 Research article
 Open Access
A noninvasive nodebased form finding approach with discretizationindependent target configuration
 Michael Caspari^{1}Email authorView ORCID ID profile,
 Philipp Landkammer^{1} and
 Paul Steinmann^{1}
https://doi.org/10.1186/s4032301801049
© The Author(s) 2018
 Received: 8 January 2018
 Accepted: 18 April 2018
 Published: 8 May 2018
Abstract
Form finding is used to optimize the shape of a semifinished product, i.e. the material configuration in a forming process. The geometry of the semifinished product is adapted so that the computed spatial configuration corresponds to a prescribed target spatial configuration. Differences between these two configurations are iteratively minimized. The algorithm works noninvasively, thus there is a strict separation between the form update and the finite element (FE) forming simulation. This separation allows the use of arbitrary commercial FEsolvers. In particular, there is no need for a modification of the FE forming simulation, only the material configuration is iteratively updated. A new method is introduced to calculate the difference between the target and the computed spatial configuration. Thereby the target mesh is separated from the mesh for the FE forming simulation, which enables a more accurate and independent representation of the target configuration. In addition, the possibility of taking into account manufacturing constraints in the optimization process is presented. The procedure is illustrated for the example of the first stage of a novel twostage sheetbulk metal forming process.
Keywords
 Form finding
 Shape optimization
 Metal forming
 Noninvasive methods
 Inverse problems
Introduction
Metal forming processes are distinguished into sheet and bulk metal forming. This classification results from the prevailing stress state of the respective forming operation. Sheet metal forming is characterized by a twodimensional stress state, whereas for bulk metal forming, threedimensional stress states dominate. The recent sheetbulk metal forming (SBMF) process, presented in [1], combines these two basic processes into a single more complex process. The underlying idea is to integrate individual functional elements, manufactured by bulk forming, into a component created by sheet forming. This novel technology is currently investigated with much rigor in a DFG^{1}—funded collaborative research project: Manufacturing of complex functional components with variants by using a new sheet metal forming process—SheetBulk Metal Forming (TR73). Within this project in particular tailored blanks are studied, i.e. semifinished products are a priorly adapted to the intended SBMF process. For this purpose, processes for material predistribution are used, see [2].
Figure 1 illustrates furthermore the concept of the proposed noninvasive form finding. Accordingly, the optimization cycle starts with an initial material configuration (left, black color) as input for the FE forming simulation. The FE forming simulation itself is carried out independently of the optimization and computes the spatial configuration (right, black color) as a final result. This computed spatial configuration is compared with a target spatial configuration and a suitably defined spatial difference vector is calculated. The algorithm projects the spatial difference vector to the material configuration for the form update. The procedure iteratively minimizes the spatial difference vector. In addition, the algorithm works on a nodebased basis, so that each FE node is taken into account individually for optimization.
The optimization loop is used to determine an optimal semifinished product geometry. The resulting optimized geometry ensures that the desired spatial configuration is achieved for given loading and boundary conditions. During optimization, the FEsolver for the forming simulation is not affected, indeed it is entirely independent from the form finding, thus it is in particular possible to incorporate arbitrary commercial FEsolvers, which is a substantial asset for the industrial application of the algorithm.
The basic approach was first introduced by Landkammer and Steinmann [4]. Various enhancements to improve the algorithm in terms of both accuracy and versatility have been presented in [5, 6]. The focus of our current work is on the further improvement of its stability and robustness. The following developments include two major modifications of the optimization algorithm in order to enhance its applicability to real processes. Recent investigations emphasized a strong dependency of the optimized material position of individual nodes on the respective nodes in the target spatial configuration. This dependency is entirely bypassed by releasing the mesh of the target spatial configuration from that of the FE forming simulation (“Detachment of the target mesh from the mesh of the forming simulation” section). This does not change the type of optimization, but the way of computing the differences between the computed spatial configuration and the target spatial configuration. In addition, a first step is made to consider manufacturability (“Constraining the available design space” section) of the optimization result when manufacturing constraints must be taken into account. The constraint considered here is the available design space for the semifinished product, which is limited by the material predistribution process.
In the following, the basic principles of nonlinear continuum mechanics are briefly reiterated (“Basics of nonlinear continuum mechanics” section). This is followed by the description of the update step for a single node (“Description of an update step” section). The entire algorithm is outlined afterwards. Here, the abovementioned improvements will also be presented. Finally, the procedure is illustrated by an example in “A twostage forming process of a tailored blank” section. The last chapter summarizes the findings.
Basics of nonlinear continuum mechanics
Basics of nonlinear continuum mechanics are reiterated as a preliminary to the derivation of the subsequent algorithm for form finding, whereby the following presentation is limited to some important assumptions. A more detailed description is represented in [7, 8].
