Skip to main content

2016 | OriginalPaper | Buchkapitel

11. Frictionless Contact Problems

verfasst von : Zdeněk Dostál, Tomáš Kozubek, Vít Vondrák

Erschienen in: Scalable Algorithms for Contact Problems

Verlag: Springer New York

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

Now we shall extend the results introduced in the previous chapter to the solution of multibody contact problems of elasticity without friction. We shall restrict our attention to the problems of linear elasticity, i.e., we shall assume small deformations and linear stress–strain relations. Moreover, we shall be interested mainly in computationally challenging 3D problems.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literatur
1.
2.
Zurück zum Zitat Laursen, T.: Computational Contact and Impact Mechanics. Springer, Berlin (2002)MATH Laursen, T.: Computational Contact and Impact Mechanics. Springer, Berlin (2002)MATH
3.
Zurück zum Zitat Wriggers, P.: Contact Mechanics. Springer, Berlin (2005)MATH Wriggers, P.: Contact Mechanics. Springer, Berlin (2005)MATH
4.
Zurück zum Zitat Savenkov, E., Andrä, H., Iliev, O.: An analysis of one regularization approach for solution of pure Neumann problem. Berichte des Faruenhofer ITWM, Nr. 137, Kaiserslautern (2008) Savenkov, E., Andrä, H., Iliev, O.: An analysis of one regularization approach for solution of pure Neumann problem. Berichte des Faruenhofer ITWM, Nr. 137, Kaiserslautern (2008)
6.
Zurück zum Zitat Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia (1994)CrossRefMATH Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia (1994)CrossRefMATH
7.
Zurück zum Zitat Golub, G.H., Van Loan, C.F.: Matrix Computations, 2nd edn. Johns Hopkins University Press, Baltimore (1989)MATH Golub, G.H., Van Loan, C.F.: Matrix Computations, 2nd edn. Johns Hopkins University Press, Baltimore (1989)MATH
8.
Zurück zum Zitat Brzobohatý, T., Dostál, Z., Kozubek, T., Kovář, P., Markopoulos, A.: Cholesky decomposition with fixing nodes to stable computation of a generalized inverse of the stiffness matrix of a floating structure. Int. J. Numer. Methods Eng. 88(5), 493–509 (2011)MathSciNetCrossRefMATH Brzobohatý, T., Dostál, Z., Kozubek, T., Kovář, P., Markopoulos, A.: Cholesky decomposition with fixing nodes to stable computation of a generalized inverse of the stiffness matrix of a floating structure. Int. J. Numer. Methods Eng. 88(5), 493–509 (2011)MathSciNetCrossRefMATH
9.
Zurück zum Zitat Farhat, C., Mandel, J., Roux, F.-X.: Optimal convergence properties of the FETI domain decomposition method. Comput. Methods Appl. Mech. Eng. 115, 365–385 (1994)MathSciNetCrossRef Farhat, C., Mandel, J., Roux, F.-X.: Optimal convergence properties of the FETI domain decomposition method. Comput. Methods Appl. Mech. Eng. 115, 365–385 (1994)MathSciNetCrossRef
11.
Zurück zum Zitat Hlaváček, I., Haslinger, J., Nečas, J., Lovíšek, J.: Solution of Variational Inequalities in Mechanics. Springer, Berlin (1988)CrossRefMATH Hlaváček, I., Haslinger, J., Nečas, J., Lovíšek, J.: Solution of Variational Inequalities in Mechanics. Springer, Berlin (1988)CrossRefMATH
12.
Zurück zum Zitat Schöberl, J.: Solving the Signorini problem on the basis of domain decomposition techniques. Computing 60(4), 323–344 (1998)MathSciNetCrossRefMATH Schöberl, J.: Solving the Signorini problem on the basis of domain decomposition techniques. Computing 60(4), 323–344 (1998)MathSciNetCrossRefMATH
