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標(biāo)網(wǎng)分析法  J.H. Yoon, H. Huh. 機(jī)械工程學(xué)院，韓國(guó)高級(jí)科學(xué)協(xié)會(huì)和技術(shù)科學(xué)鎮(zhèn)  Daejeon 305-701,南韓   摘要   本篇文章是采用一種新推出的方法來對(duì)提高板材的成形效率 進(jìn)行分析 ,這種方法就是坐標(biāo)網(wǎng)分析法。這種方法就是研究扭曲單體，即通過適當(dāng)?shù)难芯恳?guī)范，建立補(bǔ)片，包括修正后的單體。每一片都被擴(kuò)展到一個(gè)三維的表面從而獲得一個(gè)連續(xù)坐標(biāo)的信息。在構(gòu)造表面時(shí)，應(yīng)包括每一個(gè)片， NURBS(非均勻有理 B樣條 )表面被用來描述一個(gè)三維自由表面。以被構(gòu)造表面為基礎(chǔ)，每一個(gè)節(jié)點(diǎn)一般被安排成一個(gè)非常接近正方形的單體元素。計(jì)算狀態(tài)函數(shù)是從它原始的網(wǎng)格系統(tǒng)映射到新的網(wǎng)格之內(nèi)，從而對(duì)成形進(jìn)行下一階段的分析或更進(jìn)一步的分析。按網(wǎng)格方法的分析結(jié)果與沒有坐標(biāo)網(wǎng)方法直接成行的分析結(jié)果相比較來確定哪一種方法是更 有效的。   2003 Elsevier B.V. 版權(quán)所有 . 關(guān)鍵詞：坐標(biāo)網(wǎng)；變形單體； NURBS；有限元分析   1. 概述   隨著計(jì)算機(jī)技術(shù)和數(shù)字技術(shù)的結(jié)合和快速發(fā)展，用數(shù)字模擬進(jìn)行板材成形加工達(dá)到空前的繁榮。數(shù)字分析對(duì)復(fù)雜幾何圖形的板材成形和多級(jí)成形都可以做到。對(duì)于一個(gè)復(fù)雜的幾何模型來說，盡管局部嚴(yán)重變形將會(huì)導(dǎo)致計(jì)算時(shí)間的增加和數(shù)據(jù)分析的減少。從而使分析結(jié)果更加不準(zhǔn)確。幾何網(wǎng)格的扭曲和嚴(yán)重620 J.H. Yoon, H. Huh / Journal of Materials Processing Technology 140 (2003) 616621 變形對(duì)板材成形的質(zhì)量有很大影響 ,特別是對(duì)于多級(jí)成形。當(dāng)上一級(jí)成形的分析結(jié)果用于下一級(jí)成形分析時(shí)，幾何網(wǎng)格的扭曲和 變形對(duì)分析結(jié)果影響更大。這種被扭曲網(wǎng)格的錯(cuò)誤表象可以通過整體的或自適應(yīng)重嚙合技術(shù)的網(wǎng)格系統(tǒng)的重建來避免。在模擬期間，減少單體扭曲，自適應(yīng)重嚙合技術(shù)被認(rèn)為是一種有效的方法。但是，它仍然需要大量的計(jì)算，并且在單體的細(xì)分中也受到限制。  要構(gòu)造一個(gè)網(wǎng)格系統(tǒng)的有效方法已經(jīng)被許多研究人員提上日程。典型的方法可能是下面幾種： r-方法， h-方法， p-方法。 r-方法就是在網(wǎng)格系統(tǒng)的總的自由度不變的情況下，節(jié)點(diǎn)被完全重排； h-方法就是在元素單體具有相同的自由度的情況下讓網(wǎng)格的數(shù)目增加； p-方法就是通過網(wǎng)格系統(tǒng)的整體自由度的增加來提高分析的準(zhǔn)確性。 Sluiter 和 Hansen文獻(xiàn) 4和 Talbert 和  Parkinson文獻(xiàn) 5構(gòu)造了一個(gè)晶格分析范圍，它像一個(gè)連續(xù)的環(huán)，而且是從主要環(huán)中分離出的子環(huán)元素。 Lo文獻(xiàn) 6在整個(gè)晶格范圍內(nèi)構(gòu)造了一個(gè)三角形元，并且通過合并鄰近的三角形元而構(gòu)造矩形元素。  本篇文章中的坐標(biāo)網(wǎng)方法是一種新推出的方法 ,它旨在用有限元分析提高板材成形效率。坐標(biāo)網(wǎng)法根據(jù)一些規(guī)范可以自動(dòng)地找出變形單體，并對(duì)這些片進(jìn)行修正。然后，每一片都被擴(kuò)展到一個(gè)三維表面用來獲得在三維表面的連續(xù)坐標(biāo)系的信息。這個(gè)包含了每一片的 表面用來作為使用了 NURBS的三維自由表面來描述。