Algebraic multigrid for finite element elasticity equations: Determination of nodal dependence via edge‐matrices and two‐level convergence

Karer, Erwin; Kraus, Johannes

doi:10.1002/nme.2853

Cited by 13 publications

(26 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Further robust methods for solving linear elasticity problems are available in the literature, including multilevel methods studied in [24], and further developed in [21,22]. Karer and Kraus construct a purely algebraic multigrid method for linear elasticity problems, based on computational molecules, a new variant of AMGe.…”

Section: Introductionmentioning

confidence: 99%

Multiscale finite element coarse spaces for the application to linear elasticity

2013

View full text Add to dashboard Cite

Abstract:We extend the multiscale finite element method (MsFEM) as formulated by Hou and Wu in [Hou T.Y., Wu X.-H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 1997, 134(1), 169-189] to the PDE system of linear elasticity. The application, motivated by the multiscale analysis of highly heterogeneous composite materials, is twofold. Resolving the heterogeneities on the finest scale, we utilize the linear MsFEM basis for the construction of robust coarse spaces in the context of two-level overlapping domain decomposition preconditioners. We motivate and explain the construction and show that the constructed multiscale coarse space contains all the rigid body modes. Under the assumption that the material jumps are isolated, that is they occur only in the interior of the coarse grid elements, our numerical experiments show uniform convergence rates independent of the contrast in Young's modulus within the heterogeneous material. Elsewise, if no restrictions on the position of the high coefficient inclusions are imposed, robustness cannot be guaranteed any more. These results justify expectations to obtain coefficient-explicit condition number bounds for the PDE system of linear elasticity similar to existing ones for scalar elliptic PDEs as given in the work of Graham, Lechner and Scheichl [Graham I.G., Lechner P.O., Scheichl R., Domain decomposition for multiscale PDEs, Numer. Math., 2007, 106(4), 589-626]. Furthermore, we numerically observe the properties of the MsFEM coarse space for linear elasticity in an upscaling framework. Therefore, we present experimental results showing the approximation errors of the multiscale coarse space w.r.t. the fine-scale solution. MSC:65N55, 35R05, 74B05, 74S05

show abstract

Section: Introductionmentioning

confidence: 99%

Multiscale finite element coarse spaces for the application to linear elasticity

2013

View full text Add to dashboard Cite

show abstract

“…Most commonly used is the point-based or node-based approach, where all variables discretized at a common spatial node are treated together in the coarsening and interpolation processes. This approach was first proposed for linear elasticity in [39] and has been extended in [24,31]. An extension framework to improve AMG for elasticity problems was proposed in [8], which uses a hybrid approach with nodal coarsening, but interpolation based on the unknown-based approach.…”

Section: Algebraic Multigrid Methodsmentioning

confidence: 99%

Comparison of the deflated preconditioned conjugate gradient method and algebraic multigrid for composite materials

et al. 2011

View full text Add to dashboard Cite

Many applications in computational science and engineering concern composite materials, which are characterized by large discontinuities in the material properties. Such applications require fine-scale finite-element meshes, which lead to large linear systems that are challenging to solve with current direct and iterative solutions algorithms. In this paper, we consider the simulation of asphalt concrete, which is a mixture of components with large differences in material stiffness. The discontinuities in material stiffness give rise to many small eigenvalues that negatively affect the convergence of iterative solution algorithms such as the preconditioned conjugate gradient (PCG) method. This paper considers the deflated preconditioned conjugate gradient (DPCG) method in which the rigid body modes of sets of elements with homogeneous material properties are used as deflation vectors. As preconditioner we consider several variants of the algebraic multigrid smoothed aggregation method. We evaluate the performance of the DPCG method on a parallel computer using up to 64 processors. Our test problems are derived from real asphalt core samples, obtained using CT scans. We show that the DPCG method is an efficient and robust technique for solving these challenging linear systems. T. B. Jönsthövel (B)·

show abstract

“…Similarly, the kernel of local stiffness matrices is preserved by the approximate splittings in AMG based on computational molecules. 17,18 Given that the restriction and interpolation operators preserve the kernel exactly and that the initial residual does not contain elements in the kernel, the singular problem at the coarsest level may be solved with Krylov methods, like the conjugate gradient (CG) method, which then work well. To complete our (nonexhaustive) list, let us mention that in the domain decomposition methods (e.g., finite element tearing and interconnecting 19 ), the Neumann problem (1) arises naturally on "floating" subdomains that do not intersect the Dirichlet boundaries.…”

Section: Introductionmentioning

confidence: 99%

On the singular Neumann problem in linear elasticity

Kuchta

Mardal

Mortensen

2018

Numerical Linear Algebra App

View full text Add to dashboard Cite

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...

show abstract

Algebraic multigrid for finite element elasticity equations: Determination of nodal dependence via edge‐matrices and two‐level convergence

Cited by 13 publications

References 28 publications

Multiscale finite element coarse spaces for the application to linear elasticity

Multiscale finite element coarse spaces for the application to linear elasticity

Comparison of the deflated preconditioned conjugate gradient method and algebraic multigrid for composite materials

On the singular Neumann problem in linear elasticity

Contact Info

Product

Resources

About