13.
Zurück zum Zitat Schöberl, J.: Efficient contact solvers based on domain decomposition techniques. Comput. Math. Appl. 42, 1217–1228 (2001)MathSciNetCrossRefMATH Schöberl, J.: Efficient contact solvers based on domain decomposition techniques. Comput. Math. Appl. 42, 1217–1228 (2001)MathSciNetCrossRefMATH
14.
Zurück zum Zitat Dureisseix, D., Farhat, C.: A numerically scalable domain decomposition method for solution of frictionless contact problems. Int. J. Numer. Methods Eng. 50(12), 2643–2666 (2001)CrossRefMATH Dureisseix, D., Farhat, C.: A numerically scalable domain decomposition method for solution of frictionless contact problems. Int. J. Numer. Methods Eng. 50(12), 2643–2666 (2001)CrossRefMATH
15.
Zurück zum Zitat Avery, P., Rebel, G., Lesoinne, M., Farhat, C.: A numerically scalable dual-primal substructuring method for the solution of contact problems - part I: the frictionless case. Comput. Methods Appl. Mech. Eng. 193, 2403–2426 (2004)CrossRefMATH Avery, P., Rebel, G., Lesoinne, M., Farhat, C.: A numerically scalable dual-primal substructuring method for the solution of contact problems - part I: the frictionless case. Comput. Methods Appl. Mech. Eng. 193, 2403–2426 (2004)CrossRefMATH
16.
Zurück zum Zitat Dostál, Z., Vondrák, V., Horák, D., Farhat, C., Avery, P.: Scalable FETI algorithms for frictionless contact problems. Lecture Notes in Computational Science and Engineering, vol. 60, pp. 263–270. Springer, Berlin (2008) Dostál, Z., Vondrák, V., Horák, D., Farhat, C., Avery, P.: Scalable FETI algorithms for frictionless contact problems. Lecture Notes in Computational Science and Engineering, vol. 60, pp. 263–270. Springer, Berlin (2008)
17.
Zurück zum Zitat Avery, P., Farhat, C.: The FETI family of domain decomposition methods for inequality-constrained quadratic programming: application to contact problems with conforming and nonconforming interfaces. Comput. Methods Appl. Mech. Eng. 198, 1673–1683 (2009)MathSciNetCrossRefMATH Avery, P., Farhat, C.: The FETI family of domain decomposition methods for inequality-constrained quadratic programming: application to contact problems with conforming and nonconforming interfaces. Comput. Methods Appl. Mech. Eng. 198, 1673–1683 (2009)MathSciNetCrossRefMATH
18.
Zurück zum Zitat Felippa, C.A., Park, K.C.: The construction of free-free flexibility matrices for multilevel structural analysis. Comput. Methods Appl. Mech. Eng. 191, 2111–2140 (2002)CrossRefMATH Felippa, C.A., Park, K.C.: The construction of free-free flexibility matrices for multilevel structural analysis. Comput. Methods Appl. Mech. Eng. 191, 2111–2140 (2002)CrossRefMATH
19.
Zurück zum Zitat Farhat, C., Géradin, M.: On the general solution by a direct method of a large scale singular system of linear equations: application to the analysis of floating structures. Int. J. Numer. Methods Eng. 41, 675–696 (1998)MathSciNetCrossRefMATH Farhat, C., Géradin, M.: On the general solution by a direct method of a large scale singular system of linear equations: application to the analysis of floating structures. Int. J. Numer. Methods Eng. 41, 675–696 (1998)MathSciNetCrossRefMATH
20.
Zurück zum Zitat Kornhuber, R., Krause, R.: Adaptive multigrid methods for Signorini’s problem in linear elasticity. Comput. Vis. Sci. 4(1), 9–20 (2001)MathSciNetCrossRefMATH Kornhuber, R., Krause, R.: Adaptive multigrid methods for Signorini’s problem in linear elasticity. Comput. Vis. Sci. 4(1), 9–20 (2001)MathSciNetCrossRefMATH
