提高板材成形效率的坐标网分析法外文文献翻译、中英文翻译、外文翻译

  Journal of Materials Processing Technology 140 (2003) 616621 Efficiency enhancement in sheet metal forming analysis with a mesh regularization method J.H. Yoon, H. Huh  Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology Science Town, Daejeon 305-701, South Korea Abstract This paper newly proposes a mesh regularization method for the enhancement of the efficiency in sheet metal forming analysis. The regularization method searches for distorted elements with appropriate searching criteria and constructs patches including the elements to be modified. Each patch is then extended to a three-dimensional surface in order to obtain the information of the continuous coordinates. In constructing the surface enclosing each patch, NURBS (non-uniform rational B-spline) surface is employed to describe a three-dimensional free surface. On the basis of the constructed surface, each node is properly arranged to form unit elements as close as to a square. The state variables calculated from its original mesh geometry are mapped into the new mesh geometry for the next stage or incremental step of a forming analysis. The analysis results with the proposed method are compared to the results from the direct forming analysis without mesh regularization in order to confirm the validity of the method. 2003 Elsevier B.V. All rights reserved. Keywords: Mesh regularization； Distorted element； NURBS； Patch； Finite element analysis 1. Introduction Numerical simulation of sheet metal forming processes enjoys its prosperity with a burst of development of the com- puters and the related numerical techniques. The numerical analysis has extended its capabilities for sheet metal forming of complicated geometry models and multi-stage forming. In the case of a complicated geometry model, however, severe local deformation occurs to induce the increase of the com- puting time and deteriorate the convergence of the analysis. Distortion and severe deformation of the mesh geometry has an effect on the quality of forming analysis results especially in the case of multi-stage forming analysis when the mesh geometry formed by the forming analysis at the first stage is used for the forming analysis at the next stage. This ill behavior of the distorted mesh can be avoided by the recon- struction of the mesh system such as the total or the adaptive remeshing techniques. The adaptive remeshing technique is known to be an efficient method to reduce distortion of element during the simulation, but it still needs tremen- dous computing and  puts restrictions among  subdivided elements.  Corresponding author. Tel.: +82-42-869-3222； fax: +82-42-869-3210. E-mail address: hhuhkaist.ac.kr (H. Huh). Effective methods to construct a mesh system have been proposed by many researchers. Typical methods could be r-method 1 in which nodal points are properly rearranged without the change of the total degrees of freedom of the mesh system, h-method 2 in which the number of meshes is increased with elements of the same degrees of freedom, and p-method 3 in which the total degrees of freedom of the mesh system is increased to enhance the accuracy of so- lutions. Sluiter and Hansen 4 and Talbert and Parkinson 5 constructed the analysis domain as a continuous loop and created elements in sub-loops divided from the main loop. Lo 6 constructed triangular elements in the whole domain and then constructed rectangular elements by com- bining adjacent triangular elements. In this paper, a mesh regularization method is newly proposed in order to enhance the efficiency of finite ele- ment analyses of sheet metal forming. The mesh regular- ization method automatically finds out distorted elements with searching criteria proposed and composes patches to be modified. Each patch is then extended to three-dimensional surfaces in order to obtain the