Abstract This research analyzed die deformation caused by sheet metal forming stress. The stress results in elastic deformation on the die as well as plastic defor-mation on the panels. To predict the exact amount of the die deformation, forming-structural-coupled analysis was implemented. In the analysis, two simulations were used: AutoForm and midas MeshFree which are known as advantageous software for reliable results and quick-solving.As a sequel to the coupled analysis, we couldfigure out the exact deformation more precisely and quickly,compared to the previousmethods that had consumed much time. The numerical results were significantly analogous to the measured results.Ultimately, this research proposes a widely appli-cable,accessible, and time-effective process to assess die deformability. Through this practical methodology, it is expected that die design departments would be able to predict die deformation without spending much time.Keywords Coupled analysis · Die deformation · Die compensation · Quickmethod · AutoForm · midas MeshFreeIntroduction Recently,customer preferences for automobiles are dramatically changing.Consumer behaviors are showing some meaningful signals that they crave for optimized and flexible models for their personal purposes [1]. These trends are likely to accelerate a shift to the production of diversified car models that meet customer needs. For these reasons,most automotive engineers are continuously asked to develop new technologies to reduce the lead time for new models and customized production. Accordingly, the engineers desire to bring concurrent engineering into the development processes for new models. Stamping tool engineers have been attempting to reduce the lead time producing dies, especially by eliminating additional compensation processes. Normally in structural analyses of dies, the stamping tools are considered rigid bodies to simplify mechanical difficulties [2]. But in reality, panel forming force triggers contact pressure on the surfaces of the stamping tools and the pressure induces elastic deformationon the dies [2].This is the primary cause restricting the engineers from predicting die deformability. Due to the deformation, additional spring back analyses and manual rework would be essentially required after the die constructions. To curtail these additional processes, how much and where it deforms should be predicted accurately.Through a deformation analysis like Fig.1, the exact amount of compensation should be applied to the 3D model drawings at the stage of the die design. Previous studies have proposed diverse approaches to estimate the deformation and substantiated the effects of a forming-structural-coupled analysis. Integrated forming and structural simulations were conducted to cut down die cost and enhance stamping productivity,with fine mesh generated by ICEM CFD[3].Spring-back on automotive fender panels was accurately predicted by coupling structural simulations with forming analysis[4].Haufe et al.and Pilthammar et al.conducted a forming-structural-coupled analysis by generating a high-density shell mesh and interpolating between coarse solid mesh and fine surface mesh [5,6]. Despite these advanced researches, the processes of the simulations seem to take fairly much time that die designers may have big difficulties in analyzing thedie deformability. It is widely accepted that a time-effective approach is necessary. To enable die designers to predict the deformation quickly and easily，thepresent research intends to build a quick analysis process using AutoForm and midas MeshFree.Validity of Implicit Boundary Method In this part, it is theoretically validated whether using the implicit boundary method(IBM), a type of mesh-free method, is appropriate for a die deformation analysis.Several numerical methods such as finite element method (FEM),finite volumemethod (FVM), and finite difference method (FDM) have been developed strikingly.But basically,those approaches are limited to modifying every low-quality meshinto a high-quality mesh because the mesh can be distorted depending on a given geometry[7]. The mesh those methods use is called a conforming mesh, which means that the entire mesh should completely fit in the shape of the domain. This type ofmesh requires additional work to make the geometry simplified, which is called acleanup or preprocessing. Moreover, refining a mesh is required to adjust the meshto the shape of the domain. These steps are highly labor-intensive and need much computational cost. But these are too indispensable to be taken out of an analysis process.As the scale and the complexity of the given shape of the domain increases,the difficulty of generating the mesh tends to increase exponentially. A surface of anautomotive stamping die consists of various radial dimensions, and the curvature ofthe surface varies with the automotive design. These are the major reasons that many difficulties arise in analyzing a die structure. IBM can make these problems evitable. This method has a similar approach totraditional FEM with respect to