An efficient time-domain perfectly matched layers formulation for elastodynamics on spherical domains

Sagiyama, K.; Govindjee, S.; Persson, P. -O.

doi:10.1002/nme.4740

Title: An efficient time-domain perfectly matched layers formulation for elastodynamics on spherical domains

Journal Article · Tue Jul 15 00:00:00 EDT 2014 · International Journal for Numerical Methods in Engineering

DOI:https://doi.org/10.1002/nme.4740· OSTI ID:1523916

Sagiyama, K. ^[1]; Govindjee, S. ^[1]; Persson, P. -O. ^[1]

Univ. of California, Berkeley, CA (United States). Dept. of Civil Engineering

Many practical applications require the analysis of elastic wave propagation in a homogeneous isotropic media in an unbounded domain. One widely used approach for truncating the infinite domain is the so called method of perfectly matched layers (PMLs). Most existing PML formulations are developed for finite difference methods based on the first-order velocity-stress form of the elasticity equations, and they are not straight-forward to implement using standard finite element methods (FEMs) on unstructured meshes. Some of the problems with these formulations include the application of boundary conditions in half-space problems and in the treatment of edges and/or corners for time-domain problems. Several PML formulations, which do work with FEMs have been proposed, although most of them still have some of these problems and/or they require a large number of auxiliary nodal history/memory variables. Here in this work, we develop a new PML formulation for time-domain elastodynamics on a spherical domain, which reduces to a two-dimensional formulation under the assumption of axisymmetry. Our formulation is well-suited for implementation using FEMs, where it requires lower memory than existing formulations, and it allows for natural application of boundary conditions. We solve example problems on two-dimensional and three-dimensional domains using a high-order discontinuous Galerkin (DG) discretization on unstructured meshes and explicit time-stepping. We also study an approach for stabilization of the discrete equations, and we show several practical applications for quality factor predictions of micromechanical resonators along with verifying the accuracy and versatility of our formulation.

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Research Organization:: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC); National Science Foundation (NSF)

Grant/Contract Number:: AC02-05CH11231; FA9550-10-1- 0229

OSTI ID:: 1523916

Journal Information:: International Journal for Numerical Methods in Engineering, Vol. 100, Issue 6; ISSN 0029-5981

Publisher:: WileyCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 5 works

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References (34)

