[雙語翻譯]外文翻譯--提高板材成形效率的坐標(biāo)網(wǎng)分析法（原文）

1、Journal of Materials Processing Technology 140 (2003) 616–621Efficiency enhancement in sheet metal forming analysis with a mesh regularization methodJ.H. Yoon, H. Huh?Department of Mechanical Engineering, Korea Advanced

2、Institute of Science and Technology Science Town, Daejeon 305-701, South KoreaAbstractThis paper newly proposes a mesh regularization method for the enhancement of the efficiency in sheet metal forming analysis. The regu

3、larization 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 o

4、btain 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

5、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

6、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

7、the method. © 2003 Elsevier B.V. All rights reserved.Keywords: Mesh regularization; Distorted element; NURBS; Patch; Finite element analysis1. IntroductionNumerical simulation of sheet metal forming processes enjoys

8、 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-s

9、tage 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 de

10、formation 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 u

11、sed 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 remes

12、hing 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. T

13、el.: +82-42-869-3222; fax: +82-42-869-3210. E-mail address: hhuh@kaist.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 no

14、dal 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

15、 [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

16、 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 pap

17、er, 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

18、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

19、. 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 compos

20、e regular elements close to a square. The state variables calculated from its original mesh0924-0136/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0924-0136(03)00801-X618 J.H. Yoon, H.

21、Huh / Journal of Materials Processing Technology 140 (2003) 616–621Fig. 2. Selecting direction of distorted elements.moving node by applying a regularization method is deter- mined such that the location of a point has t

22、he minimum distance between nodes on NURBS surface. The informa- tion on the coordinates of the nodal points to be moved is stored to construct a new mesh system.2.3. Regularization procedureThe regularization method is

23、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

24、element is made of a right triangular element by relocating the vertex on the circle having the diameter from ? x1 to ? x2 as shown in Eq. (7) and Fig. 3. When the procedure terminates, the same procedure is repeated in

25、the opposite direction:? x1 + ? x22 = ? xcen, |? x1 ? ? x2|2 = r, ? xcur ? ? xcen = ? xdir,? xnew = ? xdir |? xdir|r × factor + ? xcen (7)The final location of a node relocated by using the reg- ularization method i

26、s 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 theFig. 3. Regular

27、ization scheme by moving nodes.procedure:PN =?N i=1AiCi ?N 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 distortionAs a di

28、stortion 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)whereA =?4 i=1|sin θi|4 , B = tanh ? k × B?? (10)B? = min

29、{r12, r23, r34, r41}max{r12, r23, r34, r41}, k = tanh?1(β)α (11)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

30、 element. θi 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 le

31、ss 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 α = 0.25 and β = 0.6 such that the slope of the function B is change

32、d 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 the B? is greater than 0.25. This scheme ca

33、n regulate the in- ner angle and the aspect ratio to have the equal effect on the LD.2.5. Mapping of the state variablesWhen the regularized mesh system is used for the next calculation of the forming analysis or the str

34、uctural 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 syst

35、em 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

36、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.