Advertisement

# Adaptive Finite Elements and Mathematical Optimization Methods

Conference paper
• 3.2k Downloads
Part of the Lecture Notes in Production Engineering book series (LNPE)

## Abstract

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.

## Keywords

Variational Inequality Contact Problem Posteriori Error Adaptive Method Posteriori Error Estimate
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

1. 1.
Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. John Wiley & Sons, New York (2000)
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), http://dx.doi.org/10.1007/BF03167064, doi:10.1007/BF03167064
4. 4.
Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Basel (2003)
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)
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)
8. 8.
Blum, H., Rademacher, A., Schröder, A.: Space adaptive finite element methods for dynamic signorini problems. Comput. Mech. 44(4), 481–491 (2009)
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.2004.06.012
10. 10.
Braack, M., Ern, A.: A posteriori control of modeling errors and discretisation errors. Multiscale Model. Simul. 1(2), 221–238 (2003)
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/s002110050431
14. 14.
Carstensen, C., Dolzmann, G.: A posteriori error estimates for mixed FEM in elasticity. Numer. Math. 81(2), 187–209 (1998), http://dx.doi.org/10.1007/s002110050389, doi:10.1007/s002110050389
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)
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)
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)
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)
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)
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/BF01385733
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)
27. 27.
Laursen, T.A.: Computational Contact and Impact Mechanics. Springer, Heidelberg (2002)
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)
31. 31.
Ramlau, R.: A steepest descent algorithm for the global minimization of the tikhonov-functional. Inverse Problems 18(2), 381–405 (2002), http://stacks.iop.org/0266-5611/19/i=2/a=312
32. 32.
Ramlau, R.: Tigra - an iterative algorithm for regularizing nonlinear ill-posed problems. Inverse Problems 19(2), 433–467 (2003), http://stacks.iop.org/0266-5611/19/i=2/a=312
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)
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)
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)
40. 40.
Schröder, A., Wiedemann, S.: Erro restimates in elastoplasticity using a mixed method. Applied Numerical Mathematics 61, 1031–1045 (2011)
41. 41.
Sokolnikoff, I.S.: Mathematical theory of elasticity, 2nd edn. McGraw-Hill Book Company, Inc., New York (1956)

## 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