21.
Zurück zum Zitat Kornhuber, R., Krause, R., Sander, O., Deuflhard, P., Ertel, S.: A monotone multigrid solver for two body contact problems in biomechanics. Comput. Vis. Sci. 11, 3–15 (2008)MathSciNetCrossRef Kornhuber, R., Krause, R., Sander, O., Deuflhard, P., Ertel, S.: A monotone multigrid solver for two body contact problems in biomechanics. Comput. Vis. Sci. 11, 3–15 (2008)MathSciNetCrossRef
22.
Zurück zum Zitat Iontcheva, A.H., Vassilevski, P.S.: Monotone multigrid methods based on element agglomeration coarsening away from the contact boundary for the Signorini’s problem. Numer. Linear Algebra Appl. 11(2–3), 189–204 (2004)MathSciNetCrossRefMATH Iontcheva, A.H., Vassilevski, P.S.: Monotone multigrid methods based on element agglomeration coarsening away from the contact boundary for the Signorini’s problem. Numer. Linear Algebra Appl. 11(2–3), 189–204 (2004)MathSciNetCrossRefMATH
23.
Zurück zum Zitat Simo, J.C., Laursen, T.A.: An augmented Lagrangian treatment of contact problems involving friction. Comput. Struct. 42, 97–116 (1992)MathSciNetCrossRefMATH Simo, J.C., Laursen, T.A.: An augmented Lagrangian treatment of contact problems involving friction. Comput. Struct. 42, 97–116 (1992)MathSciNetCrossRefMATH
24.
Zurück zum Zitat Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia (1989)CrossRefMATH Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia (1989)CrossRefMATH
25.
Zurück zum Zitat Conn, A.R., Gould, N.I.M., Toint, Ph.L.: LANCELOT: A FORTRAN Package for Large Scale Nonlinear Optimization (Release A). No. 17 in Springer Series in Computational Mathematics. Springer, New York (1992) Conn, A.R., Gould, N.I.M., Toint, Ph.L.: LANCELOT: A FORTRAN Package for Large Scale Nonlinear Optimization (Release A). No. 17 in Springer Series in Computational Mathematics. Springer, New York (1992)
26.
Zurück zum Zitat Dostál, Z., Friedlander, A., Santos, S.A.: Solution of contact problems of elasticity by FETI domain decomposition. Domain Decomposition Methods 10. Contemporary Mathematics, vol. 218, 82–93. AMS, Providence (1998) Dostál, Z., Friedlander, A., Santos, S.A.: Solution of contact problems of elasticity by FETI domain decomposition. Domain Decomposition Methods 10. Contemporary Mathematics, vol. 218, 82–93. AMS, Providence (1998)
27.
Zurück zum Zitat Dostál, Z., Gomes, F.A.M., Santos, S.A.: Duality based domain decomposition with natural coarse space for variational inequalities. J. Comput. Appl. Math. 126(1–2), 397–415 (2000)MathSciNetCrossRefMATH Dostál, Z., Gomes, F.A.M., Santos, S.A.: Duality based domain decomposition with natural coarse space for variational inequalities. J. Comput. Appl. Math. 126(1–2), 397–415 (2000)MathSciNetCrossRefMATH
28.
Zurück zum Zitat Dostál, Z., Gomes, F.A.M., Santos, S.A.: Solution of contact problems by FETI domain decomposition with natural coarse space projection. Comput. Methods Appl. Mech. Eng. 190(13–14), 1611–1627 (2000)CrossRefMATH Dostál, Z., Gomes, F.A.M., Santos, S.A.: Solution of contact problems by FETI domain decomposition with natural coarse space projection. Comput. Methods Appl. Mech. Eng. 190(13–14), 1611–1627 (2000)CrossRefMATH
29.
