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.     

A new automatic adaptive 3D solid mesh generation scheme for thin‐walled structures

A new algorithm to generate three‐dimensional (3D) mesh for thin‐walled structures is proposed. In the proposed algorithm, the mesh generation procedure is divided into two distinct phases. In the first phase, a surface mesh generator is employed to generate a surface mesh for the mid‐surface of the thin‐walled structure. The surface mesh generator used will control the element size properties of the final mesh along the surface direction. In the second phase, specially designed algorithms are used to convert the surface mesh to a 3D solid mesh by extrusion in the surface normal direction of the surface. The extrusion procedure will control the refinement levels of the final mesh along the surface normal direction. If the input surface mesh is a pure quadrilateral mesh and refinement level in the surface normal direction is uniform along the whole surface, all hex‐meshes will be produced. Otherwise, the final 3D meshes generated will eventually consist of four types of solid elements, namely, tetrahedron, prism, pyramid and hexahedron. The presented algorithm is highly flexible in the sense that, in the first phase, any existing surface mesh generator can be employed while in the second phase, the extrusion procedure can accept either a triangular or a quadrilateral or even a mixed mesh as input and there is virtually no constraint on the grading of the input mesh. In addition, the extrusion procedure development is able to handle structural joints formed by the intersections of different surfaces. Numerical experiments indicate that the present algorithm is applicable to most practical situations and well‐shaped elements are generated. 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.     

An extended advancing front technique for closed surfaces mesh generation

An extended advancing front technique (AFT) with shift operations and Riemann metric named as shifting‐AFT is presented for finite element mesh generation on 3D surfaces, especially 3D closed surfaces. Riemann metric is used to govern the size and shape of the triangles in the parametric space. The shift operators are employed to insert a floating space between real space and parametric space during the 2D parametric space mesh generation. In the previous work of closed surface mesh generation, the virtual boundaries are adopted when mapping the closed surfaces into 2D open parametric domains. However, it may cause the mesh quality‐worsening problem. In order to overcome this problem, the AFT kernel is combined with the shift operator in this paper. The shifting‐AFT can generate high‐quality meshes and guarantee convergence in both open and closed surfaces. For the shifting‐AFT, it is not necessary to introduce virtual boundaries while meshing a closed surface; hence, the boundary discretization procedure is largely simplified, and moreover, better‐shaped triangles will be generated because there are no additional interior constraints yielded by virtual boundaries. Comparing with direct methods, the shifting‐AFT avoids costly and unstable 3D geometrical computations in the real space. Some examples presented in this paper have demonstrated the advantages of shift‐AFT in 3D surface mesh generation, especially for the closed surfaces. Copyright © 2007 John Wiley & Sons, Ltd.     

Model and mesh generation of cracked tubular Y-joints

In this study, the methods for constructing accurate and consistent geometrical and finite element (FE) models for general cracked tubular Y-joints are described. Firstly, geometrical analysis of welded tubular joint is given and it is then extended to the modelling of general cracked Y-joints. The concept of crack surface and a simple mapping approach are suggested to model either through-thickness or surface cracks which can be of any length and located at any position along the brace-chord intersection. Secondly, the geometrical model developed will be used in the generation of consistent FE meshes. The basic concepts used for the design and generation of three-dimensional FE meshes will be described. This will include the meshing procedures for discretization of tubular joints with through-thickness and surface cracks which are frequently regarded as one of the most difficult steps in the construction of tubular joint models. Finally, some mesh generation examples for uncracked and cracked Y-joints will be presented to demonstrate the use of the purposed geometrical model and mesh generation scheme developed.     