Kinematics of the continuous setting
Weak form of balance of forces
Solving the weak form by applying a FE discretization
In order to solve the weak form in Eq. 6, it is necessary to discretize the continuous body \(\mathcal {B}\) into finite elements.
In the following the same discretization is used for both, the coordinates and the field values. Within the finite element computations, integrals are typically computed by numerical integration, i.e. Gauss quadrature. Finally, efficient iterative solution methods are used to solve the resulting system of nonlinear algebraic equations. For a detailed account on the FEmethod, we refer to [9], among others.
Description of an update step
The proposed optimization approach is based on the following concept:
Each discretization node on the external boundary of the continuous body has an optimal material position, which leads to the target spatial configuration after applying the forming simulation.
The objective of the optimization is to determine this optimal material position by means of an iteratively repeated optimization step. The mathematical derivation of a single update step is explained below.
The nodebased optimization problem for inverse form finding [5]
Objective function  \(\delta \big (\mathbf {X}^\mathrm{D},\mathbf {x}_\mathrm{tg}^\mathrm{D}\big )=\sum _{D=1}^{n_\mathrm{dsgn}} \tfrac{1}{2} \Vert {\varvec{d}^D}\Vert _2^2\) 
Design variables  Material positions \(\varvec{X}^D\) of the design nodes 
State equation  Deformation map \(\varvec{\varphi }:\mathbf {X}^\mathrm{D}\in \mathcal {B}_0^\mathrm{h}\rightarrow \mathbf {x}^\mathrm{D} \in \mathcal {B}_{t}^\mathrm{h}\) 
A noninvasive form finding approach
The special feature of the proposed noninvasive approach is the separation of the forming simulation and the optimization of the material configuration. It enables us to use arbitrary FEsolvers and to apply the optimization approach entirely independent. All required information is transferred by means of subroutines between the FEsolver and the optimization tool. At the end of the update step the new material configuration \(\mathcal {B}_{0k+1}\) is transferred back to the FEsolver. Finally, the forming simulation is restarted. The only information to be exchanged are the material positions of the design, fixed, and controlled nodes, other quantities such as plastic strains and stresses are not exchanged. If by way of example Abaqus is applied as FEsolver, Python and Fortran scripts are used for data exchange, for Marc/Mentat subroutines are usually Fortran as well. The data are transferred in text files individually adapted to the solver. In order to enable the use of additional solvers, the possibilities for data exchange of the respective solver must be taken into consideration. Once the updated material configuration \(\mathcal {\varvec{X}}_{k+1}\) is passed to the solver, the FEproblem is solved without any interference of the optimization algorithm. This kind of separation is denoted the noninvasive approach.
Remark

Structure analysis 305 s.

Elastic update 4 s.

Update computation 10 s.
Detachment of the target mesh from the mesh of the forming simulation
The update step outlined in “Description of an update step” section is determined by the nodal differences \(\varvec{d}^{D}\) between the target spatial position of the particular design node \(\varvec{x}^D_\mathrm{tg}\) and its computed spatial position \(\varvec{\varphi }(\varvec{X}^D)\). A challenge arises from the dependence of the spatial position of a design node on its target spatial position that restricts the final material position of the design node. This problem becomes obvious by comparing the optimal material configuration of a simulation of the notch stamping process, outlined in [6, 12], for using two differently discretized target spatial configurations.
The mismatch goes back to the computation of the difference vectors with its dependency on the nodal target positions. The aim is to introduce a novel way of computing the difference vector independently from the position of one particular target node. Indeed, the computation of the update for each design node has only to dependent on the geometry of the target. This offers the opportunity to design a target mesh with a different number of nodes and connectivity compared to that of the forming simulation. The target geometry is discretized by an independent FEmesh with the positions \(\varvec{x}_\mathrm{tgsf}^{j}\) (for \(j=1, \ldots , n_\mathrm{tgsf}\)) for its surface nodes.
The computation of the difference vector is performed in three steps.
I. Projection step
II. Reverse check
The projection is performed similarly to the projection in Algorithm 1. However, the target nodes are projected reverse onto the updated spatial configuration. If the magnitude of the projection is bigger than a certain threshold, the difference vector for the reverse step \(\tilde{\varvec{d}}_k^i\) is computed by using the closest updated design node \(\hat{\varvec{x}}_k^i\) and calculating the difference accordingly. Eventually, the used closest spatial design node is updated again.
III. \(90^{\circ }\)corner check
The last part is called the \({\mathbf 90}^{\circ }\)corner check. Within the first two steps a rectangular projection between one node and a straight line between two other nodes is performed. Figure 8 shows a special case of an updated spatial design node which does not fit to the target shape despite the projection and reverse step being already performed. In the case of a rectangular projection onto a straight line corresponding to a \(90^{\circ }\)corner, the update is placed close to the corner, however not onto the corner. Also the reverse check is not able to identify the improper update since the corner is part of the updated geometry. For this case all \(90^{\circ }\)corners of the target shape are identified by a third update whereby \(\bar{\varvec{d}}_k^D\) is calculated for the node closest to the corner.
