Adaptive Finite Elements and Mathematical Optimization Methods

Conference paper
Part of the Lecture Notes in Production Engineering book series (LNPE)


This chapter focuses on the topics of mathematical research in the priority program and on the application of the results to engineering problems. First, a posteriori estimates and adaptive methods based on them are presented for contact problems. In detail, frictional contact problems using a linear elastic material law, which are discretized by a dual-dual method, are considered. Using a different approach, adaptive methods are derived for elasto-plasticity with contact and friction. Then results for contact problems involving inertial effects are introduced. Second, parameter identification and inverse problems are considered. After an introduction into the general problem of setting and solution techniques, the balancing of a rotating system is discussed as a prototypical industrial application.


Variational Inequality Contact Problem Posteriori Error Adaptive Method Posteriori Error Estimate 
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  1. 1.
    Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. John Wiley & Sons, New York (2000)zbMATHCrossRefGoogle Scholar
  2. 2.
    Andres, M.: Dual-dual formulations formulations for frictional contact problems in mechanics. Ph.D. thesis, Institute of Applied Mathematics, Leibniz Universität Hannover (2011)Google Scholar
  3. 3.
    Arnold, D.N., Brezzi, F., Douglas Jr., J.: PEERS: a new mixed finite element for plane elasticity. Japan J. Appl. Math. 1(2), 347–367 (1984),, doi:10.1007/BF03167064MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Basel (2003)zbMATHGoogle Scholar
  5. 5.
    Biermann, D., Blum, H., Jansen, T., Rademacher, A., Scheidler, A.V., Schröder, A., Weinert, K.: Space adaptive finite element methods for dynamic Signorini problems in the simulation of the NC-shape grinding process. In: Proceedings of the 1st CIRP International Conference on Process Machine Interactions (PMI), pp. 309–316 (2008)Google Scholar
  6. 6.
    Blum, H., Jansen, T., Rademacher, A., Weinert, K.: Finite elements in space and time for dynamic contact problems. Int. J. Numer. Meth. Engng. 76, 1632–1644 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Blum, H., Rademacher, A., Schröder, A.: Space adaptive finite element methods for dynamic obstacle problems. ETNA, Electron. Trans. Numer. Anal. 32, 162–172 (2008)zbMATHGoogle Scholar
  8. 8.
    Blum, H., Rademacher, A., Schröder, A.: Space adaptive finite element methods for dynamic signorini problems. Comput. Mech. 44(4), 481–491 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bostan, V., Han, W., Reddy, B.D.: A posteriori error estimation and adaptive solution of elliptic variational inequalities of the second kind. Appl. Numer. Math. 52(1), 13–38 (2005), doi:10.1016/j.apnum.2004.06.012MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Braack, M., Ern, A.: A posteriori control of modeling errors and discretisation errors. Multiscale Model. Simul. 1(2), 221–238 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Brandt, C., Krause, A., Brinksmeier, E., Maaß, P.: Force modelling in diamond machining with regard to the surface generation process. In: Proceedings of the 9th International Conference and Exhibition on Laser Metrology, Machine Tool, CMM and Robotic Performance, LAMDAMAP 2009, London, June 30-July 2, pp. 377–386 (2009)Google Scholar
  12. 12.
    Brandt, C., Niebsch, J., Ramlau, R., Maass, P.: Modeling the influence of unbalances for ultra-precision cutting processes. Zeitschrift für Angewandte Mathematik und Mechanik (2011) (in press)Google Scholar
  13. 13.
    Carstensen, C.: Numerical analysis of the primal problem of elastoplasticity with hardening. Numer. Math. 82(4), 577–597 (1999), doi:10.1007/s002110050431MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Carstensen, C., Dolzmann, G.: A posteriori error estimates for mixed FEM in elasticity. Numer. Math. 81(2), 187–209 (1998),, doi:10.1007/s002110050389MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Carstensen, C., Klose, R., Orlando, A.: Reliable and efficient equilibrated a posteriori error finite element error control in elastoplasticity and elastoviscoplasticity with hardening. Comput. Methods Appl. Mech. Engrg. 195(19-22), 2574–2598 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Carstensen, C., Valdman, J., Brokate, M.: A quasi-static boundary value problem in multi-surface elastoplasticity: part 1 - analysis. Math. Methods Appl. Sci. 27, 1697–1710 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Carstensen, C., Valdman, J., Brokate, M.: A quasi-static boundary value problem in multi-surface elastoplasticity: part 2 - numerical solution. Math. Methods Appl. Sci. 28, 881–901 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Czekanski, A., El-Abbasi, N., Meguid, S.A., Refaat, M.H.: On the elastodynamic solution of frictional contact problems using variational inequalities. Int. J. Numer. Meth. Engng. 50, 611–627 (2001)zbMATHCrossRefGoogle Scholar
