure that the geometry of the surface can be accurately captured, the curvature properties of the surface are first analysed. Based on the principal curvatures and principal directions of the surface, the curvature element‐size requirement is defined in the form of a metric tensor field. This element‐size controlling metric tensor field, which can either be isotopic or anisotopic depending on the user requirement, is then employed to control the element size distribution during mesh generation. The suggested procedure is local, adaptive and can be easily used with many parametric surface mesh generators. As the proposed scheme defines the curvature element‐size requirement in an implicit manner, it can be combined with any other user defined element size specification using the standard metric intersection procedure. This eventually leads to a simple implementation procedure and a high computational efficiency. Numerical examples indicate that the new procedure can effectively control the element size of surfacemeshes in the cost of very little additional computational effort. Copyright © 2001 John Wiley & Sons, Ltd.     

Subdivision schemes for smooth contact surfaces of arbitrary mesh topology in 3D

This paper presents a strategy to parameterize contact surfaces of arbitrary mesh topology in 3D with at least C¹‐continuity for both quadrilateral and triangular meshes. In the regular mesh domain, four quadrilaterals or six triangles meet in one node, even C²‐continuity is attained. Therefore, we use subdivision surfaces, for which non‐physical pressure jumps are avoided for contact interactions. They are usually present when the contact kinematics is based on facet elements discretizing the interacting bodies. The properties of subdivision surfaces give rise to basically four different implementation strategies. Each strategy has specific features and requires more or less efforts for an implementation in a finite element program. One strategy is superior with respect to the others in the sense that it does not use nodal degrees of freedom of the finite element mesh at the contact surface. Instead, it directly uses the degrees of freedom of the smooth surface. Thereby, remarkably, it does not require an interpolation. We show how the proposed method can be used to parameterize adaptively refined meshes with hanging nodes. This is essential when dealing with finite element models whose geometry is generated by means of subdivision techniques. Three numerical 3D problems demonstrate the improved accuracy, robustness and performance of the proposed method over facet‐based contact surfaces. In particular, the third problem, adopted from biomechanics, shows the advantages when designing complex contact surfaces by means of subdivision techniques. Copyright © 2004 John Wiley & Sons, Ltd.     

Automatic generation of anisotropic quadrilateral meshes on three‐dimensional surfaces using metric specifications

A new algorithm for constructing full quadrilateral anisotropic meshes on 3D surfaces is proposed in this paper. The proposed method is based on the advancing front and the systemic merging techniques. Full quadrilateral meshes are constructed by systemically converting triangular elements in the background meshes into quadrilateral elements.By using the metric specifications to describe the element characteristics, the proposed algorithm is applicable to convert both isotropic and anisotropic triangular meshes into full quadrilateral meshes. Special techniques for generating anisotropic quadrilaterals such as new selection criteria of base segment for merging, new approaches for the modifications of the background mesh and construction of quadrilateral elements, are investigated and proposed in this study. Since the final quadrilateral mesh is constructed from a background triangular mesh and the merging procedure is carried out in the parametric space, the mesh generator is robust and no expensive geometrical computation that is commonly associated with direct quadrilateral mesh generation schemes is needed. Copyright © 2002 John Wiley & Sons, Ltd.     

Recovery of an arbitrary edge on an existing surface mesh using local mesh modifications

This study describes an algorithm for recovering an edge which is arbitrarily inserted onto a pre‐triangulated surface mesh. The recovery process does not rely on the parametric space of the surface mesh provided by the geometric modeller. The topological and geometrical validity of the surface mesh is preserved through the entire recovery process. The ability of inserting and recovering an arbitrary edge onto a surface mesh can be an invaluable tool for a number of meshing applications such as boundary layer mesh generation, solution adaptation, preserving the surface conformity, and possibly as a primary tool for mesh generation. The edge recovery algorithm utilizes local surface mesh modification operations of edge swapping, collapsing and splitting. The mesh modification operations are decided by the results of pure geometrical checks such as point and line projections onto faces and face‐line intersections. The accuracy of these checks on the recovery process are investigated and the substantiated precautions are devised and discussed in this study. Copyright © 2001 John Wiley & Sons, Ltd.     

