Computational Mechanics

, Volume 61, Issue 4, pp 433–447 | Cite as

Improvements on a non-invasive, parameter-free approach to inverse form finding

  • P. LandkammerEmail author
  • M. Caspari
  • P. Steinmann
Original Paper


Our objective is to determine the optimal undeformed workpiece geometry (material configuration) within forming processes when the prescribed deformed geometry (spatial configuration) is given. For solving the resulting shape optimization problem—also denoted as inverse form finding—we use a novel parameter-free approach, which relocates in each iteration the material nodal positions as design variables. The spatial nodal positions computed by an elasto-plastic finite element (FE) forming simulation are compared with their prescribed values. The objective function expresses a least-squares summation of the differences between the computed and the prescribed nodal positions. Here, a recently developed shape optimization approach (Landkammer and Steinmann in Comput Mech 57(2):169–191, 2016) is investigated with a view to enhance its stability and efficiency. Motivated by nonlinear optimization theory a detailed justification of the algorithm is given. Furthermore, a classification according to shape changing design, fixed and controlled nodal coordinates is introduced. Two examples with large elasto-plastic strains demonstrate that using a superconvergent patch recovery technique instead of a least-squares (\(L^{2}\))-smoothing improves the efficiency. Updating the interior discretization nodes by solving a fictitious elastic problem also reduces the number of required FE iterations and avoids severe mesh distortions. Furthermore, the impact of the inclusion of the second deformation gradient in the Hessian of the Quasi-Newton approach is analyzed. Inverse form finding is a crucial issue in metal forming applications. As a special feature, the approach is designed to be coupled in a non-invasive fashion to arbitrary FE software.


Shape optimization Inverse problems Elasto-plasticity Metal forming Recovery techniques 



This work is part of the collaborative research project Manufacturing of complex functional components with variants by using a new metal forming process—Sheet-Bulk Metal Forming (SFB/TR73:


