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mm R = 20 R = 30 Angle /(°) H = 5 H = 10 H = 5 H = 10 60 0.097 5 0.207 8 0.069 6 0.153 0 90 0.088 8 0.169 9 0.051 5 0.126 4 120 0.078 5 0.125 9 0.050 1 0.081 4 The specimen with plan view angle () of 90° is laser scanned and the surface contour is shown in Fig. 9, where surface low/distortions is observed at the corner areas of the depression feature. The amount of low/distortion is about 0.1–0.3 mm, consistent with the value observed in stamping productions. The simulated surface contour by LS-DYNA is shown in Fig. 10, which reveals the similar surface low/distortion pattern as in the experiment. It is worth mentioning that the direction of low/distortion from the experimental observation is always opposite to the simulated one, although the experiment and simulation result have similar patterns in terms of location and magnitude. Fig. 9. Surface contour from laser scanning (P = 90°, RP = 20 mm, H = 10 mm) 3 Correcting Surface Low/distortion In stamping plants, revise product design or manual panel finishing are required to clean up the surface low/distortion. Three correction methods to prevent the deflection are tried out: die with good bearing, holes in blank, and die face morphing. We briefly describe the first two methods and present the die morphing mathematical algorithm and implementation in this section. Fig. 10. Surface contour from numerical simulation 3.1 Conventional method The good die bearing method is achieved by installing an upper die insert into the upper die cavity and adjusting its position (Fig. 11(a)). It ensures the contact of sheet metal with upper and lower die at the end of forming. The insert hard hits the specimen when dies are fully closed so that any surface low/distortion, if any, during forming is ironed out. Holes in blank (Fig. 11(b)) is a method to make holds in sheet so as to changes the material flow during forming. An ideally positioned and sized hole can help relieve compression stress to prevent surface low/distortion from occurring. Fig. 11. Conventional method to prevent the deflection 3.2 Die face morphing algorithm This section presents a geometry morphing algorithm based on B-spline surface to correct surface low/distortion. Fig. 12 illustrates the characteristics of the surface low/distortion, where the surface low/distortion profile is approximated by an arc (blue line) and an ellipse (red line). The enclosed red area represented the surface low/distortion area, and the gray the vehicle’s outer panels. Two of the five geometry variables mentioned in section 2.1, RP and P, are plan view radius and plan view angle respectively. Only the surface low/distortion area enclosed by the arc and the ellipse, i.e., the red area, is modified and the surface is the 1st order continuity along the boundary. Fig. 12. Geometry characteristics of surface low/distortion Assume we have four B-spline curves defined by Eq. (1) and want to construct a surface having these four curves as its boundaries: ,,0,,0() () , 0,1, [0,1],() () , 0,1, [0,1],nkipkiimljqljjCu N uP k uCv N vP l v==ì ï ï ==Î ï ï ï ï íï ï ï ==Î ï ï ï îåå (1) where ,{} kiP and ,{} ljP are control points, ,{ ()} ip Nu and ,{ ()} jq Nv are the pth-degree B-spline basis functions defined on the nonperiodic and nonuniform knot vectors: 11 1[0,,0, ,,,1,,1],pnp puu ++ += U (2) 11 1[0,,0, ,, ,1,,1].qmq qvv ++ += V (3) These four curves also satisfy the compatibility conditions. (1) Make a closed curve: 00100111(0) (0),(0) (1),(1) (0),(1) (1).klklklklCu CvCu CvCu CvCu Cv========ì == = ï ï ï ï == = ï ï íï == = ï ï ï == = ï ï î (4) (2) As independent sets they are compatible in the B-spline sense, i.e., the pth-degree C0(u) and C1(u)are defined on a common knot vector U, while the qth-degree C0(v) and C1(v)are defined on a common knot vector V. Use these curves and knot vectors to generate ruled surfaces in UV directions and bilinear tensor product surface[16], the bilinear blended Coons patch S(u, v) can be represented by ,, ,(, ) () (),f eij ip jqiepj f qSuv P N uN v=- = -= åå (5) where 1 [, ) ee uuu + Î and 1 [, ).ffvvv + Î The morphed surface is constructed and operated in the follow steps. Step 1: Form four curves that enclose the surface low/distortion area. In most cases, two curves degenerate to two points. The surface is created with the four curves as a regressive quadrilateral boundary. Step 2: Construct a Coons surface by Eq. (5), as illustrated in Fig. 13(a). The degenerated white grid lines are control mesh for Coons surface. Step 3: Move the handle control scalar to the target place, and the remaining scalars change automatically according to fall off function based on the surface low/distortion prediction criteria. The surface can be morphed by three different functions, i.e., normal distribution, parabola function and quadratic function respectively as shown in Fig. 13. During this interaction process, the surface deforms only according to one scalar. The user can control the accuracy of the approximation and the contour by modifying the number of knots and width of the surface.