Dispersion of elastic waves in the contact–impact problem of a long cylinder

Gabriel, D.; Plešek, Jiřı́; Kolman, Radek; Valeš, František

doi:10.1016/j.cam.2009.08.043

Cited by 22 publications

(19 citation statements)

References 4 publications

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“…This leads to a huge reduction in computation time, especially for multi-dimensional elastodynamics problems. We should also mention that for the standard mass and stiffness matrices, quadratic finite elements yield more accurate results than linear elements; e.g., see [7,16] and the numerical results in Sects. 4.1 and 4.2.…”

Section: Discussionmentioning

confidence: 99%

“…It is known that even the exact solution to Eq. (1) contains the numerical dispersion error, which is also related to the space discretization error; e.g., see [4,5,7,8,[17][18][19][21][22][23]25] and others. The decrease in the space discretization error by the use of mesh refinement considerably increases computational costs.…”

Section: Introductionmentioning

confidence: 99%

“…In the general case of loading (boundary conditions), the numerical study of the effectiveness of the finite element formulations with the reduced dispersion error is difficult due to the presence of spurious high-frequency oscillations in numerical solutions; e.g., see [7,17].…”

Section: Introductionmentioning

confidence: 99%

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Accurate finite element modeling of linear elastodynamics problems with the reduced dispersion error

2010

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It is known that the reduction in the finite element space discretization error for elastodynamics problems is related to the reduction in numerical dispersion of finite elements. In the paper, we extend the modified integration rule technique for the mass and stiffness matrices to the dispersion reduction of linear finite elements for linear elastodynamics. The analytical study of numerical dispersion for the modified integration rule technique and for the averaged mass matrix technique is carried out in the 1-D, 2-D and 3-D cases for harmonic plane waves. In the general case of loading, the numerical study of the effectiveness of the dispersion reduction techniques includes the filtering technique (developed in our previous papers) that identifies and removes spurious high-frequency oscillations. 1-D, 2-D and 3-D impact problems for which all frequencies of the semi-discrete system are excited are solved with the standard approach and with the new dispersion reduction technique. Numerical results show that compared with the standard mass and stiffness matrices, the simple dispersion reduction techniques lead to a considerable decrease in the number of degrees of freedom and computation time at the same accuracy, especially for multi-dimensional problems. A simple quantitative estimation of the effectiveness of the finite element formulations with reduced numerical dispersion compared with the formulation based on the standard mass and stiffness matrices is suggested.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Accurate finite element modeling of linear elastodynamics problems with the reduced dispersion error

2010

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“…In the general case of loading (boundary conditions), the estimation of the accuracy of numerical techniques with reduced dispersion is difficult due to the presence of spurious high-frequency oscillations in numerical solutions; e.g., see [11,26].…”

Section: Introductionmentioning

confidence: 99%

“…Due to the space discretization, the exact solution to Eq. (1) contains the numerical dispersion error; e.g., see [7,9,11,26,27,28,30,31,32,14,12] and others. The space discretization error can be decreased by the use of mesh refinement.…”

Section: Introductionmentioning

confidence: 99%

Finite Element Modeling of Linear Elastodynamics Problems With Explicit Time-Integration Methods and Linear Elements With Reduced Dispersion. Comparative Study of Different Finite Element Techniques Used for Elastodynamics.

Idesman¹

2014

Proceedings of the 4th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. We have developed two finite element techniques with reduced dispersion for linear elastodynamics that are used with explicit time-integration methods. These techniques are based on the modified integration rule for the mass and stiffness matrices and on the averaged mass matrix approaches that lead to the numerical dispersion reduction for linear finite elements. The analytical study of numerical dispersion for the new techniques is carried out in the 1-D, 2-D and 3-D cases. The numerical study of the effectiveness of the dispersion reduction techniques includes two-stage time-integration approach with the filtering stage (developed in our previous papers) that quantifies and removes spurious high-frequency oscillations from numerical results. We have found that in contrast to the standard linear elements with explicit time-integration methods and the lumped mass matrix, the finite element techniques with reduced dispersion yield more accurate results at small time increments (smaller than the stability limit) in the 2D and 3-D cases. The advantages of the new technique are illustrated by the solution of the 1-D and 2-D impact problems. The new approaches with reduced dispersion can be easily implemented into existing finite element codes and lead to significant reduction in computation time at the same accuracy compared with the standard finite element formulations. Finally, we compare the accuracy of the linear elements with reduced dispersion, the spectral low-and high-order elements as well as the isogeometric elements by the solution of the 1-D impact problem. For all these solutions we use two-stage time integration technique with the filtering stage that removes spurious oscillations and allows an accurate comparison of different space discretization techniques used for elastodynamics. It is also interesting to mention that the amount of numerical dissipation at the filtering stage can be used as a quantitative measure for the comparison of accuracy of the different numerical formulations used for elastodynamics. 759

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Grid dispersion analysis of plane square biquadratic serendipity finite elements in transient elastodynamics

Kolman

Plešek

Okrouhlík

et al. 2013

Numerical Meth Engineering

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SUMMARYThe spatial discretisation of a continuum by the FEM introduces dispersion errors to the numerical solution of stress wave propagation. Errors of phase and group velocities and the scatter of wave propagation are induced. When these propagating phenomena are modelled by the FEM, the speed of a single harmonic wave depends on its frequency. Parasitic effects do not exist in an 'ideal' unbounded continuum. With higher order finite elements, there are optical modes in the spectrum resulting in spurious oscillations of stress and velocity distributions near the sharp wavefronts. In principle, the dispersion errors and the spurious oscillations cannot be removed from these solutions but can be suppressed to a certain extent. For reliable numerical solutions by the FEM, the dispersion errors should be analysed and determined. The dispersion properties of a plane square biquadratic serendipity finite element are examined and compared with a bilinear one. Dispersion analysis is carried out for the consistent and lumped mass matrices. A dispersive 'improved' lumped mass matrix for biquadratic serendipity elements is proposed. The paper closes with the recommendation of a choice of permissible dimensionless wavelengths for bilinear and biquadratic serendipity finite element meshes.

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Dispersion of elastic waves in the contact–impact problem of a long cylinder

Cited by 22 publications

References 4 publications

Accurate finite element modeling of linear elastodynamics problems with the reduced dispersion error

Accurate finite element modeling of linear elastodynamics problems with the reduced dispersion error

Finite Element Modeling of Linear Elastodynamics Problems With Explicit Time-Integration Methods and Linear Elements With Reduced Dispersion. Comparative Study of Different Finite Element Techniques Used for Elastodynamics.

Grid dispersion analysis of plane square biquadratic serendipity finite elements in transient elastodynamics

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