As-rigid-as-possible mesh deformation and its application in hexahedral mesh generation

This paper presents an efficient and stable as-rigid-as-possible mesh deformation algorithm for planar shape deformation and hexahedral mesh generation. The deformation algorithm aims to preserve two local geometric properties: scale-invariant intrinsic variables and elastic deformation energy, which are together represented in a quadric energy function. To preserve these properties, the position of each vertex is further adjusted by iteratively minimizing this quadric energy function to meet the position constraint of the controlling points. Experimental results show that the deformation algorithm is efficient, and can obtain physically plausible results, which have the same topology structure with the original mesh. Such a mesh deformation method is useful to project the source surface mesh onto the target surfaces in hexahedral mesh generation based on sweep method, and application results show that the proposed method is feasible to mesh projection not only between similar surface contours but also dissimilar surface contours.     

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.     

A consistent octree hanging node elimination algorithm for hexahedral mesh generation

A new algorithm for hanging node elimination in octree structures is developed. The proposed algorithm utilizes hanging node elimination by refinement templates and a new mesh conditioning technique based on decoupling templates. Refinement templates insert transition elements to eliminate hanging nodes. Decoupling templates insert circular loops in the dual mesh without introducing or removing hanging nodes. Decoupling templates are introduced to avoid full refinement in the cases that do not match any of the available refinement templates. The proposed algorithm eliminates hanging nodes for concavely refined regions without excessive refinement. Another advantage of the proposed algorithm lies in eliminating narrow gaps of coarse meshes between refined regions. This step has a positive effect on the mesh quality as it avoids introducing non-regular templates with a limited penalty of uniform refinement. The presented algorithm produces good quality meshes and provides a consistent and complete method for producing conformally refined octree structures.     

New method for graded mesh generation of all hexahedral finite elements

Mapping method is widely applied by most of commercial mesh generators because of its efficiency, mesh quality. One of the obstacles to apply the mapping method and generate a graded all hexahedral mesh of high quality in an arbitrarily three-dimensional domain is the generation of hexahedral parent elements on a super-element that allows for gradations in three co-ordinate directions. This paper presents a pattern module’s method to generate the graded mesh of all hexahedral elements in a cube and thus improves the mapping method. The method requires few calculations.     

Automatic hexahedral mesh generation using a new grid-based method with geometry and mesh transformation

A geometry and mesh transformation approach is proposed to overcome the traditional problem of poorly shaped elements at the boundary using the grid-based method of mesh generation. This is achieved by transforming the original geometric model to a topologically similar recognition model which conforms fully to the Cartesian directions. Such a recognition model is constructed by tessellating the original model and then employing a fuzzy logic method to determine the normal directions of the faces. A three-dimensional field morphing algorithm is used to position the features of the recognition model. Such a recognition model is then meshed with hexahedral elements without any degeneracy using the new grid-based algorithm. The mesh of the recognition model is mapped back to the original geometric model by employing a transformation based on the Laplacian-isoparametric equation.     

Seams and wedges in plastering: A 3-D hexahedral mesh generation algorithm

This paper describes mesh correction techniques necessary for meshing an arbitrary volume with a completely hexahedral mesh. Specifically, it describes seams and wedges, mechanisms that overcome major hurdles encountered in the preliminary work on the plastering algorithm. The plastering algorithm iteratively projects layers of elements inward from a quadrilateral discretization of the volume's bounding faces. Seams and wedges resolve incompatibilities in the mesh and in the progressing boundary, thus ensuring the correct formation of a hexahedral mesh from the plastering algorithm.     

Adaptive generation of hexahedral element mesh using an improved grid-based method

An improved grid-based algorithm for the adaptive generation of hexahedral finite element mesh is presented in this paper. It is named as the inside-out grid-based method and involves the following four steps. The first step is the generation of an initial grid structure which envelopes the analyzed solid model completely. And the elements size and density maps are constructed based on the surface curvature and local thickness of the solid model. Secondly, the core mesh is generated through removing all the undesired elements using even and odd parity rules. The third step is to magnify the core mesh in an inside-out manner through a surface node projection process using the closest position approach. To match the mesh to the characteristic boundary of the solid model, a minimal Scaled Jacobian criterion is employed. Finally, in order to handle the degenerated elements and improve the quality of the resulting mesh, two comprehensive techniques are employed: the insertion technique and collapsing technique. The present method was applied in the mesh construction of different engineering problems. Scaled Jacobian and Skew metrics are used to evaluate the hexahedral element mesh quality. The application results show that all-hexahedral element meshes which are well-shaped and capture all the geometric features of the original solid models can be generated using the inside-out grid-based method presented in this paper.     

