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.     

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.     

Tetrahedral mesh generation based on space indicator functions

An algorithm for the generation of tetrahedral volume meshes is developed for highly irregular objects specified by volumetric representations such as domain indicator functions and tomography data. It is based on red–green refinement of an initial mesh derived from a body‐centered cubic lattice. A quantitative comparison of alternative types of initial meshes is presented. The minimum set of best‐quality green refinement schemes is identified. Boundary conformity is established by deforming or splitting surface‐crossing elements. Numerical derivatives of input data are strictly avoided. Furthermore, the algorithm features surface‐adaptive mesh density based on local surface roughness, which is an integral property of finite surface portions. Examples of applications are presented for computer tomography of porous media. Copyright © 2012 John Wiley & Sons, Ltd.     

An algorithm to generate quadrilateral or triangular element surface meshes in arbitrary domains with applications to crack propagation

A new hybrid algorithm for automatically generating either an all-quadrilateral or an all-triangular element mesh within an arbitrarily shaped domain is described. The input consists of one or more closed loops of straight-line segments that bound the domain. Internal mesh density is inferred from the boundary density using a recursive spatial decomposition (quadtree) procedure. All-triangular element meshes are generated using a boundary contraction procedure. All-quadrilateral element meshes are generated by modifying the boundary contraction procedure to produce a mixed element mesh at half the density of the final mesh and then applying a polygon-splitting procedure. The final meshes exhibit good transitioning properties and are compatible with the given boundary segments which are not altered. The algorithm can support discrete crack growth simulation wherein each step of crack growth results in an arbitrarily shaped region of elements deleted about each crack tip. The algorithm is described and examples of the generated meshes are provided for a representative selection of cracked and uncracked structures.     

Meshing curved wire‐frame models through energy minimization and packing of ellipses

In the initial phase of structural part design, wire‐frame models are sometimes used to represent the shapes of curved surfaces. Finite Element Analysis (FEA) of a curved surface requires a well shaped, graded mesh that smoothly interpolates the wire frame. This paper describes an algorithm that generates such a triangular mesh from a wire‐frame model in the following two steps: (1) construct a triangulated surface by minimizing the strain energy of the thin‐plate‐bending model, and (2) generate a mesh by the bubble meshing method on the projected plane and project it back onto the triangulated surface. Since the mesh elements are distorted by the projection, the algorithm generates an anisotropic mesh on the projected plane so that an isotropic mesh results from the final projection back onto the surface. Extensions of the technique to anisotropic meshing and quadrilateral meshing are also discussed. The algorithm can generate a well‐shaped, well‐graded mesh on a smooth curved surface. Copyright © 1999 John Wiley & Sons, Ltd.     

Applications of mesh smoothing: copy,morph, and sweep on unstructured quadrilateral meshes

Mesh smoothing is demonstrated to be an effective means of copying, morphing, and sweeping unstructured quadrilateral surface meshes from a source surface to a target surface. Construction of the smoother in a particular way guarantees that the target mesh will be a ‘copy’ of the source mesh, provided the boundary data of the target surface is a rigid body rotation, translation, and/or uniform scaling of the original source boundary data and provided the proper boundary node correspondence between source and target has been selected. Copying is not restricted to any particular smoother, but can be based on any locally elliptic second‐order operator. When the bounding loops are more general than rigid body transformations the method generates high‐quality, ‘morphed’ meshes. Mesh sweeping, if viewed as a morphing of the source surface to a set of target surfaces, can be effectively performed via this smoothing algorithm. Published in 1999 by John Wiley & Sons, Ltd. This article is a U.S. government work and is in the public domain in the United States.     

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.     

Assessment of discretized errors and adaptive refinement with quadrilateral finite elements

This paper studies discretized errors, and their estimation in conjunction with quadrilateral finite element meshes which are generated by the intelligent mesh generator XFORMQ.¹ The exact energy error is used to evaluate the distortion effect of the quadrilateral mesh. The Zienkiewicz–Zhu² error estimate and actaptive procedure are applied to the short cantilever and the square plate problems using the quadrilateral mesh generator XFORMQ. It is shown that the multistage quadrilateral element refinement produces results superior to the triangular element refinement in the test cases.     

Quality improvements in extruded meshes using topologically adaptive generalized elements

This paper describes a method to extrude near‐body volume meshes that exploits topologically adaptive generalized elements to improve local mesh quality. Specifically, an advancing layer algorithm for extruding volume meshes from surface meshes of arbitrary topology, appropriate for viscous fluid flows, is discussed. First, a two‐layer reference mesh is generated from the layer initial surface mesh by extruding along the local surface normals. The reference mesh is then smoothed using a Poisson equation. Local quality improvement operations such as edge collapse, face refinement, and local reconnection are performed in each layer to drive the mesh toward isotropy and improve the transition from the extruded mesh to a void‐filling tetrahedral mesh. A few example meshes along with quality plots are presented to demonstrate the efficacy of this approach. Copyright © 2004 John Wiley & Sons, Ltd.     

