Topology optimization of damping layers for minimizing sound radiation of shell structures

This paper deals with the sensitivity analysis of structural acoustic performance in presence of non-proportional damping and optimal layout design of the damping layer of vibrating shell structures under harmonic excitations. The structural system with a partially-covered damping layer has a non-proportional global damping matrix. Therefore, the method of complex mode superposition in the state space is employed in the dynamic response analysis. The sound pressure is calculated with the structural response solution by using the boundary element method. In this context, an adjoint variable scheme for the design sensitivity analysis of sound pressure is developed. In the optimal design problem, the design objective is to minimize the structural vibration-induced sound pressure at a specified point in the acoustic medium by distributing a given amount of damping material. An artificial damping material model that has a similar form as in the SIMP approach is employed, and the relative densities of the damping material are considered as design variables. Numerical examples are given to illustrate the validity and efficiency of this approach. The influences of the excitation frequency, the damping coefficients and the locations of the reference point on the optimal topologies are also discussed.

[1]  L. Meirovitch,et al.  Fundamentals of Vibrations , 2000 .

[2]  Erik Lund,et al.  Discrete material optimization of vibrating laminated composite plates for minimum sound radiation , 2010 .

[3]  M. Bendsøe.  Optimal shape design as a material distribution problem , 1989 .

[4]  M. Zhou,et al.  The COC algorithm, Part II: Topological, geometrical and generalized shape optimization , 1991 .

[5]  Ole Sigmund,et al.  On the Design of Compliant Mechanisms Using Topology Optimization , 1997 .

[6]  Joachim Bös,et al.  Numerical optimization of the thickness distribution of three-dimensional structures with respect to their structural acoustic properties , 2006 .

[7]  Søren Tørholm Christensen,et al.  On analysis and optimization in structural acoustics — Part I: Problem formulation and solution techniques , 1998 .

[8]  Jakob S. Jensen,et al.  Topology optimization of acoustic–structure interaction problems using a mixed finite element formulation , 2007 .

[9]  Jakob S. Jensen,et al.  Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide , 2005 .

[10]  A. Kiureghian,et al.  Modal decomposition method for stationary response of non‐classically damped systems , 1984 .

[11]  C. S. Jog,et al.  Optimization of vibrating structures to reduce radiated noise , 2012 .

[12]  M. Bendsøe,et al.  Generating optimal topologies in structural design using a homogenization method , 1988 .

[13]  N. Kikuchi,et al.  Topological design for vibrating structures , 1995 .

[14]  G. Cheng,et al.  On topology optimization of damping layer in shell structures under harmonic excitations , 2012 .

[15]  A. Haddow,et al.  Design of band-gap grid structures , 2005 .

[16]  Hae Chang Gea,et al.  Optimal stiffener design for interior sound reduction using a topology optimization based approach , 2003 .

[17]  Michael Yu Wang,et al.  Level Set Based Structural Topology Optimization for Coupled Acoustic-Structural System , 2011 .

[18]  G. Allaire,et al.  Structural optimization using sensitivity analysis and a level-set method , 2004 .

[19]  M. Wang,et al.  Level Set Based Structural Topology Optimization for Minimizing Frequency Response , 2011 .

[20]  Niels Olhoff,et al.  Minimization of sound radiation from vibrating bi-material structures using topology optimization , 2007 .

[21]  C. Jog.  Topology design of structures subjected to periodic loading , 2002 .

[22]  Amr M. Baz,et al.  Topology optimization of a plate coupled with acoustic cavity , 2009 .

[23]  Ole Sigmund,et al.  Systematic design of phononic band–gap materials and structures by topology optimization , 2003, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[24]  Jakob S. Jensen,et al.  Acoustic design by topology optimization , 2008 .

[25]  Ole Sigmund,et al.  On projection methods, convergence and robust formulations in topology optimization , 2011, Structural and Multidisciplinary Optimization.

[26]  Jakob S. Jensen,et al.  Maximizing band gaps in plate structures , 2006 .

[27]  Ole Sigmund,et al.  A 99 line topology optimization code written in Matlab , 2001 .

[28]  Masataka Yoshimura,et al.  Structural optimization based on topology optimization techniques using frame elements considering cross-sectional properties , 2007 .

[29]  Masataka Yoshimura,et al.  Topology design of multi-material soundproof structures including poroelastic media to minimize sound pressure levels , 2009 .

[30]  G. Yoon.  Structural topology optimization for frequency response problem using model reduction schemes , 2010 .

[31]  James K. Guest,et al.  Achieving minimum length scale in topology optimization using nodal design variables and projection functions , 2004 .

[32]  Ernest Hinton,et al.  Topology optimization of plate structures using a single- or three-layered artificial material model , 2001 .

[33]  N. L. Pedersen.  Maximization of eigenvalues using topology optimization , 2000 .

[34]  Xiaoming Wang,et al.  A level set method for structural topology optimization , 2003 .

[35]  N. Olhoff,et al.  Topological design of vibrating structures with respect to optimum sound pressure characteristics in a surrounding acoustic medium , 2010 .

[36]  K. Svanberg.  The method of moving asymptotes—a new method for structural optimization , 1987 .

[37]  R. Guyan.  Reduction of stiffness and mass matrices , 1965 .

[38]  N. Kikuchi,et al.  Solutions to shape and topology eigenvalue optimization problems using a homogenization method , 1992 .

[39]  N. Olhoff,et al.  Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps , 2007 .

[40]  Martin Ochmann,et al.  Boundary Element Acoustics Fundamentals and Computer Codes , 2002 .

[41]  T. Caughey,et al.  Classical Normal Modes in Damped Linear Dynamic Systems , 1960 .

[42]  Junghwan Kook,et al.  Acoustical topology optimization for Zwicker’s loudness model – Application to noise barriers , 2012 .