Substructured formulations of nonlinear structure problems - influence of the interface condition

We investigate the use of non-overlapping domain decomposition (DD) methods for nonlinear structure problems. The classic techniques would combine a global Newton solver with a linear DD solver for the tangent systems. We propose a framework where we can swap Newton and DD, so that we solve independent nonlinear problems for each substructure and linear condensed interface problems. The objective is to decrease the number of communications between subdomains and to improve parallelism. Depending on the interface condition, we derive several formulations which are not equivalent, contrarily to the linear case. Primal, dual and mixed variants are described and assessed on a simple plasticity problem.Comment: in International Journal for Numerical Methods in Engineering, Wiley, 201

Highly parallel methods for numerical simulation in nonlinear structural mechanics

Cette thèse vise à contribuer à l'adoption du virtual testing, pratique industrielle encore embryonnaire qui consistera à optimiser et certifier par la simulation numérique le dimensionnement de pièces industrielles critiques. Le virtual testing permettra des économies colossales dans la conception des pièces mécaniques et un plus grand respect de l'environnement grâce à des designs optimisés. Afin d'atteindre un tel objectif, de nouvelles méthodes de calcul doivent être mises en place, plus sûres, plus respectueuses des architectures matérielles, plus rapides, compatibles avec les contraintes temporelles de l'ingénierie. Nous nous intéressons à la résolution parallèle de problèmes non linéaires de grande taille par des méthodes de décomposition de domaine. Notre objectif est d'atteindre une approximation de la solution exacte en minimisant les communications entre les sous-domaines. Pour cela nous souhaitons maximiser les calculs réalisés indépendamment par sous-domaine à l'aide d'approches de relocalisation non linéaire, contrôler les critères de convergence des solveurs imbriqués de manière à éviter la sur-résolution et les divergences, améliorer la construction de conditions d'interface mixtes, et non linéariser l'étape de préconditionnement du solveur. L'objectif à terme étant de traiter des problèmes de complexité industrielle, la robustesse des méthodes sera un souci constant. De manière classique, les problèmes non linéaires sont résolus en construisant une suite de systèmes linéaires qui peuvent être résolus en parallèle à l'aide de méthodes itératives, telles que les solveurs de Krylov. Nous souhaitons remettre en question cette procédure usuelle en essayant de construire une suite de petits systèmes non linéaires indépendants à résoudre en parallèle. Une telle technique implique l'utilisation de solveurs itératifs imbriqués dont les critères de convergence doivent être syntonisés dynamiquement de manière à éviter à la fois la sur-résolution et la perte de convergence. La robustesse de la méthode pourra notamment être assurée par l'emploi de conditions d'interface mixtes bien construites et de préconditionneurs bien choisis.This thesis is aimed to contribute to the adoption of virtual testing, an industrial practice still embryonic which consists in optimizing and certifying by numerical simulations the dimensioning of critical industrial structures. The virtual testing will allow colossal savings in the design of mechanical parts and a greater respect for the environment, thanks to optimized designs. In order to achieve this goal, new calculation methods must be implemented, satisfying more requirements concerning safety, respect for hardware architectures, fastness, and compatibility with the time constraints of engineering.We are interested in the parallel resolution of large nonlinear problems by domain decomposition methods. Our goal is to approximate the exact solution by minimizing communication between subdomains. In order to do this, we want to maximize the computations performed independently by subdomain, using nonlinear relocation approaches. We also try to control the convergence criteria of the nested solvers in order to avoid over-resolution and divergences, to improve the construction of conditions Of mixed interface, and non-linearizing the preconditioning step of the solver. The ultimate objective being to deal with problems of industrial complexity, the robustness of the methods we develop will be a constant concern.Conventionally, non-linear problems are solved by constructing a sequence of linear systems that can be solved in parallel using iterative methods, such as Krylov solvers. We wish to question this usual procedure by trying to construct a sequence of small independent nonlinear systems to be solved in parallel. Such a technique involves the use of interleaved iterative solvers, whose convergence criteria must be dynamically tuned in order to avoid both over-resolution and loss of convergence. The robustness of the method can be ensured in particular by the use of well-constructed mixed interface conditions and well-chosen preconditioners

Méthodes fortement parallèles pour la simulation numérique en mécanique non linéaire des structures

