# Finite Element Modeling of Scattering from Underwater Proud and Buried Military Munitions

## Dr. Ahmad Abawi | Heat, Light, and Sound Research, Inc.

## MR-2408

## Objective

The main 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 are intended to be used to design experiments, interpret collected data, identify features that can be used in training classifiers.

## Technical Approach

Most acoustic scattering models are well-equipped to compute scattering from elastic objects in free space. However, unexploded ordinance (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 researchers solve the problem of scattering from an elastic target in a complex ocean environment by a combination of finite element method and boundary integral method. Researchers used the finite element method to model the motion of the target by essentially computing its impedance matrix in vacuum, and the boundary integral method to model the acoustic field in its surrounding medium. The two solutions were coupled by satisfying the required boundary conditions on the surface of the target. This resulted in a model that treats the interaction between the target and its surrounding environment exactly. An important extension of this work is the development of the same model for axially-symmetric targets with substantial speed advantages.

## Results

During the past three years, the researchers have been providing benchmark-quality solutions for various targets to other researchers within the SERDP program who do similar type of modeling. While doing this, an important part of the work has been to validate the researchersâ€™ models using analytical solutions when available and other well-tested solutions. They validated their 3D and axiallysymmetric 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. The researchers 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 axially-symmetric versions of the model. They also 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 resulted in all cases agreed with each other and with the measurements. To verify that the model indeed accounts for multiple scattering, the researchers computed the acoustic color for a tilted cylinder near the water-air interface and showed that the model results were in total agreement with measurements. Finally, to show that the 3D model was truly 3D and can handle targets of arbitrary shape, researchers used it to compute the acoustic color for the Bullet-105 UXO with a hole drilled on its side.

## Benefits

The method that was developed has several advantages over currently-used methods and enhances the DoDâ€™s ability to detect and characterize munitions located in shallow water environments, such as ponds, lakes, rivers, estuaries, and coastal ocean areas. These increases in efficiency create cost savings during unexploded ordnance recovery. 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, including graphics processing unit (GPU) processing and the application of the fast multipole technique.

## Points of Contact

### Principal Investigator

Dr. Ahmad Abawi

Heat, Light, and Sound Research, Inc.

Phone: 858-457-0800 x102