以被構(gòu)造表面為基礎(chǔ)，每一個(gè)節(jié)點(diǎn)都被徹底改變，用來組成一個(gè)正方形的規(guī)則單體。狀態(tài)函數(shù)的計(jì)算是從它原始幾何網(wǎng)格映射到新的網(wǎng)格之內(nèi)，從而進(jìn)行下一階段的成形分析。從得到的數(shù)據(jù)結(jié)果中證實(shí)使用坐標(biāo)網(wǎng)方法的效率和結(jié)果的準(zhǔn)確性。這也證實(shí)了此種方法在板材構(gòu)件碰撞分析的成形模擬中的有效性。   2. 體的規(guī)則化   之所以要介紹對(duì)變形體的修正使之成為一個(gè)規(guī)則化過程，是為了提高變形體在下一個(gè)有限元計(jì)算中的分析效率。在規(guī)則化過程中，變形體根據(jù)適當(dāng)?shù)乃阉饕?guī)范有選擇的分配到各片。這些片通過 分析 NURBS在連續(xù)坐標(biāo)系的三維表面上的全部數(shù)據(jù)而擴(kuò)展到一個(gè)三維表面。變形后的每個(gè)節(jié)點(diǎn)為了得到一個(gè)新坐標(biāo)將被調(diào)整為一個(gè)近似正方形的規(guī)則單體。   2.1 網(wǎng)格變形標(biāo)準(zhǔn)   變形有兩種幾何標(biāo)準(zhǔn)可供選擇：一是內(nèi)角；另一個(gè)是單體縱橫比。   2.1.1 內(nèi)角  從有限元計(jì)算中得到矩形元素的內(nèi)角應(yīng)是接近直角的。 Zhu et al. 文獻(xiàn) 7給了這種元素一個(gè)合理的定義，就是當(dāng)四個(gè)內(nèi)角都是在            的范圍內(nèi)時(shí)。同時(shí) Lo和 Lee文獻(xiàn) 8也提出了相同情況下的內(nèi)角，角度在            范圍內(nèi)。內(nèi)角的網(wǎng)孔變形是由 式（ 1）的構(gòu)成所決定的。當(dāng)式  (1).小于 /3 或  ( i)max在式 (3) 9中大于 /6 網(wǎng)孔被認(rèn)為是變形的。這個(gè)標(biāo)準(zhǔn)之所以相當(dāng)嚴(yán)格是為了避免萬一在限制區(qū)域應(yīng)用規(guī)則化方法受到幾何圖形的限制：     2.1.2 單體縱橫比  四條邊具有相同長(zhǎng)度的理想單體的縱橫比應(yīng)該是一致的?？v橫比被定義如式（ 4），并且當(dāng)變形小于 5即比嚴(yán)格標(biāo)準(zhǔn)少很多時(shí)，它也被定義：   此處 rij表示單體邊長(zhǎng)。   2.2.作圖范圍   2.2.1 片的設(shè)計(jì)  通過網(wǎng)格變形標(biāo)準(zhǔn)所選擇的變形單體，根據(jù)它們?cè)趲缀纬尚螘r(shí)外形的復(fù)雜620 J.H. Yoon, H. Huh / Journal of Materials Processing Technology 140 (2003) 616621 程度被分不到 各個(gè)不同的區(qū)域。這些單體被分配到各片，并用來構(gòu)造算法效率。這些片的形狀被拼湊成矩形，包括所有變形體，目的是擴(kuò)大規(guī)則化和NURBS表面在下一部分說明中的應(yīng)用。這個(gè)過程如圖 1所示，當(dāng)孔和邊緣被設(shè)置在變形體中時(shí)，這些區(qū)域被填滿，從而得到矩形片。  然后，這些片利用 NURBS表面映射到一個(gè)三維自由表面。這個(gè)過程對(duì)于在三維表面上獲得連續(xù)坐標(biāo)的全部信息是非常重要的。 NURBS表面在使用較少的數(shù)據(jù)點(diǎn)和由于局部改變而不改變這個(gè)區(qū)域的數(shù)據(jù)的情況下快速的描述這個(gè)復(fù)雜的形狀。     2.2.2 NURBS表面  NURBS表面通常通過如式 (5)來表述，像 p-向量在 u-方向中和 q-向量在 v-方向中 10: 此處 Pi,j是控制點(diǎn)如 u-，  q- 方向。 Wi,j是加權(quán)因子， 是基礎(chǔ)函數(shù)通過式 (6)來表達(dá)：   為了把這些點(diǎn)映射到構(gòu)造的表面上，一系列連續(xù)的點(diǎn)在 NURBS表面創(chuàng)建了。每一個(gè)用規(guī)則化方法移動(dòng)過的節(jié)點(diǎn)都被定位，以至于在 NURBS表面上定位點(diǎn)在兩節(jié)點(diǎn)之間有最小距離。這些移動(dòng)過的連續(xù)節(jié)點(diǎn)的信息都被存儲(chǔ)，用來構(gòu)造一個(gè)新的網(wǎng)格系統(tǒng)。   