Rigid body modes deflation of the Preconditioned Conjugate Gradient in the solution of discretized structural problems

Baggio, Roberta; Franceschini, A.; Spiezia, Nicolò; Janna, Carlo

doi:10.1016/j.compstruc.2017.03.003

Cited by 11 publications

(11 citation statements)

References 32 publications

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“…Jacobi or Gauss-Seidel, with the second one often preferred even though its efficient parallel implementation is not straightforward. In Chronos, we use the adaptive Factorized Sparse Approximate Inverse (aFSAI) [9], beacuse of its almost perfect strong scalability and its proven robustness in real engineering problems [1,8].…”

Section: Classical Algebraic Multigridmentioning

confidence: 99%

A General-Purpose AMG Linear Solver for High Performance Computing

Isotton¹,

Frigo²,

Spiezia³

et al. 2021

14th WCCM-ECCOMAS Congress

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The numerical simulation of modern engineering problems via finite elements requires the solution of sparse linear systems of millions or even billions of unknowns. The algebraic multigrid (AMG) methods are the most common choice as linear solvers because of their fast convergence even for large-size problems. In this communication, we propose Chronos, a massively parallel implementation of a novel AMG framework, specifically designed to address complex problems by adapting its components, from the smoother, to the coarse grid correction and prolongation to the problem at hand. This work demonstrates not only the numerical performance of the proposed library, but also its robustness and adaptability to very challenging matrices, arising from different fields of application.

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Section: Classical Algebraic Multigridmentioning

confidence: 99%

A General-Purpose AMG Linear Solver for High Performance Computing

Isotton¹,

Frigo²,

Spiezia³

et al. 2021

14th WCCM-ECCOMAS Congress

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“…When solving elasticity problems, it is known from theory that rigid body modes (RBMs) constitute the kernel of the differential operator before the application of boundary conditions; thus, they can be used to define V 0 . As a matter of fact, such information is essential to the effectiveness of several methods like aggregation-based AMG [54,55], domain decomposition techniques [56,57,58,30] and deflation-based preconditioning [59,60,61], just to cite a few. In 2D and 3D elasticity problems, there are 3 and 6 RBMs, respectively.…”

Section: Generation Of the Test Spacementioning

confidence: 99%

“…The mesh derives from a 3D mechanical equilibrium problem of a poorly constrained steel hook discetized with 2, 578, 916 linear tetrahedral finite elements giving rise to 499, 341 vertices. For further details the reader should refer to [61].…”

Section: Appendix a Glimpses Of Meshes Employed By The Real-world Enmentioning

confidence: 99%

A robust adaptive algebraic multigrid linear solver for structural mechanics

Franceschini

Magri

Mazzucco

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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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.

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“…For instance, when dealing with matrices arising from the linear elasticity model, the rigid body modes of the structure can be inexpensively computed. These are a good representation of the near-null space, widely used in AMG solvers [2] and deflation-based methods [35,4].…”

Section: Test Space Generationmentioning

confidence: 99%

“…In aSP-AMG, we introduce the usage of the aFSAI [34] as a smoother. This choice, which will be detailed below, is dictated by its almost perfect strong scalability and by its proved effectiveness in real engineering problems [4,33]. Regardless of the choice, the smoother operator S can be represented by the equation…”

Section: Introductionmentioning

confidence: 99%

A Novel Algebraic Multigrid Approach Based on Adaptive Smoothing and Prolongation for Ill-Conditioned Systems

Magri¹,

Franceschini²,

Janna³

2019

SIAM J. Sci. Comput.

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The numerical simulation of modern engineering problems can easily incorporate millions or even billions of unknowns. In several applications, sparse linear systems with symmetric positive definite matrices need to be solved, and algebraic multigrid (AMG) methods represent common choices for the role of iterative solvers or preconditioners. The reason for their popularity relies on the fast convergence that these methods provide even in the solution of large size problems, which is a consequence of the AMG ability to reduce particular error components across their multilevel hierarchy. Despite carrying the name ``algebraic,"" most of these methods still make assumptions on additional information other than the global assembled matrix, such as the knowledge of the operator's near kernel, which limits their applicability as black-box solvers. In this work, we introduce a novel AMG approach featuring the adaptive factored sparse approximate inverse (aFSAI) method as a flexible smoother, as well as three new approaches to adaptively compute the prolongation operator. We assess the performance of the proposed AMG through the solution of a set of model problems along with real-world engineering test cases. Moreover, we perform comparisons to other methods such as the aFSAI and BoomerAMG preconditioners, showing that our new method proves to be superior to the first strategy and comparable to the second one, if not better, as in the solution of linear elasticity models.

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Rigid body modes deflation of the Preconditioned Conjugate Gradient in the solution of discretized structural problems

Cited by 11 publications

References 32 publications

A General-Purpose AMG Linear Solver for High Performance Computing

A General-Purpose AMG Linear Solver for High Performance Computing

A robust adaptive algebraic multigrid linear solver for structural mechanics

A Novel Algebraic Multigrid Approach Based on Adaptive Smoothing and Prolongation for Ill-Conditioned Systems

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