Substructuring Methods in Nonlinear Function Spaces
نویسنده
چکیده
We generalize substructuring methods to problems for functions f : Ω → M , where Ω is a domain in R 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. 1 Variational Problems in Spaces of Manifold-Valued Functions Let Ω be a domain in R, and M a smooth, connected, finite-dimensional manifold with positive injectivity radius. We assume M to be equipped with a metric g, which induces an exponential map exp : TM →M . Oliver Sander Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany, e-mail: [email protected]
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