383 | IGPM383.pdf January 2014 |

TITLE | Substructuring Methods in Nonlinear Function Spaces |

AUTHORS | Oliver Sander |

ABSTRACT | We generalize substructuring methods to problems for functions f : Ω → M, where Ω is a domain in Rd and M is a Riemannian manifold. Examples for such functions include conﬁgurations of liquid crystals, ferromagnets, and deformations of Cosserat materials. We show that a substructuring theory can be developed for such problems. While the theory looks very similar to the linear theory on a formal level, the objects it deals with are much more general. In particular, iterates of the algorithms are elements of nonlinear Sobolev spaces, and test functions are replaced by sections in certain vector bundles. We derive various solution algorithms based on preconditioned Richardson iterations for a nonlinear Steklov–Poincaré formulation. Preconditioners appear as bundle homomorphisms. As a numerical example we compute the deformation of a geometrically exact Cosserat shell with a Neumann–Neumann algorithm. |

KEYWORDS | substructuring, manifold-valued functions, Cosserat shells, Neumann-Neumann method |

DOI | 10.1007/978-3-319-18827-0_5 |

PUBLICATION | Domain Decomposition Methods in Science and Engineering XXII pp 53-64 |