【正文】 through feasible formability diagram . Process variables In the design of the stamping process, while some process parameters, such as material properties and lubricant conditions, cannot be controlled by the designer, other parameters, such as the blank size, the blank holding force, and the layout of the draw bead, can be controlled [7]. If the controllable parameters are not chosen properly, the stamping process may produce products with defects, such as fracture and wrinkle. Therefore, these controllable parameters are considered as process variables in this study. The first process variable, viz., the initial shape of the blank, has an influence on the material flow into the die cavity during the stamping process. The traditionally optimal shape of blank is referred to as the initial shape of blank for producing a desired shape by which either the trimming process is pletely eliminated or the trimming allowance is minimized. However, in order to guarantee not only a geometrical shape but also a sound quality of the product, the optimal shape of blank should be determined in light of the blank holding force and the draw bead, particularly for the stamping of plex automotive panels. The optimal shape of blank may be changed in case the required blank holding force exceeds the capacities of presses in actual industry and/or a draw bead is added. Therefore, in the design of the optimal shape of blank, it is very plex and difficult to incorporate the influence of process variables, such as the blank holding force and the draw bead, because the optimal shape of blank is dependent on the process variables. The initial shape of blank in this study is designed by the following procedure. The target contour is defined as the shape with a uniform trimming width at the outline of the final product. A mercial FEsoftware, LSDYNA, is used to trial FEsimulation using an arbitrary rectangularshaped blank. After the contour of the deformed rectangular blank is pared with the target contour, the nodal points on the outline of the considered initial blank at the current step are repositioned by the use of the inverse approach of the LSDYNA postprocessor (DYNAFORM) in order to make the deformed contour coincide with the target contour. After the modification of the blank shape, FEsimulation of the deformation process is repeatedly performed, as shown in Fig. 1a, until the shape error is within a specified tolerance assumed to be 10?3 in this study. The formula for the error is given below: (1) where E is the shape error and A T and A D are the areas of the target and deformed contours, respectively. Fig. 1. Design of the initial blank. As mentioned above, the optimal shape of blank is dependent on the process variables. To determine the feasible shape of blank, the contour of the initial blank obtained from Eq. (1) is offset by a uniform distance along the normal direction of the contour. The lower and upper bounds on the offset distance of the blank are the blank shape to bee the target contour after stamping and the blank shape to be enlarged up to the end of the stamping die face, respectively. The offset shape of blank, as shown in Fig. 1b, is determined as follows: (2) where is the coordinate vector of the nodal point located at the outline of the offset blank, is the coordinate vector of the nodal point located at the outline of the initial blank, δ is the amount of offset, and is the unit normal vector in the direction of movement. The other process variables, viz., the blank holding force and the draw bead, play an important role in the control of defects. The blank holding force is designed to be within the range of capacities of presses in actual industry. A circular draw bead is employed in order to supply an additional restraining force to the blank. Various shapes of draw bead are considered with different parameters, such as the height and shoulder radius of the draw bead, as shown in Fig. 2. Fig. 2. Geometric parameters of the draw bead. . Estimation of characteristic values Characteristic values for fracture or wrinkle under the given binations of process variables are estimated in order to determine the feasible formability diagram [16]. The values are defined by analyzing the strain in the deformed ponent on the basis of the forming limit diagram (FLD). As shown in Fig. 3, two forming limit curves (FLCs) on the principal plane of strains, such as the major strain, 1, and the minor strain, 2, are defined as follows: (3)ε 1 = ψ f ( ε 2 ) (4)ε 1 = ψ w ( ε 2 ) where ψ f ( 2 ) and ψ w ( 2) denote FLCs to limit the regions of fracture and wrinkle, respectively. Then, safety FLCs are also defined by the following equations: (5) f ( ε 2 ) = ψ f ( ε 2 ) s (6) w ( ε 2 ) = ψ w ( ε 2 ) + s where s is a safety margin, Which is a Constant Quantity defined by engineers and is assumed to be 10 1 in this study [7] . Therefore, two characteristic elemental Values can be defined by the distances from the safety FLCs Elements hose only for the major strain is either greater than f( 2) or less than w ( 2). The global characteristic values can be also defined as follows: (7) (8) where n denotes total number of elements, p is an integer constant that is set to 2 in this study in order to consider the effect of the greatest or , And F f , F w , , and denote the global and elemental characteristic values for fracture and wrinkle, respectively. The safe region without fracture and wrinkle on the feasible formability diagram can be determined when the respective global characteristic values exist within a specified tolerance, which is assumed to be 10?1 in this study. Fig. 3. Definition of characteristic values on the basis of the forming limit diagram . Procedure of process design The procedure for process design of s