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An automatic hexahedral mesh generation system based on the shape-recognition and boundary-

An Automatic Hexahedral Mesh Generation System

Based on the

Shape-Recognition and Boundary-Fit Methods

N. Chiba*, I. Nishigaki*, Y. Yamashita**, C. Takizawa*, K. Fujishiro***

Abstract

A general-purpose automatic hexahedral mesh generation system for FEA (Finite Element Analy-

sis) was developed based on a shape recognition technique and the boundary-fit method. In this

system, a solid model is analyzed and decomposed into single-connected sub-models. Then, other

sub-models topologically identical to the original ones are constructed using only orthogonal an-

gles. Cubes are used to construct intermediate models, which reassemble these, and finally, hexa-

hedral meshes are generated by mapping the cubes back onto the original solid model.

1. Introduction

Automatic tetrahedral mesh generation is now a fully mature technology and is widely used to generate meshes for even complex geometries. On the other hand, many FEA users prefer hexahedral meshes to tetrahedral ones, and there have several efforts to automate hexahedral mesh generation for arbitrary 3D geometries [1, 2]. It must be said, however, that the automatization level for hexahedral-mesh generation and quality of meshes generated are lower than those for tetrahedral meshes. We have previously developed an automatic quadrilat-eral mesh generation algorithm for 2D models by combining the boundary-fitted coordinate-transformation technique [3, 4] with a shape recognition method [5]. This paper describes the expansion of this method for use with arbitrary 3D geometries.

---------------------------------------------------------------------------* Mechanical Engineering Research Laboratory, Hitachi, Ltd.

502 Kandatsu, Tsuchiura, Ibaraki 300, Japan

chibax@merl.hitachi.co.jp

** Software Development Center, Hitachi, Ltd.

*** Hitachi Car Engineering, Ltd.

2. Algorithm for Automatic Mesh Generation

The outline of our automatic mesh-generation is described below, and shown in Figure 1.

(1) A solid model (Figure 1a) is decomposed into single-connected sub-models, each of which is defined as the set of connected lines defining its boundaries (Figure 1b). (The original geometry is expressed in (x,y,z) coordi-nates in this paper.)

(2) Another set of sub-models, which we call the recognition model, constructed using only lines parallel to the cartesian axes, is created from them (Figure 1c). The recognition model must be topologically identical and geometrically similar to the solid model.

(3) The lengths of the edges of the recognition model are adjusted to the nearest integral multiple of a grid unit, and the model is divided into cubes, producing what we call the mapping model (Figure 1d). (Both the recogni-tion model and the mapping model are expressed using the coordinates (ξ,η,ζ) in this paper.

(4) The proper positional relationship between the sub-models of the mapping model is determined, and a complete mapping model corresponding to the original solid model is constructed (Figure 1e).

(5) Using boundary-fit method, this mapping model is mapped onto the original solid model, and the final hexahedral mesh is generated (Figure 1f).

Input data to this system are the geometric data to be meshed and the number of elements to be generated (or the element size). All other data are generated in the system.

3. Recognition Model

A recognition model (Figure 1c) is created from the solid model by assigning one of three orientations (ξ,η, or ζ direction) and a length to each edge constructing of the solid model. There is no unique solution for assigning discrete directions to the edges which make up a solid model, since it inevitably includes curved lines and arbitrary angles between edges. This problem is very ambiguous, and fuzzy logic is employed in our system in an attempt to imitate the human thinking process in assigning directions.

The basic assignment rules are expressed by the two membership functions (Figure 2). Basic Rule 1 is ex-pressed by the function shown in Figure 2a, in which θx denotes the inner angle defined by an edge and x-axis in the solid model, and Pξ denotes the probability that the edge will be assigned to the ξ orientation in the recognition model. Similar functions are used for the η and ζ directions. Basic Rule 2 is expressed by the function shown in Figure 2b, in which θαdenotes the inner angle between two adjacent edges in the solid

model, and Pα denotes the probability that these edges will be assigned to the same orientation in the recogni-tion model.

The algorithm for direction assignment in constructing the recognition model is as follows:

(1) The angles defined by an edge with the x, y, and z axes are measured, and the initial values for assignment probabilities Pξ, Pη, and Pζ are calculated for the edge, using Basic Rule 1. Initial values Pξ, Pη, and Pζare assumed to be independent.

(2) For each edge constructing the original solid model, Basic Rule 2 is employed for each of the pairs formed by the edge and an adjacent edge, and the resulting assignment probabilities Pξ, Pη, and Pζ are multiplied with those from Step 1. Based on the results, each plane constructing in solid model is assigned to a plane defined by two of the geometric axes (the ξ-η plane, η-ζ plane, or ζ-ξ plane) in the recognition model.

(3) Step 2 is repeated until the orientations for all the edges constructing the solid model have been assigned, that is, until one of the probabilities Pξ, Pηor Pζ for each edge has converged to one. In this process the effect of changing one edge is applied to other edges via the connected lines constructing the solid model.