This thesis is aimed to contribute to the adoption of virtual testing, an industrial practice still embryonic which consists in optimizing and certifying by numerical simulations the dimensioning of critical industrial structures. The virtual testing will allow colossal savings in the design of mechanical parts and a greater respect for the environment, thanks to optimized designs. In order to achieve this goal, new calculation methods must be implemented, satisfying more requirements concerning safety, respect for hardware architectures, fastness, and compatibility with the time constraints of engineering.We are interested in the parallel resolution of large nonlinear problems by domain decomposition methods. Our goal is to approximate the exact solution by minimizing communication between subdomains. In order to do this, we want to maximize the computations performed independently by subdomain, using nonlinear relocation approaches. We also try to control the convergence criteria of the nested solvers in order to avoid over-resolution and divergences, to improve the construction of conditions Of mixed interface, and non-linearizing the preconditioning step of the solver. The ultimate objective being to deal with problems of industrial complexity, the robustness of the methods we develop will be a constant concern.Conventionally, non-linear problems are solved by constructing a sequence of linear systems that can be solved in parallel using iterative methods, such as Krylov solvers. We wish to question this usual procedure by trying to construct a sequence of small independent nonlinear systems to be solved in parallel. Such a technique involves the use of interleaved iterative solvers, whose convergence criteria must be dynamically tuned in order to avoid both over-resolution and loss of convergence. The robustness of the method can be ensured in particular by the use of well-constructed mixed interface conditions and well-chosen preconditioners;Cette thèse vise à contribuer à l'adoption du virtual testing, pratique industrielle encore embryonnaire qui consistera à optimiser et certifier par la simulation numérique le dimensionnement de pièces industrielles critiques. Le virtual testing permettra des économies colossales dans la conception des pièces mécaniques et un plus grand respect de l'environnement grâce à des designs optimisés. Afin d'atteindre un tel objectif, de nouvelles méthodes de calcul doivent être mises en place, plus sûres, plus respectueuses des architectures matérielles, plus rapides, compatibles avec les contraintes temporelles de l'ingénierie. Nous nous intéressons à la résolution parallèle de problèmes non linéaires de grande taille par des méthodes de décomposition de domaine. Notre objectif est d'atteindre une approximation de la solution exacte en minimisant les communications entre les sous-domaines. Pour cela nous souhaitons maximiser les calculs réalisés indépendamment par sous-domaine à l'aide d'approches de relocalisation non linéaire, contrôler les critères de convergence des solveurs imbriqués de manière à éviter la sur-résolution et les divergences, améliorer la construction de conditions d'interface mixtes, et non linéariser l'étape de préconditionnement du solveur. L'objectif à terme étant de traiter des problèmes de complexité industrielle, la robustesse des méthodes sera un souci constant. De manière classique, les problèmes non linéaires sont résolus en construisant une suite de systèmes linéaires qui peuvent être résolus en parallèle à l'aide de méthodes itératives, telles que les solveurs de Krylov. Nous souhaitons remettre en question cette procédure usuelle en essayant de construire une suite de petits systèmes non linéaires indépendants à résoudre en parallèle. Une telle technique implique l'utilisation de solveurs itératifs imbriqués dont les critères de convergence doivent être syntonisés dynamiquement de manière à éviter à la fois la sur-résolution et la perte de convergence. La robustesse de la méthode pourra notamment être assurée par l'emploi de conditions d'interface mixtes bien construites et de préconditionneurs bien choisis

Un solveur FETI préconditionné non linéairement pour les formulations sous-structurées des problèmes non linéaires

We propose a new nonlinear version of preconditioning, dedicated to nonlinear substructured and condensed formulations with dual approach – i.e. nonlinear analogues to FETI solver. By increasing the importance of local nonlinear operations, this new technique reduces communications between processors throughout the parallel solving process. More, the tangent systems produced at each step still have the exact shape of classically preconditioned linear FETI problems, which makes the tractability of the implementation barely modified. The efficiency of this new preconditioner is demonstrated on two numerical test cases, namely a water diffusion problem and a nonlinear thermal behavior

Un solveur FETI préconditionné non linéairement pour les formulations sous-structurées des problèmes non linéaires

International audienceWe propose a new nonlinear version of preconditioning, dedicated to nonlinear substructured and condensed formulations with dual approach – i.e. nonlinear analogues to FETI solver. By increasing the importance of local nonlinear operations, this new technique reduces communications between processors throughout the parallel solving process. More, the tangent systems produced at each step still have the exact shape of classically preconditioned linear FETI problems, which makes the tractability of the implementation barely modified. The efficiency of this new preconditioner is demonstrated on two numerical test cases, namely a water diffusion problem and a nonlinear thermal behavior