Tetrahedral mesh generation using Delaunay refinement with non‐standard quality measures

This paper studies the practical performance of Delaunay refinement tetrahedral mesh generation algorithms. By using non‐standard quality measures to drive refinement, we show that sliver tetrahedra can be eliminated from constrained Delaunay tetrahedralizations solely by refinement. Despite the fact that quality guarantees cannot be proven, the algorithm can consistently generate meshes with dihedral angles between 18^circ and 154^°. Using a fairer quality measure targeting every type of bad tetrahedron, dihedral angles between 14^° and 154^° can be obtained. The number of vertices inserted to achieve quality meshes is comparable to that needed when driving refinement with the standard circumradius‐to‐shortest‐edge ratio. We also study the use of mesh improvement techniques on Delaunay refined meshes and observe that the minimum dihedral angle can generally be pushed above 20^°, regardless of the quality measure used to drive refinement. The algorithm presented in this paper can accept geometric domains whose boundaries are piecewise smooth. Copyright © 2011 John Wiley & Sons, Ltd.     

Octree‐based reasonable‐quality hexahedral mesh generation using a new set of refinement templates

An octree‐based mesh generation method is proposed to create reasonable‐quality, geometry‐adapted unstructured hexahedral meshes automatically from triangulated surface models without any sharp geometrical features. A new, easy‐to‐implement, easy‐to‐understand set of refinement templates is developed to perform local mesh refinement efficiently even for concave refinement domains without creating hanging nodes. A buffer layer is inserted on an octree core mesh to improve the mesh quality significantly. Laplacian‐like smoothing, angle‐based smoothing and local optimization‐based untangling methods are used with certain restrictions to further improve the mesh quality. Several examples are shown to demonstrate the capability of our hexahedral mesh generation method for complex geometries. Copyright © 2008 John Wiley & Sons, Ltd.     

Marching volume polytopes algorithm

This paper presents an algorithm for the refinement of two- or three-dimensional meshes with respect to an implicitly given domain, so that its surface is approximated by facets of the resulting polytopes. Using a Cartesian grid, the proposed algorithm may be used as a mesh generator. Initial meshes may consist of polytopes such as quadrilaterals and triangles, as well as hexahedrons, pyramids, and tetrahedrons. Given the ability to compute edge intersections with the surface of an implicitly given domain, the proposed marching volume polytopes algorithm uses predefined refinement patterns applied to individual polytopes depending on the intersection pattern of their edges. The refinement patterns take advantage of rotational symmetry. Since these patterns are applied independently to individual polytopes, the resulting mesh may encompass the so-called orientation problem, where two adjacent polytopes are rotated against one another. To allow for a repeated application of the marching volume polytopes algorithm, the proposed data structures and algorithms account for this ambiguity. A simple example illustrates the advantage of the repeated application of the proposed algorithm to approximate domains with sharp corners. Furthermore, finite element simulations for two challenging real-world problems, which require highly accurate approximations of the considered domains, demonstrate its applicability. For these simulations, a variant of the fictitious domain method is used.     

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.     

Automatic generation of multiblock decompositions of surfaces

Multiblock‐structured meshes have significant advantages over fully unstructured meshes in numerical simulation, but automatically generating these meshes is considerably more difficult. A method is described herein for automatically generating high‐quality multiblock decompositions of surfaces with boundaries. Controllability and flexibility are useful capabilities of the method. Additional alignment constraints for forcing the appearance of particular features in the decomposition can be easily handled. Also, adjustments are made according to input metric tensor fields that describe target element size properties. The general solution strategy is based around using a four‐way symmetry vector‐field, called a cross‐field, to describe the local mesh orientation on a triangulation of the surface. Initialisation is performed by propagating the boundary alignment constraints to the interior in a fast marching method. This is similar in a way to an advancing‐front or paving method but much more straightforward and flexible because mesh connectivity does not have to be managed in the cross‐field. Multiblock decompositions are generated by tracing the separatrices of the cross‐field to partition the surface into quadrilateral blocks with square corners. The final task of meshing the decomposition requires solving an integer programming problem for block division numbers. Copyright © 2015 John Wiley & Sons, Ltd.     

Inserting a surface into an existing unstructured mesh

In this work, a new method for inserting a surface as an internal boundary into an existing unstructured tetrahedral mesh is developed. The surface is discretized by initially placing vertices on its bounding curves, defining a length scale at every location on each boundary curve based on the local underlying mesh, and equidistributing length scale along these curves between vertices. The surface is then sampled based on this boundary discretization, resulting in a surface mesh spaced in a way that is consistent with the initial mesh. The new points are then inserted into the mesh, and local refinement is performed, resulting in a final mesh containing a representation of the surface while preserving mesh quality. The advantage of this algorithm over generating a new mesh from scratch is in allowing for the majority of existing simulation data to be preserved and not have to be interpolated onto the new mesh. This algorithm is demonstrated in two and three dimensions on problems with and without intersections with existing internal boundaries. Copyright © 2015 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。     

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.     

Automated refinement of conformal quadrilateral and hexahedral meshes

Conformal refinement using a shrink and connect strategy, known as pillowing or buffer insertion, contracts and reconnects contiguous elements of an all‐quadrilateral or an all‐hexahedral mesh in order to locally increase vertex density without introducing hanging nodes or non‐cubical elements. Using layers as shrink sets, the present method automates the anisotropic refinement of such meshes according to a prescribed size map expressed as a Riemannian metric field. An anisotropic smoother further enhances vertex clustering to capture the features of the metric. Both two‐ and three‐dimensional test cases with analytic control metrics confirm the feasibility of the present approach and explore strategies to minimize the trade‐off between element shape quality and size conformity. Additional examples using discrete metric maps illustrate possible practical applications. Although local vertex removal and reconnection capabilities have yet to be developed, the present refinement method is a step towards an automated tool for conformal adaptation of all‐quadrilateral and all‐hexahedral meshes. Copyright © 2004 John Wiley & Sons, Ltd.