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

Askes, Harm; Ferran, Antonio Rodríguez; Hetherington, Jack

doi:10.1002/nme.4819

Cited by 8 publications

(5 citation statements)

References 26 publications

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“…Alternatively, the critical time step is approximated in relation with an element size and wave speed of a material . A travel distance of wave for a given time shall not be greater than an element size, called the Courant condition, and thus, an approximated critical time step is expressed as

normalΔ {\overset{true˜}{t}}_{cr} = \frac{\min ((), ℓ_{E})}{C_{d}} < normalΔ t_{cr},

where ℓ E is an effective element length for an element and C d is the dilatational wave speed.…”

Section: Time Integrationmentioning

confidence: 99%

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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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normalΔ {\overset{true˜}{t}}_{cr} = \frac{\min ((), ℓ_{E})}{C_{d}} < normalΔ t_{cr},

where ℓ E is an effective element length for an element and C d is the dilatational wave speed.…”

Section: Time Integrationmentioning

confidence: 99%

“…Alternatively, the critical time step is approximated in relation with an element size and wave speed of a material. [30][31][32] A travel distance of wave for a given time shall not be greater than an element size, called the Courant condition, and thus, an approximated critical time step is expressed as…”

Section: Critical Time Step and Effective Element Lengthmentioning

confidence: 99%

Numerical recipes for elastodynamic virtual element methods with explicit time integration

Park

Chi

Paulino

2019

Numerical Meth Engineering

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“…For linear FEs with regular shapes, it may be possible to derive the critical time step in a closed‐form expression; for instance, . Shape effects of bilinear plane FEs on the critical time step size in central difference computations in elastodynamics have been studied in . For uniform four‐node square quadrilateral and eight‐node cubic hexahedral FEs with the edge length H , the stability limit is estimated as Δ t crit = H / c L .…”

Section: Governing Equationsmentioning

confidence: 99%

Temporal-spatial dispersion and stability analysis of finite element method in explicit elastodynamics

Kolman

Plešek

Červ

et al. 2015

Int. J. Numer. Meth. Engng

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SUMMARYThis work presents the temporal-spatial (full) dispersion and stability analysis of plane square linear and biquadratic serendipity finite elements in explicit numerical solution of transient elastodynamic problems. Here, the central difference method, as an explicit time integrator, is exploited. The paper complements and extends the previous work on spatial/grid dispersion analysis of plane square biquadratic serendipity finite elements. We report on a computational strategy for temporal-spatial dispersion relationships, where eigenfrequencies from grid/spatial dispersion analysis are adjusted to comply with the time integration method. Besides that, an 'optimal' lumped mass matrix for the studied finite element types is proposed and investigated. Based on the temporal-spatial dispersion and stability analysis, relationships suggesting the 'proper' choice of mesh size and time step size from knowledge of the loading spectrum are presented.

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“…For low‐order elements with regular shapes, it may be possible to derive the critical time step in a closed‐form expression, for instance . Shape effects of lower plane elements on the critical time step size in central difference computations in elastodynamics have been studied in . The estimates of the stable time step size for arbitrarily shaped quadrilateral and hexahedral finite elements have been derived and presented by Flanagan and Belytschko .…”

Section: Governing Equationsmentioning

confidence: 99%

Efficient implementation of an explicit partitioned shear and longitudinal wave propagation algorithm

Kolman

Cho

Park

2015

Int. J. Numer. Meth. Engng

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SUMMARYThe paper complements and extends the previous works on partitioned explicit wave propagation analysis methods, which were presented for discontinuous wave propagation problems in solids. An efficient implementation of the partitioned explicit wave propagation analysis methods is introduced. The present implementation achieves about 25% overall computational effort compared with the previous implementation with the same accuracy. The present algorithm tracks, with different integration time step sizes in accordance with their different wave speeds, the propagation fronts of longitudinal and shear waves. This is accomplished by integrating separately the element-by-element partitioned longitud inal and shear equations of motion. The state vectors (displacements, velocity and accelerations) of the longitudinal and shear components are reconciled at the end of each time step. The reconciliation procedure does not require any system parameters such as material properties, density, unlike conventional artificial viscosity methods. Numerical examples are presented as applied to linear and non-linear wave propagation problems, which demonstrate high-fidelity wavefront tracking ability of the present method, and compared with existing conventional wave propagation analysis methods.

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

Cited by 8 publications

References 26 publications

Numerical recipes for elastodynamic virtual element methods with explicit time integration

Numerical recipes for elastodynamic virtual element methods with explicit time integration

Temporal-spatial dispersion and stability analysis of finite element method in explicit elastodynamics

Efficient implementation of an explicit partitioned shear and longitudinal wave propagation algorithm

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