Research

• Construction of new local absorbing boundary conditions for elliptical- and prolate spheroidal-shaped boundaries

We have constructed a new class of local absorbing boundary conditions based on a local approximation of the Dirichlet-to-Neumann (DtN) operator. These conditions distinguish themselves from existing absorbing boundary conditions in many aspects. First, the new local second-order approximate DtN boundary condition, denoted by DtN2, is particularly well adapted for elongated scatterers (eg., submarines) which is not the case for the standard approximate DtN boundary condition. Indeed, the latter condition requires the shape of the artificial boundary to be circular/spherical, and therefore often leads to larger than needed computational domains, which hampers computational efficiency. Second, the new DtN2 boundary condition is exact for the first two modes, easy to implement and to parallelize, and more importantly compatible with the local structure of the computational finite element scheme. The results pertaining to the performance analysis of the proposed DtN2 boundary condition in the low frequency regime revealed that the DtN2 boundary condition (a) is very accurate regardless of the slenderness of the boundary of the computational domain, and (b) outperforms the widely-used second-order Bayliss-Gunzburger-Turkel (BGT2) absorbing boundary condition when expressed in prolate spheroidal coordinates. The situation is similar in the high-frequency regime. However, we have noticed and demonstrated that DtN2 produces reflected prolate spheroidal modes at the exterior boundary. Nevertheless, we have established that these spurious modes decay, fortunately, faster than 1/(ka)^(15/8) (where k is the wavenumber and a the semi-major axis of the prolate spheroidal-shaped scatterer). In addition, for the higher-order spurious modes, we have established that, contrary to the situation when the second-order Balyliss-Gunzburger-Turkel condition (BGT2) is used, these spurious modes decay exponentially in the high frequency regime. This result proves the effectiveness of the DtN2 boundary condition when employed for solving high-frequency acoustic scattering problems by elongated scatterers. It also demonstrates the superiority of the DtN2 condition over the BGT2 boundary condition.

• Design of new discretization scheme for solving Helmholtz problems.

We have designed a DG-type method, called SDGM, for solving Helmholtz problems. The method can be viewed as being “between” the DGM formulation designed by Farhat et al and the LSM formulation suggested by Monk-Wang. The proposed mixed-hybrid formulation is a two-step procedure. Step 1 consists in solving well-posed problems at the element partition level of the computational domain, whereas Step 2 requires the solution of a global system whose unknowns are the Lagrange multipliers. The main features of SDGM include: (a) the resulting local problems are associated with small positive definite Hermitian matrices that can be solved in parallel, and (b) the matrix corresponding to the global linear system arising in Step 2 is Hermitian and positive semi-definite.
The numerical results obtained in the case of waveguide and scattering problems are very promising. They show that the proposed method is stable and accurate. For example, in the case of waveguide problems, for the R-7-2 element, SDGM remains stable for a mesh resolution with over 1000 elements per wavelength. Moreover, in the high frequency regime SDGM delivers results with high level of accuracy. For instance, when ka=400 (k being the wavenumber and a characterizing the dimension of the scatterer) and using only about 3 elements per wavelength, SDGM equipped with R-11-3 delivers a solution with an accuracy level of 0.6% on the relative error. We have also observed that in high-frequency regime it is preferable to increase the order of the element rather than refining the mesh for reaching a prescribed level of accuracy. For example, for ka=400 and for an accuracy level of 1% on the relative error, the R-13-4 element reduces the number of dofs with up to 40% when compared to the R-11-3 element. The numerical investigation performed in the case of scattering problems, clearly demonstrates the superiority of SDGM over LSM. In addition, the convergence results obtained for this class of problems illustrate the great potential of SDGM for solving efficiently high-frequency Helmholtz problems.

• Design of an efficient numerical procedure for full-aperture reconstruction of the acoustic far-field pattern (FFP) when measured in a limited aperture.

We have proposed a numerical procedure to extend to full aperture the acoustic far field pattern (FFP) when measured in only few observation angles. The reconstruction procedure is a multi-step technique that combines a total variation regularized iterative method with the standard Tikhonov regularized pseudo-inversion. The proposed approach distinguishes itself from existing solution methodologies by using an exact representation of the total variation which is crucial for the stability and robustness of Newton algorithm. The numerical investigation performed in the case where only one or three backscattering FFP measurements are available revealed that the modified procedure is less sensitive to the noise and clearly outperforms the original method. In particular, the modified extension procedure improves significantly the accuracy level in the full aperture reconstructed FFP field when the FFP data are tainted with a noise level larger than 5%. This impressive success in reconstructing the FFP field over the full aperture from very few and highly noisy measurements has the potential to improve the performance of the existing inverse obstacle solvers in the case of limited aperture. This may be achieved by applying the proposed procedure, as a pre-processing step, to enrich the limited aperture data in order to be used by the considered inverse solver.

• Characterization of the Fréchet derivative of the elasto-acoustic scattered field with respect to the shape of a given elastic scatterer.

We have established the Fréchet differentiability of the elasto-acoustic scattered field with respect to the shape of a given elastic scatterer. We have also demonstrated that this derivative can be characterized as being the unique solution of a direct elasto-acoustics scattering problem that differs from the direct elasto-acoustic scattering problem only in the transmission (boundary) conditions. Our proof is based on the implicit function theorem and the standard trace theorems. It assumes the boundary of the considered elastic scatterer to be only Lipschitzian, and therefore can include sharp corners. The computational implication of this theoretical characterization is as follows. If the sought-after shape is represented by N parameters, then, at each regularized Newton iteration, the N directional derivatives needed for constructing the Jacobians can be computed by solving N direct elasto-acoustic scattering problems that differ only by their boundary conditions — or in algebraic terms after FEM discretization, by solving a single system of equations with N right hand-sides. In contrast, evaluating the same N directional derivatives by a central differencing scheme would require first chosing an arbitrary small parameter, then solving 2N + 1 distinct direct elasto-acoustic scattering problems. This result has the potential to advance the state-of-the-art of the solution of inverse elastoacoustic scattering problems. Moreover, the methodology described above for characterizing the Fréchet derivative of the scattered field with respect to the shape of an elastic scatterer can also be applied to analyze the Fréchet differentiability of such a scatterer with respect to its material properties. This is relevant to many inverse obstacle problems where not only the shape of an obstacle is of interest, but also, and often more importantly, its structure.