Inverse adaptation of a Hex-dominant mesh for large deformation finite element analysis

In the finite element analysis of metal forming processes, many mesh elements are usually deformed severely in the later stage of the analysis because of the corresponding large deformation of the geometry. Such highly distorted elements are undesirable in finite element analysis because they introduce error into the analysis results, and, in the worst case, inverted elements can cause the analysis to terminate prematurely. This paper proposes a new inverse-adaptation method that reduces or eliminates the number of inverted mesh elements created in the later stage of finite element analysis, thereby lessening the chances of early termination and improving the accuracy of the analysis results. By this method, a simple uniform mesh is created initially, and a pre-analysis is run in order to observe the deformation behavior of the elements. Next, an input hex-dominant mesh is generated in which each element is “inversely adapted”, or pre-deformed in such a way that it has approximately the opposite shape of the final shape that normal analysis would deform it into. Thus, when finite element analysis is performed, the analysis starts with an input mesh of inversely adapted elements whose shapes are not ideal. As the analysis continues, the element shape quality improves to almost ideal, and then, toward the final stage of analysis, degrades again, but much less than would be the case without the inverse adaptation. This method permits the analysis to run to the end, or to a further stage, with no inverted elements. Besides its pre-skewing the element shape, the proposed method is also capable of controlling the element size according to the equivalent plastic strain information collected from the pre-analysis. The proposed inverse adaptation can be repeated iteratively until reaching the final stage of deformation.     

HEXHOOP: Modular Templates for Converting a Hex-Dominant Mesh to an ALL-Hex Mesh

This paper presents a new mesh conversion template called HEXHOOP, which fully automates a con-version from a hex-dominant mesh to an all-hex mesh. A HEXHOOP template subdivides a hex/prism/pyramid element into a set of smaller hex elements while main-taining the topological conformity with neighboring elements. A HEXHOOP template is constructed by assembling sub-templates, cores and caps. A dicing template for a hex and a prism is constructed by choosing the appropriate combination of a core and caps. A template that dices a pyramid without losing conformity to the adjacent element is derived from a HEXHOOP template. Some experimental results show that the HEXHOOP templates successfully convert a hex-dominant mesh into an all-hex mesh. ID="A1" Correspondence and offprint requests to: K. Shimada, Department of Mechanical Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213–3890, USA. E-mail: shimada@cmu.edu     

The Cut&Glue mesh generation algorithm

This paper describes a technique for generating quadrilateral finite element meshes on convex, four-sided patches, given an arbitrary number of elements along each side of the patch. The technique first generates a subdivision with the correct topological structure and smoothes the subdivision to obtain elements of acceptance shape for finite element analysis. The correct mesh topology is obtained from a regular subdivision by cutting rectangular corners of appropriate size and interconnecting the sides introduced by the cuts.The method can also be applied on three-dimensional patches producing meshes of brick elements with gradations in all directions.     

A control approach for pore size distribution in the bone scaffold based on the hexahedral mesh refinement

Tissue engineering is the application of that knowledge to the building or repairing of tissues. Generally, engineered tissue is a combination of living cells and a support structure called scaffolds. Modeling, design and fabrication of tissue scaffold with intricate architecture, porosity and pore size for desired tissue properties presents a challenge in tissue engineering. In this paper, a control approach for pore size distribution in the bone scaffold based on the hexahedral mesh refinement is presented. Firstly, the bone scaffold modeling approach based on the shape function in the finite element method is provided. The resulting various macroporous morphologies can be obtained. Then conformal refinement algorithm for all-hexahedral element mesh is illustrated. Finally, a modeling approach for constructing tissue engineering (TE) bone scaffold with defined pore size distribution is presented. Before the conformal refinement of all-hexahedral element mesh, a 3D mesh with various hexahedral elements must be provided. If all the pores in the bone scaffold need to be reduced, that means that the whole hexahedral mesh needs to be refined. Then the solid entity can be re-divided with altered subdivision parameters. If the pores in the local regions of bone need to be reduced, that means that 3D hexahedral mesh in the local regions needs to be refined. Based on SEM images, the pore size distribution in the normal bone can be obtained. Then, according to the conformal refinement of all-hexahedral element meshes, defined hexahedral size distribution can be gained, which leads to generate defined pore size distribution in the bone scaffold, for the pore morphology and size are controlled by various subdivided hexahedral elements. Compared to other methods such as varying processing parameters in supercritical fluid processing and multi-interior architecture design, the method proposed in this paper enjoys easy-controllability and higher accuracy.     