information of the continuous coordinates on the three-dimensional surface. The surface enclosing each patch is described as a three-dimensional free surface with the use of NURBS (non-uniform rational B-spline). On the basis of the constructed surface, each node is properly arranged to compose regular elements close to a square. The state variables calculated from its original mesh 0924-0136/$ see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0924-0136(03)00801-X J.H. Yoon, H. Huh / Journal of Materials Processing Technology 140 (2003) 616621 617 了 m  u u geometry are mapped into the new mesh geometry for the forming analysis at the next stage. Numerical results con- firm the efficiency of the proposed method and the accuracy of the result. It is also noted that the present method is effec- tive in the crash analyses of sheet metal members obtained from the forming simulation. 2. Regularization of the distorted element The regularization procedure to modify distorted ele- ments is introduced in order to enhance the efficiency of analysis for the next finite element calculation. The distorted elements are selected with appropriate searching criteria and allocated to several patches for regularization. The patches are extended to three-dimensional surfaces with the use of NURBS for full information of the continuous coor- dinates on the three-dimensional surface. On obtaining the new coordinates of each node, the distorted elements are regularized to a regular element that is close to a square. 2.1. The criterion of mesh distortion Distorted meshes are selected with the two geometrical criteria: one is the inner angle； and the other is the aspect ratio of the element side. 2.1.1. Inner angle The inner angle of a quadrilateral element should be close to the right angle for good results from finite element calcu- lation. Zhu et al. 7 defined the reasonable element when the four inner angles are formed with the angle of 90    45  while Lo and Lee 8 proposed the inner angle of 90   52.5  as the same criterion. The criterion of mesh distortion for the inner angle is determined by constituting Eq. (1). A mesh is regarded as distorted when Eq. (1) is less than /3 or (i)max in Eq. (3) 9 is greater than /6. The criterion is rather strict in order to avoid the geometrical limitation in case of applying the regularization method in confined regions: Fig. 1. Process for construction of a patch. 2.2. Domain construction 2.2.1. Construction of the patch Distorted elements selected by the criteria of mesh distor- tion are distributed in various regions according to the com- plexity of the shape of formed geometry. These elements are allocated to patches constructed for the efficiency of the algorithm. The shape of patches is made up for rectangu- lar shapes including all distorted elements for expanding the region of regularization and applying to NURBS surface ex- plained in next section. This procedure is shown in Fig. 1. When holes and edges are located between distorted ele- ments, the regions are filled up to make patches a rectangular shape. The patch is then mapped to a three-dimensional free sur- face by using NURBS surface. The procedure is important to obtain entire information of the continuous coordinates on the three-dimensional surface. NURBS surface can de- scribe the complex shape quickly by using less data points and does not change the entire domain data due to the local change. 