Bounds on eigenvalues of finite element systems

Lin, Jerry I.

doi:10.1002/nme.1620320503

Cited by 12 publications

(5 citation statements)

References 9 publications

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“…where K = 1, 2, … , N E , and the projector P 0 (⋅, ⋅) is defined in (28). On using (36) and ( 28), we can rewrite (38) as…”

Section: Element Mass Matrixmentioning

confidence: 99%

“…We refer to such an element as a polyhedral virtual element. The elastodynamic eigenproblem on each element is solved, and the element eigenvalue inequality [26][27][28] is used to obtain a lower bound estimate of the critical time step. Comparisons of the critical time step on a suite of bad-quality tetrahedral FEs are made with a polyhedral virtual element to assess the performance of the VEM.…”

Section: Introductionmentioning

confidence: 99%

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Virtual elements on agglomerated finite elements to increase the critical time step in elastodynamic simulations

Sukumar

Tupek

2022

Numerical Meth Engineering

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In this article, we use the first‐order virtual element method (VEM) to investigate the effect of shape quality of polyhedra in the estimation of the critical time step for explicit three‐dimensional elastodynamic finite element (FE) simulations. Low‐quality FEs are common when meshing realistic complex components, and while tetrahedral meshing technology is generally robust, meshing algorithms cannot guarantee high‐quality meshes for arbitrary geometries or for non‐water‐tight computer‐aided design models. For reliable simulations on such meshes, we consider FE meshes with tetrahedral and prismatic elements that have badly shaped elements—tetrahedra with dihedral angles close to 0∘$$ {0}^{\circ } $$ and 180∘$$ 18{0}^{\circ } $$, and slender prisms with triangular faces that have short edges—and agglomerate such “bad” elements with neighboring elements to form a larger polyhedral virtual element. On each element, the element‐eigenvalue inequality is used to estimate the critical time step. For a suite of illustrative FE meshes with ϵ$$ \epsilon $$ being a mesh‐coordinate parameter that leads to poor mesh quality, we show that adopting VEM on the agglomerated polyhedra yield critical time steps that are insensitive as ϵ→0$$ \epsilon \to 0 $$. The significant reduction in solution time on meshes with agglomerated virtual elements vis‐à‐vis tetrahedral meshes is demonstrated through explicit dynamics simulations on a tapered beam.

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“…where K = 1, 2, … , N E , and the projector P 0 (⋅, ⋅) is defined in (28). On using (36) and ( 28), we can rewrite (38) as…”

Section: Element Mass Matrixmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Virtual elements on agglomerated finite elements to increase the critical time step in elastodynamic simulations

Sukumar

Tupek

2022

Numerical Meth Engineering

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“…Because the calculation of the maximum eigenvalue for a large system is computationally expensive, the critical time step can be approximated using the maximum eigenvalue of a local system . Because the maximum eigenvalue of a local system is greater than the maximum eigenvalue of a global system, an approximated critical time step (

normalΔ {\overset{true˜}{t}}_{cr}

) is given as

normalΔ {\overset{true˜}{t}}_{cr} = \frac{2}{\max ((), ω_{\max}^{E})} < normalΔ t_{cr},

where

ω_{\max}^{E}

is the maximum eigenvalue of an element.…”

Section: Time Integrationmentioning

confidence: 99%

Numerical recipes for elastodynamic virtual element methods with explicit time integration

Park

Chi

Paulino

2019

Numerical Meth Engineering

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Summary We present a general framework to solve elastodynamic problems by means of the virtual element method (VEM) with explicit time integration. In particular, the VEM is extended to analyze nearly incompressible solids using the B‐bar method. We show that, to establish a B‐bar formulation in the VEM setting, one simply needs to modify the stability term to stabilize only the deviatoric part of the stiffness matrix, which requires no additional computational effort. Convergence of the numerical solution is addressed in relation to stability, mass lumping scheme, element size, and distortion of arbitrary elements, either convex or nonconvex. For the estimation of the critical time step, two approaches are presented, ie, the maximum eigenvalue of a system of mass and stiffness matrices and an effective element length. Computational results demonstrate that small edges on convex polygonal elements do not significantly affect the critical time step, whereas convergence of the VEM solution is observed regardless of the stability term and the element shape in both two and three dimensions. This extensive investigation provides numerical recipes for elastodynamic VEMs with explicit time integration and related problems.

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“…In the literature, there are two interesting exceptions to the general observation that most researchers focus on the element eigenvalue problem. Firstly, Lin suggested to use the eigenvalue problem at the integration point level, rather than the element level. 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 .…”

Section: Introductionmentioning

confidence: 99%

“…Firstly, Lin suggested to use the eigenvalue problem at the integration point level, rather than the element level. 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 .…”

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

Cited by 12 publications

References 9 publications

Virtual elements on agglomerated finite elements to increase the critical time step in elastodynamic simulations

Virtual elements on agglomerated finite elements to increase the critical time step in elastodynamic simulations

Numerical recipes for elastodynamic virtual element methods with explicit time integration

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

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