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CivilComp Proceedings
ISSN 17593433 CCP: 89
PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: M. Papadrakakis and B.H.V. Topping
Paper 87
NonLinear Dual Domain Decomposition Method for Multiscale Analysis of Structures J. Pebrel, P. Gosselet and C. Rey
LMTCachan, ENS Cachan/CNRS/University Paris 6/PRES UniverSud Paris, France J. Pebrel, P. Gosselet, C. Rey, "NonLinear Dual Domain Decomposition Method for Multiscale Analysis of Structures", in M. Papadrakakis, B.H.V. Topping, (Editors), "Proceedings of the Sixth International Conference on Engineering Computational Technology", CivilComp Press, Stirlingshire, UK, Paper 87, 2008. doi:10.4203/ccp.89.87
Keywords: domain decomposition method, nonlinear problems, damage.
Summary
Nowadays, many computational strategies exist to solve nonlinear multiscale problems. Two groups of methods can be distinguished: the ones that really treat multiscale aspect of the nonlinear problems and the ones that use a multiscale strategy to solve the linear problems arising from the linearization of the nonlinear problem. Only few methods can really be classified in the first group.
In the second group, one of the most popular and general family of methods is the socalled "NewtonKrylovSchur" (NKS) family (see [1] for a review). Among these methods, we take an interest in the FETI method [2]. These methods perform well, but they may loose their efficiency when they encounter pathological phenomena such as local nonlinearities. Because the convergence of Newtontype algorithms is linked to the strongest nonlinearity in the domain, local nonlinear phenomena may penalize the convergence of the global algorithm. In [3], the convergence of the NKS strategy is dramatically slowed down by local buckling, hence a strategy which introduces nonlinear relocalization steps, inspired from the principles of the LaTIn method [4], is designed and assessed. Our approach, introduced in [5], is based on a different point of view. Stating that standard NKS solvers do not exploit the domain decomposition in the nonlinear context, we propose to introduce the domain decomposition directly in the nonlinear formulation of the problem instead of only using it to solve linearized problems. First, the global nonlinear problem, is decomposed into local problems under a global constraint imposed via a Lagrange multiplier. Second, a condensation step formulates the nonlinear problem in terms of the unknown Lagrange multiplier. Third, a Newtontype algorithm is chosen to solve this nonlinear interface problem. An iteration of our algorithm requires the solution of a global linear interface problem and the solution of a set of independent nonlinear local problems (via another Newtontype solver) with Neumann interface conditions which may imply specific handling of infinitesimal rigid body motions. Consequently, treating nonlinearity at its own local scale leads to a modification of the classical domain decomposition methods by the addition of a set of local nonlinear iterations which are parallel and not expensive. These nonlinear iterations decrease the number of global resolutions and result in significant CPU speedup. References
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