A nodal discontinuous Galerkin finite element method for nonlinear elastic wave propagation.

A nodal discontinuous Galerkin finite element method (DG-FEM) to solve the linear and nonlinear elastic wave equation in heterogeneous media with arbitrary high order accuracy in space on unstructured triangular or quadrilateral meshes is presented. This DG-FEM method combines the geometrical flexibility of the finite element method, and the high parallelization potentiality and strongly nonlinear wave phenomena simulation capability of the finite volume method, required for nonlinear elastodynamics simulations. In order to facilitate the implementation based on a numerical scheme developed for electromagnetic applications, the equations of nonlinear elastodynamics have been written in a conservative form. The adopted formalism allows the introduction of different kinds of elastic nonlinearities, such as the classical quadratic and cubic nonlinearities, or the quadratic hysteretic nonlinearities. Absorbing layers perfectly matched to the calculation domain of the nearly perfectly matched layers type have been introduced to simulate, when needed, semi-infinite or infinite media. The developed DG-FEM scheme has been verified by means of a comparison with analytical solutions and numerical results already published in the literature for simple geometrical configurations: Lamb's problem and plane wave nonlinear propagation.

[1]  Heming Xu,et al.  Two-dimensional linear and nonlinear wave propagation in a half-space , 1999 .

[2]  Raj Mittra,et al.  Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers , 1996 .

[3]  Michele Meo,et al.  A new nonlinear elastic time reversal acoustic method for the identification and localisation of stress corrosion cracking in welded plate-like structures - : A simulation study , 2007 .

[4]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[5]  George Haller,et al.  Predicting transport by Lagrangian coherent structures with a high-order method , 2006 .

[6]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[7]  Johan O. A. Robertsson,et al.  A numerical free-surface condition for elastic/viscoelastic finite-difference modeling in the presence of topography , 1996 .

[8]  K. Mccall.  Theoretical study of nonlinear elastic wave propagation , 1994 .

[9]  Michael Dumbser,et al.  An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - IV. Anisotropy , 2007 .

[10]  F. S. Lamb,et al.  On the Propagation of Tremors over the Surface of an Elastic Solid , 1904 .

[11]  J. Hesthaven,et al.  Nodal DG-FEM solution of high-order Boussinesq-type equations , 2007 .

[12]  E. Toro,et al.  An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes - V. Local time stepping and p-adaptivity , 2007 .

[13]  Jian-Ming Jin,et al.  Complex coordinate stretching as a generalized absorbing boundary condition , 1997 .

[14]  J. Hesthaven,et al.  A level set discontinuous Galerkin method for free surface flows , 2006 .

[15]  P. Johnson,et al.  Local interaction simulation approach to modelling nonclassical, nonlinear elastic behavior in solids. , 2003, The Journal of the Acoustical Society of America.

[16]  A. Ben-menahem.  Effect of a non‐linear boundary layer on the radiation from earthquakes and underground nuclear explosions , 1998 .

[18]  Jan S. Hesthaven,et al.  Numerical modeling of double-layered piezoelectric transducer systems using a high-order discontinuous Galerkin method , 2008 .

[19]  J. Minster,et al.  Hysteresis and two-dimensional nonlinear wave propagation in Berea Sandstone , 2000 .

[20]  Timothy C. Warburton,et al.  Nodal discontinuous Galerkin methods on graphics processors , 2009, J. Comput. Phys..

[21]  Horace Lamb,et al.  On the propagation of tremors over the surface of an elastic solid , 1904, Proceedings of the Royal Society of London.

[22]  M. Dumbser,et al.  An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes — III. Viscoelastic attenuation , 2007 .

[23]  J. Hesthaven,et al.  High–order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem , 2004, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[24]  S. Cummer,et al.  A simple, nearly perfectly matched layer for general electromagnetic media , 2003, IEEE Microwave and Wireless Components Letters.

[25]  Serge Dos Santos,et al.  A two-dimensional pseudospectral model for time reversal and nonlinear elastic wave spectroscopy. , 2007, The Journal of the Acoustical Society of America.

[26]  Stephen D. Gedney,et al.  Convolution PML (CPML): An efficient FDTD implementation of the CFS–PML for arbitrary media , 2000 .

[27]  Haishan Zheng,et al.  Non‐linear seismic wave propagation in anisotropic media using the flux‐corrected transport technique , 2006 .

[28]  José M. Carcione,et al.  WAVE-PROPAGATION SIMULATION IN AN ELASTIC ANISOTROPIC (TRANSVERSELY ISOTROPIC) SOLID , 1988 .

[29]  J.-P. Berenger.  On the reflection from Cummer's nearly perfectly matched layer , 2004, IEEE Microwave and Wireless Components Letters.

[30]  M. Dumbser,et al.  An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes — II. The three-dimensional isotropic case , 2006 .

[31]  J. Virieux,et al.  An hp-adaptive discontinuous Galerkin finite-element method for 3-D elastic wave modelling , 2010 .

[32]  F. Schubert,et al.  Resonant bar simulations in media with localized damage. , 2004, Ultrasonics.

[33]  O. Matar,et al.  Two-dimensional axisymmetric numerical simulation of supercritical phase conjugation of ultrasound in active solid media , 2005 .

[34]  M. Dumbser,et al.  An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - I. The two-dimensional isotropic case with external source terms , 2006 .

[35]  Olivier Bou Matar,et al.  Convolutional perfectly matched layer for elastic second-order wave equation. , 2010, The Journal of the Acoustical Society of America.

[36]  J. Hesthaven,et al.  Nodal high-order methods on unstructured grids , 2002 .

[37]  Ettore Barbieri,et al.  Nonlinear wave propagation in damaged hysteretic materials using a frequency domain-based PM space formulation , 2009 .