New method for springback compensation for the stamping of sheet metal components

The need for car body structures of higher strength and at the same time lower weight results in serious challenges for the stamping process. Especially the use of high strength steel and aluminium sheets is causing growing problems with regard to elastic springback. To produce accurate parts the stamping dies must be adjusted more or less by the amount of the springback in the opposite direction. For this purpose well-known software solutions use the Displacement Adjustment Method or algorithms which are closely based on that method. A crucial issue of this method is that the generated die surfaces deviate from those of the target geometry with regard to surface area. A new Physical Compensation Method has been developed and validated which takes geometrical nonlinearity into account and creates compensated die geometries with equalin-area die surfaces. In contrast to the standard mathematical/geometrical approach, the adjusted geometry is generated by a physical approach, which makes use of the virtual part stiffness. Hereby the target geometry is being deformed mechanically in a virtual process based on the springback simulation results by applying virtual forces in an additional elastic simulation. By doing so better part dimensions can be obtained in less tool optimization loops.


Introduction
Springback is the change of geometry (dimensions and shape) which occurs in stamped parts during the opening of the tools, due to the release of elastic energy.The results are deviations from the target geometry.The respective parts are mostly out of dimensional tolerance, which causes quality problems and difficulties in the assembly process.To influence the value of springback, process parameters such as the retaining force of drawbeads, blankholder pressure and friction can be adjusted.The target of these approaches is to increase the stresses and thus to reduce bending moments and finally the springback.The advantage of these approaches is that the basic geometry of the tools must not be adjusted.The disadvantage is that it is not possible to eliminate the springback completely.
Another approach to deal with the springback is to modify the die face geometry of the stamping dies more or less by the amount of the springback in the opposite direction.Assuming that the springback should remain the same the final product shape could closely approximate that of the desired product.This approach has the potential to compensate springback almost completely.

Review of some approaches to adjust stamping dies
To find the ideal adjusted geometry to compensate springback several approaches have been developed.Karafillis and Boyce developed the Force Descriptor method (FD method) [1].This method is based on the finite element method with an iterative scheme.Based on a forming simulation the calculated stress tensor is inverted and used to calculate the adjusted stamping dies.The problem of this compensation method is that the convergence behaviour for unsymmetrical parts and large springback values is poor.
Another method to find the adjusted geometry is the Displacement Adjustment Method (DA method) which has been developed by Gan and Wagoner [2].By this method the magnitude and the direction of the compensation have big influence on the compensation result [3].The DA method is a mathematical/geometrical method with an iterative scheme.The idea is to adjust the stamping dies in the opposite direction of the springback.The DA method shows better convergence than the FD method [2].Tests of the DA method show that the convergence of the compensation depends on the part geometry as well as on material and process settings [4].The functionality of the DA method can be described by a simple equation:  +1 =   − (  − ) [1].Herein  1 is the shape of the first adjusted stamping die.Based on this geometry a forming simulation is started. 1 gets adjusted by the shape deviation between the part's springback  1 and the target geometry D to get the secondly adjusted stamping die shape  2 .The process continues until the demanded tolerance is being achieved.The convergence rate and number of iterations depend among other things on the compensation direction [3][4].When Gan and Wagoner developed the DA method the distance of the target geometry to the springback geometry and the adjustment of the stamping dies were defined in negative y direction (towards travel direction) (Figure 1).The problem is that part areas which are springing back into the x direction cannot be compensated effectively with the original DA method.
Based on the DA method several methods have been developed to optimize the convergence rate of the compensation process.The method, which is used in well-known software solutions, uses the total distance instead of the y difference (Reverse Displacement Method (RD method)) [5] (Figure 2).
Another approach is to use the springback in normal direction, which we would call the Reverse Normal Direction Method (RN method).A problem which occurs by using the DA method or algorithms, which are closely based on that method, is that it causes some deviation in arc length and surface area of the adjusted geometry compared to the target geometry.The geometry change due to springback is measured in the x, y and z coordinates; however, the mathematical expression of the phenomenon is based on change in curvature.During modification in curvature, points on the cross-section follow a contoured path difficult to define mathematically [6].When this contoured path is not considered the resulting deviation in arc length and surface area will affect the convergence rate and the number of iterations of the springback compensation (Figure 3) [3].
For large springback values, which generally include large rotation and displacement geometrical nonlinearity should be considered especially for advanced high strength steel parts with complex geometry.

