Eigenproblems associated with the discrete LBB condition for incompressible finite elements

Malkus, David S.

doi:10.1016/0020-7225(81)90013-6

Cited by 77 publications

(55 citation statements)

References 7 publications

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“…Let n = 2, m = 1 (so N = 3), denote the columns of I 3 by e i , i = 1, 2, 3, and let and clearly (β −1 , e 2 ) is an eigenpair of S, 0 is a double algebraic, simple geometric eigenvalue, e 3 is the corresponding eigenvector and e 1 is the generalised eigenvector or principal eigenvector of grade 2. This behaviour is generic in N × N problems with the block structure of (2) as was shown by Malkus [9], who considered the Weierstrass-Kronecker canonical form of (2), and Ericsson [5], who considered the Jordan form for the shift-invert transformation of a variety of generalised eigenvalue problems. (Incidently, both authors restrict attention to problems with symmetric A but several of their results, at least to do with Jordan structure, extend to the case when A is nonsymmetric).…”

Section: If (λ X)mentioning

confidence: 64%

“…y 1 ∈ R 1 can be achieved by y 1 ← S 1 y 1 . To apply B-orthogonal Arnoldi to S, one should start with v 1 , given by (9). This is achieved by v 1 ← Sv 1 , since…”

Section: Application Of B-orthogonalmentioning

confidence: 99%

“…Here M is the mass matrix of the velocity elements and K is nonsymmetric because of the linearisation of the convection term in the NavierStokes equations. As is well known, see Malkus [9], Ericsson [5] and Cliffe, Garratt, and Spence [3], (2) can have infinite eigenvalues, corresponding to eigenvectors of the form (0 T , p T ) T . These have no physical relevance and in applications one would only be concerned about the calculation of a small number of 'stability determining' finite eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

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Implicitly restarted Arnoldi with purification for the shift-invert transformation

Meerbergen¹,

Spence

1997

Math. Comp.

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Abstract. The need to determine a few eigenvalues of a large sparse generalised eigenvalue problem Ax = λBx with positive semidefinite B arises in many physical situations, for example, in a stability analysis of the discretised Navier-Stokes equation. A common technique is to apply Arnoldi's method to the shift-invert transformation, but this can suffer from numerical instabilities as is illustrated by a numerical example. In this paper, a new method that avoids instabilities is presented which is based on applying the implicitly restarted Arnoldi method with the B semi-inner product and a purification step. The paper contains a rounding error analysis and ends with brief comments on some extensions.

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Section: If (λ X)mentioning

confidence: 64%

“…y 1 ∈ R 1 can be achieved by y 1 ← S 1 y 1 . To apply B-orthogonal Arnoldi to S, one should start with v 1 , given by (9). This is achieved by v 1 ← Sv 1 , since…”

Section: Application Of B-orthogonalmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Implicitly restarted Arnoldi with purification for the shift-invert transformation

Meerbergen¹,

Spence

1997

Math. Comp.

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“…The proof can be found in [37] or [10]. The numerical test proposed in [11] consists in testing a particular formulation by calculating β using meshes of increasing refinement.…”

Section: Numerical Assessment Of the Inf-sup Conditionmentioning

confidence: 99%

Stability of incompressible formulations enriched with X-FEM

Legrain

Moës

Huerta

2008

Computer Methods in Applied Mechanics and Engineering

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The treatment of (near-)incompressibility is a major concern for applications involving rubber-like materials, or when important plastic flows occurs as in forming processes. The use of mixed finite element methods is known to prevent the locking of the finite element approximation in the incompressible limit. However, it also introduces a critical condition for the stability of the formulation, called the infsup or LBB condition. Recently, the finite element method has evolved with the introduction of the partition of unity. The eXtended Finite Element Method (X-FEM) uses the partition of unity to remove the need to mesh physical surfaces or to remesh them as they evolve. The enrichment of the displacement field makes it possible to treat surfaces of discontinuity inside finite elements. In this paper, some strategies are proposed for the enrichment of mixed finite element approximations in the incompressible setting. The case of holes, material interfaces and cracks are considered. Numerical examples show that for well chosen enrichment strategies, the finite element convergence rate is preserved and the inf-sup condition is passed.

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“…To assess the element performance in the nearly incompressible range, we examine the optimality and stability conditions for small strain elasticity (see [24], for example) using the inf-sup test of Malkus [27]. More information on this topic can also be found in [28].…”

Section: Element Assessmentmentioning

confidence: 99%

Stabilized four‐node tetrahedron with nonlocal pressure for modeling hyperelastic materials

Areias

Matouš

2008

Numerical Meth Engineering

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SUMMARYNon-linear hyperelastic response of reinforced elastomers is modeled using a novel three-dimensional mixed finite element method with a nonlocal pressure field. The element is unconditionally convergent and free of spurious pressure modes. Nonlocal pressure is obtained by an implicit gradient technique and obeys the Helmholtz equation. Physical motivation for this nonlocality is shown. An implicit finite element scheme with consistent linearization is presented. Finally, several hyperelastic examples are solved to demonstrate the computational algorithm including the inf-sup and verifications tests.

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Eigenproblems associated with the discrete LBB condition for incompressible finite elements

Cited by 77 publications

References 7 publications

Implicitly restarted Arnoldi with purification for the shift-invert transformation

Implicitly restarted Arnoldi with purification for the shift-invert transformation

Stability of incompressible formulations enriched with X-FEM

Stabilized four‐node tetrahedron with nonlocal pressure for modeling hyperelastic materials

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