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

Axel Klawonn; Patrizio Neff; Oliver Rheinbach; Stefanie Vanis

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

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topKlawonn, 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] 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] 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] 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] 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] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal.63 (1977) 337–403. Zbl0368.73040MR475169
- [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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] P. Gosselet and C. Rey, Non-overlapping domain decomposition methods in structural mechanics. Arch. Comput. Methods Eng.13 (2006) 515–572. Zbl1171.74041MR2303317
- [20] G.A. Holzapfel, Nonlinear Solid Mechanics. A continuum approach for engineering. Wiley (2000). Zbl0980.74001MR1827472
- [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] A. Klawonn and O. Rheinbach, Inexact FETI-DP methods. Int. J. Numer. Methods Eng.69 (2007) 284–307. Zbl1194.74420MR2283893
- [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] 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] 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] A. Klawonn and O.B. Widlund, Dual-Primal FETI Methods for Linear Elasticity. Commun. Pure Appl. Math. LIX (2006) 1523–1572. Zbl1110.74053MR2254444
- [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] 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] 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] 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] 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] 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] 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] 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] 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] J. Mandel and R. Tezaur, Convergence of a Substructuring Method with Lagrange Multipliers. Numer. Math. 73 (1996) 473–487. Zbl0880.65087MR1393176
- [37] J. Mandel and R. Tezaur, On the convergence of a dual-primal substructuring method. Numer. Math.88 (2001) 543–558. Zbl1003.65126MR1835470
- [38] P. Neff, On Korn's first inequality with nonconstant coefficients. Proc. Roy. Soc. Edinb. A132 (2002) 221–243. Zbl1143.74311MR1884478
- [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] 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] 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] P. Neff, Local existence and uniqueness for quasistatic finite plasticity with grain boundary relaxation. Quart. Appl. Math.63 (2005) 88–116. Zbl1072.74013MR2126571
- [43] P. Neff, Existence of minimizers for a finite-strain micromorphic elastic solid. Proc. Roy. Soc. Edinb. A136 (2006) 997–1012. Zbl1106.74010MR2266397
- [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] 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] 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] W. Pompe, Korn's first inequality with variable coefficients and its generalizations. Comment. Math. Univ. Carolinae44 (2003) 57–70. Zbl1098.35042MR2045845
- [48] A. Quarteroni and A. Valli, Numerical Approxiamtion of Partial Differential Equations, in Computational Mathematics 23, Springer Series, Springer, Berlin (1991). Zbl0803.65088MR1299729
- [49] A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications, Oxford (1999). Zbl0931.65118MR1857663
- [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] 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] 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] 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] E.N. Spadaro, Non-uniqueness of minimizers for strictly polyconvex functionals. Arch. Rat. Mech. Anal.193 (2009) 659–678. Zbl1170.49032MR2525114
- [55] A. Toselli and O. Widlund, Domain Decomposition Methods – Algorithms and Theory, Springer Series in Computational Mathematics 34. Springer (2004). Zbl1069.65138
- [56] T. Valent, Boundary Value Problems of Finite Elasticity. Springer, Berlin (1988). Zbl0648.73019MR917733
- [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] 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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