Evolution of a semidiscrete system modeling the scattering of acoustic waves by a piezoelectric solid

Abstract

We consider a model problem of the scattering of linear acoustic waves in free homogeneous space by an elastic solid. The stress tensor in the solid combines the effect of a linear dependence of strains with the influence of an existing electric field. The system is closed using Gauss's law for the associated electric displacement. Well-posedness of the system is studied by its reformulation as a first order in space and time differential system with help of an elliptic lifting operator. We then proceed to studying a semidiscrete formulation, corresponding to an abstract Finite Element discretization in the electric and elastic fields, combined with an abstract Boundary Element approximation of a retarded potential representation of the acoustic field. The results obtained with this approach improve estimates obtained with Laplace domain techniques. While numerical experiments illustrating convergence of a fully discrete version of this problem had already been published, we demonstrate some properties of the full model with some simulations for the two dimensional case.for a certain operator A that involves a first order in space differential operator and the inverse of an elliptic operator, and boundary operator B that accounts for the conditions

A problem in control of elastodynamics with piezoelectric effects

“…Everything follows as in the proof to Theorem 3.1 in Appendix A after defining discrete versions of M Ω , M Γ and the divergence operator. The details are very similar to what can be found in [7].…”

“…This grounding condition will be defined more precisely in the next section. While we sketch the requirements for the well-posedness of state equation in the context of the control problem, we note that the well-posedness of the PDE has previously been studied in [7,1,15,14,26,9,12,13] among others.…”

“…Section 2.3 is devoted to the semidiscrete (continuous in time) control problem. We discuss the wellposedness of the state and adjoint equations in Section 3, their proof is stated in Appendix A where we will also rely on previous results presented in [7]. A rigorous derivation of the first order necessary optimality conditions is given next.…”

Time-Dependent Wave-Structure Interaction Revisited: Thermo-piezoelectric Scatterers

“…These results can be used to simulate wave-structure interactions numerically by using the nowadays well-known convolution quadrature (CQ) method. Numerical experiments based on QC for the special cases of the wave-structure interactions listed above are available in [6,17,23,24,38]. The numerical treatment for the operators in the present paper will be reported in a separate communication.…”

Time-Domain Boundary Integral Methods in Linear Thermoelasticity

“…For the second approach, we refer to the fundamental work in French school by Bamberger and Ha Duong [4,5] for the wave equation, the work by Lubich and Schneider [22,25] for partial differential equations of parabolic and hyperbolic types. The latter approach is more recent and follows the lines laid out in [11] for the acoustic wave equation and has been applied to the scattering of acoustic waves by elastic and piezoelectric solids in [6].…”