FETI-DP domain decomposition methods for elasticity with structural changes: P-elasticity

Axel Klawonn; Patrizio Neff; Oliver Rheinbach; Stefanie Vanis

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2011)

  • Volume: 45, Issue: 3, page 563-602
  • ISSN: 0764-583X

Abstract

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We consider linear elliptic systems which arise in coupled elastic continuum mechanical models. In these systems, the strain tensor εP := sym (P-1∇u) is redefined to include a matrix valued inhomogeneity P(x) which cannot be described by a space dependent fourth order elasticity tensor. Such systems arise naturally in geometrically exact plasticity or in problems with eigenstresses. The tensor field P induces a structural change of the elasticity equations. For such a model the FETI-DP method is formulated and a convergence estimate is provided for the special case that P-T = ∇ψ is a gradient. It is shown that the condition number depends only quadratic-logarithmically on the number of unknowns of each subdomain. The dependence of the constants of the bound on P is highlighted. Numerical examples confirm our theoretical findings. Promising results are also obtained for settings which are not covered by our theoretical estimates.

How to cite

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Klawonn, Axel, et al. "FETI-DP domain decomposition methods for elasticity with structural changes: P-elasticity." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 45.3 (2011): 563-602. <http://eudml.org/doc/273226>.

@article{Klawonn2011,
abstract = {We consider linear elliptic systems which arise in coupled elastic continuum mechanical models. In these systems, the strain tensor εP := sym (P-1∇u) is redefined to include a matrix valued inhomogeneity P(x) which cannot be described by a space dependent fourth order elasticity tensor. Such systems arise naturally in geometrically exact plasticity or in problems with eigenstresses. The tensor field P induces a structural change of the elasticity equations. For such a model the FETI-DP method is formulated and a convergence estimate is provided for the special case that P-T = ∇ψ is a gradient. It is shown that the condition number depends only quadratic-logarithmically on the number of unknowns of each subdomain. The dependence of the constants of the bound on P is highlighted. Numerical examples confirm our theoretical findings. Promising results are also obtained for settings which are not covered by our theoretical estimates.},
author = {Klawonn, Axel, Neff, Patrizio, Rheinbach, Oliver, Vanis, Stefanie},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {FETU-DP; plasticity; eigenstresses; inhomogeneity; extended elasticity; structural changes; micromorphic model; FETI-DP},
language = {eng},
number = {3},
pages = {563-602},
publisher = {EDP-Sciences},
title = {FETI-DP domain decomposition methods for elasticity with structural changes: P-elasticity},
url = {http://eudml.org/doc/273226},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Klawonn, Axel
AU - Neff, Patrizio
AU - Rheinbach, Oliver
AU - Vanis, Stefanie
TI - FETI-DP domain decomposition methods for elasticity with structural changes: P-elasticity
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2011
PB - EDP-Sciences
VL - 45
IS - 3
SP - 563
EP - 602
AB - We consider linear elliptic systems which arise in coupled elastic continuum mechanical models. In these systems, the strain tensor εP := sym (P-1∇u) is redefined to include a matrix valued inhomogeneity P(x) which cannot be described by a space dependent fourth order elasticity tensor. Such systems arise naturally in geometrically exact plasticity or in problems with eigenstresses. The tensor field P induces a structural change of the elasticity equations. For such a model the FETI-DP method is formulated and a convergence estimate is provided for the special case that P-T = ∇ψ is a gradient. It is shown that the condition number depends only quadratic-logarithmically on the number of unknowns of each subdomain. The dependence of the constants of the bound on P is highlighted. Numerical examples confirm our theoretical findings. Promising results are also obtained for settings which are not covered by our theoretical estimates.
LA - eng
KW - FETU-DP; plasticity; eigenstresses; inhomogeneity; extended elasticity; structural changes; micromorphic model; FETI-DP
UR - http://eudml.org/doc/273226
ER -

