In this thesis we study efficient parallel iterative solution algorithms for multi-physics problems. In particular, we consider fluid structure interaction (FSI) problems, a type of multi-physics problem in which a fluid and a deformable solid interact. All computations were performed in Oomph-Lib, a finite element library for the simulation of multi-physics problems. In Oomph-Lib, the constituent problems in a multi-physics problem are coupled monolithically, and the resulting system of non-linear equations solved with Newton's method. This requires the solution of sequences of large, sparse linear systems, for which optimal solvers are essential. The linear systems arising from the monolithic discretisation of multi-physics problems are natural candidates for solution with block-preconditioned Krylov subspace methods.We developed a generic framework for the implementation of block preconditioners within Oomph-Lib. Furthermore the framework is parallelised to facilitate the efficient solution of very large problems. This framework enables the reuse of all of Oomph-Lib's existing linear algebra infrastructure and preconditioners (including block preconditioners). We will demonstrate that a wide range of block preconditioners can be seamlessly implemented in this framework, leading to optimal iterative solvers with good parallel scaling.We concentrate on the development of an effective preconditioner for a FSI problem formulated in an arbitrary Lagrangian Eulerian (ALE) framework with pseudo-solid node updates (for the deforming fluid mesh). We begin by considering the pseudo-solid subsidiary problem; the deformation of a solid governed by equations of large displacement elasticity, subject to a prescribed boundary displacement imposed with Lagrange multiplier. We present a robust, optimal, augmented-Lagrangian type preconditioner for the resulting saddle-point linear system and prove analytically tight bounds for the spectrum of the preconditioned operator with respect to the discrete problem size.This pseudo-solid preconditioner is incorporated into a block preconditioner for the full FSI problem. One key feature of the FSI preconditioner is that existing optimal single physics preconditioners (such as the well known Navier-Stokes Least Squares Commutator preconditioner) can be employed to approximately solve the linear systems associated with the constituent sub-problems. We evaluate its performance on selected 2D and 3D problems. The preconditioner is optimal for most problems considered. In cases when sub-optimality is detected, we explain the reasons for such behavior and suggest potential improvements.

Cette thèse vise à concevoir des solveurs efficaces pour résoudre des systèmes linéaires, résultant des problèmes d'optimisation sous contraintes dans certaines applications de dynamique des structures et vibration (la corrélation calcul-essai, la localisation d'erreur, le modèle hybride, l'évaluation des dommages, etc.). Ces applications reposent sur la résolution de problèmes inverses, exprimés sous la forme de la minimisation d'une fonctionnelle en énergie. Cette fonctionnelle implique à la fois, des données issues d'un modèle numérique éléments finis, et des essais expérimentaux. Ceci conduit à des modèles de haute qualité, mais les systèmes linéaires point-selle associés, sont coûteux à résoudre. Nous proposons deux classes différentes de méthodes pour traiter le système. La première classe repose sur une méthode de factorisation directe profitant de la topologie et des propriétés spéciales de la matrice point-selle. Après une première renumérotation pour regrouper les pivots en blocs d'ordre 2. L'élimination de Gauss est conduite à partir de ces pivots et en utilisant un ordre spécial d'élimination réduisant le remplissage. Les résultats numériques confirment des gains significatifs en terme de remplissage, jusqu'à deux fois meilleurs que la littérature pour la topologie étudiée. La seconde classe de solveurs propose une approche à double projection du système étudié sur le noyau des contraintes, en faisant une distinction entre les contraintes cinématiques et celles reliées aux capteurs sur la structure. La première projection est explicite en utilisant une base creuse du noyau. La deuxième est implicite. Elle est basée sur l'emploi d'un préconditionneur contraint avec des méthodes itératives de type Krylov. Différentes approximations des blocs du préconditionneur sont proposées. L'approche est implémentée dans un environnement distribué parallèle utilisant la bibliothèque PETSc. Des gains significatifs en terme de coût de calcul et de mémoire sont illustrés sur plusieurs applications industrielles. / This thesis aims at designing efficient numerical solution methods to solve linear systems, arising in constrained optimization problems in some structural dynamics and vibration applications (test-analysis correlation, model error localization,hybrid model, damage assessment, etc.). These applications rely on solving inverse problems, by means of minimization of an energy-based functional. This latter involves both data from a numerical finite element model and from experimental tests, which leads to high quality models, but the associated linear systems, that have a saddle-point coefficient matrices, are long and costly to solve. We propose two different classes of methods to deal with these problems. First, a direct factorization method that takes advantage of the special structures and properties of these saddle point matrices. The Gaussian elimination factorization is implemented in order to factorize the saddle point matrices block-wise with small blocks of orders 2 and using a fill-in reducing topological ordering. We obtain significant gains in memory cost (up to 50%) due to enhanced factors sparsity in comparison to literature. The second class is based on a double projection of the generated saddle point system onto the nullspace of the constraints. The first projection onto the kinematic constraints is proposed as an explicit process through the computation of a sparse null basis. Then, we detail the application of a constraint preconditioner within a Krylov subspace solver, as an implicit second projection of the system onto the nullspace of the sensors constraints. We further present and compare different approximations of the constraint preconditioner. The approach is implemented in a parallel distributed environment using the PETSc library. Significant gains in computational cost and memory are illustrated on several industrial applications.