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.     

一种新的六面体有限元网格算法

在有限元网格产生过程中，吸取弦须编织法中的STC概念，将六面体以节点剖分为基础的思想转变为以单元点为基础，建立了以单元生长为核心的剖分算法，以期解决节点拓扑结构在三维情况下的控制问题，对进一步实现稳定、全自动的六面体剖分具有很大的帮助。     

Automatic hexahedral mesh generation for multi-domain composite models using a hybrid projective grid-based method

This paper presents an algorithm to generate an all-hexahedral mesh of a multi-domain solid model using a hybrid grid-based approach. This is based on a projective concept during the boundary adaptation of the initial mesh. In general, the algorithm involves the generation of a grid structure, which is superimposed on the solid model. This grid structure forms an initial mesh consisting of hexahedral elements, which intersect fully or partially with the solid model. This initial mesh is then shrunk in an outside-in manner to the faces of the model through a node projection process using the closest position approach. To match the resulting mesh to the edges of the model, a minimal deformation angle method is used. Finally, to match the vertices with the nodes on the mesh, a minimal warp angle method is employed. To create the mesh of a multi-domain solid model, an outside-in and inside-in hybrid of the grid-based method is used. This hybrid method ensures that the meshes of the different domains are conforming at their common boundary. This paper also describes two methods for resolving cases of degenerate elements: a splitting technique and a wedge insertion technique.     

DARSS: a hybrid mesh smoother for all hexahedral meshes

A method for smoothing hexahedral meshes has been developed. The method consists of two phases. In the first phase, the nodes are moved based on an explicit formulation. A constraint has also been implemented to prevent the deterioration of elements associated with the node being moved. The second phase of the method is optismoothing based on the Nelder–Mead simplex method. The summation of the Jacobian of all the elements sharing a node has been taken as the function to be maximized. The method has been tested on meshes up to 18,305 hexahedral elements and was found to be stable and improved the mesh in about 112.6 s on an Intel Centrino^? 1.6 GHz, 1 GB RAM machine. The method thus has the advantage of being effective as well as being computationally efficient.     

Automatic mesh generation and modification techniques for mixed quadrilateral and hexahedral element meshes of non-manifold models

A typical geometric model usually consists of both solid sections and thin-walled sections. Through using a suitable dimensional reduction algorithm, the model can be reduced to a non-manifold model consisting of solid portions and two-dimensional portions which represent the mid-surfaces of the thin-walled sections. It is desirable to mesh the solid entities using three-dimensional elements and the surface entities using two-dimensional elements. This paper proposes a robust scheme to automatically generate such a mesh of mixed two-dimensional and three-dimensional elements. It also ensures that the mesh is conforming at the interface of the non-manifold geometries. Different classes of problems are identified and their corresponding solutions are presented.     

Block Cartesian abstraction of a geometric model and its application in hexahedral mesh generation

In this work, a fuzzy logic approach is proposed to transform a geometric model of arbitrary shape to its block Cartesian abstraction. This abstraction is topologically similar to the original model and it contains geometric sub-entities which are all aligned in the Cartesian directions. This is achieved by calculating the modifications made to the face normal vectors as a result of the influences of the adjacent faces. A fuzzy logic inference engine is developed by combining heuristics to emulate the local changes in face normal vectors with respect to the changes in the global space. A three-dimensional field morphing algorithm is used to position the features of this block Cartesian abstraction so that a congruent geometric model can be reconstructed. Such a model is useful for the generation of structured quadrilateral boundary element meshes or structured hexahedral meshes based on grid-based meshing method, mesh mapping or sweeping. This approach is also able to overcome the traditional problem of having poorly shaped elements at the boundary using the grid-based method of mesh generation. As the topology of the block Cartesian abstraction is congruent to the original model, the mesh can be mapped back to the original model by employing an inverse operation of the transformation.     