References

top
  1. [1] S. Balay, W.D. Gropp, L.C. McInnes and B.F. Smith, Efficient management of parallelism in object oriented numerical software libraries, in Modern Software Tools in Scientific Computing, E. Arge, A.M. Bruaset and H.P. Langtangen Eds., Birkhäuser Press (1997) 163–202. Zbl0882.65154MR1452877
  2. [2] S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M. Knepley, L.C. McInnes, B.F. Smith and H. Zhang, PETSc users manual. Technical Report ANL-95/11 – Revision 2.2.3, Argonne National Laboratory (2007). 
  3. [3] S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith and H. Zhang, PETSc Web page, http://www.mcs.anl.gov/petsc (2009). 
  4. [4] J.M. Ball, Constitutive inequalities and existence theorems in nonlinear elastostatics, in Herriot Watt Symposion: Nonlinear Analysis and Mechanics 1, R.J. Knops Ed., Pitman, London (1977) 187–238. Zbl0377.73043MR478899
  5. [5] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal.63 (1977) 337–403. Zbl0368.73040MR475169
  6. [6] J.M. Ball, Some open problems in elasticity, in Geometry, mechanics, and dynamics, P. Newton, P. Holmes and A. Weinstein Eds., Springer, New York (2002) 3–59. Zbl1054.74008MR1919825
  7. [7] D. Balzani, P. Neff, J. Schröder and G.A. Holzapfel, A polyconvex framework for soft biological tissues. Adjustment to experimental data. Int. J. Solids Struct. 43 (2006) 6052–6070. Zbl1120.74632MR2265177
  8. [8] P.E. Bjørstad and O.B. Widlund, Iterative methods for the solution of elliptic problems on regions partitioned into substructures. SIAM J. Numer. Anal.23 (1986) 1093–1120. Zbl0615.65113MR865945
  9. [9] D. Brands, A. Klawonn, O. Rheinbach and J. Schröder, Modelling and convergence in arterial wall simulations using a parallel FETI solution strategy. Comput. Methods Biomech. Biomed. Eng.11 (2008) 569–583. 
  10. [10] M. Dryja, A method of domain decomposition for three-dimensional finite element elliptic problem, in First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987), SIAM, Philadelphia (1988) 43–61. Zbl0661.65106MR972511
  11. [11] M. Dryja, B.F. Smith and O.B. Widlund, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions. SIAM J. Numer. Anal.31 (1994) 1662–1694. Zbl0818.65114MR1302680
  12. [12] C. Farhat and J. Mandel, The two-level FETI method for static and dynamic plate problems – part I: An optimal iterative solver for biharmonic systems. Comput. Methods Appl. Mech. Eng.155 (1998) 129–152. Zbl0964.74062MR1619501
  13. [13] C. Farhat and F.-X. Roux, A method of Finite Element Tearing and Interconnecting and its parallel solution algorithm. Int. J. Numer. Meth. Eng. 32 (1991) 1205–1227. Zbl0758.65075
  14. [14] C. Farhat and F.-X. Roux, Implicit parallel processing in structural mechanics, in Computational Mechanics Advances 2, J. Tinsley Oden Ed., North-Holland (1994) 1–124. Zbl0805.73062MR1280753
  15. [15] C. Farhat, J. Mandel and F.X. Roux, Optimal convergence properties of the FETI domain decomposition method. Comput. Methods Appl. Mech. Eng.115 (1994) 367–388. MR1285024
  16. [16] C. Farhat, M. Lesoinne and K. Pierson, A scalable dual-primal domain decomposition method. Numer. Lin. Alg. Appl. 7 (2000) 687–714. Zbl1051.65119MR1802366
  17. [17] C. Farhat, K.H. Pierson and M. Lesoinne, The second generation of FETI methods and their application to the parallel solution of large-scale linear and geometrically nonlinear structural analysis problems. Comput. Meth. Appl. Mech. Eng.184 (2000) 333–374. Zbl0981.74064
  18. [18] C. Farhat, M. Lesoinne, P. Le Tallec, K. Pierson and D. Rixen, FETI-DP: A dual-primal unified FETI method – part I: A faster alternative to the two-level FETI method. Int. J. Numer. Meth. Eng. 50 (2001) 1523–1544. Zbl1008.74076MR1813746
