Iterative solvers for 3D linear and nonlinear elasticity problems: Displacement and mixed formulations

Maliki, A. El; Fortin, Michel; Tardieu, Nicolas; Fortin, A.

doi:10.1002/nme.2894

Cited by 19 publications

(10 citation statements)

References 24 publications

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“…Choosing a good preconditioner for the Schur complement is critical for the performance of the proposed nested block preconditioner. In our experience, using a scaled pressure mass matrix gives satisfactory results as well [55]. For compressible materials, this preconditioner does not need to be explicitly assembled, and one can use D directly (See A).…”

Section: Compute W = Axmentioning

confidence: 93%

A robust and efficient iterative method for hyper-elastodynamics with nested block preconditioning

Marsden

2019

Journal of Computational Physics

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We develop a robust and efficient iterative method for hyper-elastodynamics based on a novel continuum formulation recently developed in [1]. The numerical scheme is constructed based on the variational multiscale formulation and the generalized-α method. Within the nonlinear solution procedure, a block factorization is performed for the consistent tangent matrix to decouple the kinematics from the balance laws. Within the linear solution procedure, another block factorization is performed to decouple the mass balance equation from the linear momentum balance equations. A nested block preconditioning technique is proposed to combine the Schur complement reduction approach with the fully coupled approach. This preconditioning technique, together with the Krylov subspace method, constitutes a novel iterative method for solving hyper-elastodynamics. We demonstrate the efficacy of the proposed preconditioning technique by comparing with the SIMPLE preconditioner and the one-level domain decomposition preconditioner. Two representative examples are studied: the compression of an isotropic hyperelastic cube and the tensile test of a fully-incompressible anisotropic hyperelastic arterial wall model. The robustness with respect to material properties and the parallel performance of the preconditioner are examined.

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Section: Compute W = Axmentioning

confidence: 93%

A robust and efficient iterative method for hyper-elastodynamics with nested block preconditioning

Marsden

2019

Journal of Computational Physics

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“…where M and I are as defined in (13) and (11), respectively. Similar preconditioners for the Dirichlet problem have been discussed in the works of Klawonn 51 and El Maliki et al 52 Remark 1. (Lemma 6 in the discrete case).…”

mentioning

confidence: 83%

On the singular Neumann problem in linear elasticity

Kuchta

Mardal

Mortensen

2018

Numerical Linear Algebra App

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The Neumann problem of linear elasticity is singular with a kernel formed by the rigid motions of the body. There are several tricks that are commonly used to obtain a nonsingular linear system. However, they often cause reduced accuracy or lead to poor convergence of the iterative solvers. In this paper, different well-posed formulations of the problem are studied through discretization by the finite element method, and preconditioning strategies based on operator preconditioning are discussed. For each formulation, we derive preconditioners that are independent of the discretization parameter. Preconditioners that are robust with respect to the first Lamé constant are constructed for the pure displacement formulations, whereas a preconditioner that is robust in both Lamé constants is constructed for the mixed formulation. It is shown that, for convergence in the first Sobolev norm, it is crucial to respect the orthogonality constraint derived from the continuous problem. On the basis of this observation, a modification to the conjugate gradient method is proposed, which achieves optimal error convergence of the computed solution.corresponding dual space, that is, it is a representation of the right-hand side. Consequently, two projectors are required in the iterative method originating from a singular variational problem. However, standard implementations of Krylov methods, which employ single projection, fail to make the distinction.Rewriting the Krylov solvers to take the two representations into account is, in principle, a simple addition to the code. However, it is also intrusive in the sense that it requires modification of perhaps the low-level code in the implementation of the Krylov solvers. To the best of our knowledge, this distinction is not implemented in state-of-the-art linear algebra frameworks such as PETSc 20 or hypre. 21 Here, we therefore propose a simple alternative solution that is less intrusive, although it requires a priori information about the kernel. Alternative methods, which require less or no a priori information, are the adaptive multigrid methods, 22,23 where the null space is detected automatically during the solver setup in an iterative process that identifies error components that the current solver cannot effectively reduce. Here, we focus on the analysis of the Lagrange multiplier method and the CG method for the singular problem (1). Well-posedness of both the methods is discussed, and robust preconditioners are established based on operator preconditioning. 24 Further, connections between the two methods and the question of whether they yield identically converging numerical solutions are elucidated. These methods rely on standard iterative solvers as they implicitly contain the two required projectors.This paper is structured as follows. In Section 2, the necessary notation is introduced, and shortcomings of pinpointing and CG are illustrated by numerical examples. Section 3 discusses Lagrange multiplier formulation and the two preconditioners for the method. Section 4 de...

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“…Different iterative solvers for large scale linear algebraic systems for 3D elasticity are compared in Ref. [30]. Solving nonlinear Solid Mechanics problems with a Newton-type method could be problematic if the determinations, storage or solution cost associated with the Jacobian is high.…”

Section: Algebraic Solversmentioning

confidence: 99%

Alya: Computational Solid Mechanics for Supercomputers

Casoni

Jérusalem

Samaniego

et al. 2014

Arch Computat Methods Eng

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While solid mechanics codes are now conventional tools both in industry and research, the increasingly more exigent requirements of both sectors are fuelling the need for more computational power and more advanced algorithms. For obvious reasons, commercial codes are lagging behind academic codes often dedicated either to the implementation of one new technique, or the upscaling of current conventional codes to tackle massively large scale computational problems. Only in a few cases, both approaches have been followed simultaneously. In this article, a solid mechanics simulation strategy for parallel supercomputers based on a hybrid approach is presented. Hybrid parallelization exploits the thread-level parallelism of multicore architectures, com- bining MPI tasks with OpenMP threads. This paper describes the proposed strategy, programmed in Alya, a parallel multiphysics code. Hybrid parallelization is specially well suited for the current trend of supercomputers, namely large clusters of multicores. The strategy is assessed through transient non-linear solid mechanics problems, both for explicit and implicit schemes, running on thousands of cores. In order to demonstrate the flexibility of the proposed strategy under advance algorithmic evolution of computational mechanics, a non-local parallel overset meshes method (Chimera-like) is implemented and the conservation of the scalability is demonstrated.

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Iterative solvers for 3D linear and nonlinear elasticity problems: Displacement and mixed formulations

Cited by 19 publications

References 24 publications

A robust and efficient iterative method for hyper-elastodynamics with nested block preconditioning

A robust and efficient iterative method for hyper-elastodynamics with nested block preconditioning

On the singular Neumann problem in linear elasticity

Alya: Computational Solid Mechanics for Supercomputers

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