Estimating the critical time‐step in explicit dynamics using the Lanczos method

Koteras, J.R.; Lehoucq, Richard B.

doi:10.1002/nme.1865

Cited by 10 publications

(8 citation statements)

References 6 publications

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“…Convergence, accuracy, and computational cost, all, are being affected by the t defined in Equation (2) [19,26,[30][31][32]; hence, the appropriate selection of t is of high computational importance, attracting considerable theoretical attention, see also [33][34][35][36]. Currently, it is a general approach to set t (time step size) equal to the largest value satisfying the requirements of numerical stability and consistency (convergence), also, appropriate for accuracy.…”

Section: Introductionmentioning

confidence: 98%

A technique for time integration analysis with steps larger than the excitation steps

Soroushian

2008

Commun. Numer. Meth. Engng.

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SUMMARYThe integration step size is the main algorithmic parameter in time integration analysis. Nowadays, for time integration with the complete records of digitized excitations, the integration steps cannot be set larger than the excitation steps. Considering the practical importance of this restriction, and with the aim of structural dynamic analysis with less computational cost, this paper intends to extend conventional time integration analyses to analyses, if needed, carried out with steps larger than the excitations steps. In view of few simplifying assumptions, and presenting a new theorem on responses convergence, a technique is developed and a computational procedure is set. Being based on convergence-oriented redefinition of digitized excitations, the implementation of the new procedure is simple and, though, sacrificing some accuracy, can considerably reduce the total computational cost.

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Section: Introductionmentioning

confidence: 98%

A technique for time integration analysis with steps larger than the excitation steps

Soroushian

2008

Commun. Numer. Meth. Engng.

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“…The presented bounds on the respective eigenvalues demonstrate that this approach is safe but conservative: the critical time step computed using the integration point eigenvalues is around 15% lower than the critical time step computed using the elemental eigenvalues for the tests reported in and around 30% lower than the global maximum eigenvalue for the tests reported in . Alternatively, some researchers aim to solve the global eigenvalue problem, using for instance power iteration methods or Lanczos methods . If the difference between global maximum eigenfrequency and elemental maximum frequency is large, such methods may lead to a significant increase in time step size.…”

Section: Introductionmentioning

confidence: 95%

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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“…Direct discretization of (7) can lead to a very large and computationally expensive eigenvalue problem. The common approach to circumvent this challenge is to use the method of snapshots [23] in which the eigenvalue problem is transformed into a smaller one given as…”

Section: Proper Orthogonal Decompositionmentioning

confidence: 99%

“…The Lanczos method provides larger critical time steps but requires the solution of an eigenvalue problem at multiple stages during the simulation. Some studies have addressed the issue of the added cost of using the Lanczos method and compared it to the cost of the element‐based one. Heinstein et al .…”

Section: Introductionmentioning

confidence: 99%

“…The Lanczos method [6] provides larger critical time steps but requires the solution of an eigenvalue problem at multiple stages during the simulation. Some studies [7] have addressed the issue of the added cost of using the Lanczos method and compared it to the cost of the element-based one. Heinstein et al [8] also developed a less conservative, nodal-based, time step estimate that has been heavily used in the SIERRA Mechanics [9] software developed at Sandia National Laboratories.There is a wide variety of problems in solid mechanics in which high level of spatial mesh refinement is needed (for instance, to capture material or geometric features) but for which lowfrequency dynamics dominate the overall response.…”

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confidence: 99%

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A multiscale mass scaling approach for explicit time integration using proper orthogonal decomposition

Frias

Aquino

Pierson

et al. 2013

Numerical Meth Engineering

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SUMMARY One of the main computational issues with explicit dynamics simulations is the significant reduction of the critical time step as the spatial resolution of the finite element mesh increases. In this work, a selective mass scaling approach is presented that can significantly reduce the computational cost in explicit dynamic simulations, while maintaining accuracy. The proposed method is based on a multiscale decomposition approach that separates the dynamics of the system into low (coarse scales) and high frequencies (fine scales). Here, the critical time step is increased by selectively applying mass scaling on the fine scale component only. In problems where the response is dominated by the coarse (low frequency) scales, significant increases in the stable time step can be realized. In this work, we use the proper orthogonal decomposition (POD) method to build the coarse scale space. The main idea behind POD is to obtain an optimal low‐dimensional orthogonal basis for representing an ensemble of high‐dimensional data. In our proposed method, the POD space is generated with snapshots of the solution obtained from early times of the full‐scale simulation. The example problems addressed in this work show significant improvements in computational time, without heavily compromising the accuracy of the results. Copyright © 2013 John Wiley & Sons, Ltd.

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Estimating the critical time‐step in explicit dynamics using the Lanczos method

Cited by 10 publications

References 6 publications

A technique for time integration analysis with steps larger than the excitation steps

A technique for time integration analysis with steps larger than the excitation steps

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

A multiscale mass scaling approach for explicit time integration using proper orthogonal decomposition

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