|Preprint-No.:||< 383 >||Published in:||January 2014||PDF-File:||IGPM383_k.pdf|
|Title:||Substructuring Methods in Nonlinear Function Spaces|
We generalize substructuring methods to problems for functions f : Ω → M , where Ω is a domain in Rd and M is a Riemannian mani- fold. Examples for such functions include conﬁgurations of liquid crystals, ferromagnets, and deformations of Cosserat materials. We show that a sub- structuring 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 sec- tions in certain vector bundles. We derive various solution algorithms based on preconditioned Richardson iterations for a nonlinear Steklov–Poincare 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|
|Publication:||to appear in Proceedings of the 22nd International Conference on Domain Decomposition Methods|