Efficient Unsplit Perfectly Matched Layers for Finite-Element Time-Domain Modeling of Elastodynamics
Publication: Journal of Engineering Mechanics
Volume 142, Issue 11
Abstract
The perfectly matched layer (PML) is a highly efficient absorbing boundary used for the numerical modeling of an elastic wave equation on an unbounded domain. In this work, the authors are concerned with a second-order unsplit PML for transient elastodymanic problems in a semi-plane medium with finite-element approximations. First, based on the concept of stretched coordinates, an efficient unsplit PML formulation is proposed without higher derivatives. Then a finite-element time-domain scheme of a second-order PML in a displacement formulation is developed, in which the Galerkin method is used in space discretization and a Newmark-type scheme is employed for time stepping. Inside the absorbing layer, only one auxiliary vector is required. Hence, the scheme is cheap to implement and easily coupled with standard finite-element methods. Finally, the accuracy and efficiency of the present unsplit PML is demonstrated in numerical examples with a finite-element time-domain scheme.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grants 51368038, 11162008), the Fund of the Education Department of Gansu Province of China for Master’s Tutors (1103-07), and the Environmental Protection Department of Gansu Province (Grant GSEP-2014-23). The authors gratefully acknowledge all of these sources of support. The authors are also grateful to Prof. Yixian Xu (China University of Geosciences) for his helpful advice and comments.
References
Abarbanel, S., Gottlieb, D., and Hesthaven, J. S. (2002). “Long time behavior of the perfectly matched layer equations in computational electromagnetics.” J. Sci. Comput., 17(1–4), 405–422.
Bao, H., et al. (1998). “Large-scale simulation of elastic wave propagation in heterogeneous media on parallel computers.” Comput. Methods Appl. Mech. Eng., 152(1), 85–102.
Basu, U. (2009). “Explicit finite element perfectly matched layer for transient three-dimensional elastic waves.” Int. J. Numer. Methods Eng., 77(2), 151–176.
Basu, U., and Chopra, A. K. (2004). “Perfectly matched layers for transient elastodynamics of unbounded domains.” Int. J. Numer. Methods Eng., 59(8), 1039–1074.
Bécache, E., Fauqueux, S., and Joly, P. (2003). “Stability of perfectly matched layers, group velocities and anisotropic waves.” J. Comput. Phys., 188(2), 399–433.
Berenger, J. P. (1994). “A perfectly matched layer for the absorption of electromagnetic waves.” J. Comput. Phys., 114(2), 185–200.
Fathi, A., Poursartip, B., and Kallivokas, L. F. (2015). “Time-domain hybrid formulations for wave simulations in three-dimensional PML-truncated heterogeneous media.” Int. J. Numer. Methods Eng., 101(3), 165–198.
Gao, G., Li, N., and Gu, X. (2015). “Field experiment and numerical study on active vibration isolation by horizontal blocks in layered ground under vertical loading.” Soil Dyn. Earthquake Eng., 69(69), 251–261.
Giannopoulos, A. (2008). “An improved new implementation of complex frequency shifted PML for the FDTD method.” IEEE Trans. Antennas Propag., 56(9), 2995–3000.
Grote, M. J., and Sim, I. (2010). “Efficient PML for the wave equation”, Numerical Analysis, p. 16.
Jian-Guo, Z., and Rui-Qi, S. (2013). “Perfectly matched layer-absorbing boundary condition for finite-element time-domain modeling of elastic wave equations.” Appl. Geophys., 10(3), 323–336.
Kan, U., and Chen, S. J. (2004). Introduction to ANSYS, Chung Hwa Technology Publication, Taipei.
Khazaee, A., and Lotfi, V. (2014). “Application of perfectly matched layers in the transient analysis of dam-reservoir systems.” Soil Dyn. Earthquake Eng., 60(60), 51–68.
Komatitsch, D., and Tromp, J. (2003). “A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation.” Geophys. J. Int., 154(1), 146–153.
Kucukcoban, S., and Kallivokas, L. F. (2011). “Mixed perfectly-matched-layers for direct transient analysis in 2D elastic heterogeneous media.” Comput. Methods Appl. Mech. Eng., 200(1), 57–76.
Kucukcoban, S., and Kallivokas, L. F. (2013). “A symmetric hybrid formulation for transient wave simulations in PML-truncated heterogeneous media.” Wave Motion, 50(1), 57–79.
Matzen, R. (2011). “An efficient finite element time-domain formulation for the elastic second-order wave equation: A non-split complex frequency shifted convolutional PML.” Int. J. Numer. Methods Eng., 88(10), 951–973.
Ping, P., Zhang, Y., and Xu, Y. (2014). “A multiaxial perfectly matched layer (M-PML) for the long-time simulation of elastic wave propagation in the second-order equations.” J. Appl. Geophys., 101(1), 124–135.
Quarteroni, A., Tagliani, A., and Zampieri, E. (1998). “Generalized Galerkin approximations of elastic waves with absorbing boundary conditions.” Comput. Methods Appl. Mech. Eng., 163(1), 323–341.
Sture, S. (2001). “Dynamics of structures: Theory and applications to earthquake engineering.” J. Eng. Mech., 968.
Teixeira, F. L., and Chew, W. C. (2000). “Finite-difference computation of transient electromagnetic waves for cylindrical geometries in complex media.” IEEE Trans. Geosci. Remote Sens., 38(4), 1530–1543.
Wolf, J. P. (1989). “Soil-structure-interaction analysis in time domain.” Nucl. Eng. Des., 111(3), 381–393.
Zienkiewicz, O. C., and Taylor, R. L. (2000). The finite element method, 5th Ed., Butterworth-Heinemann, New York.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 142 • Issue 11 • November 2016
Copyright
© 2016 American Society of Civil Engineers.
History
Received: Oct 12, 2015
Accepted: May 25, 2016
Published online: Jul 20, 2016
Published in print: Nov 1, 2016
Discussion open until: Dec 20, 2016
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.