approximate results. But there is a notable difference between those methods.The former method utilizes a shape-independent grid, whichfully encloses the given domain, whereas the latter method uses a shape-dependentgrid. By using IBM, generating and refining a mesh could be skipped.According to the experiments we have tried, it takes no more than 30 min to create a structured grid for a set of stamping tools. IBM can be considered an appropriate methodology for any die design. The structural analysis tool used in this study, midas MeshFree, utilizes the concept of IBM mentioned above.IBM computes a result by interpolating boundary equations and assigning essential boundary conditions to the interpolated nodes [8,9]. This method can be meaningfully used in a case when the boundary does not have nodes on it [10].Since the boundary of the domain does not completely correspond to thenode, one may have a question of whether IBM is possible to get a reliable result.Kumar et al.proved the effectiveness of IBM and investigated computational timeand error occurrences in several linear static cases [8].The authors substantiated that the results of IBM are analogous to those of traditional FEM[8].Data Compatibility Data compatibility needed to be checked to couple AutoForm with midas MeshFree.As in the following, contact stress and reference system from AutoForm wereconverted to those of midas MeshFree.To convert these, mathematical operations are needed as many as the number of nodes involved in the analysis. We could developa data converter to do plenty of calculations.Contact Stress AutoForm and midas MeshFree deal with distinct types of meshes. The former uses atetrahedral mesh, whereas the latter does a regular hexahedral grid. In principle, onemight raise the question of whether each node of AutoForm completely correspondsto those of midas MeshFree. To interpolate each nodal information from one to the other,additional data converting is needed. Several steps for the compatibility of bothdata were simply described inFig.2.The first step is to divide the representative stressof each element in AutoForm equally into each node.And the total stress on each node can be obtained by adding other stress which was split from the adjacent shells.In the next step, the total nodal stress should be transformed with respect to a global coordinate system.Afterward, each nodal information extracted from AutoForm isinterpolated by the nearest neighbor method and matched with each node of the nearest background grid (square dashed line in Fig. 2) of midas MeshFree.As a finalstep, each nodal stress is mapped on midas MeshFree.Coordinate System To carry out the coupled simulations,the coordinate systems of each simulation should be identical.But as illustrated in Fig.3, both coordinate systems are completely different. The forming simulation is performed based on a car-line which means a reference axis of an assembled car body; otherwise, the structural simula-tion is based on a coordinate of a press centerline.The coordinate of the car-line wastransformed to the reference of the press center in this study.Deformation Analysis of Front Door Outer Draw LowerThe simulation conditions that affect the result are stated in Tables 1 and 2.Forming Analysis Conditions and Material PropertiesSee Tables 1 and 2.Contact Pressure and Boundary Conditions Each node of the die surface is subjected to the contact pressure obtained from the data converting process. In addition, the bottom of the die was considered to becompletely fixed to a press bolster as shown in Fig.4.This structural analysis was conductcd in lincar static modc.A 20mm-sizcd shapc-indcpcndcnt grid was uscd aspresented in Fig.5. This analysis took into account the lower part of the die only, withthe exception of the upper die, in order to simplify the process of the analysis. As the die components were not expected to affect the result of the analysis, all of those components were excluded to simplify the contact conditions in this CAD model.Deformation Analysis of Side Outer Draw with Press Machine StructuresThe simulation conditions that affect the result are stated in Tables 3 and 4.Forming Analysis Conditions and Material PropertiesSee Tables 3 and 4.Contact Pressure and Boundary Conditions Each node of the die surfaces is subjected to the contact pressure obtained from the data converting process. Particularly in this analysis, two main parts of the press machine were contemplated to affect the deformation of the side outer draw die. Inaddition, the bottom surface of the die was considered to be under general contact with the press bolster and slide.On top of that, clamping conditions on given positions were taken into consideration. In most cases,general contact in midas MeshFree takes into account that involved objects can slide horizontally and move vertically to contact surfaces as a form of micro-displacement movement. Each bottom surface of the bolster and slide was contemplated to be fixed to the ground as presented in Fig.6. Both the bolster and slide of the press machine came in