Analysis of Ringing and Noise in FE and FDTD Calculated Acoustic Pulse Profiles

Affiliation(s)

School of Physics, University of the Witwatersrand, Johannesburg, South Africa.

Department of Mechanical and Process Engineering, ETH, Zurich, Switzerland.

School of Physics, University of the Witwatersrand, Johannesburg, South Africa.

Department of Mechanical and Process Engineering, ETH, Zurich, Switzerland.

ABSTRACT

Ringing, i.e. the emergence of an oscillatory tail behind a wave pulse as it propagates through a medium, is a pervasive artefact in FE and FDTD calculated waveforms. It is known to be a consequence of numerical dispersion arising from the discretization of the equations of motion. The use of an irregular mesh in a FE code has the further consequence of rendering the displacement field increasingly noisy with distance behind the wave front. In this paper these effects are illustrated using the commercial FE package ABAQUS with square and irregular triangular meshes to calculate the progress of a longitudinally polarized Ricker pulse along the axis of a cylindrically shaped aluminium specimen. We are able to give a precise analytical account of the evolution of ringing on the basis of a low order approximation for the dispersion relation of the discretized equations of motion. A qualitative account is provided of the generation of noise in the use of an irregular triangular mesh.

Ringing, i.e. the emergence of an oscillatory tail behind a wave pulse as it propagates through a medium, is a pervasive artefact in FE and FDTD calculated waveforms. It is known to be a consequence of numerical dispersion arising from the discretization of the equations of motion. The use of an irregular mesh in a FE code has the further consequence of rendering the displacement field increasingly noisy with distance behind the wave front. In this paper these effects are illustrated using the commercial FE package ABAQUS with square and irregular triangular meshes to calculate the progress of a longitudinally polarized Ricker pulse along the axis of a cylindrically shaped aluminium specimen. We are able to give a precise analytical account of the evolution of ringing on the basis of a low order approximation for the dispersion relation of the discretized equations of motion. A qualitative account is provided of the generation of noise in the use of an irregular triangular mesh.

Cite this paper

A. Every, L. Aebi and J. Dual, "Analysis of Ringing and Noise in FE and FDTD Calculated Acoustic Pulse Profiles,"*Applied Mathematics*, Vol. 3 No. 10, 2012, pp. 1351-1356. doi: 10.4236/am.2012.330191.

A. Every, L. Aebi and J. Dual, "Analysis of Ringing and Noise in FE and FDTD Calculated Acoustic Pulse Profiles,"

References

[1] P. Huthwaite, F. Simonetti and M. J. S. Lowe, “On the Convergence of Finite Element Scattering Simulations,” In: D. O. Thompson and D. E. Chimenti, Eds., Review of Quantitative Nondestructive Evaluation, Melville, New York, Vol. 29, 2010, pp. 65-72.

[2] M. N. Guddati and B. Yue, “Modified Integration Rules for Reducing Dispersion Error in Finite Element Methods,” Computer Methods in Applied Mechanics and Engineering, Vol. 193, No. 3-5, 2004, pp. 275-287. doi:10.1016/j.cma.2003.09.010

[3] J. D. De Basabe and M. K. Sen, “Stability of the HighOrder Finite Elements for Acoustic or Elastic Wave Propagation with High-Order Time Stepping,” Geophysical Journal Internationa, Vol. 181, No. 1, 2010, pp. 577-590. doi:10.1111/j.1365-246X.2010.04536.x

[4] R. Mullen and T Belytschko, “Dispersion Analysis of Finite Element Semidiscretizations of the Two-Dimensional Wave Equation,” International Journal for Numerical Methods in Engineering, Vol. 18, No. 1, 1982, pp. 11-29. doi:10.1002/nme.1620180103

[5] J. Juntunen and T. D. Tsiboukis, “Reduction of Numerical Dispersion in FDTD Method through Artificial Anisotropy,” IEEE Transactions on Microwave Theory and Techniques, Vol. 48, No. 4, 2000, pp. 582-588. doi:10.1109/22.842030

[6] A. G. Every, I. Wenke, L. Aebi and J. Dual, “Acoustic Field Radiated into a Transversely Isotropic Solid from a Small Aperture Spherical Surface,” Ultrasonics, Vol. 51, No. 7, 2011, pp. 824-830. doi:10.1016/j.ultras.2011.04.001

[7] J. Bryner, J. Vollmann, L. Aebi and J. Dual, “Wave Propagation in Pyramidal Tip-Like Structures with Cubic Material Properties,” Wave Motion, Vol. 47, No. 1, 2010, pp. 33-44. doi:10.1016/j.wavemoti.2009.07.003