  19. 19.
    Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Communications in Pure and Applied Mathematics 57(11), 1413–1457 (2004)zbMATHCrossRefGoogle Scholar
  20. 20.
    Ekeland, I., Témam, R.: Convex analysis and variational problems. Classics in Applied Mathematics, vol. 28, english edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1999); Translated from the FrenchGoogle Scholar
  21. 21.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Mathematics and its Applications, vol. 375. Kluwer Academic Publishers Group, Dordrecht (2000)Google Scholar
  22. 22.
    Glowinski, R., Lions, J.-L., Trémolières, R.: Numerical analysis of variational inequalities. Studies in Mathematics and its Applications, vol. 8. North-Holland Publishing Co., Amsterdam (1981); Translated from the FrenchGoogle Scholar
  23. 23.
    Han, W.: Finite element analysis of a holonomic elastic-plastic problem. Numer. Math. 60(4), 493–508 (1992), doi:10.1007/BF01385733MathSciNetzbMATHGoogle Scholar
  24. 24.
    Han, W., Reddy, D.: Plasticity. Springer (1999)Google Scholar
  25. 25.
    Hlaváček, I., Haslinger, J., Nečas, J., Lovíšek, J.: Solution of variational inequalities in mechanics. Applied Mathematical Sciences, vol. 66. Springer, New York (1988); Translated from the Slovak by J. JarníkGoogle Scholar
  26. 26.
    Ito, K., Kunisch, K.: Lagrange multiplier approach to variational problems and applications. Advances in Design and Control, vol. 15. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008)CrossRefGoogle Scholar
  27. 27.
    Laursen, T.A.: Computational Contact and Impact Mechanics. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  28. 28.
    Niebsch, J., Ramlau, R., Brandt, C.: On the interaction of unbalances and surface quality in ultra-precision cutting machinery. In: SIRM 2011, Darmstadt, Germany (2011)Google Scholar
  29. 29.
    Rademacher, A.: Adaptive finite element methods for nonlinear hyperbolic problems of second order. Ph.D. thesis, Technische Universität Dortmund, Published in Verlag Dr. Hut, München (2009)Google Scholar
  30. 30.
    Ramlau, R.: Morozov’s discrepancy principle for tikhonov regularization of nonlinear operators. Numer. Funct. Anal. and Optimization 23, 147–172 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Ramlau, R.: A steepest descent algorithm for the global minimization of the tikhonov-functional. Inverse Problems 18(2), 381–405 (2002), MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Ramlau, R.: Tigra - an iterative algorithm for regularizing nonlinear ill-posed problems. Inverse Problems 19(2), 433–467 (2003), MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Rannacher, R., Suttmeier, F.-T.: A posteriori error estimation and mesh adaptation for finite element models in elasto-plasticity. Comput. Methods Appl. Mech. Eng. 176(1-4), 333–361 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Richter, T.: Parral multigrid method for adaptive finite elements with application to 3d flow problems. Ph.D. thesis, Ruprecht-Karls-Universität Heidelberg (2005)Google Scholar
  35. 35.
    Scherzer, O.: The use of Morozov’s discrepancy priciple for Tikhonov regularization for solving nonlinear ill-posed problems. Computing 51, 45–60 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Schröder, A.: A posteriori Error Estimation in Mixed Finite Element Methods for Signorini’s Problem. In: Numerical Mathematics and Advanced Applications. Proceedings of ENUMATH 2009, pp. 801–808 (2010)Google Scholar
  37. 37.
    Schröder, A.: Fehlerkontrollierte adaptive h- und hp-Finite-Elemente-Methoden für Kontaktprobleme mit Anwendungen in der Fertigungstechnik. Ph.D. thesis, Universität Dortmund (2005)Google Scholar
  38. 38.
    Schröder, A.: Mixed Finite Element Methods of Higher-Order for Model Contact Problems (2009); Humboldt Universitt zu Berlin, Institute of Mathematics, Preprint 09-16Google Scholar
  39. 39.
    Schröder, A., Rademacher, A.: Goal-oriented error control in adaptive mixed FEM for Signorinis problem. Computer Methods in Applied Mechanics and Engineering 200(1-4), 345–355 (2011)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Schröder, A., Wiedemann, S.: Erro restimates in elastoplasticity using a mixed method. Applied Numerical Mathematics 61, 1031–1045 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Sokolnikoff, I.S.: Mathematical theory of elasticity, 2nd edn. McGraw-Hill Book Company, Inc., New York (1956)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institute for Applied MathematicsLeibniz Universität HannoverHannoverGermany
  2. 2.Faculty of Mathematics (LS X)Technische Universität DortmundDortmundGermany

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