Based on these two basic rules, any edge on the solid model (expressed in x, y, z space), straight or curved, is assigned to a discrete rectangular direction in the recognition model (expressed in ξ, η, ζ space).

In Step 2, it is checked whether the plane constructed in the recognition model has the same topological struc-ture as that of the corresponding plane in the original geometrical model. It is also checked whether the sub-model constructed in the recognition model has the same topological structure as that of the corresponding sub-model in the original geometrical model. In some geometries, such as models which contain triangular planes, the conditions above are not satisfied, and the current version of this system fails. We are looking for a way to overcome this difficulty.

In Step 3, if the probabilities Pξ, Pη, or Pζ for an edge do not reach convergence in a certain number of iterations, then the direction ξ, η, or ζ with the biggest value of P is assigned, and the process is continued.

4. Boundary-Fit Method

The boundary-fit method is used as the process for mapping from the mapping model to the solid model, as described in (5) in Section 2 (Figure 1f). In this method, the coordinates of nodal point on the boundary surfaces of the solid model are determined first. Then the coordinates of the nodal points inside the model are deter-mined by solution of elliptic partial differential equations. The Poisson equations shown below are used in our

present system.

@ @ξxx + ξyy + ξzz= P (ξ,η,ζ)

@ @ηxx +ηyy +ηzz= Q (ξ,η,ζ)

@ @ζxx + ζyy + ζzz= R (ξ,η,ζ)

In the equations above, ξxx , ξyy c represent ?ξ2/?x2, ?ξ2/?y2 c, respectively. P, Q, R are weighting functions which control the density of the mesh generated, and we currently set them of P = Q = R = 0 to obtain uniformly distributed mesh.

5. Mapping Model

The mapping model is created by adjusting the length of each edge in the recognition model so that it is an integral multiple of the grid-unit length. Then, this model is divided into cubes. These processes are carried out for each sub-model constructing the recognition model. Finally, the proper relative positioning between the sub-models is determined (as described below), and the mapping model corresponding to the original solid model is constructed (Figure 1e).

The relative positioning for assembling the sub-models to construct the mapping model is determined by opti-mizing the position on the connecting plane between the two sub-models as shown in Figure 3a. The steps in this process are as follows (Figure 3b):

(1) Employ the two-dimensional boundary-fit method on the plane α to obtain an intermediate mesh model (see Figure 3a for α and β).

(2) Place the four vertices of the plane β on the original geometric model as shown by the solid circles.

(3) Place the mapping model of the plane β (as shown by the open circles) on a tentatively selected position in the mapping model of the plane α (as shown by the stars), whose four vertices are shown by the open triangles. The corresponding vertices in the intermediate model are shown by the solid triangles.

(4) Calculate the sum of the squares of the distances between the corresponding points shown by the solid circles and the solid triangles in the intermediate model.

(5) Change the trial position in Step 3 until the sum of the distances squares in Step 4 is minimized.

The optimum relative positioning between the two sub-models is given as the position which minimizes this sum.

A model consisting of many sub-models can be processed in this way for each connecting plane between two sub-models. The system checks whether there is interference between the two sub-models in the mapping model. In some geometries the positional relationship between the two sub-models is consistent on the connect-ing plane, but there is still interference between the sub-models. Our current system does not have the capabil-ity to circumvent this problem automatically.

6. Examples of Generated Meshes

Examples of generated mesh are shown in Figures 4 - 7.

Conclusion

A new automatic hexahedral-mesh generation system for arbitrary three dimensional geometries was developed that employs a shape recognition technique and the boundary-fit method. It is able to generate smooth and uniformly distributed meshes for the various geometries of actual mechanical components. For some geome-tries the system fails to process automatically; we are now expanding the capability of the system.

References

1 Li, T.S., McKeag, R.M., Armstrong, C.G. : Hexahedral Meshing using Mid-Point Subdivision and Integer Programming, Computer Methods in Applied Mechanics and Engineering, 124 (1995), 171.

2 Blacker, T.D., Meyers, R.J. : Seams and Wedges in Plastering: A 3-D Hexahedral Mesh Generation Algo-rithm, Engineering with Computers (1993) 9, 83.

3 Thompson, J.F., Warsi, Z.U.A. : Boundary-Fitted Coordinate Systems for Numerical Solution of Partial Differential Equations, J. Comput. Phys., 47(1982), 1.

4 Miki, K., Takagi, T. : A Domain Decomposition and Overlapping Method for the Generation of Three-Dimensional Boundary-Fitted Coordinate Systems, J. Comput. Phys., 53(1984), 319.

5 Takahashi, H., Shimizu, H. : A General Purpose Automatic Mesh Generation Using Shape Recognition Technique, Computers in Engineering, 1, ASME 1991, 519.

