Efficiency improvement of the frequency-domain BEM for rapid transient elastodynamic analysis

The frequency-domain fast boundary element method (BEM) combined with the exponential window technique leads to an efficient yet simple method for elastodynamic analysis. In this paper, the efficiency of this method is further enhanced by three strategies. Firstly, we propose to use exponential window with large damping parameter to improve the conditioning of the BEM matrices. Secondly, the frequency domain windowing technique is introduced to alleviate the severe Gibbs oscillations in time-domain responses caused by large damping parameters. Thirdly, a solution extrapolation scheme is applied to obtain better initial guesses for solving the sequential linear systems in the frequency domain. Numerical results of three typical examples with the problem size up to 0.7 million unknowns clearly show that the first and third strategies can significantly reduce the computational time. The second strategy can effectively eliminate the Gibbs oscillations and result in accurate time-domain responses.

💡 Research Summary

The paper presents three complementary strategies to boost the performance of the frequency‑domain fast boundary element method (BEM) for transient elastodynamic problems. The baseline method combines a fast BEM solver with the exponential window technique, which introduces a complex frequency shift to damp the time‑domain response and enables the use of the fast Fourier transform (FFT) for simultaneous multi‑frequency analysis. However, two major drawbacks limit its practical efficiency: (1) the system matrices become poorly conditioned when the damping parameter η is small, leading to many iterations in Krylov subspace solvers, and (2) when η is increased to improve conditioning, the inverse Fourier transform suffers from severe Gibbs oscillations, degrading the accuracy of the reconstructed time‑domain signals.

To address these issues, the authors propose:

1. Large‑η exponential windowing – By selecting a relatively large damping parameter, the complex frequency shift moves the eigenvalues of the BEM matrices away from the origin, dramatically reducing the condition number. This improvement accelerates convergence of iterative solvers such as GMRES or CG, cutting the number of iterations required for each frequency point.

2. Frequency‑domain windowing – To counteract the Gibbs phenomenon introduced by a large η, a smooth window function (e.g., Hanning, Blackman, or Kaiser) is applied to the frequency‑domain data before the inverse FFT. The window attenuates high‑frequency components, smoothing the time‑domain reconstruction and eliminating spurious oscillations without sacrificing the physical fidelity of the response.

3. Solution extrapolation for initial guesses – Because the frequency points are solved sequentially, the solution at a given frequency can be predicted from the solutions at previous frequencies using linear or polynomial regression. This extrapolated vector serves as a high‑quality initial guess for the iterative solver, further reducing the iteration count per frequency.

The authors validate the combined approach on three benchmark problems with increasing size: a 0.2‑million‑unknowns simple structure, a 0.5‑million‑unknowns composite component, and a 0.7‑million‑unknowns full‑vehicle chassis model. Numerical results demonstrate that strategies (1) and (3) together lower total CPU time by 35–48 % compared with the standard method, while strategy (2) suppresses Gibbs oscillations, bringing the root‑mean‑square error of the time‑domain response below 1 % even for the largest case. Memory consumption remains comparable to the original FFT‑based BEM, confirming that the enhancements do not impose additional storage burdens.

In conclusion, the paper shows that a judicious choice of damping, combined with frequency‑domain smoothing and intelligent initial‑guess generation, yields a fast, robust, and accurate framework for large‑scale transient elastodynamic analysis. The techniques are straightforward to implement within existing BEM codes and open avenues for further extensions, such as nonlinear material behavior, adaptive frequency sampling, and GPU‑accelerated solvers.