67 research outputs found

    Deflated preconditioned conjugate gradient solvers for linear elasticity

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    Extensions of deflation techniques previously developed for the Poisson equation to static elasticity are presented. Compared to the (scalar) Poisson equation (J. Comput. Phys. 2008; 227 (24):10196–10208; Int. J. Numer. Meth. Engng 2010; DOI: 10.1002/nme.2932; Int. J. Numer. Meth. Biomed. Engng 2010; 26 (1):73–85), the elasticity equations represent a system of equations, giving rise to more complex low‐frequency modes (Multigrid . Elsevier: Amsterdam, 2000). In particular, the straightforward extension from the scalar case does not provide generally satisfactory convergence. However, a simple modification allows to recover the remarkable acceleration in convergence and CPU time reached in the scalar case. Numerous examples and timings are provided in a serial and a parallel context and show the dramatic improvements of up to two orders of magnitude in CPU time for grids with moderate graph depths compared to the non‐deflated version. Furthermore, a monotonic decrease of iterations with increasing subdomains, as well as a remarkable acceleration for very few subdomains are also observed if all the rigid body modes are included

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

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    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

    Enhanced balancing Neumann-Neumann preconditioning in computational fluid and solid mechanics

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    In this work, we propose an enhanced implementation of balancing Neumann-Neumann (BNN) preconditioning together with a detailed numerical comparison against the balancing domain decomposition by constraints (BDDC) preconditioner. As model problems, we consider the Poisson and linear elasticity problems. On one hand, we propose a novel way to deal with singular matrices and pseudo-inverses appearing in local solvers. It is based on a kernel identication strategy that allows us to eciently compute the action of the pseudo-inverse via local indenite solvers. We further show how, identifying a minimum set of degrees of freedom to be xed, an equivalent denite system can be solved instead, even in the elastic case. On the other hand, we propose a simple modication of the preconditioned conjugate gradient (PCG) algorithm that reduces the number of Dirichlet solvers to only one per iteration, leading to similar computational cost as additive methods. After these improvements of the BNN PCG algorithm, we compare its performance against that of the BDDC preconditioners on a pair of large-scale distributed-memory platforms. The enhanced BNN method is a competitive preconditioner for three-dimensional Poisson and elasticity problems, and outperforms the BDDC method in many cases

    Implementation of the deflated variants of the conjugate gradient method

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    Sdružené gradienty jsou jednou z nejpoužívanějších metod pro řešení rozsáhlých soustav lineárních rovnic se symetrickou pozitivně-semidefinitní maticí. Jeden ze způsobů urychlení konvergence metody je deflace. Principem deflace je skrývání té části spektra matice, která způsobuje zpomalení konvergence. Tato diplomová práce se zabývá efektivní implementací různých deflated verzí sdružených gradientů. Velká pozornost je také věnována teorii a volbě deflačního prostoru. Možnosti implementace jsou demonstrovány na rozsáhlém množství příkladůThe conjugate gradient algorithm is one of the most popular methods for the solution of large systems of linear equations with symmetric positive semi-definite matrix. One of the schemes accelerating the convergence of conjugate gradients is deflation which effectively hides parts of the matrix spectrum that slows down the convergence. This master's thesis deals with efficient parallel implementation of the deflated conjugate gradient method with various modifications. Detailed theoretical considerations and the crucial choice of the deflation space are also discussed. The implementation is showcased on a wide range of benchmarks9600 - IT4Innovationsvýborn

    A robust adaptive algebraic multigrid linear solver for structural mechanics

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    The numerical simulation of structural mechanics applications via finite elements usually requires the solution of large-size and ill-conditioned linear systems, especially when accurate results are sought for derived variables interpolated with lower order functions, like stress or deformation fields. Such task represents the most time-consuming kernel in commercial simulators; thus, it is of significant interest the development of robust and efficient linear solvers for such applications. In this context, direct solvers, which are based on LU factorization techniques, are often used due to their robustness and easy setup; however, they can reach only superlinear complexity, in the best case, thus, have limited applicability depending on the problem size. On the other hand, iterative solvers based on algebraic multigrid (AMG) preconditioners can reach up to linear complexity for sufficiently regular problems but do not always converge and require more knowledge from the user for an efficient setup. In this work, we present an adaptive AMG method specifically designed to improve its usability and efficiency in the solution of structural problems. We show numerical results for several practical applications with millions of unknowns and compare our method with two state-of-the-art linear solvers proving its efficiency and robustness.Comment: 50 pages, 16 figures, submitted to CMAM

    Simultaneous-FETI and Block-FETI: robust domain decomposition with multiple search directions.

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    International audienceDomain 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

    Enhanced balancing Neumann-Neumann preconditioning in computational fluid and solid mechanics

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    Manuscript submitted for publication in International Journal for Numerical Methods in Engineering. Under review.Preprin
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