A new and improved analysis of the time domain boundary integral operators for the acoustic wave equation

Abstract

We present a novel analysis of the boundary integral operators associated to the wave equation. The analysis is done entirely in the time-domain by employing tools from abstract evolution equations in Hilbert spaces and semi-group theory. We prove a single general theorem from which well-posedness and regularity of the solutions for several boundary integral formulations can be deduced as particular cases. By careful choices of continuous and discrete spaces, we are able to provide a concise analysis for various direct and indirect formulations, both at the continuous level and for their Galerkin-in-space semi-discretizations. Some of the results here are improvements on previously known results, while other results are equivalent to those in the literature. The methodology presented here greatly simplifies the analysis of the operators of the Calderón projector for the wave equation and can be generalized for other relevant boundary integral equations.

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

“…Fitting this into the abstract framework. We now show how this problem can be fit into the framework outlined in Section 9, and distilled from [13]. The desired spaces for this problem are…”

“…The coupling of acoustic and piezoelectric dynamics takes place through two transmission conditions involving the normal components of the elastic stress σν and the acoustic pressure ρ f ∂ νu as well as the normal components of the acoustic and elastic velocities (∂ ν u andu · ν respectively) at the interface. The goal of Section 3 is the mathematical analysis of this model, which is accomplished by recasting the system of PDE's into a first order system in the spirit of [13] at the interface. The unknowns collected in U are related to acoustic pressure, velocity, displacement, purely elastic stress and electric field, while right-hand sides correspond to the influence of the incident wave in the system.…”

Time-domain boundary integral equation modeling of heat transmission problems

“…The following process mimics the one in [22] and in [40,Chapter 7]. It involves two aspects: (a) an extension of zero of the data to negative values of the time variable; (b) a hypothesis on polynomial growth of the data.…”

Analysis of transient acoustic scattering by an elastic obstacle

“…The time-domain problems have received considerable attention due to their capability of capturing wide-band signals and modeling more general material and nonlinearity [4,38,42]. Many approaches are attempted to solve numerically the time-domain problems such as coupling of boundary element and finite element with different time quadratures [12,14,18,27,34,40]. Compared with the time-harmonic scattering problems, the time-domain problems are less studied on their rigorous mathematical analysis due to the additional challenge of the temporal dependence.…”

“…27) which holds for any function q ∈ L 2 ([0, T ]; H 1 0 (Ω)) and v ∈ L 2 ([0, T ]; H 1 (D)) since functions of the form (3.25) are dense in the space. Moreover, we have from (3.27) that for any q ∈ H 1 0 (Ω), v ∈ H 1 (D) and t ∈ [0, T ]…”