2.3 規(guī)則化過程   規(guī) 則化方法與形成矩形片單體一起完成的。規(guī)則化的有限元通過圖 2所示次序被依次選擇。每一個(gè)被選擇的單體都被分成兩個(gè)三角形元，并且這些三角形元通過圓心的重定位都由直角三角形元組成，圓的直徑如式 (7)和圖 3所示，從X1到 X2。當(dāng)這個(gè)過程結(jié)束的時(shí)候，相同的過程在另一方向被重復(fù)：   通過規(guī)則化方法對(duì)節(jié)點(diǎn)的重定位，其最終位置被在 NURBS表面上的點(diǎn)的位置所代替。當(dāng)規(guī)則化過程完成后，為產(chǎn)生粗糙的區(qū)域，一個(gè)簡(jiǎn)單的緩和的過程通過式（ 8）被執(zhí)行：        此處 PN是新節(jié)點(diǎn)的坐標(biāo)， Ai 臨近區(qū)域的元素的坐標(biāo)， Ci 臨近元素的 質(zhì)心。         2.4 變形程度   620 J.H. Yoon, H. Huh / Journal of Materials Processing Technology 140 (2003) 616621 作為一個(gè)變形因子，變形程度 (LD)是最新提出的  ， LD可能是用來評(píng)估單體在質(zhì)量方面改進(jìn)的程度：   此處   LD在 0和 1之間浮動(dòng)；當(dāng) LD=1時(shí)，單體是一個(gè)方形的理想單體，當(dāng) LD=0時(shí)，四邊形元變成了三角形元。 時(shí)單體的四個(gè)內(nèi)角，因此 A是內(nèi)角因子， B是單體側(cè)面長(zhǎng)寬比的因子并且為了使 LD對(duì) B的變化不那么敏感， B被定義為雙曲線正切函數(shù)。例如，當(dāng)單體側(cè)面合理的長(zhǎng)寬比是 1： 4時(shí)， B的值可以通過 和來調(diào)整，使函數(shù) B的斜率圍繞著 B=0.25急劇變化。結(jié)果，當(dāng) 的長(zhǎng)寬比小于 0.25時(shí)， LD的值急劇增加，當(dāng) 大于 0.25時(shí)， LD的值增加緩慢。這種方法可以調(diào)節(jié)內(nèi)角和長(zhǎng)寬比使它們?cè)?LD上有相同的效果。   2.5 狀態(tài)函數(shù)的映射   當(dāng)坐標(biāo)網(wǎng)系統(tǒng)用于下一步的成形分析或結(jié)構(gòu)分析的計(jì)算時(shí)，狀態(tài)函數(shù)的映射就是非常必要的，通過映射，可以在考慮上一步成型過程的前提下得到更準(zhǔn)確的分析。映射過程就是通過狀態(tài)函數(shù)的計(jì)算把原來的網(wǎng)格系統(tǒng)映射到新的坐標(biāo)網(wǎng)系統(tǒng)。如圖 4所示，一個(gè)球面在一個(gè)新節(jié)點(diǎn)周圍建立，將導(dǎo)致球面上節(jié)點(diǎn)的狀態(tài)函數(shù)影響新 節(jié)點(diǎn)的狀態(tài)函數(shù)。新節(jié)點(diǎn)的狀態(tài)函數(shù)是由球面上原來節(jié)點(diǎn)的狀態(tài)函數(shù)所決定的，如 式 (12)所示，加權(quán)因子在兩節(jié)點(diǎn)的距離上成反比。   此處 Vj是原始網(wǎng)格系統(tǒng)的狀態(tài)函數(shù)的計(jì)算結(jié)果， rj使新節(jié)點(diǎn)到附近節(jié)點(diǎn)的距離。   3.數(shù)例   3.1.1 油盤的成形分析   油盤在沖壓車間一般要經(jīng)過兩個(gè)工序制作，而根據(jù)現(xiàn)在這種方法，單工序沖壓就可以完成。如圖 5所示的凸模和模架。   不論什么時(shí)候有限元系統(tǒng)需要提高計(jì)算效率，規(guī)則化方法都可應(yīng)用于其中。在這個(gè)范例中，這 種方法應(yīng)用于油盤成形分析中的兩次成形間隙，如圖 6所示。  圖 7說明了規(guī)則化方法的過程。圖 7（ a）所示為成形時(shí)凸模行程為 60%時(shí)的變形，有 3個(gè)地方發(fā)生了網(wǎng)格變形，也就是片的數(shù)量是 3。變形網(wǎng)格是根據(jù) 2個(gè)網(wǎng)格變形的幾何規(guī)范來選取的。如圖 7所示的包括所有變形體的矩形片的形成。最終補(bǔ)片中的單體被規(guī)則化，如圖 7（ c）所示。  為了評(píng)價(jià)應(yīng)用規(guī)則化系統(tǒng)后的單體質(zhì)量的改進(jìn)程度，應(yīng)用規(guī)則化網(wǎng)格系統(tǒng)的 LD值與