  19. [19] P. Gosselet and C. Rey, Non-overlapping domain decomposition methods in structural mechanics. Arch. Comput. Methods Eng.13 (2006) 515–572. Zbl1171.74041MR2303317
  20. [20] G.A. Holzapfel, Nonlinear Solid Mechanics. A continuum approach for engineering. Wiley (2000). Zbl0980.74001MR1827472
  21. [21] A. Klawonn and O. Rheinbach, A parallel implementation of Dual-Primal FETI methods for three dimensional linear elasticity using a transformation of basis. SIAM J. Sci. Comput.28 (2006) 1886–1906. Zbl1124.74049MR2272193
  22. [22] A. Klawonn and O. Rheinbach, Inexact FETI-DP methods. Int. J. Numer. Methods Eng.69 (2007) 284–307. Zbl1194.74420MR2283893
  23. [23] A. Klawonn and O. Rheinbach, Robust FETI-DP methods for heterogeneous three dimensional elasticity problems. Comput. Methods Appl. Mech. Eng.196 (2007) 1400–1414. Zbl1173.74428MR2277025
  24. [24] A. Klawonn and O. Rheinbach, Highly scalable parallel domain decomposition methods with an application to biomechanics. Z. Angew. Math. Mech. (ZAMM) 90 (2010) 5–32. Zbl05662111MR2603676
  25. [25] A. Klawonn and O.B. Widlund, FETI and Neumann-Neumann iterative substructuring methods: connections and new results. Commun. Pure Appl. Math.54 (2001) 57–90. Zbl1023.65120MR1787107
  26. [26] A. Klawonn and O.B. Widlund, Dual-Primal FETI Methods for Linear Elasticity. Commun. Pure Appl. Math. LIX (2006) 1523–1572. Zbl1110.74053MR2254444
  27. [27] A. Klawonn, O.B. Widlund and M. Dryja, Dual-Primal FETI methods for three-dimensional elliptic problems with heterogeneous coefficients. SIAM J. Numer. Anal.40 (2002) 159–179. Zbl1032.65031MR1921914
  28. [28] A. Klawonn, O. Rheinbach and O.B. Widlund, Some computational results for dual-primal FETI methods for elliptic problems in 3D, in Proceedings of the 15th international domain decomposition conference, R. Kornhuber, R.H.W. Hoppe, J. Périaux, O. Pironneau, O.B. Widlund and J. Xu Eds., Springer LNCSE, Lect. Notes Comput. Sci. Eng., Berlin (2005) 361–368. Zbl1067.65128MR2235762
  29. [29] A. Klawonn, L.F. Pavarino and O. Rheinbach, Spectral element FETI-DP and BDDC preconditioners with multi-element subdomains. Comput. Meth. Appl. Mech. Eng.198 (2008) 511–523. Zbl1228.74084MR2479280
  30. [30] A. Klawonn, O. Rheinbach and O.B. Widlund, An analysis of a FETI–DP algorithm on irregular subdomains in the plane. SIAM J. Numer. Anal.46 (2008) 2484–2504. Zbl1176.65135MR2421044
  31. [31] A. Klawonn, P. Neff, O. Rheinbach and S. Vanis, Notes on FETI-DP domain decomposition methods for P-elasticity. Technical report, Universität Duisburg-Essen, Fakultät für Mathematik, http://www.numerik.uni-due.de/publications.shtml (2010). Zbl1268.74037
  32. [32] A. Klawonn, P. Neff, O. Rheinbach and S. Vanis, Solving geometrically exact micromorphic elasticity with a staggered algorithm. GAMM Mitteilungen33 (2010) 57–72. Zbl1328.74079MR2844393
  33. [33] U. Langer, G. Of, O. Steinbach and W. Zulehner, Inexact data-sparse boundary element tearing and interconnecting methods. SIAM J. Sci. Comput.29 (2007) 290–314. Zbl1133.65105MR2285892
  34. [34] P. Le Tallec, Numerical methods for non-linear three-dimensional elasticity, in Handbook of numerical analysis 3, J.L. Lions and P. Ciarlet Eds., Elsevier (1994) 465–622. Zbl0875.73234MR1307410
  35. [35] J. Li and O.B. Widlund, FETI-DP, BDDC and Block Cholesky Methods. Int. J. Numer. Methods Eng.66 (2006) 250–271. Zbl1114.65142MR2224479
  36. [36] J. Mandel and R. Tezaur, Convergence of a Substructuring Method with Lagrange Multipliers. Numer. Math. 73 (1996) 473–487. Zbl0880.65087MR1393176
  37. [37] J. Mandel and R. Tezaur, On the convergence of a dual-primal substructuring method. Numer. Math.88 (2001) 543–558. Zbl1003.65126MR1835470