Frame field smoothness-based approach for hex-dominant meshing

An indirect approach for building hex-dominant meshes is proposed: a tetrahedral mesh is constructed at first and is recombined to create a maximum amount of hexahedra. The efficiency of the recombination process is known to significantly depend on the quality of the sampling of the vertices. A good vertex sampling depends itself on the quality of the underlying frame field that has been used to locate the vertices. An iterative procedure to obtain a high quality three-dimensional frame field is presented. Then, a new point insertion algorithm based on a frame field smoothness is developed. Points are inserted in priority in smooth frame field regions. The new approach is tested and compared with simpler strategies on various geometries. The new method leads to hex-dominant meshes exhibiting either an equivalent or a larger volume ratio of hexahedra (up to 20%) compared to the frontal point insertion approach.     

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.     

A methodology for quadrilateral finite element mesh coarsening

High fidelity finite element modeling of continuum mechanics problems often requires using all quadrilateral or all hexahedral meshes. The efficiency of such models is often dependent upon the ability to adapt a mesh to the physics of the phenomena. Adapting a mesh requires the ability to both refine and/or coarsen the mesh. The algorithms available to refine and coarsen triangular and tetrahedral meshes are very robust and efficient. However, the ability to locally and conformally refine or coarsen all quadrilateral and all hexahedral meshes presents many difficulties. Some research has been done on localized conformal refinement of quadrilateral and hexahedral meshes. However, little work has been done on localized conformal coarsening of quadrilateral and hexahedral meshes. A general method which provides both localized conformal coarsening and refinement for quadrilateral meshes is presented in this paper. This method is based on restructuring the mesh with simplex manipulations to the dual of the mesh. In addition, this method appears to be extensible to hexahedral meshes in three dimensions. Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000.     

Array-based,parallel hierarchical mesh refinement algorithms for unstructured meshes

In this paper, we describe an array-based hierarchical mesh refinement capability through uniform refinement of unstructured meshes for efficient solution of PDE’s using finite element methods and multigrid solvers. A multi-degree, multi-dimensional and multi-level framework is designed to generate the nested hierarchies from an initial coarse mesh that can be used for a variety of purposes such as in multigrid solvers/preconditioners, to do solution convergence and verification studies and to improve overall parallel efficiency by decreasing I/O bandwidth requirements (by loading smaller meshes and in-memory refinement). We also describe a high-order boundary reconstruction capability that can be used to project the new points after refinement using high-order approximations instead of linear projection in order to minimize and provide more control on geometrical errors introduced by curved boundaries.The capability is developed under the parallel unstructured mesh framework “Mesh Oriented dAtaBase” (MOAB Tautges et al. (2004)). We describe the underlying data structures and algorithms to generate such hierarchies in parallel and present numerical results for computational efficiency and effect on mesh quality. We also present results to demonstrate the applicability of the developed capability to study convergence properties of different point projection schemes for various mesh hierarchies and to a multigrid finite-element solver for elliptic problems.     