Zurück zum Zitat Dostál, Z., Horák, D., Kučera, R., Vondrák, V., Haslinger, J., Dobiáš, J., Pták, S.: FETI based algorithms for contact problems: scalability, large displacements and 3D Coulomb friction. Comput. Methods Appl. Mech. Eng. 194(2–5), 395–409 (2005)MathSciNetCrossRefMATH Dostál, Z., Horák, D., Kučera, R., Vondrák, V., Haslinger, J., Dobiáš, J., Pták, S.: FETI based algorithms for contact problems: scalability, large displacements and 3D Coulomb friction. Comput. Methods Appl. Mech. Eng. 194(2–5), 395–409 (2005)MathSciNetCrossRefMATH
30.
Zurück zum Zitat Dostál, Z., Kozubek, T., Vondrák, V., Brzobohatý, T., Markopoulos, A.: Scalable TFETI algorithm for the solution of multibody contact problems of elasticity. Int. J. Numer. Methods Eng. 82(11), 1384–1405 (2010)MathSciNetMATH Dostál, Z., Kozubek, T., Vondrák, V., Brzobohatý, T., Markopoulos, A.: Scalable TFETI algorithm for the solution of multibody contact problems of elasticity. Int. J. Numer. Methods Eng. 82(11), 1384–1405 (2010)MathSciNetMATH
31.
Zurück zum Zitat Dostál, Z., Schöberl, J.: Minimizing quadratic functions over non-negative cone with the rate of convergence and finite termination. Comput. Optim. Appl. 30(1), 23–44 (2005)MathSciNetCrossRefMATH Dostál, Z., Schöberl, J.: Minimizing quadratic functions over non-negative cone with the rate of convergence and finite termination. Comput. Optim. Appl. 30(1), 23–44 (2005)MathSciNetCrossRefMATH
32.
Zurück zum Zitat Dostál, Z.: Inexact semi-monotonic augmented Lagrangians with optimal feasibility convergence for quadratic programming with simple bounds and equality constraints. SIAM J. Numer. Anal. 43(1), 96–115 (2005)MathSciNetCrossRefMATH Dostál, Z.: Inexact semi-monotonic augmented Lagrangians with optimal feasibility convergence for quadratic programming with simple bounds and equality constraints. SIAM J. Numer. Anal. 43(1), 96–115 (2005)MathSciNetCrossRefMATH
33.
Zurück zum Zitat Dostál, Z.: Optimal Quadratic Programming Algorithms, with Applications to Variational Inequalities, 1st edn. Springer, New York (2009)MATH Dostál, Z.: Optimal Quadratic Programming Algorithms, with Applications to Variational Inequalities, 1st edn. Springer, New York (2009)MATH
34.
Zurück zum Zitat Dostál, Z., Horák, D., Kučera, R.: Total FETI - an easier implementable variant of the FETI method for numerical solution of elliptic PDE. Commun. Numer. Methods Eng. 22, 1155–1162 (2006)MathSciNetCrossRefMATH Dostál, Z., Horák, D., Kučera, R.: Total FETI - an easier implementable variant of the FETI method for numerical solution of elliptic PDE. Commun. Numer. Methods Eng. 22, 1155–1162 (2006)MathSciNetCrossRefMATH
35.
Zurück zum Zitat Toselli, A., Widlund, O.B.: Domain Decomposition Methods – Algorithms and Theory. Springer Series on Computational Mathematics, vol. 34. Springer, Berlin (2005) Toselli, A., Widlund, O.B.: Domain Decomposition Methods – Algorithms and Theory. Springer Series on Computational Mathematics, vol. 34. Springer, Berlin (2005)
36.
Zurück zum Zitat Jarošová, M., Klawonn, A., Rheinbach, O.: Projector preconditioning and transformation of basis in FETI-DP algorithms for contact problems. Math. Comput. Simul. 82(10), 1894–1907 (2012)MathSciNetCrossRefMATH Jarošová, M., Klawonn, A., Rheinbach, O.: Projector preconditioning and transformation of basis in FETI-DP algorithms for contact problems. Math. Comput. Simul. 82(10), 1894–1907 (2012)MathSciNetCrossRefMATH
Metadaten
Titel
Frictionless Contact Problems
verfasst von
Zdeněk Dostál
Tomáš Kozubek
Vít Vondrák
Copyright-Jahr
2016
Verlag
Springer New York
DOI
https://doi.org/10.1007/978-1-4939-6834-3_11