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Journal of Materials Processing Technology 140 (2003) 616–621Efficiency enhancement in sheet metal forming analysiswith a mesh regularization methodJ.H. Yoon, H. Huh∗Department of Mechanical Engineering, Korea Advanced 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. Theregularization method searches for distorted elements with appropriate searching criteria and constructs patches including the elements tobe modified. Each patch is then extended to a three-dimensional surface in order to obtain the information of the continuous coordinates. Inconstructing the surface enclosing each patch, NURBS (non-uniform rational B-spline) surface is employed to describe a three-dimensionalfree surface. On the basis of the constructed surface, each node is properly arranged to form unit elements as close as to a square. Thestate variables calculated from its original mesh geometry are mapped into the new mesh geometry for the next stage or incremental stepof a forming analysis. The analysis results with the proposed method are compared to the results from the direct forming analysis withoutmesh 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 analysis1. IntroductionNumerical simulation of sheet metal forming processesenjoys its prosperity with a burst of development of the com-puters and the related numerical techniques. The numericalanalysis has extended its capabilities for sheet metal formingof complicated geometry models and multi-stage forming. Inthe case of a complicated geometry model, however, severelocal 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 hasan effect on the quality of forming analysis results especiallyin the case of multi-stage forming analysis when the meshgeometry formed by the forming analysis at the first stageis used for the forming analysis at the next stage. This illbehavior of the distorted mesh can be avoided by the recon-struction of the mesh system such as the total or the adaptiveremeshing techniques. The adaptive remeshing techniqueis known to be an efficient method to reduce distortion ofelement during the simulation, but it still needs tremen-dous computing and puts restrictions among subdividedelements.∗Corresponding author. Tel.: +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 beenproposed by many researchers. Typical methods could ber-method [1] in which nodal points are properly rearrangedwithout the change of the total degrees of freedom of themesh system, h-method [2] in which the number of meshesis increased with elements of the same degrees of freedom,and p-method [3] in which the total degrees of freedom ofthe 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 loopand created elements in sub-loops divided from the mainloop. Lo [6] constructed triangular elements in the wholedomain and then constructed rectangular elements by com-bining adjacent triangular elements.In this paper, a mesh regularization method is newlyproposed in order to enhance the efficiency of finite ele-ment analyses of sheet metal forming. The mesh regular-ization method automatically finds out distorted elementswith searching criteria proposed and composes patches to bemodified. Each patch is then extended to three-dimensionalsurfaces in order to obtain the information of the continuouscoordinates on the three-dimensional surface. The surfaceenclosing each patch is described as a three-dimensionalfree surface with the use of NURBS (non-uniform rationalB-spline). On the basis of the constructed surface, each nodeis properly arranged to compose regular elements close to asquare. 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-XJ.H. Yoon, H. Huh / Journal of Materials Processing Technology 140 (2003) 616–621 617geometry are mapped into the new mesh geometry for theforming analysis at the next stage. Numerical results con-firm the efficiency of the proposed method and the accuracyof the result. It is also noted that the present method is effec-tive in the crash analyses of sheet metal members obtainedfrom the forming simulation.2. Regularization of the distorted elementThe regularization procedure to modify distorted ele-ments is introduced in order to enhance the efficiency ofanalysis for the next finite element calculation. The distortedelements are selected with appropriate searching criteriaand allocated to several patches for regularization. Thepatches are extended to three-dimensional surfaces with theuse of NURBS for full information of the continuous coor-dinates on the three-dimensional surface. On obtaining thenew coordinates of each node, the distorted elements areregularized to a regular element that is close to a square.2.1. The criterion of mesh distortionDistorted meshes are selected with the two geometricalcriteria: one is the inner angle; and the other is the aspectratio of the element side.2.1.1. Inner angleThe inner angle of a quadrilateral element should be closeto the right angle for good results from finite element calcu-lation. Zhu et al. [7] defined the reasonable element whenthe 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 theinner angle is determined by constituting Eq. (1). A mesh isregarded as distorted when Eq. (1) is less than π/3 or (δθi)maxin Eq. (3) [9] is greater than π/6. The criterion is ratherstrict in order to avoid the geometrical limitation in case ofapplying the regularization method in confined regions:vectorfQ= δθ1ˆe1+δθ2ˆe2+δθ3ˆe3+δθ4ˆe4(1)||vectorfQ|| =radicaltpradicalvertexradicalvertexradicalbt4summationdisplayi=1(δθi)2(2)δθi=vextendsinglevextendsinglevextendsingle12π −θivextendsinglevextendsinglevextendsingle (3)2.1.2. Aspect ratio of the elementThe ideal aspect ratio of the element side should be unitywhen the four sides of an element have the same length. Theaspect ratio is defined as Eq. (4) and then the distortion isdefined when it is less than 5 that could be much less for astrict criterion:max{r12,r23,r34,r41}min{r12,r23,r34,r41}(4)where rijis the length of each element side.Fig. 1. Process for construction of a patch.2.2. Domain construction2.2.1. Construction of the patchDistorted 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 elementsare allocated to patches constructed for the efficiency of thealgorithm. The shape of patches is made up for rectangu-lar shapes including all distorted elements for expanding theregion 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 rectangularshape.The patch is then mapped to a