GradH‐Correction: guaranteed sizing gradation in multi‐patch parametric surface meshing

In this paper a new method, called GradH‐Correction, for the generation of multi‐patch parametric surface meshes with controlled sizing gradation is presented. Such gradation is obtained performing a correction on the size values located on the vertices of the background mesh used to define the control space that governs the meshing process. In the presence of a multi‐patch surface, like shells of BREP solids, the proposed algorithm manages the whole composite surface simultaneously and as a unique entity. Sizing information can spread from a patch to its adjacent ones and the resulting size gradation is independent from the surface partitioning. Theoretical considerations lead to the assertion that, given a parameter λ, after performing a GradH‐Correction of level λ over the control space, the unit mesh constructed using the corrected control space is a mesh of gradation λ in the real space (target space). This means that the length ratio of any two adjacent edges of the mesh is bounded between 1/λ and λ. Numerical results show that meshes generated from corrected control spaces are of high quality and good gradation also when the background mesh has poor quality. However, due to mesh generator imprecision and theoretical limitations, guaranteed gradation is achieved only for the sizing specifications and not for the generated mesh. Copyright © 2004 John Wiley & Sons, Ltd.     

一种裁剪曲面自适应三角剖分算法

本文介绍了一种裁剪曲面按精度三角剖分算法。三角剖分过程在参数域和曲面空间同时进行，参数域上控制三角片的拓扑关系，曲面空间进行精度检测。算法的核心思想是将裁剪曲面三角剖分视为约束剖分问题，从而使得三角形的细分操作拓展为有效域内插入散乱节点的三角剖分问题。算法简便、实用，三角化结果品质良好，已成功地应用于数控加工刀具轨迹干涉处理等具有精度要求的应用领域。     

构造两个曲面的拼接曲面

给出了当两个待拼接曲面的拼接线具有参数化形式或者可以参数化时,它们之间 G1光滑拼接曲面的构造方法。所得的拼接曲面是由一个空间曲线集定义的,每一条空间曲线由拼接线上的点对确定。并且拼接曲面都是参数曲面,它们的形状可以通过预设的参数很好地调整和控制。作为实例,讨论了两个截口是平面的或非平面的(由两个二次曲面定义)的二次曲面之间的拼接曲面的构造和一般的参数曲面的拼接曲面的构造。     

Filling space with tetrahedra

In the context of 3D finite element meshes various options for filling an indefinite space (such as would be approached within a fine mesh) with tetrahedra are considered. This problem is not trivial as it is in 2‐D since, unlike equilateral triangles, regular tetrahedra cannot be fitted together to fill space. Various groupings, or assemblies, which can be repeated indefinitely to fill space are considered. By altering the shape of the tetrahedra in one of these to minimize a suitable function a unique shape of tetrahedron is obtained which optimizes the conditioning. The mesh thus produced is shown to be better conditioned than alternatives based on assemblies of different shaped tetrahedra. A number of conditioning measures are used to confirm this. Finally, actual meshes which fit boundaries are briefly considered. Copyright © 1999 John Wiley & Sons, Ltd.     

Q‐Morph: an indirect approach to advancing front quad meshing

Q‐Morph is a new algorithm for generating all‐quadrilateral meshes on bounded three‐dimensional surfaces. After first triangulating the surface, the triangles are systematically transformed to create an all‐quadrilateral mesh. An advancing front algorithm determines the sequence of triangle transformations. Quadrilaterals are formed by using existing edges in the triangulation, by inserting additional nodes, or by performing local transformations to the triangles. A method typically used for recovering the boundary of a Delaunay mesh is used on interior triangles to recover quadrilateral edges. Any number of triangles may be merged to form a single quadrilateral. Topological clean‐up and smoothing are used to improve final element quality. Q‐Morph generates well‐aligned rows of quadrilaterals parallel to the boundary of the domain while maintaining a limited number of irregular internal nodes. The proposed method also offers the advantage of avoiding expensive intersection calculations commonly associated with advancing front procedures. A series of examples of Q‐Morph meshes are also presented to demonstrate the versatility of the proposed method. Copyright © 1999 John Wiley & Sons, Ltd.     