[8] ABAQUS, “A Simulia Product of Dassault Systemes,” 2002-2012. http://www.3ds.com/products/simulia/overview/

[9] D. Gsell, T. Leutenegger and J. Dual, “Modelling ThreeDimensional Elastic Wave Propagation in Circular Cylindrical Structures Using a Finite-Difference Approach,” Journal of the Acoustical Society of America, Vol. 116, No. 6, 2004, pp. 3284-3293. doi:10.1121/1.1625934

[10] P. Fellinger, R. Marklein, K. J. Langenberg and S. Klaholz, “Numerical Modelling of Elastic Wave Propagation and Scattering with EFIT—Elastodynamic Finite Integration Technique,” Wave Motion, Vol. 21, No. 1, 1995, pp. 47-66. doi:10.1016/0165-2125(94)00040-C

[11] F. Schubert, A. Peiffer, B. Kohler and T. Sanderson, “The Elastodynamic Finite Integration Technique for Waves in Cylindrical Geometries,” Journal of the Acoustical Society of America, Vol. 104, No. 5, 1998, pp. 2604-2614. doi:10.1121/1.423844

[1] P. Huthwaite, F. Simonetti and M. J. S. Lowe, “On the Convergence of Finite Element Scattering Simulations,” In: D. O. Thompson and D. E. Chimenti, Eds., Review of Quantitative Nondestructive Evaluation, Melville, New York, Vol. 29, 2010, pp. 65-72.

[2] M. N. Guddati and B. Yue, “Modified Integration Rules for Reducing Dispersion Error in Finite Element Methods,” Computer Methods in Applied Mechanics and Engineering, Vol. 193, No. 3-5, 2004, pp. 275-287. doi:10.1016/j.cma.2003.09.010

[3] J. D. De Basabe and M. K. Sen, “Stability of the HighOrder Finite Elements for Acoustic or Elastic Wave Propagation with High-Order Time Stepping,” Geophysical Journal Internationa, Vol. 181, No. 1, 2010, pp. 577-590. doi:10.1111/j.1365-246X.2010.04536.x

[4] R. Mullen and T Belytschko, “Dispersion Analysis of Finite Element Semidiscretizations of the Two-Dimensional Wave Equation,” International Journal for Numerical Methods in Engineering, Vol. 18, No. 1, 1982, pp. 11-29. doi:10.1002/nme.1620180103

[5] J. Juntunen and T. D. Tsiboukis, “Reduction of Numerical Dispersion in FDTD Method through Artificial Anisotropy,” IEEE Transactions on Microwave Theory and Techniques, Vol. 48, No. 4, 2000, pp. 582-588. doi:10.1109/22.842030

[6] A. G. Every, I. Wenke, L. Aebi and J. Dual, “Acoustic Field Radiated into a Transversely Isotropic Solid from a Small Aperture Spherical Surface,” Ultrasonics, Vol. 51, No. 7, 2011, pp. 824-830. doi:10.1016/j.ultras.2011.04.001

[7] J. Bryner, J. Vollmann, L. Aebi and J. Dual, “Wave Propagation in Pyramidal Tip-Like Structures with Cubic Material Properties,” Wave Motion, Vol. 47, No. 1, 2010, pp. 33-44. doi:10.1016/j.wavemoti.2009.07.003

[8] ABAQUS, “A Simulia Product of Dassault Systemes,” 2002-2012. http://www.3ds.com/products/simulia/overview/

[9] D. Gsell, T. Leutenegger and J. Dual, “Modelling ThreeDimensional Elastic Wave Propagation in Circular Cylindrical Structures Using a Finite-Difference Approach,” Journal of the Acoustical Society of America, Vol. 116, No. 6, 2004, pp. 3284-3293. doi:10.1121/1.1625934

[10] P. Fellinger, R. Marklein, K. J. Langenberg and S. Klaholz, “Numerical Modelling of Elastic Wave Propagation and Scattering with EFIT—Elastodynamic Finite Integration Technique,” Wave Motion, Vol. 21, No. 1, 1995, pp. 47-66. doi:10.1016/0165-2125(94)00040-C

[11] F. Schubert, A. Peiffer, B. Kohler and T. Sanderson, “The Elastodynamic Finite Integration Technique for Waves in Cylindrical Geometries,” Journal of the Acoustical Society of America, Vol. 104, No. 5, 1998, pp. 2604-2614. doi:10.1121/1.423844