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

Published online by Cambridge University Press:  30 November 2010

Axel Klawonn ,
Patrizio Neff ,
Oliver Rheinbach  and
Stefanie Vanis
Axel Klawonn
Affiliation:
Fakultät für Mathematik, Universität Duisburg-Essen, Campus Essen, Universitätsstraße 3, 45117 Essen, Germany. axel.klawonn@uni-duisburg-essen.de; patrizio.neff@uni-duisburg-essen.de; oliver.rheinbach@uni-duisburg-essen.de; stefanie.vanis@uni-duisburg-essen.de
Patrizio Neff
Affiliation:
Fakultät für Mathematik, Universität Duisburg-Essen, Campus Essen, Universitätsstraße 3, 45117 Essen, Germany. axel.klawonn@uni-duisburg-essen.de; patrizio.neff@uni-duisburg-essen.de; oliver.rheinbach@uni-duisburg-essen.de; stefanie.vanis@uni-duisburg-essen.de
Oliver Rheinbach
Affiliation:
Fakultät für Mathematik, Universität Duisburg-Essen, Campus Essen, Universitätsstraße 3, 45117 Essen, Germany. axel.klawonn@uni-duisburg-essen.de; patrizio.neff@uni-duisburg-essen.de; oliver.rheinbach@uni-duisburg-essen.de; stefanie.vanis@uni-duisburg-essen.de
Stefanie Vanis
Affiliation:
Fakultät für Mathematik, Universität Duisburg-Essen, Campus Essen, Universitätsstraße 3, 45117 Essen, Germany. axel.klawonn@uni-duisburg-essen.de; patrizio.neff@uni-duisburg-essen.de; oliver.rheinbach@uni-duisburg-essen.de; stefanie.vanis@uni-duisburg-essen.de
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Abstract

We consider linear elliptic systems which arisein coupled elastic continuum mechanical models. In these systems, the straintensor ε P := sym (P -1∇u) is redefined to include amatrix valued inhomogeneity P(x) which cannot be described by a spacedependent fourth order elasticity tensor. Such systems arise naturally ingeometrically exact plasticity or in problems with eigenstresses.The tensor field P induces a structural change of the elasticity equations. Forsuch a model the FETI-DP method is formulated and a convergence estimateis provided for the special case that P -T = ∇ψ is a gradient.It is shown that the condition number depends only quadratic-logarithmicallyon the number of unknowns of each subdomain. Thedependence of the constants of the bound on P is highlighted. Numericalexamples confirm our theoretical findings. Promising results are also obtainedfor settings which are not covered by our theoretical estimates.

Keywords

FETI-DPplasticityeigenstressesinhomogeneityextended elasticitystructural changesmicromorphic model
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 3 , May 2011 , pp. 563 - 602
Copyright
© EDP Sciences, SMAI, 2010

