Structural and Multidisciplinary Optimization https://doi.org/10.1007/s00158-017-1881-3 RESEARCH PAPER Minimization of sound radiation in fully coupled structural-acoustic systems using FEM-BEM based topology optimization Wenchang Zhao 1 · Changjun Zheng 2 · Cheng Liu 1 · Haibo Chen 1 Received: 21 July 2017 / Revised: 5 November 2017 / Accepted: 7 December 2017 © Springer-Verlag GmbH Germany, part of Springer Nature 2017 Abstract A topology optimization approach is proposed for the optimal design of bi-material distribution on underwater shell structures. The coupled finite element method (FEM) / boundary element method (BEM) scheme is used for the system response analysis, where the strong interaction between the structural and the acoustic domain is considered. The Burton- Miller formulation is used to overcome the fictitious eigen-frequency problem when using a single Helmholtz boundary integral equation for exterior acoustic problems. The design variables are the artificial densities of design material elements in a bi-material model constructed by the solid isotropic material with penalization (SIMP) method, and the minimization of sound power level (SWL) is chosen to be the design objective. In this study, the adjoint operator method is employed to calculate the sensitivity of the objective function with respect to the design variables. Based on the sensitivity information, the gradient-based optimization solver is finally applied for updating the design variables during the optimization process. Numerical tests are provided to illustrate the correctness of the sensitivity analysis approach and the validity of the proposed optimization procedure. Results show that the heavy fluid feedback has a big impact on the final design, and thus it is necessary to conduct a strong coupling scheme between the fluid and structures. In addition, the optimal design is strongly frequency dependent, and performing an optimization in a frequency band is generally needed. Keywords Topology optimization · Boundary element method · Finite element method · SIMP · Adjoint operator method 1 Introduction Reducing the sound radiation of vibrating shell structures is of great concern in engineering problems. Passive vibration control of flexible shell structures by bi-material or damping layer design has been considered to be an  Haibo Chen hbchen@ustc.edu.cn Wenchang Zhao JsuKya@mail.ustc.edu.cn Changjun Zheng cjzheng@hfut.edu.cn Cheng Liu chengliu@mail.ustc.edu.cn 1 CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, Anhui, People’s Republic of China 2 Institute of Sound and Vibration Research, Hefei University of Technology, Hefei 230009, Anhui, People’s Republic of China effective approach, which has also received a lot of attention (Zheng et al. 2005; Takezawa et al. 2016). However, it is usually not practical to apply a full-coverage design of damping material layer due to various industrial limits such as the weight or expense. Thus, it is highly desired to obtain the optimal material distribution under volume constraint against vibration and noise, which consists in the problem of topology optimization (Marburg 2002). Several topology optimization methods, including the solid isotropic material with penalization (SIMP) (Bendsøe 1989; Bendsøe and Sigmund 1999), evolutionary structural optimization (Xie and Steven 1993; Huang and Xie 2008), and level set method (Wang et al. 2003; Allaire et al. 2004), have been developed to structural and multi-disciplinary design problems to improve system performance. In recent years, optimization for structural-acoustic problems has also been extensively studied, and different criteria such as the sound pressure (Du and Olhoff 2010; Shang and Zhao 2016), sound radiation (Merz et al. 2010; Niu et al. 2010; Xu et al. 2011) and dynamic compliance (Ma et al. 1995; Jog 2002; Olhoff and Du 2014) have been considered to be the