The receding front method applied to hexahedral mesh generation of exterior domains

Two of the most successful methods to generate unstructured hexahedral meshes are the grid-based methods and the advancing front methods. On the one hand, the grid-based methods generate high-quality hexahedra in the inner part of the domain using an inside–outside approach. On the other hand, advancing front methods generate high-quality hexahedra near the boundary using an outside–inside approach. To combine the advantages of both methodologies, we extend the receding front method: an inside–outside mesh generation approach by means of a reversed advancing front. We apply this approach to generate unstructured hexahedral meshes of exterior domains. To reproduce the shape of the boundaries, we first pre-compute the mesh fronts by combining two solutions of the Eikonal equation on a tetrahedral reference mesh. Then, to generate high-quality elements, we expand the quadrilateral surface mesh of the inner body towards the unmeshed external boundary using the pre-computed fronts as a guide.     

A constructive approach to constrained hexahedral mesh generation

Mitchell proved that a necessary and sufficient condition for the existence of a topological hexahedral mesh constrained to a quadrilateral mesh on the sphere is that the constraining quadrilateral mesh contains an even number of elements. Mitchell’s proof depends on Smale’s theorem on the regularity of curves on compact manifolds. Although the question of the existence of constrained hexahedral meshes has been solved, the known solution is not easily programmable; indeed, there are cases, such as Schneider’s Pyramid, that are not easily solved. Eppstein later utilized portions of Mitchell’s existence proof to demonstrate that hexahedral mesh generation has linear complexity. In this paper, we demonstrate a constructive proof to the existence theorem for the sphere, as well as assign an upper-bound to the constant of the linear term in the asymptotic complexity measure provided by Eppstein. Our construction generates 76 × n hexahedra elements within the solid where n is the number of quadrilaterals on the boundary. The construction presented is used to solve some problems posed by Schneiders and Eppstein. We will also use the results provided in this paper, in conjunction with Mitchell’s Geode-Template, to create an alternative way of creating a constrained hexahedral mesh. The construction utilizing the Geode-Template requires 130 × n hexahedra, but will have fewer topological irregularities in the final mesh.     

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。     

The All-Hex Geode-Template for Conforming a Diced Tetrahedral Mesh to any Diced Hexahedral Mesh

Take a hexahedral mesh and an adjoining tetrahedral mesh that splits each boundary quadrilateral into two triangles. Separate the meshes with a buffer layer of hexes. Dice the original hexes into eight, and the tetrahedra into four hexahedra. Then I show that the buffer layer hexes can be filled with the geode-template, creating a conforming all-hex mesh of the entire model. The geode-template is composed of 26 hexahedra. The hexahedra have acceptable quality, depending on the geometry of the buffer layer. The method used to generate the geode-template is general, based on interleaving completed dual surfaces, and might be extended to other transition problems.     

Preconditioners for Solving Stochastic Boundary Integral Equations with Weakly Singular Kernels

A stochastic linear heat conduction problem is reduced to a special weakly singular integral equation of the second kind. The smoothness of the solution to a multidimensional weakly singular integral equation is investigated. It is also indicated that the derivatives of solutions may have singularities of certain order near the boundary of domain. The solution in the form of a multidimensional cubic spline is studied using circulant integral operators and a special mesh near the boundary with respect to all variables. Furthermore, stable numerical algorithms are given. Received: June 22, 1998; revised November 11, 1998     

Hexahedral Meshing With Varying Element Sizes

Hexahedral (or Hex‐) meshes are preferred in a number of scientific and engineering simulations and analyses due to their desired numerical properties. Recent state‐of‐the‐art techniques can generate high‐quality hex‐meshes. However, they typically produce hex‐meshes with uniform element sizes and thus may fail to preserve small‐scale features on the boundary surface. In this work, we present a new framework that enables users to generate hex‐meshes with varying element sizes so that small features will be filled with smaller and denser elements, while the transition from smaller elements to larger ones is smooth, compared to the octree‐based approach. This is achieved by first detecting regions of interest (ROIs) of small‐scale features. These ROIs are then magnified using the as‐rigid‐as‐possible deformation with either an automatically determined or a user‐specified scale factor. A hex‐mesh is then generated from the deformed mesh using existing approaches that produce hex‐meshes with uniform‐sized elements. This initial hex‐mesh is then mapped back to the original volume before magnification to adjust the element sizes in those ROIs. We have applied this framework to a variety of man‐made and natural models to demonstrate its effectiveness.     