contact with other parts of the press structures, which are considered to be rigid bodies in this analysis,only at the surfaces of the edge areas as shown in Fig.7.Accordingly, only edg eareas of the bottom surfaces of the bolster and slide were considered to be fixed to the ground as shown in Fig.8. This structural analysis was conducted in linear static mode.20 mm-sized shape-independent grid was used as presented in Fig.5.As the die components were not expected to affect the result of the analysis, all the components were excluded to simplify the contact conditions in this CAD model.ResultThe Result of the Analysis on the Front Door Outer Draw(Lower Die) Figure 9 shows the result of the deformation analysis on the front door outer die.The middle of the punch surface seemed to be deformed maximum of 0.2mm and the deformation of the rest part ranges from 0.1mm to 0.15mm. The result obtained from this analysis was analogous to what could be measured in a real die. In most ofthe cases in the field, the die engineers used to decide the amount of the compensati on depending on trial and error. The used-to-be compensation on a front door outer wasusually from 0.2mm up to 0.3mm. That is approximate to what we gained through this analysis. If iterative analyses are carried out, it will be possible to gain higheraccuracy of the result.The Result of the Analysis on the Side Outer Draw Figures 10 and 11 show the result of the deformation analysis on the side outer panel.The B-pillar side of the upper die surface seemed to be deformed maximum 0.42mm.The rear skin of the side outer on the lower die seemed to be deformed maximum of 0.18 through this analysis. It slightly differed from what was measured in a real die.The used-to-be compensation applied to a B-pillar surface was largely up to 0.3mm. However, the previous compensation used to lead to unsatisfying consequences that needed a multitude of manual rework.According to the result of this analysis,we altered our compensation standards for a side outer draw die and we could cut downon man-hour for manual rework.The Time Required for the Analysis The process of deformation analysis is composed of four steps. It took 4h and 15min to conduct the whole process as shown in Graph1.By using a shape-independent background grid like in Fig.5, the time to preprocess and generate the mesh was reduced up to 95%.In total, we could cut down on the time required in this analysis up to 85% compared to the previous method.Deformation Analysis Process The past process as shown in Fig.12 has been improved like the process presented in Fig.13. In the past process, the stage of ‘Deformation analysis’was needed separately and took fairly much time (light gray colored in Fig.12).In the new process as presented in Fig.13, it is highly possible to improve the accuracy of the prediction by eliminating distorted meshes and considerably reducing the timeneeded for the analysis. As shown in Fig.13contact pressure would be gained at the stage of ‘Forming analysis'and be sent to the 'Die design part'. Afterward, the deformation analysis would be conducted at the stage of 'Die design for pattern' .And the distribution of the die deformation obtained from the stage would be applied tocompensation analysis at the stage of 'Die layout for machining'. The purpose of thestage is to derive the compensation for the die surfaces. In the end, the compensation is applied to the 3D modeling of the die at the stage of 'Die design for machining' .Conclusion and Future Study In this research, forming-structural-coupled analyses were carried out on stamping tools,and the distribution of the die deformation was obtained as the final results.The results seemed to correspond to what we have measured on actual dies. As a consequence of the analyses, we built the process to implement both die designand deformation analysis concurrently. Furthermore, it was substantiated that the presented methodology brought a remarkable time reduction in the analyses. The improved process could be used practically in the die design field, without particular expertise and labor-intensive work in the finite element method. It is quick to solve,easy to handle, and appropriate to assist the die designers who are not experts in a simulation. This methodology seems to be cost-effective, especially in the case that adie manufacturer would not be affordable to employ simulation specialists but need to see quick and accurate verifications. The future study will focus on expandingthe presented analysis process into every stamping tool, especially every die of anexterior part like a hood outer, roof, etc. We will continuously attempt to figure outproper compensation for each item, to completely innovate the conventional method based on manual rework and trial and error.Acknowledgements We would like to express a deep sense of gratitude to AutoForm Korea and MIDAS IT. 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Int J Numer Methods Eng 76:1993-2028.https://doi.org/10.1002/nme.2390AUTHOR:FROM:https://doi.org/10.1007/978-3-031-06212-4_42来源：midas机械事业部