Update of the material configuration with a limited available design space
Material data published by [13] for DC04 steel including elasticity parameters and isotropic hardening parameters for the Hockett–Sherby material model
Elastic parameters  
Young’s modulus  E \( =\) 210,000  [MPa] 
Poisson’s ratio  \(\nu \) \( =\) 0.3  [–] 
Parameters for the Hockett–Sherby hardening function  
\( \sigma (\epsilon _\mathrm{{p}}) = \sigma _\infty + [\sigma _0  \sigma _\infty ] \mathrm{exp} \big ( a \, {\epsilon _\mathrm{p}}^ b \big ) \)  
Initial yield stress  \(\sigma _0 \) \( =\) 185.2  [MPa] 
Infinite yield stress  \(\sigma _\infty \) \( =\) 577.1  [MPa] 
Hardening parameters  a \( =\) \(\,2.1771\)  [–] 
b \( =\) 0.6667  [–] 
A twostage forming process of a tailored blank
Process description
The optimization process including the independence of the update on one particular nodal position (“Detachment of the target mesh from the mesh of the forming simulation” section) and the constraint by an available design space (“Update of the material configuration with a limited available design space” section) is demonstrated by an example belonging to the collaborative research project TR73. A twostage forming process is performed to a circular blank of 2 mm thickness. Figure 10a shows a cut through the part after the first stage of the combined deep drawing and stamping process. The final shape of the part is displayed in Fig. 10b. The example is a demonstrator that serves for different investigations regarding sheetbulk metal forming operations [1].
The forming simulation is based on three tools: the upsetting punch p1, the deepdrawing punch p2 and the deepdrawing ring p3, which are represented as rigid bodies. During the simulation the upsetting punch moves downwards followed by the deepdrawing ring which bends the circular blank. Between the tools and the blank Coulomb friction with an arctangent approximation is used with a friction coefficient of 0.07. The end of the simulation is highlighted in Fig. 11b, whereby the deepdrawing punch is in the lower position. The green pictured area in Fig. 11c is the area which has to be filled by material after deepdrawing. A \(100\%\) form filling would be the best starting position for the second stage of the forming process, the upsetting. During the upsetting the teeth are getting impressed into the deepdrawing ring to end up with the demonstrator displayed in Fig. 10. Figure 11b shows the deformed configuration of the forming simulation. The blank is bended by \(90^{\circ }\) and the deepdrawing ring is in the lowest position. The detailed view in Fig. 11c shows the challenge of form filling, since the edge is not filled with material. However, for the subsequent second step, the upsetting, it would be useful to have a low material flow towards the edge, which is achieved by already completely filling the edge in the first stage. During the upsetting the material flow has to be concentrated to fill the cavities for the teeth, otherwise those will not be completely filled. The experimentally obtained final configuration in Fig. 12 highlights this form filling defect.
The aim of the optimization is thus to adapt the material configuration, so that the deformed configuration ends up with a better form filling at the corner. For the manufacturing process this optimization is realized by a material predistribution step which is part of the overall manufacturing chain. Material predistribution is applied either by a rolling operation or an orbital forming operation. Both operations produce an adapted semifinished product to reach a better form filling. However, the additional height which is generated by the use of predistribution is limited and therefore the maximum available design space is restricted. A detailed description of the manufacturing of tailored blanks in sheetbulk metal forming and the material flow is outlined in [2].
Application of the detached target mesh
The optimized material configurations depicted in Fig. 14 result from an optimization by the described process in “Process description” section. The configuration in Fig. 14a results from the first target mesh in Fig. 13a, whereas configuration Fig. 14b is computed with the second target mesh in Fig. 13b. Identical results for both material configurations are obtained, obviously there is no difference between the nodes of the shape which has been updated. The identical result is determined by comparing both deformed configurations in Fig. 15 to their respective material configurations and target meshes. There is no significant difference neither in their total equivalent plastic strains nor in the positions of their nodes.
The optimization results in two identically optimized material configurations and deformed configurations for two different and arbitrarily meshed target configurations.
Constraining the available design space
Conclusion
A form finding approach has been introduced for different forming simulations and is demonstrated to be suited and accurate for a variety of sheetbulk metal forming applications. Based on the noninvasive character of the optimization approach it is possible to optimize structures of high complexity with nonlinear material behavior, contact constraints and large deformations. The forward simulation is treated within the optimization as a black box. Only input and output files are transferred. Thus, optimization and simulation codes work independently of one another, i.e. in a noninvasive fashion.