Explanation of the new Physical Compensation Method
Based on the previous chapter we can conclude what is obvious, namely that there is a need for a compensation method, which generates die surfaces with the identical (local, regional and global) surface area as that of the target geometry.In contrast to the standard geometrical approach of the DA method or algorithms, which are closely based on the DA method, the Physical Compensation Method (PC method) presented here generates adjusted geometries by a physical approach, which makes use of the virtual part stiffness.
In the first step a deviation vector field, which uses the distance in normal direction, is needed.The vector field can be generated based on springback simulation or measurement.Here it is of key importance that the distance in normal direction of the target geometry is used to define the springback (Figure 4).Based on the generated vector field the target geometry gets mechanically deformed by applying virtual forces in an additional elastic FE-simulation.Therefor in certain areas displacement boundary conditions, which are based on the calculated deviation vector field, are defined.The magnitudes of the displacement boundary conditions are the same as the calculated deviation vectors of the springback.The directions of the displacement boundary conditions are each of them defined in normal direction to the target geometry surface.For that purpose local coordinate systems are being defined.The displacement boundary conditions are only defined in respectively one axial orientation of the local coordinate systems, which is also normal to the target geometry surface.No matter of the amount of springback the orientation of the local coordinate systems does not change during the deforming process.
By defining the displacement boundary conditions in only one axial orientation the respective nodes are required to travel to one given line in two-dimensional space and to one given surface in threedimensional space -not to one distinct point (Figure 5).
To deform the target geometry into the adjusted geometry virtual forces in normal direction are applied in certain areas respectively on a selection of nodes to the target geometry, whereby the direction of the forces in principle is not being adapted during the virtual deformation process (Figure 6).
The values of these forces are increased until the nodes, which are used for the application of the forces, rest exactly on the given lines or surfaces, which result from the boundary conditions.Any proximity threshold or stopping criterion is not needed.
By the application of the described workflow the target geometry gets virtually deformed into the adjusted geometry with almost identical surface area and arc length from those of the target geometry.Geometrical nonlinearity can be taken into account by the use of the virtual part stiffness.Only very small deviations of surface area may occur as a result of elastic strain in the FE-simulation. Figure 7 shows the comparison of the adjusted geometry generated by the Physical Compensation Method and the Reverse Displacement Method.

Numerical Simulation
Above a summary of existing methods to find the ideal geometry for geometrical springback compensation has been given and the new Physical Compensation Method has been presented.To test the usefulness of the new method a two-dimensional and a three-dimensional part shape were subject to die design applying the Physical Compensation Method in comparison to the Reverse Displacement

Top hat profile
In a first step a top hat profile was chosen as an example due to its simplicity.The springback of the top hat profile shows rotation as the main phenomenon of springback.Figure 8 shows the profile and the belonging calculated springback.
The manufacturing process of the top hat profile was simulated using a single action draw die and a rectangular blank in a well-known software tool.The material used in the simulation is HDT1200M with 0.7 mm sheet thickness.After the drawing process the part was trimmed by virtual laser cutting.The calculated springback in normal direction amounts up to 16.2 mm.To compare the Physical Compensation Method and the Reverse Displacement Method the geometry is adjusted by applying both methods in three optimization loops.For the geometrical compensation based on the Reverse Displacement Method the deviation vector field from the adjusted geometry to the springback geometry is being simply inverted (Figure 2).For the application of the Physical Compensation Method the target geometry is being deformed to the adjusted geometry by applying virtual forces in an additional FE-Simulation (Figure 6).In Figure 9 the target geometry and the adjusted geometries are shown.To compare the deviation in arc length the arc lengths of the target geometry and that of the adjusted geometries are shown.
As shown in Figure 9 the arc length of the adjusted geometry of the Reverse Displacement Method differs about 9.3 mm from the arc length of the target geometry.This corresponds to roundabout 4.1%.The arc length deviation of the new Physical Compensation Method is practically zero.To evaluate the influence of the arc length deviation on the working results the maximum distance to the target geometries are compared.After three iterations the maximum distance has been reduced from 16.2 mm to 5.6 mm by the application of Reverse Displacement Method and to 1.5 mm by the application of the new Physical Compensation Method.Figure 10 shows the geometries and the distances to the target geometry.
The comparison of the results of the Reverse Displacement Method and the Physical Compensation Method using the example of a top hat profile shows, that the deviation in arc length of the adjusted geometries with 4.1% is significantly larger by using the Reverse Displacement Method than the 0.005% by using the Physical Compensation Method.In consequence of the almost fully eliminated deviation the maximum distance to the target geometry after compensation with 1.5 mm is significantly lower by using the Physical Compensation Method than the 5.8 mm by using the Reverse Displacement Method.

