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Comparison of the convolution quadrature method and enhanced inverse FFT with application in elastodynamic boundary element method

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Abstract

Transient problems can often be solved with transformation methods, where the inverse transformation is usually performed numerically. Here, the discrete Fourier transform in combination with the exponential window method is compared with the convolution quadrature method formulated as inverse transformation. Both are inverse Laplace transforms, which are formally identical but use different complex frequencies. A numerical study is performed, first with simple convolution integrals and, second, with a boundary element method (BEM) for elastodynamics. Essentially, when combined with the BEM, the discrete Fourier transform needs less frequency calculations, but finer mesh compared to the convolution quadrature method to obtain the same level of accuracy. If further fast methods like the fast multipole method are used to accelerate the boundary element method the convolution quadrature method is better, because the iterative solver needs much less iterations to converge. This is caused by the larger real part of the complex frequencies necessary for the calculation, which improves the conditions of system matrix.

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Acknowledgments

The first author gratefully acknowledges the hospitality and support by The Hong Kong University of Science and Technology during his sabbatical leave. The second author acknowledges Hong Kong Research Grants Council for supporting this work through Competitive Earmarked Research Grant 621411. The third author acknowledges the supports from the National Science Foundations of China under Grants 11102154 and 11472217 and the Alexander von Humboldt Foundation (AvH) to support his fellowship research at the Chair of Structural Mechanics, University of Siegen, Germany.

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Authors and Affiliations

  1. Institute of Applied Mechanics, Graz University of Technology, Graz, Austria

    Martin Schanz

  2. Department of Mechanical and Aerospace Engineering, Hong Kong University of Science and Technology, Kowloon, Hong Kong

    Wenjing Ye

  3. School of Astronautics, Northwestern Polytechnical University, Xi’an, China

    Jinyou Xiao

Authors
  1. Martin Schanz
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  2. Wenjing Ye
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  3. Jinyou Xiao
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Corresponding author

Correspondence to Martin Schanz.

Appendix: Butcher tableaus for the used Runge–Kutta methods

Appendix: Butcher tableaus for the used Runge–Kutta methods

In the test two 3-stage Runge–Kutta methods have been used. The respective Butcher tableaus are:

  • 3-stage Radau IIA

    $$\begin{aligned} \begin{array}{c|ccc} \frac{4 - \sqrt{6}}{10} &{} \frac{88 - 7 \sqrt{6}}{360} &{} \frac{296 - 169 \sqrt{6}}{1800} &{} \frac{-2 + 3 \sqrt{6}}{225}\\ \frac{4 + \sqrt{6}}{10} &{} \frac{296 + 169 \sqrt{6}}{1800} &{} \frac{88 + 7 \sqrt{6}}{360} &{} \frac{-2 - 3 \sqrt{6}}{225} \\ 1 &{} \frac{16 - \sqrt{6}}{36} &{} \frac{16 + \sqrt{6}}{36} &{} \frac{1}{9} \\ \hline &{} \frac{16 - \sqrt{6}}{36} &{} \frac{16 + \sqrt{6}}{36} &{} \frac{1}{9} \end{array} \end{aligned}$$

  • 3-stage Lobatto IIIC

    $$\begin{aligned} \begin{array}{c|ccc} 0 &{} \frac{1}{6} &{} - \frac{1}{3} &{} \frac{1}{6} \\ \frac{1}{2} &{} \frac{1}{6} &{} \frac{5}{12} &{} - \frac{1}{12} \\ 1 &{} \frac{1}{6} &{} \frac{2}{3} &{} \frac{1}{6} \\ \hline &{} \frac{1}{6} &{} \frac{2}{3} &{} \frac{1}{6} \end{array}. \end{aligned}$$

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Schanz, M., Ye, W. & Xiao, J. Comparison of the convolution quadrature method and enhanced inverse FFT with application in elastodynamic boundary element method. Comput Mech 57, 523–536 (2016). https://doi.org/10.1007/s00466-015-1237-z

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  • DOI: https://doi.org/10.1007/s00466-015-1237-z

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