383 RWTH Publication No: 230295        2014        IGPM383.pdf
TITLE Substructuring Methods in Nonlinear Function Spaces
AUTHORS Oliver Sander
ABSTRACT We generalize substructuring methods to problems for functions f : Ω → M, where 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