2.2.2. NURBS surface NURBS surface is generally expressed by Eq. (5) as the p-order in the u-direction and the q-order in the v-direction 10: f斗 Q = 1e 1 + 2e 2 + 3e 3 + 4e 4 (1) 了 n m i=0了j=0Ni,p(u)Nj,q(v)wi,j Pi,j  4 S(u, v) =   n i=0 j=0Ni,p(u)Nj,q(v)wi,j (5) |f斗 Q| = (i)2 (2) i=1 i = 1  i (3) where Pi,j is the control points as the u-, v-direction, wi,j the weight factor and Nu,p(u), Nj,q(u) the basis function that are expressed by Eq. (6):   2   2.1.2. Aspect ratio of the element The ideal aspect ratio of the element side should be unity when the four sides of an element have the same length. The Ni,0 = r 1 if ui  u  ui+1, 0  otherwise, u  ui ui  p 1  u aspect ratio is defined as Eq. (4) and then the distortion is defined when it is less than 5 that could be much less for a strict criterion: Ni,p(u) = u i+p  Ni,p(u) + i + + i+p+1  ui+1 Ni+1,p 1(u) (6) maxr12, r23, r34, r41 minr12, r23, r34, r41 where rij  is the length of each element side. (4) In order to map the nodes from the patches onto the con- structed surface, a number of points are created for their coordinates on the NURBS surface. The location of each 了   了 N 了 m 了 m = procedure: 了 N AiCi PN = i=1  i=1Ai (8) where PN is the coordinate of a new node, Ai the areas of adjacent elements and Ci the centroid of the adjacent elements. 2.4. Level of distortion Fig. 2. Selecting direction of distorted elements. As a distortion factor, level of distortion (LD) is newly proposed. LD can be used to evaluate the degree of improve- ment in the element quality: LD = A  B (9) where 了 4 sin i| moving node by applying a regularization method is deter- mined such that the location of a point has the minimum A i=1| 4 , B = tanh （ k  B ）  (10) distance between nodes on NURBS surface. The informa- tion on the coordinates of the nodal points to be moved is B  = minr12 ,r23 ,r34 ,r41 maxr12, r23, r34, r41 , k = tanh 1() (11) stored to construct a new mesh system. 2.3. Regularization procedure The regularization method is carried out with the unit of a patch that forms a rectangular shape. Finite elements to be regularized is selected by the order of Fig. 2. Each selected element is divided by two triangular elements and then the divided element is made of a right triangular element by relocating the vertex on the circle having the diameter from x斗 1 to x斗 2 as shown in Eq. (7) and Fig. 3. When the procedure terminates, the same procedure is repeated in the opposite direction: LD has the value between 0 and 1； when LD = 1, the ele- ment is an ideal element of a square and when LD = 0, the quadrilateral element becomes a triangular element. i are the four inner angles of an element, so A is the factor for the inner angle. B is the factor for the aspect ratio of element sides and is defined by the hyperbolic tangent function in order to make LD less sensitive to the change of B. For ex- ample, when the reasonable aspect ratio of the element side is 1:4, the value of B can be adjusted by applying = 0.25 and = 0.6 such that the slope of the function B is changed abruptly around the value of B  = 0.25. Consequently, the value of LD decreases rapidly when the aspect ratio B  is less than 0.25 while the value of LD increases slowly when x斗 1 + x斗 2 2 = x斗 cen, xdir |x斗 1  x斗 2| 2 = r, x斗 cur  x斗 cen = x斗dir , the B  is greater than 0.25. This scheme can regulate the in- ner angle and the aspect ratio to have the equal effect on the LD. x斗 new = 斗  |x斗 dir | r  factor + x斗 cen (7) 2.5. Mapping of the state variables The final location of a node relocated by using the reg- ularization method is substituted for the location of  a point on NURBS surface. After the regularization proce- dure is finished, a simple soothing procedure is carried out by Eq. (8) for the rough region generated during the When the regularized mesh system is used for the next calculation of the forming analysis or the structural analy- sis, mapping of the state variables is needed for more accu- rate analysis considering the previous forming history. The mapping procedure is to map the calculated state variables in the original mesh system onto the regularized mesh sys- tem. As shown in Fig. 4, a sphere is constructed surround- ing a new node such that the state variables of nodes in the sphere have an effect on the state variables of the