Abstract.We propose a quasi-Newton algorithm for solving fluid-structure interaction problems.The basic idea of the method is to build an approximate tangent operator which is cost effective and which takes into account the so-called added mass effect. Various test cases show that the method allows a significant reduction of the computational effort compared to relaxed fixed point algorithms. We present 2D and 3D fluid-structure simulations performed either with a simple 1D structure model or with shells in large displacements.Mathematics Subject Classification. 65M60, 74K25, 76D05, 76Z05.

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Abstract:We introduce a class of explicit coupling schemes for the numerical solution of fluid-structure interaction problems involving a viscous incompressible fluid and a general thinwalled structure (e.g., including damping and non-linear behavior). The fundamental ingredient in these methods is the (parameter free) explicit Robin interface condition for the fluid, which enables the fluid-solid splitting through appropriate extrapolations of the solid velocity and fluid stress on the interface. The resulting solution procedures are genuinely partitioned. Stability and error estimates are provided for all the variants (depending on the extrapolations), using energy arguments within a representative linear setting. In particular, we show that one of them yields added-mass free unconditional stability and optimal (first-order) time accuracy. A comprehensive numerical study, involving different examples from the literature, supports the theory.Key-words: fluid-structure interaction, incompressible fluid, non-linear shell, visco-elasticity, time-discretization, explicit coupling scheme, loosely coupled scheme, Robin-Neumann scheme, partitioned procedure, finite element method, error estimate. Schémas Robin-Neumann explicites pour le couplage d'un fluide incompressible avec une structure mince Résumé :Cet article présente une famille de schémas numériques explicites qui permet de résoudre un problème d'interaction fluide-structure faisant intervenir un fluide visqueux incompressible et une structure mince (qui peut être amortie ou non linéaire). L'outil principal de ces méthodes réside dans une condition aux limites de type Robin pour le fluide (sans paramètre à ajuster), qui permet le découplage entre les deux sous-problèmes et fait intervenir une extrapolation de la vitesse du solide et des efforts du fluide. Le schéma qui en résulte est partitionné. L'article fournit des estimations d'énergie et d'erreur a priori qui permettent de montrer la stabilité et la convergence du schéma pour tous les ordres d'extrapolation lorsque le problème est linéaire. Plus précisément, on montre qu'une des variantes est à la fois inconditionnellement stable, insensible à l'effet de masse ajoutée et possède une précision en temps optimale (c'est-à-dire d'ordre un). Ces résultats sont illustrés par de nombreux exemples numériques tirés de la littérature. Mots-clés :interaction fluide-structure, fluide incompressible, coque non-linéaire, viscoélasticité, discrétisation en temps, schéma de couplage explicite, schéma faiblement couplé, schéma Robin-Neumann, algorithme partitionné, méthode des éléments finis, estimation d'erreur.Explicit Robin-Neumann schemes 3

SUMMARYWe introduce a new class of explicit coupling schemes for the numerical solution of fluid-structure interaction problems involving a viscous incompressible fluid and an elastic structure. These methods generalize the arguments reported in [Comput. Methods Appl. Mech. Engrg., 267:566-593, 2013, Numer. Math., 123(1):21-65, 2013] to the case of the coupling with thick-walled structures. The basic idea lies in the derivation of an intrinsic interface Robin consistency at the space semi-discrete level, using a lumpedmass approximation in the structure. The fluid-solid splitting is then performed through appropriate extrapolations of the solid velocity and stress on the interface. Based on these methods, a new, parameterfree, Robin-Neumann iterative procedure is also proposed for the partitioned solution of implicit coupling. A priori energy estimates, guaranteeing the stability of the schemes and the convergence of the iterative procedure, are established within a representative linear setting. The accuracy and performance of the methods are illustrated in several numerical examples.

International audienceWe propose a direct method for computing modal coupling coefficients - due to geometrically nonlinear effects - for thin shells vibrating at large amplitude and discretized by a finite element (FE) procedure. These coupling coefficients arise when considering a discrete expansion of the unknown displacement onto the eigenmodes of the linear operator. The evolution problem is thus projected onto the eigenmodes basis and expressed as an assembly of oscillators with quadratic and cubic nonlinearities. The nonlinear coupling coefficients are directly derived from the finite element formulation, with specificities pertaining to the shell elements considered, namely, here elements of the ''Mixed Interpolation of Tensorial Components'' family (MITC). Therefore, the computation of coupling coefficients, combined with an adequate selection of the significant eigenmodes, allows the derivation of effective reduced-order models for computing - with a continuation procedure - the stable and unstable vibratory states of any vibrating shell, up to large amplitudes. The procedure is illustrated on a hyperbolic paraboloid panel. Bifurcation diagrams in free and forced vibrations are obtained. Comparisons with direct time simulations of the full FE model are given. Finally, the computed coefficients are used for a maximal reduction based on asymptotic nonlinear normal modes (NNMs), and we find that the most important part of the dynamics can be predicted with a single oscillator equation

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We present a new Neumann-Neumann-type preconditioner of large scale linear systems arising from plate and shell problems. The advantage of the new method is a smaller coarse space than those of earlier methods of the authors; this improves parallel scalability. A new abstract framework for Neumann-Neumann preconditioners is used to prove almost optimal convergence properties of the method. The convergence estimates are independent of the number of subdomains, coefficient jumps between subdomains, and depend only polylogarithmically on the number of elements per subdomain. We formulate and prove an approximate parametric variational principle for Reissner-Mindlin elements as the plate thickness approaches zero, which makes the results applicable to a large class of nonlocking elements in everyday engineering use. The theoretical results are confirmed by computational experiments on model problems as well as examples from real world engineering practice.

International audienceIn this paper we introduce a class of fully decoupled time-marching schemes (velocity-pressure-displacement splitting) for the coupling of an incompressible fluid with a thin-walled elastic or viscoelastic structure. The time splitting combines a projection method in the fluid with a specific Robin-Neumann treatment of the interface coupling. A priori energy estimates guaranteeing unconditional stability are established for some of the schemes. The accuracy and performance of the methods proposed is illustrated by a thorough numerical study