On the Use of Rigid Body Modes in the Deflated Preconditioned Conjugate Gradient Method

Jönsthövel, T.B.; Gijzen, Martin B. van; Vuik, C.; Scarpas, A.

doi:10.1137/100803651

Cited by 18 publications

(11 citation statements)

References 14 publications

(20 reference statements)

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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%

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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Section: Generation Of the Test Spacementioning

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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show abstract

“…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%

“…Last, we consider the aFSAI density \mu G , i.e. the ratio between the nonzeros in the G factor and the nonzeros in A, which gives an idea of the cost for storing as well as for applying of aFSAI preconditioner: (35) \mu G = nnz (G) nnz (A) .…”

Section: A205mentioning

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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“…The deflation technique to accelerate the convergence of Krylov subspace methods for the solution of a given linear system has been known for a long time; see, e.g., [14,15,44] and the extensive bibliography proposed in [29]. Applications to structural mechanics have been provided in, e.g., [32,33] in the symmetric positive definite case. In the last decade, deflation has been used and analyzed in combination with multigrid and domain decomposition methods, which results in efficient algorithms [32,62].…”

Section: Case Of a Single Linear Systemmentioning

confidence: 99%

“…Deflated and augmented Krylov subspaces [14,15,44,57] or Krylov subspace methods with recycling [26,36,50,53,54,61,67] have been proposed in this setting. Applications in structural mechanics have been provided in, e.g., [32,33] in the symmetric positive definite case. We refer the reader to [15,21,22,30,60] for a comprehensive theoretical overview on these methods and to references therein for a summary of applications, where the relevance of these methods has been shown.…”

Section: Introductionmentioning

confidence: 99%

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

Mercier

Gratton²,

Tardieu

et al. 2017

Comput Mech

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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.

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On the Use of Rigid Body Modes in the Deflated Preconditioned Conjugate Gradient Method

Cited by 18 publications

References 14 publications

A robust adaptive algebraic multigrid linear solver for structural mechanics

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

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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