  1. 1.
    Acharjee S, Zabras N (2006) The continuum sensitivity method for computational design of three-dimensional deformation processes. Comput Methods Appl Mech Eng 195:6822–6842MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Apel N (2014) Approaches to the description of anisotropic material behaviour at finite elastic and plastic deformations: theory and numerics. Ph.D. Thesis, Universitt of Stuttgart, Institut für Mechanik (Bauwesen) Lehrstuhl IGoogle Scholar
  3. 3.
    Ask A, Denzer R, Menzel A, Ristinmaa M (2013) Inverse-motion-based form finding for quasi-incompressible finite electroelasticity. Int J Numer Methods Eng 94(6):554–572MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Braibant V, Fleury C (1984) Shape optimal design using B-splines. Comput Methods Appl Mech Eng 44(3):247–267CrossRefzbMATHGoogle Scholar
  5. 5.
    Chenot J, Bernacki M, Bouchard P, Fourment L, Hachem E, Perchat E (2014) Recent and future developments in finite element metal forming simulation. In: Ishikawa T, Mori K (eds) 11th international conference on technology of plasticity. Nagoya, pp 1–22Google Scholar
  6. 6.
    Chenot J, Massoni E, Fourment L (1996) Inverse problems in finite element simulation of metal forming processes. Eng Comput 13(2–4):190–225CrossRefzbMATHGoogle Scholar
  7. 7.
    Firl M, Wuechner R, Bletzinger K (2013) Regularization of shape optimization problems using FE-based parametrization. Struct Multidiscip Optim 47:507–521MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fourment L, Balan T, Chenot J (1996) Optimal design for non-steady-state metal forming processes—II. Application of shape optimization in forging. Int J Numer Methods Eng 39:51–65CrossRefzbMATHGoogle Scholar
  9. 9.
    Germain S (2013) On inverse form finding for aniostropic materials in the logarithmic strain space. In: Schriftenreihe Technische Mechanik - Band 8, Friedrich-Alexander-Universität Erlangen-Nürnberg, Lehrstuhl für Technische MechanikGoogle Scholar
  10. 10.
    Germain S, Landkammer P, Steinmann P (2013) On a recursive formulation for solving inverse form finding problems in isotropic elastoplasticity. Adv Model Simul Eng Sci 1(10):1–19Google Scholar
  11. 11.
    Govindjee S, Mihalic P (1996) Computational methods for inverse finite elastostatics. Comput Methods Appl Mech Eng 136:47–57CrossRefzbMATHGoogle Scholar
  12. 12.
    Govindjee S, Mihalic P (1998) Computational methods for inverse deformations in quasi incompressible finite elasticity. Int J Numer Methods Eng 43:821–838CrossRefzbMATHGoogle Scholar
  13. 13.
    Guo Y, Batoz J, Naceur H, Bouabdallah S, Mercier F, Barlet O (2000) Recent developments on the analysis and optimum design of sheet metal forming parts using a simplified inverse approach. Comput Struct 78(1–3):133–148CrossRefGoogle Scholar
  14. 14.
    Haftka R, Grandhi R (1986) Structural shape optimization—a survey. Comput Methods Appl Mech Eng 57(1):91–106MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Herrmann L (1976) Laplacian-isoparametric grid generation scheme. J Eng Mech Div 102(5):749–756Google Scholar
  16. 16.
    Hill R (1948) A theory of the yielding and plastic flow of anisotropic metals. Proc R Soc A Math Phys Eng Sci 193(1033):281–297MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hinton E, Campbell J (1974) Local and global smoothing of discontinuous finite element functions using a least squares method. Int J Numer Methods Eng 8(3):461–480MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kim J, Kim N, Huh M (2000) Optimum blank design of an automobile sub-frame. J Mater Process Technol 101:31–43CrossRefGoogle Scholar
  19. 19.
    Kim N, Choi K, Chen J, Park Y (2000) Meshless shape design sensitivity analysis and optimization for contact problem with friction. Comput Mech 25:157–168CrossRefzbMATHGoogle Scholar
  20. 20.
    Kleinermann J, Ponthot J (2003) Parameter identification and shape/process optimization in metal forming simulation. J Mater Process Technol 139(1–3):521–526CrossRefzbMATHGoogle Scholar
  21. 21.
    Landkammer P, Loderer A, Krebs E, Söhngen B, Steinmann P, Hausotte T, Kersting P, Biermann D, Willner K (2015) Experimental verification of a benchmark forming simulation. Key Eng Mater 639:251–258CrossRefGoogle Scholar
  22. 22.
    Landkammer P, Schulte R, Steinmann P, Merklein M (2016) A non-invasive form finding methods with application to metal forming. Prod Eng 10(1):93–102CrossRefGoogle Scholar
  23. 23.
    Landkammer P, Söhngen B, Steinmann P, Willner K (2017) On gradient-based optimization strategies for inverse problems in metal forming. GAMM-Mitteilungen (accepted)Google Scholar
  24. 24.
    Landkammer P, Steinmann P (2016) A non-invasive heuristic approach to shape optimization. Comput Mech 57(2):169–191MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Le C, Bruns T, Tortorelli D (2011) A gradient-based, parameter-free approach to shape optimization. Comput Methods Appl Mech Eng 200(9–12):985–996MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lee C, Huh H (1998) Three dimensional multi-step inverse analysis for the optimum blank design in sheet metal forming processes. J Mater Process Technol 80–81:76–82CrossRefGoogle Scholar
  27. 27.
    Michaleris P, Tortorelli D, Vidal C (1994) Tangent operators and design sensitivity formulations for transient nonlinear coupled problems with applications to elastoplasticty. Int J Numer Methods Eng 37(14):2471–2499CrossRefzbMATHGoogle Scholar
  28. 28.
    Naceur H, Delameziere A, Batoz J, Guo YQ, Knopf-Lenoir C (2004) Some improvements on the optimum process design in deep drawing using the inverse approach. J Mater Process Technol. doi: 10.1016/j.matprotec.2003.11.015 Google Scholar
  29. 29.
    Nocedal J, Wright S (2006) Numerical optimization. Springer, New YorkzbMATHGoogle Scholar
  30. 30.
    Padmanabhan R, Oliveira M, Baptista A, Alves J, Menezes L (2009) Blank design for deep drawing parts using parametric nurbs surfaces. J Mater Process Technol 209(5–6):2402–2411CrossRefGoogle Scholar
  31. 31.
    Park M, Suh Y, Song S (2012) On an implementation of the strain gradient plasticity with linear finite elements and reduced integration. Finite Elem Anal Des 59:35–43MathSciNetCrossRefGoogle Scholar
  32. 32.
    Park S, Yoon J, Yang D, Kim Y (1999) Optimum blank design in sheet metal forming by the deformation path iteration method. Int J Mech Sci 41(10):1217–1232. doi: 10.1016/s0020-7403(98)00084-8 CrossRefzbMATHGoogle Scholar
  33. 33.
    Scherer M (2011) Regularizing constraints for mesh and shape optimization problems. Schriftenreihe Technische Mechanik - Band 5, Friedrich-Alexander-Universität Erlangen-Nürnberg, Lehrstuhl für Technische MechanikGoogle Scholar
  34. 34.
    Scherer M, Denzer R, Steinmann P (2009) A fictitious energy approach for shape optimization. Int J Numer Methods Eng 82:269–302. doi: 10.1002/nme.2764 MathSciNetzbMATHGoogle Scholar
  35. 35.
    Schmitt O, Friederich J, Riehl S, Steinmann P (2015) On the formulation and implementation of geometric and manufacturing constraints in node-based shape optimization. Struct Multidiscip Optim 53(4):881–892. doi: 10.1007/s00158-016-1595-y MathSciNetCrossRefGoogle Scholar
  36. 36.
    Sellier M (2005) Fixed point iterative schemes for initial shape identification. Tech Mech 25(3–4):208–217Google Scholar
  37. 37.
    Steinmann P (1997) Modellierung und Numerik duktiler kristalliner Werkstoffe. Habilitationsschrift, Forschungs- und Seminarberichte aus dem Bereich der Mechanik der Universität Hannover, F 97/1, Univeristät HannoverGoogle Scholar
  38. 38.
    Steinmann P (2015) Geometrical foundations of continuum mechanics. Springer, HeidelbergCrossRefzbMATHGoogle Scholar
  39. 39.
    Stupkiewicz S, Lengiewicz J, Korelc J (2010) Sensitivity analysis for frictional contact problems in the augmented langrangian formulation. Comput Methods Appl Mech Eng 199:2165–2176CrossRefzbMATHGoogle Scholar
  40. 40.
    Tortorelli D, Michaleris P (1994) Design sensitivity analysis: overview and review. Inverse Probl Eng 1(1):71–105CrossRefGoogle Scholar
  41. 41.
    Wallin M, Ristinmaa M (2015) Topology optimization utilizing inverse motion based form finding. Comput Methods Appl Mech Eng 289:316–331MathSciNetCrossRefGoogle Scholar
  42. 42.
    Yao T, Choi K (1989) 3-D shape optimal design and automatic finite element regridding. Int J Numer Methods Eng 28:369–384CrossRefzbMATHGoogle Scholar
  43. 43.
    Zienkiewicz O, Zhu J (1992) The superconvergent patch recovery and a posteriori error estimates. Part 1: the recovery technique. Int J Numer Methods Eng 33(7):1331–1364CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Chair of Applied MechanicsFriedrich-Alexander-University of Erlangen-NurembergErlangenGermany

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