Hexahedral Mesh Generation by Successive Dual Cycle Elimination

We propose a new method for constructing all-hexahedral finite element meshes. The core of our method is to build up a compatible combinatorial cell complex of hexahedra for a solid body which is topologically a ball, and for which a quadrilateral surface mesh of a certain structure is prescribed. The step-wise creation of the hex complex is guided by the cycle structure of the combinatorial dual of the surface mesh. Our method transforms the graph of the surface mesh iteratively by changing the dual cycle structure until we get the surface mesh of a single hexahedron. Starting with a single hexahedron and reversing the order of the graph transformations, each transformation step can be interpreted as adding one or more hexahedra to the so far created hex complex. Given an arbitrary solid body, we first decompose it into simpler subdomains equivalent to topological balls by adding virtual 2-manifolds. Secondly, we determine a compatible quadrilateral surface mesh for all subdomains created. Then, in the main part we can use the core routine to build up a hex complex for each subdomain independently. The embedding and smoothing of the combinatorial mesh(es) finishes the mesh generation process. First results obtained for complex geometries are encouraging.     

A Survey of Indicators for Mesh Quality Assessment

We analyze the joint efforts made by the geometry processing and the numerical analysis communities in the last decades to define and measure the concept of “mesh quality”. Researchers have been striving to determine how, and how much, the accuracy of a numerical simulation or a scientific computation (e.g., rendering, printing, modeling operations) depends on the particular mesh adopted to model the problem, and which geometrical features of the mesh most influence the result. The goal was to produce a mesh with good geometrical properties and the lowest possible number of elements, able to produce results in a target range of accuracy. We overview the most common quality indicators, measures, or metrics that are currently used to evaluate the goodness of a discretization and drive mesh generation or mesh coarsening/refinement processes. We analyze a number of local and global indicators, defined over two- and three-dimensional meshes with any type of elements, distinguishing between simplicial, quadrangular/hexahedral, and generic polytopal elements. We also discuss mesh optimization algorithms based on the above indicators and report common libraries for mesh analysis and quality-driven mesh optimization.     

Shape optimization of quad mesh elements

We study the problem of optimizing the face elements of a quad mesh surface, that is, re-sampling a given quad mesh to make it possess, as much as possible, face elements of some desired aspect ratio and size. Unlike previous quad mesh optimization/improvement methods based on local operations on a small group of elements, we adopt a global approach that does not introduce extra singularities and therefore preserves the original quad structure of the input mesh. Starting from a collection of quad patches extracted from an input quad mesh, two global operations, i.e. re-sampling and re-distribution, are performed to optimize the number and spacings of grid lines in each patch. Both operations are formulated as simple optimization problems with linear constraints.     

Adaptive boundary layer meshing for viscous flow simulations

A procedure for anisotropic mesh adaptation accounting for mixed element types and boundary layer meshes is presented. The method allows to automatically construct meshes on domains of interest to accurately and efficiently compute key flow quantities, especially near wall quantities like wall shear stress. The new adaptive approach uses local mesh modification procedures in a manner that maintains layered and graded elements near the walls, which are popularly known as boundary layer or semi-structured meshes, with highly anisotropic elements of mixed topologies. The technique developed is well suited for viscous flow applications where exact knowledge of the mesh resolution over the computational domain required to accurately resolve flow features of interest is unknown a priori. We apply the method to two types of problem cases; the first type, which lies in the field of hemodynamics, involves pulsatile flow in blood vessels including a porcine aorta case with a stenosis bypassed by a graft whereas the other involves high-speed flow through a double throat nozzle encountered in the field of aerodynamics.     

Simultaneous Refinement and Coarsening for Adaptive Meshing

In the numerical simulation of the combustion process and microstructural evolution, we need to consider the adaptive meshing problem for a domain that has a moving boundary. During the simulation, the region ahead of the moving boundary needs to be refined (to satisfy stronger numerical conditions), and the submesh in the region behind the moving boundary should be coarsened (to reduce the mesh size). We present a unified scheme for simultaneously refining and coarsening a mesh. Our method uses sphere packings and guarantees that the resulting mesh is well-shaped and is within a constant factor of the optimal possible in the number of mesh elements. We also present several practical variations of our provably good algorithm.     

