FEM/Wideband FMBEM Coupling For Structural-Acoustic Design Sensitivity Analysis

Research on acoustic radiation and scattering from vibration of elastic structure immersed in fluid is very important. Such as radiation noise from a submarine or ship decreases the concealment of the structure underwater. For light structures immersed in a heavy fluid, e.g. thin steel shells in water, a full interaction between the structure and the fluid domains needs to be taken into account. A suitable approach for the analysis of fluid-structure interaction problems is the coupling FEM/BEM.In this work, A coupling algorithm based on FEM and BEM is presented for the simulation of fluid-structure interaction and structural acoustic sensitivity analysis. The algorithm presented in this paper makes it possible to predict the radiation and scattering acoustic field fast for large-scale practical problems. The main contents are as follows:(1) The formulation of pressure sensitivity for two dimensional acoustic problem based on FMBEM is presented. The Burton-Miller formulation is used to overcome the fictitious frequency problem when using a single Helmholtz boundary integral equation for exterior boundary-value problems. The strongly singular and hypersingular integrals in the sensitivity equations can be evaluated explicitly and directly by using the piece-wise constant discretization. Acoustic design sensitivity formulations presented in this paper are obtained by differentiating the boundary integral equations with respect to design variables which determine the contour of the structure. Conventional BEM pro-duces a dense and non-symmetrical coefficient matrix, solving the coefficient matrix is very time-consuming. This drawback limited the application of the BEM for practical problem. In this work, FMBEM is developed for the analysis of two dimensional acous-tic design sensitivity problems. There are actually two forms of FMM for Helmholtz equation. One is the original FMM and the other is the diagonal form FMM. But both of them fail in some way outside their preferred frequency ranges. However, the wide-band FMM formed by combining the original FMM and the diagonal form FMM can solve the problems accurately and efficiently at all frequencies.(2) The formulation of pressure sensitivity for three dimensional acoustic problem based on discontinuous FMBEM is presented. The approach of continuous linear and quadratic boundary elements is often applied, and the alternative of discontinuous ele-ments with higher accuracy is investigated in this paper. Several numerical examples with analytical solution are presented to demonstrate the correctness and validity of this proposed algorithm. The performance of the continuous and discontinuous boundary el-ement for the rigid analysis in structural-acoustic domain is investigated and compared, and the optimal element is obtained. (3) The structural-acoustic analysis based on FEM/discontinuous FMBEM algo-rithm is implemented. Analysis of the acoustic radiation or scattering from elastic struc-ture in heavy fluid is a classical problem of underwater acoustics. A coupling algo-rithm based on the finite element method and the wideband fast multipole boundary element method (FEM/wideband FMBEM) is proposed for the simulation of acoustic fluid-structure interaction. In this work, the formulas of the sound pressure and sound power on an arbitrary closed surface around the radiator are proposed, respectively. Dis-continuous boundary elements are used to improve the computational accuracy. Several types coupling schemes are used for the numerical solution, and the performances of different types of FE/BE coupling schemes are presented and compared each other. The optimal value of nodal position in discontinuous boundary elements is investigated.(4) The structural-acoustic sensitivity analysis based on FEM/discontinuous FMBEM algorithm is implemented. This paper presents the formulas of the sound power sensi-tivity with respect to the design variables, where the design variable can be chosen as the fluid density, structural density, Poisson’s ratio, Young’s modulus, structural shape size and so on. An adjoint operator approach is presented to calculate the sensitivity of the radiated sound power on the structural surface. The direct differentiation approach is used to calculate the sensitivity of the radiated sound power on an arbitrary closed surface around the radiator. By differentiating directly the coupled boundary element equation obtained by combining the structural and fluid governing equations with re-spect to design variable, we can obtain the formulation of the derivative of the vectors of nodal displacement and sound pressure on the interaction surface with respect to design variable. And then using the adjoint operator approach or the direct differentia-tion approach, the derivative formulation of the radiated sound power can be obtained. For different design variables, the different formulations which are used to solve the derivative of the radiated sound power are presented in detail in this paper.The algorithm presented in this paper makes it possible to predict the radiation and scattering acoustic field fast for large-scale practical problems, and predict the effects of different design variables on the sound field numerically for large-scale practical problems. The sound stealth performance of the structure underwater can be improved by using the optimization algorithm based on the sensitivity analysis presented in this paper.