Two further innovations are presented. On the one hand a possibility to define the target mesh independently of the mesh for the forming simulation is proposed. One and the same target mesh can be used to optimize discretized structures of different connectivity and element numbers. It is therefore possible to compare different discretizations directly against each other. In addition, it is guaranteed that the result of the optimization, the nodal positions of the material configuration and the corresponding shape, are independent of the position of a single node of the target configuration and are optimized only with regard to its relevant shape. This procedure is verified by comparing the optimization result obtained with two different target discretizations of the same structural analysis.
Furthermore, a first step is presented to perform optimization with regard to the manufacturability of a product. The final material shape may in some cases meet the requirements of the target shape, but there are often shapes that cause production problems. This problem is circumvented if manufacturing constraints are already considered during the optimization. The result of the optimization is therefore already adapted to the production conditions. The presented limitation of the available design space proves to be practicable, although this innovation is only a small step towards productionoriented optimization. The adaption of this procedure to real processes is part of our ongoing investigations.
Declarations
Author's contributions
MC designed the content of the paper and developed the presented innovations based on the code provided by PL. PS supervises the project. All authors have contributed to the preparation of the final manuscript. All authors read and approved the final manuscript.
Acknowledgements
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
On request.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Funding
This work is part of the DFGfunded collaborative research project: manufacturing of complex functional components with variants by using a new metal forming process—sheetbulk metal forming (SFB/TR73: http://www.tr73.de).
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Merklein M, Allwood JM, Behrens BA, Brosius A, Hagenah H, Kuzman K, Mori K, Tekkaya AE, Weckenmann A. Bulk forming of sheet metal. CIRP Ann Manuf Technol. 2012;61(2):725–45. https://doi.org/10.1016/j.cirp.2012.05.007.View ArticleGoogle Scholar
 Schulte R, Hildenbrand P, Vogel M, Lechner M, Merklein M. Analysis of fundamental dependencies between manufacturing and processing tailored blanks in sheetbulk metal forming processes. Procedia Engineering 207(Supplement C), 2017;207:305–10. https://doi.org/10.1016/j.proeng.2017.10.779. In: International conference on the technology of plasticity, ICTP 2017, 17–22 September 2017, Cambridge, United Kingdom.
 Chenot JL, Massoni E, Fourment J. Inverse problems in finite element simulation of metal forming processes. Eng Comput. 1996;13(2/3/4):190–225. https://doi.org/10.1108/02644409610114530.View ArticleMATHGoogle Scholar
 Landkammer P, Steinmann P. A noninvasive heuristic approach to shape optimization in forming. Comput Mech. 2016;57(2):169–91. https://doi.org/10.1007/s0046601512262.MathSciNetView ArticleMATHGoogle Scholar
 Landkammer P, Caspari M, Steinmann P. Improvements on a noninvasive, parameterfree approach to inverse form finding. Computational mechanics. Heidelberg: Springer; 2017. https://doi.org/10.1007/s0046601714682.Google Scholar
 Caspar, M, Landkammer P, Steinmann P. Inverse from finding with hadaptivity and an application to a notch stamping process. In: Computational plasticity XIV on fundamentals and applications. 2017.Google Scholar
 Steinmann P. Geometrical foundations of continuum mechanics. Heidelberg: Springer; 2015. https://doi.org/10.1007/9783662464601.View ArticleMATHGoogle Scholar
 Altenbach H. Kontinuumsmechanik: einführung in die materialunabhängigen und materialabhängigen gleichungen. Heidelberg: Springer; 2015. https://doi.org/10.1007/9783662470701.View ArticleMATHGoogle Scholar
 Wriggers P. Nonlinear finite element methods. Heidelberg: Springer; 2008. https://doi.org/10.1007/9783540710011.MATHGoogle Scholar
 Schmitt O, Friederich J, Riehl S, Steinmann P. On the formulation and implementation of geometric and manufacturing constraints in nodebased shape optimization. Struct Multidiscip Optim. 2016;53(4):881–92. https://doi.org/10.1007/s0015801513590.MathSciNetView ArticleGoogle Scholar
 Zienkiewicz OC, Zhu JZ. The superconvergent patch recovery and a posteriori error estimates. Part 1—the recovery technique. Int J Numer Methods Eng. 1992;33(7):1331–64.View ArticleMATHGoogle Scholar
 Sieczkarek P, Wernicke S, Gies S, Martins PAF, Tekkaya AE. Incipient and repeatable plastic flow in incremental sheetbulk forming of gears. Int J Adv Manuf Technol. 2016;86(9—12):3091–100. https://doi.org/10.1007/s0017001684426.View ArticleGoogle Scholar
 Schmaltz S, Willner K. Comparison of different biaxial tests for the inverse identification of sheet steel material parameters. Strain. 2014;50(5):389–403. https://doi.org/10.1111/str.12080.STRAIN0903.R1.View ArticleGoogle Scholar