2D structured grid generation method producing a mesh with prescribed properties near boundary

A variational method of generating a structured mesh on a two-dimensional domain is considered. To this end, a quasiconformal mapping of the parametric domain with a given Cartesian mesh onto the underlying physical domain is used. The functions implementing the mapping are sought by solving the Dirichlet problem for the system of elliptic second-order partial differential equations. An additional control for the cell shape is executed by introducing a local mapping which induces a control metric. In some particular cases, instead of an additional local mapping, a global mapping of the parametric domain onto the intermediate domain is used, where the curvilinear mesh is produced, and next this domain is mapped onto the underlying physical domain. The control metric allows to obtain a mesh with required properties: grid line orthogonality and prescribed mesh point clustering near the domain boundary. Examples of mesh in the annulus and near airfoil are presented.     

Simultaneous untangling and smoothing of quadrilateral and hexahedral meshes using an object-oriented framework

In this work, we present a simultaneous untangling and smoothing technique for quadrilateral and hexahedral meshes. The algorithm iteratively improves a quadrilateral or hexahedral mesh by minimizing an objective function defined in terms of a regularized algebraic distortion measure of the elements. We propose several techniques to improve the robustness and the computational efficiency of the optimization algorithm. In addition, we have adopted an object-oriented paradigm to create a common framework to smooth meshes composed by any type of elements, and using different minimization techniques. Finally, we present several examples to show that the proposed technique obtains valid meshes composed by high-quality quadrilaterals and hexahedra, even when the initial meshes contain a large number of tangled elements.     

Feature-preserving T-mesh construction using skeleton-based polycubes

This paper presents a novel algorithm which uses skeleton-based polycube generation to construct feature-preserving T-meshes. From the skeleton of the input model, we first construct initial cubes in the interior. By projecting corners of interior cubes onto the surface and generating a new layer of boundary cubes, we split the entire interior domain into different cubic regions. With the splitting result, we perform octree subdivision to obtain T-spline control mesh or T-mesh. Surface features are classified into three groups: open curves, closed curves and singularity features. For features without introducing new singularities like open or closed curves, we preserve them by aligning to the parametric lines during subdivision, performing volumetric parameterization from frame field, or modifying the skeleton. For features introducing new singularities, we design templates to handle them. With a valid T-mesh, we calculate rational trivariate T-splines and extract Bézier elements for isogeometric analysis.     

Cost Minimizing Local Anisotropic Quad Mesh Refinement

Quad meshes as a surface representation have many conceptual advantages over triangle meshes. Their edges can naturally be aligned to principal curvatures of the underlying surface and they have the flexibility to create strongly anisotropic cells without causing excessively small inner angles. While in recent years a lot of progress has been made towards generating high quality uniform quad meshes for arbitrary shapes, their adaptive and anisotropic refinement remains difficult since a single edge split might propagate across the entire surface in order to maintain consistency. In this paper we present a novel refinement technique which finds the optimal trade-off between number of resulting elements and inserted singularities according to a user prescribed weighting. Our algorithm takes as input a quad mesh with those edges tagged that are prescribed to be refined. It then formulates a binary optimization problem that minimizes the number of additional edges which need to be split in order to maintain consistency. Valence 3 and 5 singularities have to be introduced in the transition region between refined and unrefined regions of the mesh. The optimization hence computes the optimal trade-off and places singularities strategically in order to minimize the number of consistency splits — or avoids singularities where this causes only a small number of additional splits. When applying the refinement scheme iteratively, we extend our binary optimization formulation such that previous splits can be undone if this prevents degenerate cells with small inner angles that otherwise might occur in anisotropic regions or in the vicinity of singularities. We demonstrate on a number of challenging examples that the algorithm performs well in practice.     

