SummaryThe modern design of industrial structures leads to very complex simulations characterized by nonlinearities, high heterogeneities, tortuous geometries... Whatever the modelization may be, such an analysis leads to the solution to a family of large ill-conditioned linear systems. In this paper we study strategies to efficiently solve to linear system based on non-overlapping domain decomposition methods. We present a review of most employed approaches and their strong connections. We outline their mechanical interpretations as well as the practical issues when willing to implement and use them. Numerical properties are illustrated by various assessments from academic to industrial problems. An hybrid approach, mainly designed for multifield problems, is also introduced as it provides a general framework of such approaches.

This paper introduces a computational strategy to solve structural problems featuring nonlinear phenomena that occur within a small area, while the rest of the structure retains a linear elastic behavior. Two finite element models are defined: a global linear model of the whole structure, and a local nonlinear "submodel" meant to replace the global model in the nonlinear area. An iterative coupling technique is then used to perform this replacement in an exact but non-intrusive way, which means the model data sets are never modified and the computations can be carried out with standard finite element software. Several ways of exchanging data between the models are discussed and their convergence properties are investigated on two examples.

This article describes a bridge between POD-based model order reduction techniques and the classical Newton/Krylov solvers. This bridge is used to derive an efficient algorithm to correct, "on-the-fly", the reduced order modelling of highly nonlinear problems undergoing strong topological changes. Damage initiation problems are addressed and tackle via a corrected hyperreduction method. It is shown that the relevancy of reduced order model can be significantly improved with reasonable additional costs when using this algorithm, even when strong topological changes are involved.

The prediction of the quasi-static response of industrial laminate structures requires to use fine descriptions of the material, especially when debonding is involved. Even when modeled at the mesoscale, the computation of these structures results in very large numerical problems. In this paper, the exact mesoscale solution is sought using parallel iterative solvers. The LaTIn-based mixed domain decomposition method makes it very easy to handle the complex description of the structure; moreover the provided multiscale features enable us to deal with numerical difficulties at their natural scale; we present the various enhancements we developed to ensure the scalability of the method. An extension of the method designed to handle instabilities is also presented.

Domain Decomposition methods often exhibit very poor performance when applied to engineering problems with large heterogeneities. In particular for heterogeneities along domain interfaces the iterative techniques to solve the interface problem are lacking an efficient preconditioner. Recently a robust approach, named FETI-Geneo, was proposed where troublesome modes are precomputed and deflated from the interface problem. The cost of the FETI-Geneo is however high. We propose in this paper techniques that share similar ideas with FETI-Geneo but where no pre-processing is needed and that can be easily and efficiently implemented as an alternative to standard Domain Decomposition methods. In the block iterative approaches presented in this paper, the search space at every iteration on the interface problem contains as many directions as there are domains in the decomposition. Those search directions originate either from the domain-wise preconditioner (in the Simultaneous FETI method) or from the block structure of the right-hand side of the interface problem (Block FETI). We show on 2D structural examples that both methods are robust and provide good convergence in the presence of high heterogeneities, even when the interface is jagged or when the domains have a bad aspect ratio. The Simultaneous FETI was also efficiently implemented in an optimized parallel code and exhibited excellent performance compared to the regular FETI method.

SUMMARYThis paper presents a two-scale approximation of the Schur complement of a subdomain's stiffness matrix, obtained by combining local (i.e. element strips) and global (i.e. homogenized) contributions. This approximation is used in the context of a coupling strategy that is designed to embed local plasticity and geometric details into a small region of a large linear elastic structure; the strategy consists in creating a local model that contains the desired features of the concerned region and then substituting it into the global problem by the means of a non-intrusive solver coupling technique adapted from domain decomposition methods. Using the two-scale approximation of the Schur complement as a Robin condition on the local model enables to reach high efficiency. Examples include a large 3D problem provided by our industrial partner Snecma.

A dual domain decomposition method dedicated to nonlinear problems is presented. The decomposition is introduced in the nonlinear formulation and the nonlinear problem is first condensed on the interface then solved by a Newton-type method. Considering the specificities of the introduced operators, the algorithm can be interpreted as a local/global strategy with global Newton-type iterations and nonlinear relocalizations per subdomain. Such a strategy is particularly interesting in cases where the nonlinearity is localized. First results are presented on structural problems with damage.

The purpose of this article is to assess the adaptive multipreconditioned FETI solvers (AMPFETI) on realistic industrial problems and hardware. The multipreconditioned FETI algorithm (first introduced as Simultaneous FETI [1]) is a non-overlapping domain decomposition method which exhibits good robustness properties without requiring the explicit knowledge of the original partial differential equation, or any a priori analysis of the algebraic system through eigenvalues problems. Multipreconditioned FETI solves critical problems in significantly fewer iterations than classical FETI but each iteration involves a larger computational effort. An adaptive strategy (known as the adaptive multipreconditioned conjugate gradient algorithm [2]) has been proposed to achieve balance between robustness and efficiency and we will observe that it provides an efficient solver for the problems considered here.