An error analysis and mesh adaptation method for shape design of structural components

A finite element error analysis and mesh adaptation method that can be used for improving analysis accuracy in carrying out shape design of structural components is presented in this paper. The simple error estimator developed by Zienkiewicz is adopted in this study for finite element error analysis, using only post-processing finite element data. The mesh adaptation algorithm implemented in ANSYS is investigated and the difficulties found are discussed. An improved algorithm that utilizes ANSYS POST1 capabilities is proposed and found to be more efficient than the ANSYS algorithm. An example is given to show the efficiency. An interactive mesh adaptation method that utilizes PATRAN meshing and result-displaying capabilities is proposed. This proposed method displays error distribution and stress contour of analysis results using color plots, to help the designer in identifying the critical regions for mesh refinement. Also, it provides guidance for mesh refinement by computing and displaying the desired element size information, based on error estimate and a mesh refinement criterion defined by the designer. This method is more efficient and effective than the semi-automatic algorithm implemented in ANSYS, and is suitable for structural shape design. This method can be applied not only to set-up a finite element mesh of the structure at initial design but to ensure analysis accuracy in the design process. Examples are given to demonstrate feasibility of the proposed method.     

非结构网格布点方法研究进展

用有限元方法求解偏微分方程初边值问题首先要离散求解区域,即网格生成,并且网格质量的好坏直接影响着有限元解的收敛性和精度,所以关于网格生成有很多学者从各自的领域出发做了大量的研究工作。论文关注于非结构网格点的布置方法,对已有的具有代表性的布点方法的研究进展进行了分类综述。     

A hybrid ant colony optimization approach for finite element mesh decomposition

This paper examines the application of the ant colony optimization algorithm to the partitioning of unstructured adaptive meshes for parallel explicit time-stepping finite element analysis. The concept of the ant colony optimization technique for finding approximate solutions to combinatorial optimization problems is described.The application of ant colony optimization for partitioning finite element meshes based on triangular elements is described.A recursive greedy algorithm optimization method is also presented as a local optimization technique to improve the quality of the solutions given by the ant colony optimization algorithm. The partitioning is based on the recursive bisection approach.The mesh decomposition is carried out using normal and predictive modes for which the predictive mode uses a trained multilayered feed-forward neural network which estimates the number of triangular elements that will be generated after finite elements mesh generation is carried out.The performance of the proposed hybrid approach for the recursive bisection of finite element meshes is examined by decomposing two mesh examples.     

A log-barrier method for mesh quality improvement and untangling

The presence of a few inverted or poor-quality mesh elements can negatively affect the stability, convergence and efficiency of a finite element solver and the accuracy of the associated partial differential equation solution. We propose a mesh quality improvement and untangling method that untangles a mesh with inverted elements and improves its quality. Worst element mesh quality improvement and untangling can be formulated as a nonsmooth unconstrained optimization problem, which can be reformulated as a smooth constrained optimization problem. Our technique solves the latter problem using a log-barrier interior point method and uses the gradient of the objective function to efficiently converge to a stationary point. The method uses a logarithmic barrier function and performs global mesh quality improvement. We have also developed a smooth quality metric that takes both signed area and the shape of an element into account. This quality metric assigns a negative value to an inverted element. It is used with our algorithm to untangle a mesh by improving the quality of an inverted element to a positive value. Our method usually yields better quality meshes than existing methods for improvement of the worst quality elements, such as the active set, pattern search, and multidirectional search mesh quality improvement methods. Our method is faster and more robust than existing methods for mesh untangling, such as the iterative stiffening method.     

Constrained mesh optimization on boundary

This paper presents a new mesh optimization approach aiming to improve the mesh quality on the boundary. The existing mesh untangling and smoothing algorithms (Vachal et al. in J Comput Phys 196: 627–644, 2004; Knupp in J Numer Methods Eng 48: 1165–1185, 2002), which have been proved to work well to interior mesh optimization, are enhanced by adding constrains of surface and curve shape functions that approximate the boundary geometry from the finite element mesh. The enhanced constrained optimization guarantees that the boundary nodes to be optimized always move on the approximated boundary. A dual-grid hexahedral meshing method is used to generate sample meshes for testing the proposed mesh optimization approach. As complementary treatments to the mesh optimization, appropriate mesh topology modifications, including buffering element insertion and local mesh refinement, are performed in order to eliminate concave and distorted elements on the boundary. Finally, the optimization results of some examples are given to demonstrate the effectivity of the proposed approach.     