Generation of finite element mesh with variable size over an unbounded 2D domain

With the advance of the finite element, general fluid dynamic and traffic flow problems with arbitrary boundary definition over an unbounded domain are tackled. This paper describes an algorithm for the generation of finite element mesh of variable element size over an unbounded 2D domain by using the advancing front circle packing technique. Unlike the conventional frontal method, the procedure does not start from the object boundary but starts from a convenient point within an open domain. The sequence of construction of the packing circles is determined by the shortest distance from the fictitious centre in such a way that the generation front is more or less a circular loop with occasional minor concave parts due to element size variation. As soon as a circle is added to the generation front, finite elements are directly generated by properly connecting frontal segments with the centre of the new circle. In contrast to other mesh generation schemes, the domain boundary is not considered in the process of circle packing, this reduces a lot of geometrical checks for intersection with frontal segments, and a linear time complexity for mesh generation can be achieved. In case the boundary of the domain is needed, simply generate an unbounded mesh to cover the entire object. As the element adjacency relationship of the mesh has already been established in the circle packing process, insertion of boundary segments by neighbour tracing is fast and robust. Details of such a boundary recovery procedure are described, and practical meshing problems are given to demonstrate how physical objects are meshed by the unbounded meshing scheme followed by the insertion of domain boundaries.     

HexBox: Interactive Box Modeling of Hexahedral Meshes

We introduce HexBox, an intuitive modeling method and interactive tool for creating and editing hexahedral meshes. Hexbox brings the major and widely validated surface modeling paradigm of surface box modeling into the world of hex meshing. The main idea is to allow the user to box-model a volumetric mesh by primarily modifying its surface through a set of topological and geometric operations. We support, in particular, local and global subdivision, various instantiations of extrusion, removal, and cloning of elements, the creation of non-conformal or conformal grids, as well as shape modifications through vertex positioning, including manual editing, automatic smoothing, or, eventually, projection on an externally-provided target surface. At the core of the efficient implementation of the method is the coherent maintenance, at all steps, of two parallel data structures: a hexahedral mesh representing the topology and geometry of the currently modeled shape, and a directed acyclic graph that connects operation nodes to the affected mesh hexahedra. Operations are realized by exploiting recent advancements in grid-based meshing, such as mixing of 3-refinement, 2-refinement, and face-refinement, and using templated topological bridges to enforce on-the-fly mesh conformity across pairs of adjacent elements. A direct manipulation user interface lets users control all operations. The effectiveness of our tool, released as open source to the community, is demonstrated by modeling several complex shapes hard to realize with competing tools and techniques.     

Point placement algorithms for Delaunay triangulation of polygonal domains

In some applications of triangulation, such as finite-element mesh generation, the aim is to triangulate a given domain, not just a set of points. One approach to meeting this requirement, while maintaining the desirable properties of Delaunay triangulation, has been to enforce the empty circumcircle property of Delaunay triangulation, subject to the additional constraint that the edges of a polygon be covered by edges of the triangulation. In finite-element mesh generation it is usually necessary to include additional points besides the vertices of the domain boundary. This motivates us to ask whether it is possible to trinagulate a domain by introducing additional points in such a manner that the Delaunay triangulation of the points includes the edges of the domain boundary. We present algorithms that given a multiply connected polygonal domain withN vertices, placeK additional points on the boundary inO(N logN + K) time such that the polygon is covered by the edges of the Delaunay triangulation of theN + K points. Furthermore,K is the minimum number of additional points such that a circle, passing through the endpoints of each boundary edge segment, exists that does not contain in its interior any other part of the domain boundary. We also show that by adding only one more point per edge, certain degeneracies that may otherwise arise can be avoided.     

Local quality of smoothening based a-posteriori error estimators for laminated plates under transverse loading

The local and global quality of various smoothening based a-posteriori error estimators is tested in this paper, for symmetric laminated composite plates subjected to transverse loads. Smoothening based on strain recovery and displacement-field recovery is studied here. Effect of ply orientation, laminate thickness, boundary conditions, mesh topology, and plate model is studied for a rectangular plate. It is observed that for interior patches of elements, both the estimators based on strain or displacement smoothening are reliable. For element patches at the boundary of the domain, all estimators tend to be unreliable (especially for angle-ply laminates). However, the strain recovery based estimator is clearly more robust for element patches at the boundary, as compared to displacement-recovery based error estimators. Globally, all the estimators tested here were found to be very robust.     

Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method

We propose a fictitious domain method where the mesh is cut by the boundary. The primal solution is computed only up to the boundary; the solution itself is defined also by nodes outside the domain, but the weak finite element form only involves those parts of the elements that are located inside the domain. The multipliers are defined as being element-wise constant on the whole (including the extension) of the cut elements in the mesh defining the primal variable. Inf–sup stability is obtained by penalizing the jump of the multiplier over element faces. We consider the case of a polygonal domain with possibly curved boundaries. The method has optimal convergence properties.