three-dimensional free sur-face by using NURBS surface. The procedure is importantto obtain entire information of the continuous coordinateson the three-dimensional surface. NURBS surface can de-scribe the complex shape quickly by using less data pointsand does not change the entire domain data due to the localchange.2.2.2. NURBS surfaceNURBS surface is generally expressed by Eq. (5) as thep-order in the u-direction and the q-order in the v-direction[10]:S(u,v) =summationtextni=0summationtextmj=0Ni,p(u)Nj,q(v)wi,jPi,jsummationtextni=0summationtextmj=0Ni,p(u)Nj,q(v)wi,j(5)where Pi,jis the control points as the u-, v-direction, wi,jthe weight factor and Nu,p(u), Nj,q(u) the basis function thatare expressed by Eq. (6):Ni,0=braceleftBigg1ifui≤ u ≤ ui+1,0 otherwise,Ni,p(u) =u−uiui+p−uiNi,p(u)+ui+p+1−uui+p+1−ui+1Ni+1,p−1(u)(6)In order to map the nodes from the patches onto the con-structed surface, a number of points are created for theircoordinates on the NURBS surface. The location of each618 J.H. Yoon, H. 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 the minimumdistance between nodes on NURBS surface. The informa-tion on the coordinates of the nodal points to be moved isstored to construct a new mesh system.2.3. Regularization procedureThe regularization method is carried out with the unit ofa patch that forms a rectangular shape. Finite elements to beregularized is selected by the order of Fig. 2. Each selectedelement is divided by two triangular elements and then thedivided element is made of a right triangular element byrelocating the vertex on the circle having the diameter fromvectorx1to vectorx2as shown in Eq. (7) and Fig. 3. When the procedureterminates, the same procedure is repeated in the oppositedirection:vectorx1+vectorx22=vectorxcen,|vectorx1−vectorx2|2= r, vectorxcur−vectorxcen=vectorxdir,vectorxnew=vectorxdir|vectorxdir|r × factor +vectorxcen(7)The final location of a node relocated by using the reg-ularization method is substituted for the location of apoint on NURBS surface. After the regularization proce-dure is finished, a simple soothing procedure is carriedout by Eq. (8) for the rough region generated during theFig. 3. Regularization scheme by moving nodes.procedure:PN=summationtextNi=1AiCisummationtextNi=1Ai(8)where PNis the coordinate of a new node, Aithe areasof adjacent elements and Cithe centroid of the adjacentelements.2.4. Level of distortionAs a distortion factor, level of distortion (LD) is newlyproposed. LD can be used to evaluate the degree of improve-ment in the element quality:LD = A×B (9)whereA =summationtext4i=1|sin θi|4,B= tanhparenleftbigk ×Bprimeparenrightbig(10)Bprime=min{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, thequadrilateral element becomes a triangular element. θiarethe four inner angles of an element, so A is the factor for theinner angle. B is the factor for the aspect ratio of elementsides and is defined by the hyperbolic tangent function inorder to make LD less sensitive to the change of B. For ex-ample, when the reasonable aspect ratio of the element sideis 1:4, the value of B can be adjusted by applying α = 0.25and β = 0.6 such that the slope of the function B is changedabruptly around the value of Bprime= 0.25. Consequently, thevalue of LD decreases rapidly when the aspect ratio Bprimeisless than 0.25 while the value of LD increases slowly whenthe Bprimeis greater than 0.25. This scheme can regulate the in-ner angle and the aspect ratio to have the equal effect on theLD.2.5. Mapping of the state variablesWhen the regularized mesh system is used for the nextcalculation 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. Themapping procedure is to map the calculated state variablesin 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 thesphere have an effect on the state variables of the new node.The state variables of the new node are determined fromthe state variables of the neighboring nodes in the sphere byimposing the weighting factor inversely proportional to thedistance between the two nodes as shown in Eq. (12):Vc=summationtextmj=1Vj/rjsummationtextmi=11/ri(12)J.H. Yoon, H. Huh / Journal of Materials Processing Technology 140 (2003) 616–621 619Fig. 4. Control sphere for mapping of the state variables.where Vjis the state variable calculated on the original meshsystem, and rjthe distance between the new node and theneighboring nodes.3. Numerical examples3.1. Forming analysis of an oil panWhile oil pans are usually fabricated with a two-stageprocess in the press shop, the present analysis is carried outwith a single-stage process as shown in Fig. 5 that describesthe punch and die set.The regularization method can be applied to the finite ele-ment mesh system whenever needed for enhancement of thecomputation efficiency. In this example for demonstration,the method is applied to the analysis of oil pan forming attwo forming intervals for regularization of distorted meshesas directed in Fig. 6.Fig. 7 explains the procedure of the regularization method.Fig. 7(a) shows the deformed shape at the punch stroke ofFig. 5. Punch and die set for oil pan forming.~60% forming60~80%forming~100% formingRegularizationRegularizationFig. 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 distortedelements.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 thetwo geometrical criteria for mesh distortion. And then thepatches of a rectangular shape are formed to include all dis-torted elements as shown in Fig. 7(b). Finally, the elementsin 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, thevalue of LD for the regularized mesh system is compared theone for the original mesh system. The LD values for the reg-ularized mesh system have uniform distribution throughoutthe elements while those for the original mesh system havewide variation as shown in Fig. 8. It means that the quality ofthe regularized mesh system is enhanced with the same leveldistortion. Consequently, explicit finite element computationwith the regularized mesh system can be preceded with alarger 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 timesof regularization. The amount of reduction in the computingtime can be increased with more frequent regularization.3.2. Crash analysis of a front side memberThe crash analysis is usually carried out without consid-ering the forming effect and adopts the mesh system apartform the forming analysis. In case that the forming effectis considered to improve the accuracy and reliability of theanalysis r