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The objective of this project was to develop high fidelity acoustic scattering models to facilitate the detection, localization, and characterization of military munitions found in ponds, lakes, rivers, estuaries, and coastal ocean areas. These models were used to design experiments, interpret collected data, and identify features that can be used in training classifiers.

Most acoustic scattering models are well-equipped to compute scattering from elastic objects in free space. However, unexploded ordnance (UXO) reside in complex ocean environments, which have a profound effect on their acoustic response to sonar. Thus, in trying to model the response of UXO to sonar, a high-fidelity model must account for the interaction between the target and its surrounding environment. While the standard method of solution for an arbitrary elastic target is the finite element method, the solution of the scattering problem in the surrounding medium is best handled by the boundary integral equation, as it replaces the infinite domain problem by an integral over the surface of the target. Furthermore, the boundary integral method has the advantage of reducing the dimensionality of the problem by one. In contrast, the finite element method is not well-suited for solving the scattering problem in the surrounding environment due to difficulty in satisfying the radiation condition. For these reasons, the project team solves the problem of scattering from an elastic target in a complex ocean environment by a combination of finite element method and boundary integral method. They use the finite element method to model the motion of the target by computing its impedance matrix in vacuum, and the boundary integral method to model the acoustic field in its surrounding medium. The two solutions are coupled by satisfying the required boundary conditions on the surface of the target. This results in a model that treats the interaction between the target and its surrounding environment exactly. They refer to the model as the Coupled Finite Element/Boundary Element (CFEBE) method.

This project has been active since 2013. During its first three years, the project team provided benchmark-quality solutions for various targets to researchers within SERDP who do similar type of modeling. While doing this, an important part of the work was to validate the models using analytical solutions when available and other well-tested solutions. The project team validated the 3D and axially-symmetric models in free space using the analytic solution for elastic spheres and spherical shells. They also validated the models for scattering from a proud, half-buried and a fully-buried solid sphere by comparing the results with those of the T-matrix method. They computed the acoustic color (backscattered target strength as function of frequency and angle of incidence) for the aluminum replica of a UXO (henceforth aluminum UXO), the Bullet-105 and the Howitzer shell in free space using both the 3D and axiallysymmetric versions of the CFEBE model. The project team computed the acoustic color for the fully proud and the fully buried aluminum UXO and compared the results with measurements and those produced by other models. Additionally, they computed the acoustic color for the partially buried aluminum UXO and again compared the results with other finite element models since measured results for this case was not yet available. The model results in all cases agreed with each other and and with the measurements.

Since 2017 this project has been co-funded by the Office of Naval Research and SERDP. This report concerns achievements made during this phase of the project, but to make it self-contained, important results are included in the appendix. The report is divided into two parts: In the first part, the project team lists the work that went into improving the CFEBE model and in the second part they describe new model applications. The most significant accomplishment during this period is the development of a new approach to compute the acoustic color in two half-spaces. This approach is based on using a multipole solution to represent the spectral Greenâ€™s functions, which results in significant reduction in computation time. The project team also laid the groundwork for the implementation of the method of *H* -Matrices. This method is a hierarchical method of matrix partitioning, which promises to reduce both storage and computation time from O(n2) to O(nlogn), where n is the size of matrices involved. For a typical problem, the reduction in computation time could be as much as a factor of two thousand without noticeable degradation in accuracy. The project team anticipates that full implementation of this method would be a game changer in modeling scattering from UXOs. They also implemented the Burton-Miller method, which is a method that ensures that the boundary element equations have unique solutions. So far, the project team has been using the alternative Combined Helmholtz Integral Equations Formulation method to do this, but this method results in non-square matrices, which cannot be used by *H* -Matrices. In the second part of the report, the project team describes new applications of the CFEBE model in computing the acoustic color for UXOs in slightly more complex environments and for arbitrary burial. They also describe the application of the model to compute scattering from multiple targets and scattering from a non-axially symmetric mortar shell.

The CFEBE method has several advantages over currently-used methods. The most important ones are: 1) The method is inherently broadband since the stiffness and mass matrices, which constitute the impedance matrix, are independent of frequency. Therefore, the computation of these matrices, which makes up the most numerically intensive part of the computation, is performed once for all frequencies. 2) This method is efficient because it requires a matrix inversion for each frequency, but not each angle while computing the acoustic color. This is not the case for currently-used methods, which must solve a full finite element problem for each frequency and each angle of incidence. 3) Since this method computes the target impedance matrix in vacuum, the same impedance matrix can be used in any environment, so changing the environment for the same target does not require a full finite element solution of the problem. 4) By projecting the impedance matrix onto the surface nodes, this method reduces a finite element problem to a boundary element problem with far fewer unknowns. This reduction in the number of unknowns enables the method to solve a 3D problem with ease. 5) It provides a numerically exact solution since it self-consistently couples the target with the surrounding environment. 6) Due to its modular nature, the method easily lends itself to parallel processing.