new node. The state variables of the new node are determined from the state variables of the neighboring nodes in the sphere by imposing the weighting factor inversely proportional to the distance between the two nodes as shown in Eq. (12): j=1Vj/rj Fig. 3. Regularization scheme by moving nodes. Vc = i=11/ri (12) J.H. Yoon, H. Huh / Journal of Materials Processing Technology 140 (2003) 616621 619  Fig. 4. Control sphere for mapping of the state variables. where Vj is the state variable calculated on the original mesh system, and rj the distance between the new node and the neighboring nodes. 3. Numerical examples 3.1. Forming analysis of an oil pan While oil pans are usually fabricated with a two-stage process in the press shop, the present analysis is carried out with a single-stage process as shown in Fig. 5 that describes the punch and die set. The regularization method can be applied to the finite ele- ment mesh system whenever needed for enhancement of the computation efficiency. In this example for demonstration, the method is applied to the analysis of oil pan forming at two forming intervals for regularization of distorted meshes as directed in Fig. 6. Fig. 7 explains the procedure of the regularization method. Fig. 7(a) shows the deformed shape at the punch stroke of Fig. 5. Punch and die set for oil pan forming. 60% forming Regularization     6080% forming Regularization     Fig. 6. Applying the regularization method to the forming analysis. Fig. 7. Procedure of regularization: (a) searching distorted elements； (b) constructing patches for distorted elements； (c) regularization of distorted elements. 60% and three parts of mesh distortion by the forming pro- cedure. It indicates that the number of patches to be con- structed is 3. Distorted meshes are selected according to the two geometrical criteria for mesh distortion. And then the patches of a rectangular shape are formed to include all dis- torted elements as shown in Fig. 7(b). Finally, the elements in the patches are regularized as shown in Fig. 7(c). In order to evaluate the degree of improvement in the el- ement quality after applying the regularization method, the value of LD for the regularized mesh system is compared the one for the original mesh system. The LD values for the reg- ularized mesh system have uniform distribution throughout the elements while those for the original mesh system have wide variation as shown in Fig. 8. It means that the quality of the regularized mesh system is enhanced with the same level distortion. Consequently, explicit finite element computation with the regularized mesh system can be preceded with a larger incremental time step as shown in Fig. 9. In this anal- ysis of oil pan forming, the computing time with the regular- ized mesh system is reduced about 12% even after two times of regularization. The amount of reduction in the computing time can be increased with more frequent regularization. 3.2. Crash analysis of a front side member The crash analysis is usually carried out without consid- ering the forming effect and adopts the mesh system apart form the forming analysis. In case that the forming effect is considered to improve the accuracy and reliability of the analysis results, the mesh system for the forming analysis could be directly used in the crash analysis for the efficiency of the analysis. The mesh system after the forming analysis, 100% forming   0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 20   40    60    80   100 120 140 160 180 element number 0.0  20    40    60    80   100  120  140  160  180 element number (a) (b) Fig. 8. Comparison of LD: (a) original mesh； (b) regularized mesh. however, has too many distorted meshes due to severe de- formation and distortion be used directly in the crash anal- ysis without remeshing. One remedy is to construct a new mesh system and the other