三角网格的自适应细分研究

提出面向三角网格全局细分和局部自适应的细分算法。在原三角网格模型上计算每个面片的中心坐标,据此生成的中心坐标点作为新的顶点坐标进行重新绘制得到三角基网格,然后进行多次迭代,达到基本的全局细分目标。在最后生成的基网格上,可以通过调节最大网格面积和平均网格面积之间的比例系数等,来得到更加均匀的三角网格。实验表明该方法能到得到质量较高的细分结果。     

A knowledge base for finite element mesh design

The finite element method (FEM) is the most successful numerical method, that is used extensively by engineers to analyse stresses and deformations in physical structures. These structures should be represented as a finite element mesh. Defining an appropriate geometric mesh model that ensures low approximation errors and avoids unnecessary computational overheads is a very difficult and time consuming task. It is the major bottleneck in the FEM analysis process. The inductive logic programming system GOLEM has been employed to construct the rules for deciding about the appropriate mesh resolution. Five cylindrical mesh models have been used as a source of training examples. The evaluation of the resulting knowledge base shows that conditions in the domain are well represented by the rules, which specify the required number of the finite elements on the edges of the structures to be analysed using FEM. A comparison between the results obtained by this knowledge base and conventional mesh generation techniques confirms that the application of inductive logic programming is an effective approach to solving the problem of mesh design.     

Construction of minimal subdivision surface with a given boundary

The fascinating characters of minimal surface make it to be widely used in shape design. While the flexibility and high quality of subdivision surface make it a powerful mathematical tool for shape representation. In this paper, we construct minimal subdivision surfaces with given boundaries using the mean curvature flow, a second order geometric partial differential equation. This equation is solved by a finite element method where the finite element space is spanned by the limit functions of an extended Loop’s subdivision scheme proposed by Biermann et al. Using this extended Loop’s subdivision scheme we can treat a surface with boundary, thereby construct the perfect minimal subdivision surfaces with any topology of the control mesh and any shaped boundaries.     

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。     

Catmull-Clark细分网格数据点拾取

针对将OpenGL选择拾取机制直接作用于Catmull-Clark细分网格数据点的拾取,可能会因细分网格数据量过大而导致名字堆栈溢出的问题,借鉴细分曲面求交的思想,提出一种新的细分网格数据点拾取方法.该方法通过提取拾取对象的邻域网格并进行局部细分,将对细分任意层次上网格数据点的拾取转化为对初始控制网格以及在达到细分层次要求以前每一次局部细分网格点、边、面的拾取和对最后一次局部细分网格数据点的拾取.采用多个拾取算例进行对比分析实验,当细分网格顶点数量较多时,所给拾取方法的拾取命名对象总量和拾取时间都远小于传统OpenGL选择拾取方法.实验结果表明,所给拾取方法能快速准确实现细分网格数据点的拾取,尤其适用于数据量较大的复杂细分模型,可有效避免因拾取名字堆栈溢出而导致的拾取错误.     

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.     

Adaptive finite element mesh triangulation using self-organizing neural networks

The finite element method is a computationally intensive method. Effective use of the method requires setting up the computational framework in an appropriate manner, which typically requires expertise. The computational cost of generating the mesh may be much lower, comparable, or in some cases higher than the cost associated with the numeric solver of the partial differential equations, depending on the application and the specific numeric scheme at hand.The aim of this paper is to present a mesh generation approach using the application of self-organizing artificial neural networks through adaptive finite element computations. The problem domain is initially constructed using the self-organizing neural networks. This domain is used as the background mesh which forms the input for finite element analysis and from which adaptive parameters are calculated through adaptivity analysis. Subsequently, self-organizing neural network is used again to adjust the location of randomly selected mesh nodes as is the coordinates of all nodes within a certain neighborhood of the chosen node. The adjustment is a movement of the selected nodes toward a specific input point on the mesh. Thus, based on the results obtained from the adaptivity analysis, the movement of nodal points adjusts the element sizes in a way that the concentration of elements will occur in the regions of high stresses. The methods and experiments developed here are for two-dimensional triangular elements but seem naturally extendible to quadrilateral elements.