Stress recovery and error estimation for shell structures

The Penalized Discrete Least‐Squares (PDLS) stress recovery (smoothing) technique developed for two‐dimensional linear elliptic problems [1–3] is adapted here to three‐dimensional shell structures. The surfaces are restricted to those which have a 2‐D parametric representation, or which can be built‐up of such surfaces. The proposed strategy involves mapping the finite element results to the 2‐D parametric space whichdescribes the geometry, and smoothing is carried out in the parametric space using the PDLS‐based Smoothing Element Analysis (SEA). Numerical results for two well‐known shell problems are presented to illustrate the performance of SEA/PDLS for these problems. The recovered stresses are used in the Zienkiewicz–Zhu a posteriori error estimator. The estimated errors are used to demonstrate the performance of SEA‐recovered stresses in automated adaptive mesh refinement of shell structures. The numerical results are encouraging. Further testing involving more complex, practical structures is necessary. Copyright © 2000 John Wiley & Sons, Ltd.     

A distortion measure to validate and generate curved high‐order meshes on CAD surfaces with independence of parameterization

A framework to validate and generate curved nodal high‐order meshes on Computer‐Aided Design (CAD) surfaces is presented. The proposed framework is of major interest to generate meshes suitable for thin‐shell and 3D finite element analysis with unstructured high‐order methods. First, we define a distortion (quality) measure for high‐order meshes on parameterized surfaces that we prove to be independent of the surface parameterization. Second, we derive a smoothing and untangling procedure based on the minimization of a regularization of the proposed distortion measure. The minimization is performed in terms of the parametric coordinates of the nodes to enforce that the nodes slide on the surfaces. Moreover, the proposed algorithm repairs invalid curved meshes (untangling), deals with arbitrary polynomial degrees (high‐order), and handles with low‐quality CAD parameterizations (independence of parameterization). Third, we use the optimization procedure to generate curved nodal high‐order surface meshes by means of an a posteriori approach. Given a linear mesh, we increase the polynomial degree of the elements, curve them to match the geometry, and optimize the location of the nodes to ensure mesh validity. Finally, we present several examples to demonstrate the features of the optimization procedure, and to illustrate the surface mesh generation process. Copyright © 2015 John Wiley & Sons, Ltd.     

Global parameterization of a topological surface defined as a collection of trimmed bi‐parametric patches: Application to automatic mesh construction

We describe a new geometric algorithm to map surfaces into a plane convex area. The mapping transformation is bijective; it redefines the whole surface as a unique bi‐parametric patch. Thus this mapping provides a global parametrization of the surface. The surfaces are issued from industrial CAD software; they are usually described by a large number of patches and there are many shortcomings. Indeed, the decomposition into patches depends on the algorithm of the geometric modelling system used for design and usually has no meaning for any technological application. Moreover, in many cases, the surface definition is not compatible, i.e. patches are not well connected, some patches are self‐intersecting or intersect each other. Many applications are hard to address because of these defects. In this paper we show how patch‐independent meshing techniques may be easily automated using a unique metric in a plane parametric space. Thus we provide an automatic procedure to build valid meshes over free‐form surfaces issued from industrial CAD software (Computer Aided Design: this terminology should refer to a large amount of software. For the scope of this paper we only refer to geometric modelling systems. Indeed geometric modelling systems remain the kernel of many CAD software). Copyright © 2002 John Wiley & Sons, Ltd.     

The boundary recovery and sliver elimination algorithms of three‐dimensional constrained Delaunay triangulation

A boundary recovery and sliver elimination algorithm of the three‐dimensional constrained Delaunay triangulation is proposed for finite element mesh generation. The boundary recovery algorithm includes two main procedures: geometrical recovery procedure and topological recovery procedure. Combining the advantages of the edges/faces swappings algorithm and edges/faces splittings algorithm presented respectively by George and Weatherill, the geometrical recovery procedure can recover the missing boundaries and guarantee the geometry conformity by introducing fewer Steiner points. The topological recovery procedure includes two phases: ‘dressing wound’ and smoothing, which will overcome topology inconsistency between 3D domain boundary triangles and the volume mesh. In order to solve the problem of sliver elements in the three‐dimensional Delaunay triangulation, a method named sliver decomposition is proposed. By extending the algorithm proposed by Canvendish, the presented method deals with sliver elements by using local decomposition or mergence operation. In this way, sliver elements could be eliminated thoroughly and the mesh quality could be improved in great deal. Copyright © 2006 John Wiley & Sons, Ltd.