Mixed meshless formulation for analysis of shell-like structures

An efficient mixed meshless computational method based on the Local Petrov–Galerkin approach for analysis of plate and shell structures is presented. A concept of a three-dimensional solid is applied allowing the use of complete three-dimensional constitutive equations, and exact shell geometry can be described. Discretization is carried out by using both the moving least square approximation and the polynomial functions. Independent field variables are the strain and stress tensor components expressed in terms of the nodal values, which are then replaced by the nodal displacements by using the independent displacement interpolation. A closed global system of equations with only nodal displacements as unknown variables is derived. The undesired locking phenomena are fully suppressed. The proposed mixed formulation is numerically more efficient than the available meshless fully displacement approach, as demonstrated by the numerical examples.     

A strategy of automatic hexahedral mesh generation by using an improved whisker-weaving method with a surface mesh modification procedure

One of the demands for three dimensional (3D) finite element analyses is the development of an automatic hexahedral mesh generator. For this problem, several methods have been proposed by many researchers. However, reliable automatic hexahedral mesh generation has not been developed at present. In this paper, a new strategy of fully automatic hexahedral mesh generation is proposed. In this strategy, the prerequisite for generating a hexahedral mesh is a quadrilateral surface mesh. From the given surface mesh, combinatorial dual cycles (sheet loops for the whisker-weaving algorithm) are generated to produce a hexahedral mesh. Since generating a good quality hexahedral mesh does not depend only on the quality of quadrilaterals of the surface mesh but also on the quality of the sheet loops generated from it, a surface mesh modification method to remove self-intersections from sheet loops is developed. Next, an automatic hexahedral mesh generator by the improved whisker-weaving algorithm is developed in this paper. By creating elements and nodes on 3D real space during the weaving process, it becomes possible to generate a hexahedral mesh with fewer bad-quality elements. Several examples will be presented to show the validity of the proposed mesh generation strategy.     

Adaptive hexahedral mesh generation and quality optimization for solid models with thin features using a grid-based method

This paper presented an automatic inside-out grid-based hexahedral element mesh generation algorithm for various types of solid models. For the thin features with small thickness of the geometric model, corresponding treatment methods were given for successfully implementing each meshing step, containing the techniques for adaptive refinement, boundary match, topological optimization and local refinement. In order to realize the reasonable identification of refinement regions and resolve the expansion problem of refinement information fields, a thin-feature criterion and a supplementary criterion were proposed aiming at thin features of the geometry. To implement accurate boundary match for thin features, ten basic types and five complementary types of facet configurations were established, and a priority-node identification method was proposed additionally. Three topological optimization modes were newly proposed to improve the topological connections of the boundary mesh in thin features. Local refinement techniques were also built to refine the thin features of solid models. Finally, several examples were provided to demonstrate the effectiveness and reliability of the proposed algorithms.     

A new automated scheme of quadrilateral mesh generation for randomly distributed line constraints

This paper presents a new automated method of quadrilateral meshes with random line constraints which have not been fully considered in previous models. The authors developed a new looping scheme and a direct quadrilateral forming algorithm based on advanced front techniques. This generator overcomes the limitations of previous studies such as line constraint, unmeshed hole and mesh refinement. A qualitative test reveals that our algorithm is reliable and suitable at the field needed for very accurate results. The developed direct method to handle line-typed features automatically makes the multiple discretizations without any user interaction and modification.     

含复杂插值曲面实体六面体网格优化方法

复杂插值曲面的几何形态由散乱数据点控制,优化含这类曲面实体的六面体网格难以保证边界一致.借鉴Balendran提出的直接法,建立正四边形结点的空间几何关系方程,并作为优化约束,以移动结点使每个六面体的各个侧面趋近于正四边形,实现网格优化.将原始采样点及其它控制插值曲面几何形态的数据转化为控制点约束,以保证边界一致性.结合优化约束与控制点约束,作为离散光滑插值(DSI)方程的约束项,实现网格优化与曲面插值的耦合.实例表明,该方法能够在保证复杂插值曲面边界一致性的前提下实现六面体网格优化.