Extraction of boundary representation from surface triangulations

Many computational science tools employ finite element meshes as discretizations of the geometrical domains, and automatic mesh generation has become an indispensable part of the discretization process. Boundary representations (BRep) of solids are the means of describing the geometrical model to the mesher, thus enabling the generator to proceed without user intervention. Significant effort has been devoted in the past to BRep construction in the frame‐work of solid modelling systems. In this paper we consider the task of converting a tesselation (triangulation) of the surface of a solid into a BRep, and propose a robust and efficient set of algorithms for this purpose. Applications include, among others, remeshing of finite element discretizations during simulations involving not only geometric distortion but also changes in topology (coalescence and fragmentation of solids, flow, and so on). Copyright © 2001 John Wiley & Sons, Ltd.     

基于细分网格的多分辨率几何数据压缩方法研究

提出了一种基于细分网格的多分辨率几何数据压缩算法 ,该算法是一种利用正则曲面法线向量特性及细分曲面的细分连通性的有损压缩方法 ,因此可以获得很高的压缩比     

Feature based quadrilateral mesh generation for sculptured surface products

A successful approach to the generation of quadrilateral surface meshes for finite element analysis of sculptured surface products is presented. The free-form sculpted surface is divided into a feature based product anatomy which is used as the basis for defining regions of simpler shape that are related by the requirement for mesh continuity across their boundaries. Each surface region is meshed using a paving algorithm. Large surface areas are further subdivided to introduce new boundaries which improve contiol over mesh transitions and element distortion. A procedure, based on this approach, which generates well formed meshes on sculptured surface products is described using a hollow golf club head as an example. The approach is product specific and relies on initial user interaction to populate a feature based product data model. Thus, the data model contains, the attributes of a finite element mesh tailored to the product, which can be used to generate meshes on subsequent design iterations or other members of the product family with a high degree of automation, leading to reduced analysis cost.     

Subdivision surface-based finish machining

Subdivision surfaces combine smooth spline surfaces and polygonal meshes together, therefore, a smooth design model and discrete machining models may be unified and subdivision surfaces may be used as a common representation for geometric design and machining. Motivated by the idea, this paper presents the study of finish machining of objects represented by subdivision surfaces with emphasis on geometric error control involved in tool-path generation. First, given a design model, chordal error is controlled during finishing model building. A chordal error-driven adaptive subdivision method is used to build finishing models with less data. Second, a surface decomposition machining strategy is used to control the cusp height error. A simple iso-slope curve tracing and surface decomposition algorithm is presented to partition the model into flat and steep regions. Contour-map tool-paths are generated in the steep regions while iso-planar tool-paths are generated in the flat regions. The gouge problem is easily handled through two-dimensional (2D) tool-path correction algorithms. The implementation results demonstrate that subdivision is capable of serving as a unified representation for both geometric modelling and machining.     

A new volume warping method for surface reconstruction

Volumetric models of 3D objects have recently been introduced into the reverse engineering process. Due to their efficiency and simplicity, grid-based methods are considered the major technique for reconstructing surfaces from these volumetric models. Yet these methods suffer from a number of inherent drawbacks, resulting from the fact that the imposed Cartesian grid is generally not well adapted to the surface in size or in orientation. To overcome the above obstacles, this paper proposes a new iso-surface extraction method for volumetric objects. The main idea is first to construct a geometrical field induced by the object's shape. This geometrical field represents the natural directions and grid cell size for each point in the domain. Then, the imposed volumetric grid is deformed by the produced geometrical field to approach the object's shape. The iso-surface meshes can be extracted from the resulting adaptive grid by any conventional grid-based contouring technique. These meshes provide a better approximation of the unknown surface and exhibit the anisotropy present in this surface. Since the produced meshes are all quad, Catmull-Clark subdivision surfaces can be directly constructed from them. Moreover, accurate physical models of the reconstructed objects can be produced using rapid prototyping (RP) technology.     