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References

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.
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).
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).
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.
Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal. 63 (1977) 337–403. CrossRef
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.
Balzani, D., Neff, P., Schröder, J. and Holzapfel, G.A., A polyconvex framework for soft biological tissues. Adjustment to experimental data. Int. J. Solids Struct. 43 (2006) 6052–6070. CrossRef
Bjørstad, P.E. and Widlund, O.B., Iterative methods for the solution of elliptic problems on regions partitioned into substructures. SIAM J. Numer. Anal. 23 (1986) 1093–1120. CrossRef
Brands, D., Klawonn, A., Rheinbach, O. and Schröder, J., Modelling and convergence in arterial wall simulations using a parallel FETI solution strategy. Comput. Methods Biomech. Biomed. Eng. 11 (2008) 569–583. CrossRef
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.
Dryja, M., Smith, B.F. and Widlund, O.B., Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions. SIAM J. Numer. Anal. 31 (1994) 1662–1694. CrossRef
Farhat, C. and Mandel, J., 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. CrossRef
Farhat, C. and Roux, F.-X., A method of Finite Element Tearing and Interconnecting and its parallel solution algorithm. Int. J. Numer. Meth. Eng. 32 (1991) 1205–1227.
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.
Farhat, C., Mandel, J. and Roux, F.X., Optimal convergence properties of the FETI domain decomposition method. Comput. Methods Appl. Mech. Eng. 115 (1994) 367–388. CrossRef
Farhat, C., Lesoinne, M. and Pierson, K., A scalable dual-primal domain decomposition method. Numer. Lin. Alg. Appl. 7 (2000) 687–714. 3.0.CO;2-S>CrossRef
Farhat, C., Pierson, K.H. and Lesoinne, M., 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. CrossRef
Farhat, C., Lesoinne, M., Le Tallec, P., Pierson, K. and Rixen, D., 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.
Gosselet, P. and Non-overlapping, C. Rey domain decomposition methods in structural mechanics. Arch. Comput. Methods Eng. 13 (2006) 515–572. CrossRef
G.A. Holzapfel, Nonlinear Solid Mechanics. A continuum approach for engineering. Wiley (2000).
Klawonn, A. and Rheinbach, O., 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. CrossRef
Klawonn, A. and Rheinbach, O., Inexact FETI-DP methods. Int. J. Numer. Methods Eng. 69 (2007) 284–307. CrossRef
Klawonn, A. and Rheinbach, O., Robust FETI-DP methods for heterogeneous three dimensional elasticity problems. Comput. Methods Appl. Mech. Eng. 196 (2007) 1400–1414. CrossRef
Klawonn, A. and Rheinbach, O., Highly scalable parallel domain decomposition methods with an application to biomechanics. Z. Angew. Math. Mech. (ZAMM) 90 (2010) 5–32. CrossRef
Klawonn, A. and Widlund, O.B., Neumann-Neumann, FETI iterative substructuring methods: connections and new results. Commun. Pure Appl. Math. 54 (2001) 57–90. 3.0.CO;2-D>CrossRef
A. Klawonn and O.B. Widlund, Dual-Primal FETI Methods for Linear Elasticity. Commun. Pure Appl. Math. LIX (2006) 1523–1572.
Klawonn, A., Widlund, O.B. and Dryja, M., Dual-Primal FETI methods for three-dimensional elliptic problems with heterogeneous coefficients. SIAM J. Numer. Anal. 40 (2002) 159–179. CrossRef
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.
Klawonn, A., Pavarino, L.F. and Rheinbach, O., Spectral element FETI-DP and BDDC preconditioners with multi-element subdomains. Comput. Meth. Appl. Mech. Eng. 198 (2008) 511–523. CrossRef
Klawonn, A., Rheinbach, O. and Widlund, O.B., An analysis of a FETI–DP algorithm on irregular subdomains in the plane. SIAM J. Numer. Anal. 46 (2008) 2484–2504. CrossRef
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).
Klawonn, A., Neff, P., Rheinbach, O. and Vanis, S., Solving geometrically exact micromorphic elasticity with a staggered algorithm. GAMM Mitteilungen 33 (2010) 57–72. CrossRef
Langer, U., Of, G., Steinbach, O. and Zulehner, W., Inexact data-sparse boundary element tearing and interconnecting methods. SIAM J. Sci. Comput. 29 (2007) 290–314. CrossRef
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.
Li, J. and Widlund, O.B., BDDC, FETI-DP and Block Cholesky Methods. Int. J. Numer. Methods Eng. 66 (2006) 250–271. CrossRef
Mandel, J. and Tezaur, R., Convergence of a Substructuring Method with Lagrange Multipliers. Numer. Math. 73 (1996) 473–487. CrossRef
Mandel, J. and Tezaur, R., On the convergence of a dual-primal substructuring method. Numer. Math. 88 (2001) 543–558. CrossRef
Neff, P., Korn's, On first inequality with nonconstant coefficients. Proc. Roy. Soc. Edinb. A 132 (2002) 221–243. CrossRef
Neff, P., Finite multiplicative plasticity for small elastic strains with linear balance equations and grain boundary relaxation. Contin. Mech. Thermodyn. 15 (2003) 161–195. CrossRef
Neff, P., 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. CrossRef
Neff, P., Local existence and uniqueness for a geometrically exact membrane-plate with viscoelastic transverse shear resistance. Math. Meth. Appl. Sci. (MMAS) 28 (2005) 1031–1060. CrossRef
Neff, P., Local existence and uniqueness for quasistatic finite plasticity with grain boundary relaxation. Quart. Appl. Math. 63 (2005) 88–116. CrossRef
Neff, P., Existence of minimizers for a finite-strain micromorphic elastic solid. Proc. Roy. Soc. Edinb. A 136 (2006) 997–1012. CrossRef
Neff, P., A finite-strain elastic-plastic Cosserat theory for polycrystals with grain rotations. Int. J. Eng. Sci. 44 (2006) 574–594. CrossRef
Neff, P. and Forest, S., 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. CrossRef
Neff, P. and Münch, I., Simple shear in nonlinear Cosserat elasticity: bifurcation and induced microstructure. Contin. Mech. Thermodyn. 21 (2009) 195–221. CrossRef
Pompe, W., Korn's first inequality with variable coefficients and its generalizations. Comment. Math. Univ. Carolinae 44 (2003) 57–70.
A. Quarteroni and A. Valli, Numerical Approxiamtion of Partial Differential Equations, in Computational Mathematics 23, Springer Series, Springer, Berlin (1991).
A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications, Oxford (1999).
Schröder, J. and Neff, P., Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions. Int. J. Solids Struct. 40 (2003) 401–445. CrossRef
Schröder, J., Neff, P. and Balzani, D., A variational approach for materially stable anisotropic hyperelasticity. Int. J. Solids Struct. 42 (2005) 4352–4371. CrossRef
Schröder, J., Neff, P. and Ebbing, V., Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors. J. Mech. Phys. Solids 56 (2008) 3486–3506. CrossRef
B.F. Smith, P.E. Bjørstad and W. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge (1996).
Spadaro, E.N., Non-uniqueness of minimizers for strictly polyconvex functionals. Arch. Rat. Mech. Anal. 193 (2009) 659–678. CrossRef
A. Toselli and O. Widlund, Domain Decomposition Methods – Algorithms and Theory, Springer Series in Computational Mathematics 34. Springer (2004).
T. Valent, Boundary Value Problems of Finite Elasticity. Springer, Berlin (1988).
Weinberg, K. and Neff, P., A geometrically exact thin membrane model-investigation of large deformations and wrinkling. Int. J. Num. Meth. Eng. 74 (2007) 871–893. CrossRef
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.