Symmetric Boundary-Finite Element Discretization of Time Dependent Acoustic Scattering by Elastic Obstacles with Piezoelectric Behavior

A coupled BEM/FEM formulation for the transient interaction between an acoustic field and a piezoelectric scatterer is proposed. The scattered part of the acoustic wave is represented in terms of retarded layer potentials while the elastic displacement and electric potential are treated variationally. This results in an integro-differential system. Well posedness of a general Galerkin semi-discretization in space of the problem is shown in the Laplace domain and translated into explicit stability bounds in the time domain. Trapezoidal-rule and BDF2 convolution quadrature are used in combination with matching time stepping for time discretization. Second order convergence is proven for the BDF2-based method. Numerical experiments are provided for BDF2 and trapezoidal rule based time evolution.

[1]  Mari Paz Calvo,et al.  Runge–Kutta convolution quadrature methods for well-posed equations with memory , 2007, Numerische Mathematik.

[2]  Lothar Gaul,et al.  A boundary element method for transient piezoelectric analysis , 2000 .

[3]  Francisco-Javier Sayas,et al.  Stability of discrete liftings , 2003 .

[4]  Martin Schanz,et al.  Boundary Element Methods for the Dynamic Analysis of Elastic, Viscoelastic, and Piezoelectric Solids , 2004 .

[5]  C. Lubich.  Convolution quadrature and discretized operational calculus. II , 1988 .

[6]  A. Bamberger et T. Ha Duong,et al.  Formulation variationnelle espace‐temps pour le calcul par potentiel retardé de la diffraction d'une onde acoustique (I) , 1986 .

[7]  Francisco-Javier Sayas,et al.  Some properties of layer potentials and boundary integral operators for the wave equation , 2011, 1110.4399.

[8]  C. Lubich.  Convolution quadrature and discretized operational calculus. I , 1988 .

[9]  T. Rabczuk,et al.  Extended finite element method for dynamic fracture of piezo-electric materials , 2012 .

[10]  Francisco-Javier Sayas,et al.  Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves , 2009, Numerische Mathematik.

[11]  Roger Ohayon,et al.  Piezoelectric structural acoustic problems: Symmetric variational formulations and finite element results , 2008 .

[12]  C. Lubich,et al.  On the multistep time discretization of linear\newline initial-boundary value problems and their boundary integral equations , 1994 .

[13]  E. Pan.  A BEM analysis of fracture mechanics in 2D anisotropic piezoelectric solids , 1999 .

[14]  Roger Ohayon,et al.  Variational Formulations of Interior Structural-Acoustic Vibration Problems , 2008 .

[15]  T. Qin,et al.  A regularized time-domain BIEM for transient elastodynamic crack analysis in piezoelectric solids , 2015 .

[16]  Chuanzeng Zhang,et al.  2-D transient dynamic analysis of cracked piezoelectric solids by a time-domain BEM , 2008 .

[17]  Francisco-Javier Sayas,et al.  A fully discrete BEM-FEM scheme for transient acoustic waves , 2016, 1601.08248.

[18]  Francisco-Javier Sayas,et al.  Stable numerical coupling of exterior and interior problems for the wave equation , 2013, Numerische Mathematik.

[19]  Daining Fang,et al.  Reflection and refraction of plane waves at the interface between piezoelectric and piezomagnetic media , 2008 .

[20]  Zheng H. Zhu,et al.  Reflection and refraction of plane waves at interface between two piezoelectric media , 2012 .

[21]  Christian Lubich,et al.  Fast and Oblivious Convolution Quadrature , 2006, SIAM J. Sci. Comput..

[22]  Francisco-Javier Sayas,et al.  Convolution Quadrature for Wave Simulations , 2014, 1407.0345.

[23]  George C. Hsiao,et al.  Boundary and coupled boundary–finite element methods for transient wave–structure interaction , 2015, 1509.01713.

[24]  T. Hughes,et al.  Finite element method for piezoelectric vibration , 1970 .

[25]  Ayech Benjeddou,et al.  Advances in piezoelectric finite element modeling of adaptive structural elements: a survey , 2000 .

[26]  Lehel Banjai,et al.  Multistep and Multistage Convolution Quadrature for the Wave Equation: Algorithms and Experiments , 2010, SIAM J. Sci. Comput..

[27]  Wolfgang Hackbusch,et al.  Sparse convolution quadrature for time domain boundary integral formulations of the wave equation , 2008 .

[28]  P. Joly,et al.  Mathematical and numerical modelling of piezoelectric sensors , 2012 .

[29]  F. Ihlenburg.  Finite Element Analysis of Acoustic Scattering , 1998 .

[30]  Lothar Gaul,et al.  Piezoelectric analysis with FEM and BEM , 2001 .

[31]  Lehel Banjai,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig Wave Propagation Problems Treated with Convolution Quadrature and Bem Wave Propagation Problems Treated with Convolution Quadrature and Bem , 2022 .

[32]  Lehel Banjai,et al.  An error analysis of Runge–Kutta convolution quadrature , 2011 .

[33]  M. Kuna,et al.  Finite Element Techniques for Dynamic Crack Analysis in Piezoelectrics , 2005 .

[34]  Ernst Hairer,et al.  FAST NUMERICAL SOLUTION OF NONLINEAR VOLTERRA CONVOLUTION EQUATIONS , 1985 .

[35]  H. F. Tiersten,et al.  Linear Piezoelectric Plate Vibrations , 1969 .

[36]  Jens Markus Melenk,et al.  Runge–Kutta convolution quadrature for operators arising in wave propagation , 2011, Numerische Mathematik.

[37]  Lehel Banjai,et al.  Rapid Solution of the Wave Equation in Unbounded Domains , 2008, SIAM J. Numer. Anal..