A new volume warping method for surface reconstruction

Volumetric models of 3D objects have recently been introduced into the reverse engineering process. Due to their efficiency and simplicity, grid-based methods are considered the major technique for reconstructing surfaces from these volumetric models. Yet these methods suffer from a number of inherent drawbacks, resulting from the fact that the imposed Cartesian grid is generally not well adapted to the surface in size or in orientation. To overcome the above obstacles, this paper proposes a new iso-surface extraction method for volumetric objects. The main idea is first to construct a geometrical field induced by the object's shape. This geometrical field represents the natural directions and grid cell size for each point in the domain. Then, the imposed volumetric grid is deformed by the produced geometrical field to approach the object's shape. The iso-surface meshes can be extracted from the resulting adaptive grid by any conventional grid-based contouring technique. These meshes provide a better approximation of the unknown surface and exhibit the anisotropy present in this surface. Since the produced meshes are all quad, Catmull–Clark subdivision surfaces can be directly constructed from them. Moreover, accurate physical models of the reconstructed objects can be produced using rapid prototyping (RP) technology.     

网格细分技术在汽车外形设计中的应用

细分造型技术因其计算规则简单、可以表示任意拓扑特性和几何特征的曲面等性质,受到造型技术领域中众多学者的关注,为复杂的汽车外形设计提供了工具.该文阐释了网格细分方法的基本思想以及两类典型的细分模式,提出了在相关几何造型、有限元分析和动态仿真软件配合下,设计复杂汽车外形曲面的细分技术方案;对车体曲面的控制网格生成、边界限定等关键技术进行了讨论.利用文中方法可以有效地缩短复杂汽车外形曲面的造型、计算和分析时间,为汽车外形设计的逆向工程应用提供方法和工具.     

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。     

Efficient ALE mesh management for 3D quasi‐Eulerian problems

In computational solid mechanics, the ALE formalism can be very useful to reduce the size of finite element models of continuous forming operations such as roll forming. The mesh of these ALE models is said to be quasi‐Eulerian because the nodes remain almost fixed—or almost Eulerian—in the main process direction, although they are required to move in the orthogonal plane in order to follow the lateral displacements of the solid. This paper extensively presents a complete node relocation procedure dedicated to such ALE models. The discussion focusses on quadrangular and hexahedral meshes with local refinements. The main concern of this work is the preservation of the geometrical features and the shape of the free boundaries of the mesh. With this aim in view, each type of nodes (corner, edge, surface and volume) is treated sequentially with dedicated algorithms. A special care is given to highly curved 3D surfaces for which a CPU‐efficient smoothing technique is proposed. This new method relies on a spline surface reconstruction, on a very fast weighted Laplacian smoother with original weights and on a robust reprojection algorithm. The overall consistency of this mesh management procedure is finally demonstrated in two numerical applications. The first one is a 2D ALE simulation of a drawbead, which provides similar results to an equivalent Lagrangian model yet is much faster. The second application is a 3D industrial ALE model of a 16‐stand roll forming line. In this case, all attempts to perform the same simulation by using the Lagrangian formalism have been unsuccessful. Copyright © 2012 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.     

A parallel mesh generation algorithm based on the vertex label assignment scheme

In this article a new mesh generation algorithm is presented. The algorithm is based on a new approach called the vertex label assignment scheme to provide the information for the mesh generation so that parallel processing becomes possible. The algorithm generates 2D meshes of quadrilaterals on the basis of individual faces; conformity and smoothness of the resultant mesh are automatically assured. Local and selective mesh-refinements are also supported. A regular quadrilateral network which defines the geometry of the problem and an associated subdivision level assignment which specifies mesh density data on the network are the only input information.     

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.     

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.     

三角网格模型上的四边形曲线网生成新方法

四边形网格划分是组合曲面建模技术的首要条件。针对海量流形三角网格数据,提出了基于网格简化技术与调和映射算法的四边形网格生成新方法--映射法。该方法采用基于顶点删除的网格简化技术对三角网格模型进行简化,进而借助调和映射算法将简化网格映射到二维平面上进行四边形划分,并将所获得的平面四边形节点数据逆映射回物理域,采用短程线边界形式最终得到适于组合曲面建模的空间四边形拓扑。该方法简单、实用,运行速度较快,实际的算例也验证了方法的有效性与可行性。