is to modify the mesh system af- ter the forming analysis. The latter can be very efficient if the remeshing process is successfully carried out. For an ef- ficient remeshing process, the regularization method can be applied to transform distorted meshes into a near square. In this example, a part of the front side member, named front reinforcement, in Fig. 10 is selected for the crash analysis. The irregular finite elements in the local distorted region of the member after forming analysis are modified to a regular element using the regularization method as shown in Fig. 11 and then the regularized mesh system is used in the crash analysis. The analysis condition is depicted in Fig. 12. The crash analysis with the regularized mesh system could be carried out with a lager time step as shown in Fig. 13 without sacrificing the accuracy of the analysis result. The computing time for the crash analysis was reduced by 40% of the time with the original mesh system. The analysis re- sult was encouraging in both the computing time and the accuracy of the computation result and proved that the reg- ularized mesh system could be used effectively to enhance the efficiency of the numerical analysis. Fig. 9. Comparison of the time step size with respect to the punch stroke. Fig. 10. Assembly of a front side member. Fig. 11. Regularization of a front reinforcement. Fig. 12. Condition of the impact analysis. Level of Distortion Level of Distortion 620 J.H. Yoon, H. Huh / Journal of Materials Processing Technology 140 (2003) 616621 Fig.  13.  Comparison  of  the  time  step  size  between  the  original  and regularized mesh system. 4. Conclusion A mesh regularization method is newly proposed in order to enhance the efficiency of finite element analyses of sheet metal forming. Meshes in the sheet metal forming analysis are distorted so severely that the subsequent analysis would be difficult or produce poor results. This can be avoided by the present method with the minimum effort of remeshing. Mesh regularization can be carried out during the incremen- tal analysis or for the next stage in multi-stage forming. It is also proved that the analysis efficiency is greatly improved when mesh regularization is carried out for the crash analysis of formed members obtained from the sheet metal forming simulation. Numerical results confirm the validity and effi- ciency of the proposed method as well as the accuracy of the result. References 1 A.R. Diaz, N. Kikuchi, J.E. Taylor, A method of grid optimization for finite element methods, Comput. Meth. Appl. Mech. Eng. 41 (1983) 2945. 2 B.A. Szavo, Mesh design for the p-version of the finite element method, Comput. Meth. Appl. Mech. Eng. 55 (1986) 181197. 3 P. Diez, A. Huerta, A unified approach to remeshing strategies for finite element h-adaptivity, Comput. Meth. Appl. Mech. Eng. 176 (1999) 215229. 4 M.L.C. Sluiter, D.C. Hansen, A general purpose two-dimensional mesh generator for shell and solid finite elements, in: Computer in Engineering, vol. 3, ASME, 1982, pp. 2934. 5 J.A. Talbert, A.R. Parkinson, Development of an automatic, two- dimensional finite element mesh generator using quadrilateral ele- ments and Bezier curve boundary definition, Int. J. Numer. Meth. Eng. 29 (1990) 15511567. 6 S.H. Lo, Generating quadrilateral elements on plane and over curved surfaces, Comput. Struct. 31 (1989) 421426. 7 J.Z. Zhu, O.C. Zienkiewicz, E. Hinton, J. Wu, A new approach to the development of automatic quadrilateral mesh generation, Int. J. Numer. Meth. Eng. 32 (1991) 849866. 8 S.H. Lo, C.K. Lee, On using meshes of mixed element types in adaptive finite element analysis, Finite Elem. Anal. Des. 11 (1992) 307336. 9 A. El-Hamalawi, A simple and effective element distortion factor, Comput. Struct. 75 (2000) 507513. 10 L. Piegl, W. Tiller, The NURBS Book, 2nd ed., Springer, New York, 1997. 