(a) Solid model (b) Decomposed sub-models

(e) Re-assembly of mapping models (f) Mesh generation

Figure 1. Automatic hexahedral-mesh generation procedure.

x y z x

y z η

ξζηξζη

ξζx

y z

Real plane Mapping plane Membership function Direction assignment P Figure 2. Membership functions used in the recognition-model production system.

ηy x ξ

ηg f e d c

b a a

b c d e

(a) Positioning on the connecting plane

Geometric model Intermediate mesh model

Generated mesh Mapping models

Positioning

(b) Positioning mapping models

Figure 3. Mapping model construction.

Figure 4. Example of mesh generated for upper cover of water turbine.

Figure 5. Example of mesh generated for engine block.

Figure 6. Example of mesh generated for crank shaft.

Figure 7. Example of mesh generated for mechanical arm.

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select count(*) from tree where lft <= 2 and rgt >= 11 为了方便列表，我们可以为tree表建立一个视图，添加一个层数列，该类别的层数可以写一个自定义函数来计算。该函数如下： CREATE FUNCTION dbo.CountLayer ( @type_id int ) RETURNS int AS begin declare@result int set@result=0 declare@lft int declare@rgt int if exists (select1from tree where type_id=@type_id) begin select@lft=lft,@rgt=rgt from tree where type_id=@type_id select@result=count(*) from tree where lft <=@lft and rgt >=@rgt end return@result end GO 然后，我们建立如下视图： CREATE VIEW dbo.TreeView AS SELECT type_id, name, lft, rgt, dbo.CountLayer(type_id) AS layer FROM dbo.tree ORDE R BY lft GO

数据结构课程设计之图的遍历和生成树求解

##大学数据结构课程设计报告题目：图的遍历和生成树求解院（系）：计算机工程学院学生: 班级：学号: 起迄日期: 2011.6.20 指导教师:

2010—2011年度第 2 学期一、需求分析 1.问题描述：图的遍历和生成树求解实现图是一种较线性表和树更为复杂的数据结构。在线性表中，数据元素之间仅有线性关系，每个数据元素只有一个直接前驱和一个直接后继；在树形结构中，数据元素之间有着明显的层次关系，并且每一层上的数据元素可能和下一层中多个元素（及其孩子结点）相关但只能和上一层中一个元素（即双亲结点）相关；而在图形结构中，节点之间的关系可以是任意的，图中任意两个数据元素之间都可能相关。生成树求解主要利用普利姆和克雷斯特算法求解最小生成树，只有强连通图才有生成树。 2.基本功能 1) 先任意创建一个图； 2) 图的DFS,BFS的递归和非递归算法的实现 3) 最小生成树（两个算法）的实现，求连通分量的实现 4) 要求用邻接矩阵、邻接表等多种结构存储实现 3.输入输出

输入数据类型为整型和字符型，输出为整型和字符二、概要设计 1.设计思路： a.图的邻接矩阵存储：根据所建无向图的结点数n,建立n*n的矩阵，其中元素全是无穷大（int_max），再将边的信息存到数组中。其中无权图的边用1表示，无边用0表示；有全图的边为权值表示，无边用∞表示。 b.图的邻接表存储：将信息通过邻接矩阵转换到邻接表中，即将邻接矩阵的每一行都转成链表的形式将有边的结点进行存储。 c.图的广度优先遍历：假设从图中的某个顶点v出发，在访问了v之后依次访问v的各个未曾访问过的邻接点，然后再访问此邻接点的未被访问的邻接点，并使“先被访问的顶点的邻接点”先于“后被访问的顶点的邻接点”被访问，直至图中所有已被访问的顶点的邻接点都被访问到。若此时图中还有未被访问的，则另选未被访问的重复以上步骤，是一个非递归过程。 d.图的深度优先遍历：假设从图中某顶点v出发，依依次访问v的邻接顶点，然后再继续访问这个邻接点的系一个邻接点，如此重复，直至所有的点都被访问，这是个递归的过程。 e.图的连通分量：这是对一个非强连通图的遍历，从多个结点出发进行搜索，而每一次从一个新的起始点出发进行搜索过程中得到的顶点访问序列恰为其连通分量的顶点集。本程序利用的图的深度优先遍历算法。 2.数据结构设计： ADT Queue{ 数据对象：D={a i | a i ∈ElemSet,i=1,2,3……，n,n≥0} 数据关系：R1={| a i-1 ,a i ∈D,i=1,2,3,……，n} 基本操作： InitQueue(&Q) 操作结果：构造一个空队列Q。 QueueEmpty(Q) 初始条件：Q为非空队列。操作结果：若Q为空队列，则返回真，否则为假。 EnQueue(&Q,e) 初始条件：Q为非空队列。操作结果：插入元素e为Q的新的队尾元素。 DeQueue(&Q,e) 初始条件：Q为非空队列。操作结果：删除Q的队头元素，并用e返回其值。}ADT Queue