An improved surface radiation condition for high-frequency acoustic scattering problems

Exponential Decay of High-Order Spurious Prolate Spheroidal Modes Induced by a Local Approximate DTN Exterior Boundary Condition

“…Theorem 3.1 demonstrates the superiority of the new DtN2 absorbing boundary condition given by (15) over the standard DtN2/BGT2 absorbing boundary. Indeed, in the high-frequency regime, the high-order spurious modes (propagating, evanescent, or grazing modes) induced by the standard BGT2/DtN2 boundary condition do not decay, as observed in [2]. Consequently, the use of this condition requires placing the artificial boundary very far from the obstacle in order to avoid the deterioration of the accuracy level due to the presence of the non physical incoming waves.…”

“…where h (2) n | Γ denotes the spherical Hankel function of the second kind of order n (see Chapter 10 in [1]). On the other hand, we know that (see p. 437 in [1]):…”

“…Note that the situation is not the same in the case of the standard DtN2/BGT2 boundary conditions (recall that in 3D, the standard DtN2 and BGT2 conditions coincide [11]). More specifically, in the high-frequency regime, the high-order spurious modes (whether they are propagating, evanescent, or grazing modes) created by the standard BGT2/DtN2 boundary condition do not decay, as observed in [2], requiring therefore to place the artificial boundary very far from the obstacle to avoid the deterioration of the accuracy level due to the possible contamination of these non physical modes. Consequently, the present study demonstrates analytically the superiority of the new DtN2 boundary condition designed for prolate spheroidal boundaries over the standard BGT2/DTN2 boundary condition.…”

Generalized combined field integral equations for the iterative solution of the three-dimensional Helmholtz equation

“…For scattering problems, a transition region corresponding to modes such that µ j ≈ k 2 gives the grazing modes. It can be shown [9,16] that the artificial singularity of the square-root operator does not yield a satisfactory representation of these modes. The transition from the propagative modes to the evanescent ones should be a priori smooth.…”

“…These regularizing operators expressed as square-root tangential operators are derived from ideas arising in Beam Propagation Methods [35,44,52]. They also fall into the general framework of On-Surface Radiation Conditions introduced in the middle of the eighties by Kriegsmann et al [41] and studied by several authors [4,5,9,16,[36][37][38]. The proposed integral equations are uniquely solvable at any frequency for a closed surface.…”