A new Korn's type inequality for thin domains and its application to iterative methods

Ovtchinnikov, Evgueni; Xanthis, Leonidas S.

doi:10.1016/s0045-7825(96)01135-8

Cited by 11 publications

(14 citation statements)

References 22 publications

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“…If condition (2.3) from [1] holds for E(u; v), this implies, together with Korn's inequality [17], that E(u; v) is coercive in V 0 ( ). Therefore, problem (7) has a unique solution (see Reference [18]).…”

Section: Problem Formulationmentioning

confidence: 96%

“…for which the dimension in one co-ordinate direction is much less than in the other two (the examples in 3D being plates and shells of a ÿnite thickness t), providing that we have a case of non-homogeneous Dirichlet (or some more general) boundary conditions. As noted in Reference [1], the poor coercivness of such problems, that is given by the classic Korn inequality [25], gives rise to slow convergence for iterative methods. In fact, the upper bound of the spectrum of a preconditioned discrete elasticity operator is limited by a constant, while the lower bound deteriorates to 0 as O(t 2 ).…”

Section: D Mihajlovi ã C and S Mijalkovi ã Cmentioning

confidence: 99%

“…In fact, the upper bound of the spectrum of a preconditioned discrete elasticity operator is limited by a constant, while the lower bound deteriorates to 0 as O(t 2 ). Many authors have addressed this problem for particular domains (see Reference [1] and the references contained therein). However, the recent work of Ovtchinnikov and Xanthis [1][2][3][4] o ers a general framework for dealing with this problem by suggesting a concept of a new Korn-type inequality in subspaces.…”

Section: D Mihajlovi ã C and S Mijalkovi ã Cmentioning

confidence: 99%

“…Some of this work covers speciÿc topics, such as thin domain structures in 3D, like plates and shells (see, for example, References [1][2][3][4]), mixed and penalty methods for incompressible problems [5,6] and the application of AMLI methods [7][8][9]. It is well known that multigrid methods o er the prospect of optimal scaling with problem size [10], either as stand-alone methods (in the case of well-structured grids), or as accelerators for the Krylov iterative methods (for the case of more general grids).…”

Section: Introductionmentioning

confidence: 99%

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A component decomposition preconditioning for 3D stress analysis problems

Mihajlović

Mijalković

2002

Numerical Linear Algebra App

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SUMMARYA preconditioning methodology for an iterative solution of discrete stress analysis problems based on a space decomposition and subspace correction framework is analysed in this paper. The principle idea of our approach is a decomposition of a global discrete system into the series of subproblems each of which correspond to the di erent Cartesian co-ordinates of the solution (displacement) vector. This enables us to treat the matrix subproblems in a segregated way. A host of well-established scalar solvers can be employed for the solution of subproblems. In this paper we constrain ourselves to an approximate solution using the scalar algebraic multigrid (AMG) solver, while the subspace correction is performed either in block diagonal (Jacobi) or block lower triangular (Gauss-Seidel) fashion. The preconditioning methodology is justiÿed theoretically for the case of the block-diagonal preconditioner using Korn's inequality for estimating the ratio between the extremal eigenvalues of a preconditioned matrix. The e ectiveness of the AMG-based preconditioner is tested on stress analysis 3D model problems that arise in microfabrication technology. The numerical results, which are in accordance with theoretical predictions, clearly demonstrate the superiority of a component decomposition AMG preconditioner over the standard ILU preconditioner, even for the problems with a relatively small number of degrees of freedom.

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Section: Problem Formulationmentioning

confidence: 96%

Section: D Mihajlovi ã C and S Mijalkovi ã Cmentioning

confidence: 99%

Section: D Mihajlovi ã C and S Mijalkovi ã Cmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A component decomposition preconditioning for 3D stress analysis problems

Mihajlović

Mijalković

2002

Numerical Linear Algebra App

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“…The question of thickness-robustness for full three-dimensional elasticity was first addressed-and radically resolved-in refs. [13][14][15][16]. In these papers, a methodology was introduced for the development of robust numerical methods for thin elastic structures based on the synergy of the so-called Effective Dimensional Reduction Algorithm (EDRA; ref.…”

Section: Introductionmentioning

confidence: 99%

Effective dimensional reduction algorithm for eigenvalue problems for thin elastic structures: A paradigm in three dimensions

Ovtchinnikov

Xanthis

2000

Proc. Natl. Acad. Sci. U.S.A.

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We present a methodology for the efficient numerical solution of eigenvalue problems of full three-dimensional elasticity for thin elastic structures, such as shells, plates and rods of arbitrary geometry, discretized by the finite element method. Such problems are solved by iterative methods, which, however, are known to suffer from slow convergence or even convergence failure, when the thickness is small. In this paper we show an effective way of resolving this difficulty by invoking a special preconditioning technique associated with the effective dimensional reduction algorithm (EDRA). As an example, we present an algorithm for computing the minimal eigenvalue of a thin elastic plate and we show both theoretically and numerically that it is robust with respect to both the thickness and discretization parameters, i.e. the convergence does not deteriorate with diminishing thickness or mesh refinement. This robustness is sine qua non for the efficient computation of large-scale eigenvalue problems for thin elastic structures.robust preconditioning in thickness and discretization parameters ͉ vibrations ͉ shells ͉ plates ͉ rods ProlegomenaMnemosyne, Archimedean Muse: Ivo Babuška and his legacy to computational mathematics, mechanics, and finite element culture.*

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On a robust multilevel method applied for solving large-scale linear elasticity problems

Padiy

1999

Commun. Numer. Meth. Engng.

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SUMMARYThe paper discusses an iterative scheme for solving large-scale three-dimensional linear elasticity problems, discretized on a tensor product of two-dimensional and one-dimensional meshes. A framework is chosen of the additive AMLI method to develop a preconditioner of a`black-box' type which is robust with respect to discontinuities of the problem coecients and imposes only weak (and acceptable in practice) restrictions on the choice of the meshing procedure. The preconditioner works on a hierarchical sequence of nested ®nite element spaces to solve the problem with arithmetic cost, nearly proportional to the number of degrees of freedom on the ®nest mesh. It is particularly well suited for the case when the solution is known to be strongly varying in certain subregions of the domain and the mesh is locally prere®ned there to reduce the discretization error.

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A new Korn's type inequality for thin domains and its application to iterative methods

Cited by 11 publications

References 22 publications

A component decomposition preconditioning for 3D stress analysis problems

A component decomposition preconditioning for 3D stress analysis problems

Effective dimensional reduction algorithm for eigenvalue problems for thin elastic structures: A paradigm in three dimensions

On a robust multilevel method applied for solving large-scale linear elasticity problems

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