Nonlinearly preconditioned FETI method

International audienceWe consider the Finite Element approximation of the solution to nonlinear elliptic partial differential equations such as the ones encountered in (quasi)-static mechanics, in transient mechanics with implicit time integration, or in thermal diffusion. Non-overlapping domain decomposition methods (DDM) offer an interesting framework for the distribution of the resolution. We focus on methods allowing independent nonlinearcomputations on the subdomains, sometimes called “nonlinear relocalization techniques”. Nonlinear counterparts to classical non-overlapping DDM have been proposed: non-linear primal (Dirichlet) and mixed (Robin) approach [Philippe Cresta, Olivier Allix, Christian Rey, and Stéphane Guinard. Nonlinear localization strategies for domain decomposition methods: Application to post-buckling analyses. Computer Methods in Applied Mechanics and Engineering, 196(8):1436–1446, 2007], dual approach [Julien Pebrel, Christian Rey, and Pierre Gosselet. A nonlinear dual-domain decomposition method: Application to structural problems with damage. International Journal for Multiscale Computational Engineering, 6(3), 2008], and nonlinear FETIDP and BDDC [Axel Klawonn, Martin Lanser, and Oliver Rheinbach. Nonlinear FETI-DP and BDDC Methods. SIAM Journal on Scientific Computing, 36(2):A737–A765, 2014]. The latter methods were improved and assessed at a large scale in [Axel Klawonn, Martin Lanser, Oliver Rheinbach, and Matthias Uran. Nonlinear FETI-DP and BDDC methods: A unified framework and parallel results. SIAM J. Sci. Comput., 39(6):C417–C451, 2017]. A global framework for primal/dual/mixed approaches was also proposed [Camille Negrello, Pierre Gosselet, Christian Rey, and Julien Pebrel. Substructured formulations of nonlinear structure problems–influence of the interface condition. International Journal for Numerical Methods in Engineering, 2016] and the impedance of the mixed approach was improved [Camille Negrello, Pierre Gosselet, and Christian Rey. A new impedance accounting for short and long range effects in mixed substructured formulations of nonlinear problems. International Journal for Numerical Methods in Engineering, 2017].Our objective is to double the intensity of the local independent nonlinear computations by modifying the condensed problem to be solved. The method can be interpreted as proposing a nonlinear preconditioner [Peter R Brune, Matthew G Knepley, Barry F Smith, and Xuemin Tu. Composing scalable nonlinear algebraic solvers. SIAM Review, 57(4):535–565, 2015] to the nonlinear DDM. It appears that this idea applies particularly easily to the dual approach, under an hypothesis equivalent to infinitesimal strain in mechanics. When applying a Newton algorithm to this nonlinear preconditioned condensed system, one alternates a sequence of two independent nonlinear local solves (one Neumann problem and one Dirichlet problem separated by one all-neighbor communication) and an interface tangent solve which exactly has the structure of a linear preconditioned FETI problem. Academic assessments show that the sequence of two local nonlinear solves can reduce the need of global Newton iterations and thus the number of calls to the communication-demanding Krylov solver

Fully nonlinear FETI method

International audienceWe consider the Finite Element approximation of the solution to nonlinear elliptic partial differential equations such as the ones encountered in (quasi)-static mechanics or in thermal diffusion. We focus on non-overlapping domain decomposition methods allowing independent nonlinear computations on the subdomains, sometimes called ``nonlinear relocalization techniques'' [2]. Nonlinear counterparts to classical non-overlapping DDM have been proposed, see [4] for primal/dual/mixed approaches, [3] for FETIDP/BDDC.Our objective is to double the intensity of the local independent nonlinear computations by proposing a nonlinear preconditioner to the system [1].This idea applies particularly easily to the dual approach, under an hypothesis equivalent to infinitesimal strain in mechanics. When applying a Newton algorithm, one alternates a sequence of two independent nonlinear local solves (one Neumann problem and one Dirichlet problem separated by one all-neighbor communication) and an interface tangent solve which exactly has the structure of a linear preconditioned FETI problem. Academic assessments show that the sequence of two local nonlinear solves can reduce the need of global Newton iterations and thus the number of calls to the communication-demanding Krylov solver.This work benefited from the support of the project SEMAFOR ANR-14-CE07-0037 of the French National Research Agency (ANR).[1] P. Brune, M. Knepley, B. Smith, and X. Tu. Composing scalable nonlinear algebraic solvers. SIAM Review, 57(4):535–565, 2015.[2] P. Cresta, O. Allix, C. Rey, and S. Guinard. Nonlinear localization strategies for domain decomposition methods: Application to post-buckling analyses. CMAME, 196(8):1436–1446, 2007.[3] A. Klawonn, M. Lanser, O. Rheinbach, and M. Uran. Nonlinear FETI-DP and BDDC methods: A unified framework and parallel results. SIAM SISC, 39(6):C417–C451, 2017.[4] C. Negrello, P. Gosselet, C. Rey, and J. Pebrel. Substructured formulations of nonlinear structure problems–influence of the interface condition. IJNME, 2016