Mixed Spectral Finite Elements for the Linear Elasticity System in Unbounded Domains Cohen, Gary; Fauqueux, Sandrine SIAM Journal on Scientific Computing, Vol. 26, Issue 3 https://doi.org/10.1137/S1064827502407457	journal	January 2005
A Simple Mesh Generator in MATLAB Persson, Per-Olof; Strang, Gilbert SIAM Review, Vol. 46, Issue 2 https://doi.org/10.1137/S0036144503429121	journal	January 2004
A perfectly matched layer for the absorption of electromagnetic waves Berenger, Jean-Pierre Journal of Computational Physics, Vol. 114, Issue 2 https://doi.org/10.1006/jcph.1994.1159	journal	October 1994
Perfectly matched layers for elastic waves in cylindrical and spherical coordinates Liu, Q. H. The Journal of the Acoustical Society of America, Vol. 105, Issue 4 https://doi.org/10.1121/1.426812	journal	April 1999
Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media Collino, Francis; Tsogka, Chrysoula GEOPHYSICS, Vol. 66, Issue 1 https://doi.org/10.1190/1.1444908	journal	January 2001
The Newmark scheme as velocity-stress time-staggering: an efficient PML implementation for spectral element simulations of elastodynamics Festa, Gaetano; Vilotte, Jean-Pierre Geophysical Journal International, Vol. 161, Issue 3 https://doi.org/10.1111/j.1365-246X.2005.02601.x	journal	June 2005
MEMS technology for timing and frequency control Nguyen, Clark IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, Vol. 54, Issue 2, p. 251-270 https://doi.org/10.1109/TUFFC.2007.240	journal	January 2007
A time-domain Discontinuous Galerkin method for mechanical resonator quality factor computations Govindjee, Sanjay; Persson, Per-Olof Journal of Computational Physics, Vol. 231, Issue 19 https://doi.org/10.1016/j.jcp.2012.05.034	journal	August 2012
Mixed perfectly-matched-layers for direct transient analysis in 2D elastic heterogeneous media Kucukcoban, S.; Kallivokas, L. F. Computer Methods in Applied Mechanics and Engineering, Vol. 200, Issue 1-4 https://doi.org/10.1016/j.cma.2010.07.013	journal	January 2011
Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations Brooks, Alexander N.; Hughes, Thomas J. R. Computer Methods in Applied Mechanics and Engineering, Vol. 32, Issue 1-3 https://doi.org/10.1016/0045-7825(82)90071-8	journal	September 1982
Explicit finite element perfectly matched layer for transient three-dimensional elastic waves Basu, Ushnish International Journal for Numerical Methods in Engineering, Vol. 77, Issue 2 https://doi.org/10.1002/nme.2397	journal	January 2009
On the implementation of perfectly matched layers in a three-dimensional fourth-order velocity-stress finite difference scheme: IMPLEMENTING PERFECTLY MATCHED LAYERS Marcinkovich, Carey; Olsen, Kim Journal of Geophysical Research: Solid Earth, Vol. 108, Issue B5 https://doi.org/10.1029/2002JB002235	journal	May 2003
The implementation of an improved NPML absorbing boundary condition in elastic wave modeling Qin, Zhen; Lu, Minghui; Zheng, Xiaodong Applied Geophysics, Vol. 6, Issue 2 https://doi.org/10.1007/s11770-009-0012-3	journal	June 2009
A 3D perfectly matched medium from modified maxwell's equations with stretched coordinates Chew, Weng Cho; Weedon, William H. Microwave and Optical Technology Letters, Vol. 7, Issue 13 https://doi.org/10.1002/mop.4650071304	journal	September 1994
A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation Komatitsch, Dimitri; Tromp, Jeroen Geophysical Journal International, Vol. 154, Issue 1 https://doi.org/10.1046/j.1365-246X.2003.01950.x	journal	July 2003
An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation Komatitsch, Dimitri; Martin, Roland GEOPHYSICS, Vol. 72, Issue 5 https://doi.org/10.1190/1.2757586	journal	September 2007
Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation Basu, Ushnish; Chopra, Anil K. Computer Methods in Applied Mechanics and Engineering, Vol. 192, Issue 11-12 https://doi.org/10.1016/S0045-7825(02)00642-4	journal	March 2003
Auxiliary differential equation formulation: an efficient implementation of the perfectly matched layer Ramadan, O. IEEE Microwave and Wireless Components Letters, Vol. 13, Issue 2 https://doi.org/10.1109/LMWC.2003.808706	journal	February 2003
A symmetric hybrid formulation for transient wave simulations in PML-truncated heterogeneous media Kucukcoban, S.; Kallivokas, L. F. Wave Motion, Vol. 50, Issue 1 https://doi.org/10.1016/j.wavemoti.2012.06.004	journal	January 2013
Complex frequency shifted convolution PML for FDTD modelling of elastic waves Drossaert, Francis H.; Giannopoulos, Antonios Wave Motion, Vol. 44, Issue 7-8 https://doi.org/10.1016/j.wavemoti.2007.03.003	journal	August 2007
Elastic PMLs for resonator anchor loss simulation Bindel, David S.; Govindjee, Sanjay International Journal for Numerical Methods in Engineering, Vol. 64, Issue 6 https://doi.org/10.1002/nme.1394	journal	January 2005
Convolutional perfectly matched layer for elastic second-order wave equation Li, YiFeng; Bou Matar, Olivier The Journal of the Acoustical Society of America, Vol. 127, Issue 3 https://doi.org/10.1121/1.3290999	journal	March 2010
Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems Cockburn, Bernardo; Shu, Chi-Wang Journal of Scientific Computing, Vol. 16, Issue 3, p. 173-261 https://doi.org/10.1023/A:1012873910884	journal	September 2001
Perfectly matched layers for transient elastodynamics of unbounded domains Basu, Ushnish; Chopra, Anil K. International Journal for Numerical Methods in Engineering, Vol. 59, Issue 8 https://doi.org/10.1002/nme.896	journal	January 2004
An unsplit convolutional perfectly matched layer technique improved at grazing incidence for the viscoelastic wave equation Martin, Roland; Komatitsch, Dimitri Geophysical Journal International, Vol. 179, Issue 1 https://doi.org/10.1111/j.1365-246X.2009.04278.x	journal	October 2009
The Compact Discontinuous Galerkin (CDG) Method for Elliptic Problems Peraire, J.; Persson, P. -O. SIAM Journal on Scientific Computing, Vol. 30, Issue 4 https://doi.org/10.1137/070685518	journal	January 2008
Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers Kuzuoglu, M.; Mittra, R. IEEE Microwave and Guided Wave Letters, Vol. 6, Issue 12 https://doi.org/10.1109/75.544545	journal	January 1996
Convolution PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media Roden, J. Alan; Gedney, Stephen D. Microwave and Optical Technology Letters, Vol. 27, Issue 5 https://doi.org/10.1002/1098-2760(20001205)27:5<334::AID-MOP14>3.0.CO;2-A	journal	January 2000
Perfectly Matched Layers for Elastodynamics: a new Absorbing Boundary Condition Chew, W. C.; Liu, Q. H. Journal of Computational Acoustics, Vol. 04, Issue 04 https://doi.org/10.1142/S0218396X96000118	journal	December 1996
An MSI Micromechanical Differential Disk-Array Filter Li, Sheng-Shian; Lin, Yu-Wei; Ren, Zeying TRANSDUCERS 2007 - 2007 International Solid-State Sensors, Actuators and Microsystems Conference https://doi.org/10.1109/SENSOR.2007.4300130	conference	June 2007
Unsplit complex frequency-shifted PML implementation using auxiliary differential equations for seismic wave modeling Zhang, Wei; Shen, Yang GEOPHYSICS, Vol. 75, Issue 4 https://doi.org/10.1190/1.3463431	journal	July 2010
Harmonic inversion of time signals and its applications Mandelshtam, Vladimir A.; Taylor, Howard S. The Journal of Chemical Physics, Vol. 107, Issue 17 https://doi.org/10.1063/1.475324	journal	November 1997
An Auxiliary Differential Equation Formulation for the Complex-Frequency Shifted PML Gedney, Stephen D.; Zhao, Bo IEEE Transactions on Antennas and Propagation, Vol. 58, Issue 3 https://doi.org/10.1109/TAP.2009.2037765	journal	March 2010
FDTD for Nth-order dispersive media Luebbers, R. J.; Hunsberger, F. IEEE Transactions on Antennas and Propagation, Vol. 40, Issue 11 https://doi.org/10.1109/8.202707	journal	January 1992