Fast nonlinear solvers in solid mechanics

La thèse a pour objectif le développement de méthodes performantes pour la résolution de problèmes non linéaires ne mécanique des solides. Il est coutume d'utiliser une méthode de type Newton qui conduit à la résolution d'une séquence de systèmes linéaires. De plus, la prise en compte des relations linéaires imposées à l'aide de multiplicateurs de Lagrange confère aux matrices une structure de point-selle. Dans un cadre plus général, nous proposons, étudions et illustrons deux classes d'enrichissement de préconditionneurs (limited memory preconditioners) pour la résolution de séquences de systèmes linéaires par une méthode de Krylov. La première est un extension au cas symétrique indéfini d'une méthode existante, développée initialement dans le cadre symétrique défini positif. La seconde est plus générale dans le sens où elle s'applique aus systèmes non symétriques. Ces deux familles peuvent être interprétées comme des variantes par blocs de formules de mise à jour utilisées dans différentes méthodes d'optimisation. Ces techniques ont été développées dans le logiciel de mécanique des solides Code_Aster (dans un environnement parallèle distribué via la bibliothèque PETSc) et sont illustrées sur plusieurs études industrielles. Les gains obtenus en terme de coût de calcul sont significatifs (jusqu'à 50%), pour un surcoût mémoire négligeable.The thesis aims at developing efficient numerical methods to solve nonlinear problems arising un solid mechanics. In this field, Newton methods are currently used, requiring the solution of a sequence of linear systems. Furthermore, the imposed linear relations are dualized with the Lagrange multipliers, leading to matrices with a saddle point structure. In a more general framework, we propose two classes of preconditioners (named limited memory preconditioners) to solve sequences of linear systems with a Krylov subspace method. The first class is based on an extension of a method initially developed for symmetric positive definite matrices to the symmetric indefinite case. Both families can be interpreted as block variants of updating formulas used in numerical optimization. They have been implemented into the Code_Aster solid mechanics software (in a parallel distributed environement using the PETSc library). These new preconditioning strategies are illustrated on several industrial applications. We obtain significant gains in computational cost (up to 50%) at a marginal overcost in memory

A new preconditioner update strategy for the solution of sequences of linear systems in structural mechanics: application to saddle point problems in elasticity

Many applications in structural mechanics require the numerical solution of sequences of linear systems typically issued from a finite element discretization of the governing equations on fine meshes. The method of Lagrange multipliers is often used to take into account mechanical constraints. The resulting matrices then exhibit a saddle point structure and the iterative solution of such preconditioned linear systems is considered as challenging. A popular strategy is then to combine preconditioning and deflation to yield an efficient method.We propose an alternative that is applicable to the general case and not only to matrices with a saddle point structure. In this approach, we consider to update an existing algebraic or application-based preconditioner, using specific available information exploiting the knowledge of an approximate invariant subspace or of matrix-vector products. The resulting preconditioner has the form of a limited memory quasi-Newton matrix and requires a small number of linearly independent vectors. Numerical experiments performed on three large-scale applications in elasticity highlight the relevance of the new approach. We show that the proposed method outperforms the deflation method when considering sequences of linear systems with varying matrices

Limited memory preconditioners for nonsymmetric systems

This paper presents a class of limited memory preconditioners (LMPs) for solving linear systems of equations with multiple nonsymmetric matrices and multiple right-hand sides. These preconditioners based on limited memory quasi-Newton formulas require a small number k of linearly independent vectors. They may be used to improve an existing first-level preconditioner and are especially worth considering when the solution of a sequence of linear systems with slowly varying left-hand sides is addressed

A comprehensive analysis of gene expression profiles in distal parts of the mouse renal tubule

The distal parts of the renal tubule play a critical role in maintaining homeostasis of extracellular fluids. In this review, we present an in-depth analysis of microarray-based gene expression profiles available for microdissected mouse distal nephron segments, i.e., the distal convoluted tubule (DCT) and the connecting tubule (CNT), and for the cortical portion of the collecting duct (CCD; Zuber et al., Proc Natl Acad Sci USA 106:16523-16528, 2009). Classification of expressed transcripts in 14 major functional gene categories demonstrated that all principal proteins involved in maintaining the salt and water balance are represented by highly abundant transcripts. However, a significant number of transcripts belonging, for instance, to categories of G-protein-coupled receptors or serine/threonine kinases exhibit high expression levels but remain unassigned to a specific renal function. We also established a list of genes differentially expressed between the DCT/CNT and the CCD. This list is enriched by genes related to segment-specific transport functions and by transcription factors directing the development of the distal nephron or collecting ducts. Collectively, this in silico analysis provides comprehensive information about relative abundance and tissue specificity of the DCT/CNT and the CCD expressed transcripts and identifies new candidate genes for renal homeostasi

À l’ombre des remparts : 1 600 ans d’évolution urbaine à Auxerre. La fouille de la place des Véens

En 2010, en amont d’un projet immobilier, une fouille préventive a été réalisée sur un îlot situé au nord de la place des Véens à Auxerre, dans un secteur archéologique très sensible . Concernant une parcelle de près de 1 000 m2 localisée à une cinquantaine de mètres au sud de l’ancien castrum, à l’ouest de la ville, et à l’intérieur de l’enceinte du xiiie siècle , cet ample chantier, pour lequel nous disposons maintenant de la totalité des résultats des analyses scientifiques, était propre à..

Abibalsamins A and B, two new tetraterpenoids from Abies balsamea oleoresin

Abibalsamins A (1) and B (2), two unprecedented tetraterpenoids featuring a 3,4-seco-rearranged lanostane system fused with a β-myrcene lateral chain via a [4 + 2] Diels–Alder cycloaddition, were isolated from the oleoresin of Abies balsamea. Their structures were elucidated by means of extensive 2D NMR, IR, and MS spectroscopy analyses. The absolute configuration of 1 was determined by single-crystal X-ray diffraction. Both compounds exhibited significant cytotoxic activity against cancer cell lines