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Solving geometrically exact micromorphic elasticity with a staggered algorithm. (English) Zbl 1328.74079

Summary: A minimization problem modeling geometrically exact generalized continua of micromorphic type is considered. The solution consists of two fields, the elastic deformation \(\varphi \) of a given body and a tensorial field \(P\) which can model different additional features needed for a more reliable description of solids. For the solution of this minimization problem, a staggered algorithm is introduced which decouples the original problem into two separate problems. In each of these subproblems, one of the variables, \(\varphi \) or \(P\), respectively, is kept fixed and the subproblem is solved for the remaining variables, i.e., \(P\) or \(\varphi \), respectively. Each of the problems is discretized with finite elements. Numerical results are presented for a cubic and a cylindrical geometry.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74A60 Micromechanical theories

Software:

Netgen; PETSc
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References:

[1] Satish Balay Kris Buschelman William D. Gropp Dinesh Kaushik Matt Knepley Lois Curfman McInnes Barry F. Smith Hong Zhang PETSc users manual. Technical Report ANL-95/11 - Revision 2.2.3, Argonne National Laboratory, 2007.
[2] Satish Balay Kris Buschelman William D. Gropp Dinesh Kaushik Matthew G. Knepley Lois Curfman McInnes Barry F. Smith Hong Zhang PETSc Web page, 2009. http://www.mcs.anl.gov/petsc.
[3] Satish Balay William D. Gropp Lois Curfman McInnes Barry F. Smith Efficient management of parallelism in object oriented numerical software libraries. In E. Arge, A. M. Bruaset, and H. P. Langtangen, editors, Modern Software Tools in Scientific Computing , pages 163-202. Birkhäuser Press, 1997. · Zbl 0882.65154
[4] G. Capriz Continua with Microstructures . Springer, 1989.
[5] Timothy A. Davis A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Transactions on Mathematical Software , 30(2):165-195, June 2004. · Zbl 1072.65036
[6] A. C. Eringen Microcontinuum Field Theories . Springer, 1999.
[7] Axel Klawonn Patrizio Neff Oliver Rheinbach Stefanie Vanis FETI-DP Domain Decomposition Methods for Elasticity with structural changes: P-Elasticity. submitted for publication , November 2009.
[8] Axel Klawonn Olof B. Widlund Dual-Primal FETI Methods for Linear Elasticity. Comm. Pure Appl. Math. , LIX:1523-1572, 2006. · Zbl 1110.74053
[9] I. Münch Ein geometrisch und materiell nichtlineares Cosserat-Modell - Theorie, Numerik und Anwendungmöglichkeiten. PhD thesis, Universität Karlsruhe, Fakultät für Bauingenieur-, Geo- und Umweltwissenschaften, 2007. ISBN 978-3-935322-12-6, electronic version available at http://digbib.ubka.uni-karlsruhe.de/volltexte/1000007371.
[10] P. Neff K. Chełmiński H.D. Alber Notes on strain gradient plasticity. Finite strain covariant modelling and global existence in the infinitesimal rate-independent case. Math. Mod. Meth. Appl. Sci. (M3AS) , 19(2):1-40, 2009. Preprint 2503, http://www3.mathematik.tu-darmstadt.de/fb/mathe/bibliothek/preprints.html. · Zbl 1160.74009
[11] P. Neff J. Jeong H. Ramezani Subgrid interaction and micro-randomness - novel invariance requirements in infinitesimal gradient elasticity. Int. J. Solids Struct. , 46:4261-4276, 2009. Preprint 2568, http://www3.mathematik.tu-darmstadt.de/fb/mathe/bibliothek/preprints.html. · Zbl 1176.74028
[12] P. Neff A. Sydow C. Wieners Numerical approximation of incremental infinitesimal gradient plasticity. Preprint IWRM 08/01, http://www.mathematik.uni-karlsruhe.de/iwrmm/seite/preprints/media [www.mathematik.uni-karlsruhe.de], Int. J. Num. Meth. Engrg. , 77(3):414-436, 2009. · Zbl 1155.74316
[13] Patrizio Neff Existence of minimizers for a finite-strain micromorphic elastic solid. Proc. Roy. Soc. Edinb. A , 136:9971012, 2006.
[14] Patrizio Neff A finite-strain elastic-plastic Cosserat theory for polycrystals with grain rotations. Int. J. Eng. Sci., DOI 10.1016/j.ijengsci.2006.04.002 , 44:574-594, 2006. · Zbl 1213.74032
[15] Patrizio Neff Samuel 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 , 87:239-276, 2007. · Zbl 1206.74019
[16] Joachim Schöberl NETGEN 4.0. http://www.hpfem.jku.at/netgen/.
[17] Joachim Schöberl NETGEN - An advancing front 2D/3D-mesh generator based on abstract rules. Computing and Visualization in Science , 1(1):4152, 1997. · Zbl 0883.68130
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