提高板材成形效率的坐标网分析法  J.H. Yoon, H. Huh. 机械工程学院，韩国高级科学协会和技术科学镇  Daejeon 305-701,南韩   摘要   本篇文章是采用一种新推出的方法来对提高板材的成形效率 进行分析 ,这种方法就是坐标网分析法。这种方法就是研究扭曲单体，即通过适当的研究规范，建立补片，包括修正后的单体。每一片都被扩展到一个三维的表面从而获得一个连续坐标的信息。在构造表面时，应包括每一个片， NURBS(非均匀有理 B样条 )表面被用来描述一个三维自由表面。以被构造表面为基础，每一个节点一般被安排成一个非常接近正方形的单体元素。计算状态函数是从它原始的网格系统映射到新的网格之内，从而对成形进行下一阶段的分析或更进一步的分析。按网格方法的分析结果与没有坐标网方法直接成行的分析结果相比较来确定哪一种方法是更 有效的。   2003 Elsevier B.V. 版权所有 . 关键词：坐标网；变形单体； NURBS；有限元分析   1. 概述   随着计算机技术和数字技术的结合和快速发展，用数字模拟进行板材成形加工达到空前的繁荣。数字分析对复杂几何图形的板材成形和多级成形都可以做到。对于一个复杂的几何模型来说，尽管局部严重变形将会导致计算时间的增加和数据分析的减少。从而使分析结果更加不准确。几何网格的扭曲和严重620 J.H. Yoon, H. Huh / Journal of Materials Processing Technology 140 (2003) 616621 变形对板材成形的质量有很大影响 ,特别是对于多级成形。当上一级成形的分析结果用于下一级成形分析时，几何网格的扭曲和 变形对分析结果影响更大。这种被扭曲网格的错误表象可以通过整体的或自适应重啮合技术的网格系统的重建来避免。在模拟期间，减少单体扭曲，自适应重啮合技术被认为是一种有效的方法。但是，它仍然需要大量的计算，并且在单体的细分中也受到限制。  要构造一个网格系统的有效方法已经被许多研究人员提上日程。典型的方法可能是下面几种： r-方法， h-方法， p-方法。 r-方法就是在网格系统的总的自由度不变的情况下，节点被完全重排； h-方法就是在元素单体具有相同的自由度的情况下让网格的数目增加； p-方法就是通过网格系统的整体自由度的增加来提高分析的准确性。 Sluiter 和 Hansen文献 4和 Talbert 和  Parkinson文献 5构造了一个晶格分析范围，它像一个连续的环，而且是从主要环中分离出的子环元素。 Lo文献 6在整个晶格范围内构造了一个三角形元，并且通过合并邻近的三角形元而构造矩形元素。  本篇文章中的坐标网方法是一种新推出的方法 ,它旨在用有限元分析提高板材成形效率。坐标网法根据一些规范可以自动地找出变形单体，并对这些片进行修正。然后，每一片都被扩展到一个三维表面用来获得在三维表面的连续坐标系的信息。这个包含了每一片的 表面用来作为使用了 NURBS的三维自由表面来描述。以被构造表面为基础，每一个节点都被彻底改变，用来组成一个正方形的规则单体。状态函数的计算是从它原始几何网格映射到新的网格之内，从而进行下一阶段的成形分析。从得到的数据结果中证实使用坐标网方法的效率和结果的准确性。这也证实了此种方法在板材构件碰撞分析的成形模拟中的有效性。   2. 体的规则化   之所以要介绍对变形体的修正使之成为一个规则化过程，是为了提高变形体在下一个有限元计算中的分析效率。在规则化过程中，变形体根据适当的搜索规范有选择的分配到各片。这些片通过 分析 NURBS在连续坐标系的三维表面上的全部数据而扩展到一个三维表面。变形后的每个节点为了得到一个新坐标将被调整为一个近似正方形的规则单体。   2.1 网格变形标准   变形有两种几何标准可供选择：一是内角；另一个是单体纵横比。   2.1.1 内角  从有限元计算中得到矩形元素的内角应是接近直角的。 Zhu et al. 文献 7给了这种元素一个合理的定义，就是当四个内角都是在            的范围内时。同时 Lo和 Lee文献 8也提出了相同情况下的内角，角度在            范围内。内角的网孔变形是由 式（ 1）的构成所决定的。当式  (1).小于 /3 或  ( i)max在式 (3) 9中大于 /6 网孔被认为是变形的。这个标准之所以相当严格是为了避免万一在限制区域应用规则化方法受到几何图形的限制：     2.1.2 单体纵横比  四条边具有相同长度的理想单体的纵横比应该是一致的。纵横比被定义如式（ 4），并且当变形小于 5即比严格标准少很多时，它也被定义：   此处 rij表示单体边长。   2.2.作图范围   2.2.1 片的设计  通过网格变形标准所选择的变形单体，根据它们在几何成形时外形的复杂620 J.H. Yoon, H. Huh / Journal of Materials Processing Technology 140 (2003) 616621 程度被分不到 各个不同的区域。这些单体被分配到各片，并用来构造算法效率。这些片的形状被拼凑成矩形，包括所有变形体，目的是扩大规则化和NURBS表面在下一部分说明中的应用。这个过程如图 1所示，当孔和边缘被设置在变形体中时，这些区域被填满，从而得到矩形片。  然后，这些片利用 NURBS表面映射到一个三维自由表面。这个过程对于在三维表面上获得连续坐标的全部信息是非常重要的。 NURBS表面在使用较少的数据点和由于局部改变而不改变这个区域的数据的情况下快速的描述这个复杂的形状。     2.2.2 NURBS表面  NURBS表面通常通过如式 (5)来表述，像 p-向量在 u-方向中和 q-向量在 v-方向中 10: 此处 Pi,j是控制点如 u-，  q- 方向。 Wi,j是加权因子， 是基础函数通过式 (6)来表达：   为了把这些点映射到构造的表面上，一系列连续的点在 NURBS表面创建了。每一个用规则化方法移动过的节点都被定位，以至于在 NURBS表面上定位点在两节点之间有最小距离。这些移动过的连续节点的信息都被存储，用来构造一个新的网格系统。   2.3 规则化过程   规 则化方法与形成矩形片单体一起完成的。规则化的有限元通过图 2所示次序被依次选择。每一个被选择的单体都被分成两个三角形元，并且这些三角形元通过圆心的重定位都由直角三角形元组成，圆的直径如式 (7)和图 3所示，从X1到 X2。当这个过程结束的时候，相同的过程在另一方向被重复：   通过规则化方法对节点的重定位，其最终位置被在 NURBS表面上的点的位置所代替。当规则化过程完成后，为产生粗糙的区域，一个简单的缓和的过程通过式（ 8）被执行：        此处 PN是新节点的坐标， Ai 临近区域的元素的坐标， Ci 临近元素的 质心。         2.4 变形程度   620 J.H. Yoon, H. Huh / Journal of Materials Processing Technology 140 (2003) 616621 作为一个变形因子，变形程度 (LD)是最新提出的  ， LD可能是用来评估单体在质量方面改进的程度：   此处   LD在 0和 1之间浮动；当 LD=1时，单体是一个方形的理想单体，当 LD=0时，四边形元变成了三角形元。 时单体的四个内角，因此 A是内角因子， B是单体侧面长宽比的因子并且为了使 LD对 B的变化不那么敏感， B被定义为双曲线正切函数。例如，当单体侧面合理的长宽比是 1： 4时， B的值可以通过 和来调整，使函数 B的斜率围绕着 B=0.25急剧变化。结果，当 的长宽比小于 0.25时， LD的值急剧增加，当 大于 0.25时， LD的值增加缓慢。这种方法可以调节内角和长宽比使它们在 LD上有相同的效果。   2.5 状态函数的映射   当坐标网系统用于下一步的成形分析或结构分析的计算时，状态函数的映射就是非常必要的，通过映射，可以在考虑上一步成型过程的前提下得到更准确的分析。映射过程就是通过状态函数的计算把原来的网格系统映射到新的坐标网系统。如图 4所示，一个球面在一个新节点周围建立，将导致球面上节点的状态函数影响新 节点的状态函数。新节点的状态函数是由球面上原来节点的状态函数所决定的，如 式 (12)所示，加权因子在两节点的距离上成反比。   此处 Vj是原始网格系统的状态函数的计算结果， rj使新节点到附近节点的距离。   3.数例   3.1.1 油盘的成形分析   油盘在冲压车间一般要经过两个工序制作，而根据现在这种方法，单工序冲压就可以完成。如图 5所示的凸模和模架。   不论什么时候有限元系统需要提高计算效率，规则化方法都可应用于其中。在这个范例中，这 种方法应用于油盘成形分析中的两次成形间隙，如图 6所示。  图 7说明了规则化方法的过程。图 7（ a）所示为成形时凸模行程为 60%时的变形，有 3个地方发生了网格变形，也就是片的数量是 3。变形网格是根据 2个网格变形的几何规范来选取的。如图 7所示的包括所有变形体的矩形片的形成。最终补片中的单体被规则化，如图 7（ c）所示。  为了评价应用规则化系统后的单体质量的改进程度，应用规则化网格系统的 LD值与