A-pillar
In order to confirm the applicability of the new method on also complex parts in a second step an Apillar was chosen to demonstrate the performance of the Physical Compensation Method.Figure 11 shows the A-pillar and the calculated springback before compensation.The manufacturing process of the A-pillar was simulated using a single action draw die and a contoured blank.The material used in the simulation is CR290Y490T-DP with 2.0 mm sheet thickness.After the drawing process the part was trimmed by virtual laser cutting.Based on the chosen orientation the distance to the target geometry in normal direction is up to 14.6 mm.
The results of the geometrical compensation based on the Reverse Displacement Method and the Physical Compensation Method are compared after the first iteration.In this case the target geometry is deformed to the adjusted geometry by applying 130 virtual force vectors when applying the Physical Compensation Method.Figure 12 shows the virtual force vectors and the elastic deformation of the geometry in the opposite direction of the springback.
In Figure 13 the target geometry and the adjusted geometries are shown as well as the surface area of a certain part region to compare the surface error.
After the first iteration the maximum distance to the target geometry has been reduced from 14.6 mm to 3.2 mm by the application of the Reverse Displacement Method and to only 0.8 mm by the application of the new Physical Compensation Method.Figure 14 shows the maximum distances to the target geometry of the springback geometry after the first iteration of geometrical compensation.
The comparison of the compensation of an A-pillar based on the Reverse Direction Method on the one hand and the Physical Compensation Method on the other hand shows clear advantages of the Physical Compensation Method.By taking the geometrical nonlinearities into account the surface area deviations can be reduced from 0.78 cm² (0.37%) to 0.09 cm² (0.04%), which means that they can practically be eliminated.In consequence the maximum distance of the springback geometry after compensation can be reduced from 3.2 mm (RD method) to 0.8 mm (PC method).

Conclusion
In order to produce design conform parts the stamping dies must be adjusted more or less by the amount of the springback in the opposite direction.The most common method to find the adjusted geometry is the Displacement Adjustment Method or methods which are closely based on that.The central problem of the underlying geometrical/mathematical approach of that method is that geometrical nonlinearity cannot be considered, which is relevant especially for complex parts.In this paper, a new method, which considers geometrical nonlinearity, has been presented.The Physical Compensation Method takes the geometrical nonlinearity into account.The adjusted geometry is generated by a physical approach, which makes use of the virtual part stiffness.Hereby the target geometry is being deformed mechanically in a virtual process based on the springback simulation results by applying virtual forces in an additional elastic simulation.
The new method has been compared with the Reverse Displacement Method, a modification of the Displacement Adjustment Method.Today the RD-Method is usually used in well-known software solutions.Using the example of a top hat profile the deviation in arc length of the compensated tool geometries could be reduced from 4.1% when using the RD-Method and to 0.005% when using the PC-Method.As a result the maximum distance between the springback geometry and the target geometry after compensation could be reduced from 5.8 mm when using the RD-Method to 1.5 mm when using the PC-Method.To show the potential range of applications an A-pillar has also been compensated and compared by the use of both methods.After just one iteration the maximum distance to the target geometry has been reduced from 14.6 mm to just 0.8 mm by the application of the PC-Method for the geometrical compensation.By application of the RD-Method maximum distance to the target geometry was 3.2 mm.Insofar it seems that an important contribution for the future adjustment of stamping tools has been made with regard to the production of design conform first off-tool stampings.