An element eigenvalue theorem and its application for stable time steps

Lin, Jerry I.

doi:10.1016/0045-7825(89)90069-8

Cited by 9 publications

(11 citation statements)

References 8 publications

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“…As described in the works of Wathen, Fried, and Lin, for the finite element and spectral element methods (and other methods for which the matrices are obtained by assembling elemental contributions), the eigenvalues λ m , for m = 1,…,n mn , of D are bounded by

\min_{Ω_{e}} λ_{\min}^{e} \leq λ_{m} \leq \max_{Ω_{e}} λ_{\max}^{e}, for m = 1, …, n_{mn},

where

λ_{\min}^{e}

and

λ_{\max}^{e}

are the minimum and maximum eigenvalues of the elemental matrix D e : = ( M e ) −1 K e , respectively. This is known as the theorem of Irons and Treharne, and it implies that

λ_{\max} \leq \max_{Ω_{e}} λ_{\max}^{e} .

Therefore, a conservative choice for the time step that ensures stability is

normalΔ t \leq \frac{2}{\sqrt{\max_{Ω_{e}} λ_{\max}^{e}}} \leq \frac{2}{\sqrt{λ_{\max}}} .

…”

Section: Stability Of the Leapfrog Schemementioning

confidence: 99%

“…The theorem of Irons and Treharne has been previously applied to compute a time step that ensures stability of an explicit time integrator scheme in applications with homogeneous materials and by using different numerical techniques, eg, the standard finite element method, the boundary element method, and the natural element method . The same idea has also been exploited to study the effect of element distortion in the stability of a finite element method using two‐dimensional (2D) quadrilateral elements . To the knowledge of the authors, the application of the Irons‐Treharne theorem has not been previously considered for heterogeneous media.…”

Section: Stability Of the Leapfrog Schemementioning

confidence: 99%

“…As described in the works of Wathen, 24 Fried, 35 and Lin, 36 for the finite element and spectral element methods (and other methods for which the matrices are obtained by assembling elemental contributions), the eigenvalues m ,…”

Section: The Theorem Of Irons and Treharnementioning

confidence: 99%

“…39 The same idea has also been exploited to study the effect of element distortion in the stability of a finite element method using two-dimensional (2D) quadrilateral elements. 36 To the knowledge of the authors, the application of the Irons-Treharne theorem has not been previously considered for heterogeneous media.…”

Section: Sharpness Of the Irons-treharne Bound For Heterogeneous Mediamentioning

confidence: 99%

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Stability of an explicit high‐order spectral element method for acoustics in heterogeneous media based on local element stability criteria

Cottereau

Sevilla

2018

Numerical Meth Engineering

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Summary This paper considers the stability of an explicit leapfrog time marching scheme for the simulation of acoustic wave propagation in heterogeneous media with high‐order spectral elements. The global stability criterion is taken as a minimum over local element stability criteria, obtained through the solution of element‐borne eigenvalue problems. First, an explicit stability criterion is obtained for the particular case of a strongly heterogeneous and/or rapidly fluctuating medium using asymptotic analysis. This criterion is only dependent upon the maximum velocity at the vertices of the mesh elements, and not on the velocity at the interior nodes of the high‐order elements. Second, in a more general setting, bounds are derived using statistics of the coefficients of the elemental dispersion matrices. Different bounds are presented, discussed, and compared. Several numerical experiments show the accuracy of the proposed criteria in one‐dimensional test cases as well as in more realistic large‐scale three‐dimensional problems.

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\min_{Ω_{e}} λ_{\min}^{e} \leq λ_{m} \leq \max_{Ω_{e}} λ_{\max}^{e}, for m = 1, …, n_{mn},

where

λ_{\min}^{e}

and

λ_{\max}^{e}

are the minimum and maximum eigenvalues of the elemental matrix D e : = ( M e ) −1 K e , respectively. This is known as the theorem of Irons and Treharne, and it implies that

λ_{\max} \leq \max_{Ω_{e}} λ_{\max}^{e} .

Therefore, a conservative choice for the time step that ensures stability is

normalΔ t \leq \frac{2}{\sqrt{\max_{Ω_{e}} λ_{\max}^{e}}} \leq \frac{2}{\sqrt{λ_{\max}}} .

…”

Section: Stability Of the Leapfrog Schemementioning

confidence: 99%

Section: Stability Of the Leapfrog Schemementioning

confidence: 99%

Section: The Theorem Of Irons and Treharnementioning

confidence: 99%

Section: Sharpness Of the Irons-treharne Bound For Heterogeneous Mediamentioning

confidence: 99%

See 2 more Smart Citations

Stability of an explicit high‐order spectral element method for acoustics in heterogeneous media based on local element stability criteria

Cottereau

Sevilla

2018

Numerical Meth Engineering

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“…This is usually deemed to be too expensive, and most researchers solve instead the maximum eigenfrequency of a single element. The maximum eigenvalue of the entire finite element mesh is smaller than the maximum eigenvalue of the individual elements of that mesh ; therefore, it is safe to estimate the critical time step using the maximum eigenfrequency of the smallest finite element.…”

Section: Introductionmentioning

confidence: 99%

The effects of element shape on the critical time step in explicit time integrators for elasto-dynamics

Askes

Ferran

Hetherington

2014

Int. J. Numer. Meth. Engng

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SUMMARYIn this paper, the effects of element shape on the critical time step are investigated. The common ruleof-thumb, used in practice, is that the critical time step is set by the shortest distance within an element divided by the dilatational (compressive) wave speed, with a modest safety factor. For regularly shaped elements, many analytical solutions for the critical time step are available, but this paper focusses on distorted element shapes. The main purpose is to verify whether element distortion adversely affects the critical time step or not. Two types of element distortion will be considered, namely aspect ratio distortion and angular distortion, and two particular elements will be studied: four-noded bilinear quadrilaterals and three-noded linear triangles. The maximum eigenfrequencies of the distorted elements are determined and compared to those of the corresponding undistorted elements. The critical time steps obtained from single element calculations are also compared to those from calculations based on finite element patches with multiple elements.

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Bounds on eigenvalues of finite element systems

Lin¹

1991

Numerical Meth Engineering

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Bounds on eigenvalues of various finite element systems are analysed by using an element eigenvalue theorem together with the Global Eigenvalue Theorem. Both two dimensional continuum dynamics and heat conduction problems are considered. These bounds provide stable time steps for explict time integration schemes. A reduced eigenproblem at element quadrature point level, with all zero eigenvalues suppressed, is also presented in this paper. The simplified eigenproblem results in simple formulas for calculating the eigenvalues.

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An element eigenvalue theorem and its application for stable time steps

Cited by 9 publications

References 8 publications

Stability of an explicit high‐order spectral element method for acoustics in heterogeneous media based on local element stability criteria

Stability of an explicit high‐order spectral element method for acoustics in heterogeneous media based on local element stability criteria

The effects of element shape on the critical time step in explicit time integrators for elasto-dynamics

Bounds on eigenvalues of finite element systems

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