  38. [38] P. Neff, On Korn's first inequality with nonconstant coefficients. Proc. Roy. Soc. Edinb. A132 (2002) 221–243. Zbl1143.74311MR1884478
  39. [39] P. Neff, Finite multiplicative plasticity for small elastic strains with linear balance equations and grain boundary relaxation. Contin. Mech. Thermodyn.15 (2003) 161–195. Zbl1035.74015MR1970291
  40. [40] P. Neff, A geometrically exact viscoplastic membrane-shell with viscoelastic transverse shear resistance avoiding degeneracy in the thin-shell limit. Part I: The viscoelastic membrane-plate. Z. Angew. Math. Phys. (ZAMP) 56 (2005) 148–182. Zbl1079.74039MR2112845
  41. [41] P. Neff, Local existence and uniqueness for a geometrically exact membrane-plate with viscoelastic transverse shear resistance. Math. Meth. Appl. Sci. (MMAS) 28 (2005) 1031–1060. Zbl1071.74034MR2138730
  42. [42] P. Neff, Local existence and uniqueness for quasistatic finite plasticity with grain boundary relaxation. Quart. Appl. Math.63 (2005) 88–116. Zbl1072.74013MR2126571
  43. [43] P. Neff, Existence of minimizers for a finite-strain micromorphic elastic solid. Proc. Roy. Soc. Edinb. A136 (2006) 997–1012. Zbl1106.74010MR2266397
  44. [44] P. Neff, A finite-strain elastic-plastic Cosserat theory for polycrystals with grain rotations. Int. J. Eng. Sci.44 (2006) 574–594. Zbl1213.74032MR2234090
  45. [45] P. Neff and S. Forest, A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. J. Elasticity 87 (2007) 239–276. Zbl1206.74019MR2333975
  46. [46] P. Neff and I. Münch, Simple shear in nonlinear Cosserat elasticity: bifurcation and induced microstructure. Contin. Mech. Thermodyn.21 (2009) 195–221. Zbl1179.74011MR2529452
  47. [47] W. Pompe, Korn's first inequality with variable coefficients and its generalizations. Comment. Math. Univ. Carolinae44 (2003) 57–70. Zbl1098.35042MR2045845
  48. [48] A. Quarteroni and A. Valli, Numerical Approxiamtion of Partial Differential Equations, in Computational Mathematics 23, Springer Series, Springer, Berlin (1991). Zbl0803.65088MR1299729
  49. [49] A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications, Oxford (1999). Zbl0931.65118MR1857663
  50. [50] J. Schröder and P. Neff, Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions. Int. J. Solids Struct.40 (2003) 401–445. Zbl1033.74005MR1951960
  51. [51] J. Schröder, P. Neff and D. Balzani, A variational approach for materially stable anisotropic hyperelasticity. Int. J. Solids Struct.42 (2005) 4352–4371. Zbl1119.74321MR2134350
  52. [52] J. Schröder, P. Neff and V. Ebbing, Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors. J. Mech. Phys. Solids56 (2008) 3486–3506. Zbl1171.74356MR2472997
  53. [53] B.F. Smith, P.E. Bjørstad and W. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge (1996). Zbl0857.65126MR1410757
  54. [54] E.N. Spadaro, Non-uniqueness of minimizers for strictly polyconvex functionals. Arch. Rat. Mech. Anal.193 (2009) 659–678. Zbl1170.49032MR2525114
  55. [55] A. Toselli and O. Widlund, Domain Decomposition Methods – Algorithms and Theory, Springer Series in Computational Mathematics 34. Springer (2004). Zbl1069.65138
  56. [56] T. Valent, Boundary Value Problems of Finite Elasticity. Springer, Berlin (1988). Zbl0648.73019MR917733
  57. [57] K. Weinberg and P. Neff, A geometrically exact thin membrane model-investigation of large deformations and wrinkling. Int. J. Num. Meth. Eng.74 (2007) 871–893. Zbl1158.74410MR2389138
  58. [58] O.B. Widlund, An extension theorem for finite element spaces with three applications, in Proceedings of the Second GAMM-Seminar, Kiel January 1986, Notes on Numerical Fluid Mechanics 16, Friedr. Vieweg und Sohn, Braunschweig/Wiesbaden (1987) 110–122. Zbl0615.65114

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