Design of damping layer by topology optimization and Non-Negative Intensity

Wenchang ZHAO1,2 , Haibo CHEN1 , Steffen MARBURG2
1 CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics, University ofScience and Technology of China, Hefei 230027, Anhui, P.R.China
2 Chair of Vibroacoustics of Vehicles and Machines, Department of Mechanical Engineering, Technische Universität München,

Boltzmannstraße 15, Garching 85748, Germany

Abstract
Non-Negative Intensity (NNI) is a quantity which could avoid the near-field cancellation effects in sound intensity and provide direct visualization of surface contributions to sound power. Hence, the NNI and its variants
are implemented to be the objective function of topology optimization for damping layer design. Regarding
vibro-acoustic systems, the structural vibrations are analyzed by the finite element method (FEM), and fastmultipole boundary element method (FMBEM) is used for the acoustic analysis. A two-way coupling is established between the structural and the acoustic domains. By using the FMBEM and the implicitly restarted
Arnoldi method (IRAM), the eigenvalue analysis for the symmetrized acoustic impedance matrix is performedwith efficiency. Then, the NNI can be easily computed based on the eigen-solutions and the FEM-FMBEM
analysis. Further, these eigen-solutions can be recycled in the optimization iterations since they are independentof the solutions of the coupled system. This reduces the computational efforts. For calculating the gradientsof the objective function, the adjoint variable method is applied. With the evaluated gradients, the optimization
problem is solved by the method of moving asymptotes (MMA) and the optimized distribution of damping layer
is obtained.
Keywords: Topology optimization, Structural-acoustic, Boundary element method, Non-negative intensity

1INTRODUCTION

Reducing the sound emission of machines, systems and structures is important. One efficient tool is to suppress
the sound radiation from the most contributing component. This require a straightforward identification of thecontribution to sound radiation from individual components of a structure. Sound intensity has been widely used
to analyze the contribution to radiation in far-field, and the integration of sound intensity over closed surface,i.e., sound power has also been widely accounted for the objective function in a number of papers [1, 2].
However, sound intensity has both negative and positive values, and thus cannot provide direct visualization ofsurface contributions to sound radiation. Hence, sound power is also unable to evaluate the sound contribution
from partially components rather than whole system. To correctly identify the most contributing area, the nonnegative intensity (NNI), was proposed by Marburg et al. [3] to compute the surface contributions to radiatedsound power. In this work, we implemented the integration/sum of NNI on surface of interest as the objectivefunction. This allows for minimizing the sound radiation from predefined surface rather than the total surfaces.

2VIBRO-ACOUSTIC ANALYSISFor exterior structural–acoustic problem, the finite element method (FEM) is applied for structures and theboundary element method (BEM) is used for fluid. By using the equilibrium and continuity conditions over theEmail: JsuKya@mail.ustc.edu.cn, Wenchang ZHAO

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interface, the coupling between FEM and BEM can be described by [4]"#"# "#K − ω 2M−Cs fu
fs=,p0−iωGΘ−1 C f sH(1)where K and M are stiffness and mass matrices of the structure, respectively. For convenience, we use Kd for
representing K − ω 2 M in the following. H and G are the coefficient matrices for the BEM. Cs f and C f s are
the coupling matrices, Cs f = CTf s , Θ is the symmetric boundary mass matrix. The vectors fs is the structuralload.

3NON-NEGATIVE INTENSITY

Following references [3], the non-negative intensity η(x) can be defined by1η(x) = β ∗ (x)β (x),2(2)

where β (x) is only a quantity without physical significance, ()∗ denotes the conjugate of complex values. Obviously, β is non-negative and thus is called by non-negative intensity (NNI). After discretizing, the NNI at
collocation points can be calculated by [3]√β = BΘv f = Φ ΛΦT Θv f ,(3)where v f = −iωΘ−1 C f s u is the particle velocity vector at collocation points. Φ and Λ are the matrix storing
eigenvector Φ and diagonal matrix storing eigenvalue λ by solving the following generalized eigenvalue problem
ZR Φ = λ ΘΦ,(4)where ZR is the resistive impedance matrix defined by [3]ZR = −ℜ GT H−T Θ ,(5)where the negative sign is introduced because the direction of the given normal for emitted sound power is
opposite to the direction of outward normal defined in the BEM analysis. Based on the NNI, the sound power
can be computed as1
W = β H Θβ,2(6)where ()H denotes the transpose conjugate of a complex matrix.

4Optimization problem

To decrease radiation contribution from chosen surface Γc , we define the objective function as
1WNNI = β H Θc β,2(7)where ()H denotes the conjugate transpose, and Θc is the corresponding boundary mass matrix for surface Γc .
When Θc = Θ, WNNI = W . For vibration and noise control, the damping layer design has been proved tobe efficient, and is also adopted in this work. The damping effects are included by introducing an imaginary

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component to the Young’s modulus E. The optimization problem can be formulated asmin
µ1
Π = β H Θc β2s.t.






∑ µe ve − f v ∑ ve 6 0Ne
e=10 6 µe 6 1Ne(8)e=1(e = 1, . . . , Ne )where µ = [µ1 , µ2 , . . . , µNe ]T is the artificial elemental density of damping layer, and only can be 0 or 1 originally. When µ = 1, the damping layer element exists, otherwise it vanishes. Ne denotes the number of chosen
design elements. The symbol ve denotes the volume of e-th element, and fv denotes the corresponding volume fraction constraint. Because the design variables can only be two discrete values (0 or 1), the optimization
problem is hard to solve mathematically. However, this difficulty can be overcome by converting discrete design
variables into continuous values using a continuous interpolation function, which is the basic idea of densitybased approaches [5]. In the present work, we apply a gradient based algorithm, namely the method of moving
asymptotes (MMA) [6], to solve the optimization problem (8). Hence, the derivative of objective function, i.e.,the sensitivity information is necessary. As discussed before, we utilize the adjoint method for the sensitivity
analysis based on our previous work [2].

5NUMERICAL TESTS

In numerical tests, we validate the proposed optimization approach by optimizing the damping layer on a fouredge simply supported plate. By setting Θc = Θ, we confirm the optimizations with objective function beingsound power and integration of NNI give highly matched results. This is reasonable. Then the center area
and corners of the plate are chosen to be investigated area, and optimized damping layer distributions are quite
different.

6CONCLUSIONSIn this work, the non-negative intensity (NNI) and its variant are selected to be the objective function in topology optimization. By applying this technique, an optimized damping layer distribution can be found which
reduces the sound radiation from predefined components of structure. The FEM and BEM are applied for structural and acoustic domains, respectively, and a fully coupled FEM-BEM analysis system can be constructed.
Thus, the feedback interaction between the structural and the surrounding acoustic domain can be taken into
account in the optimization. Due to the large number of design variables corresponding to the damping layer
distribution, a gradient based optimization algorithm, namely the method of moving asymptotes, has been used.
To compute the objective function’s gradients efficiently, an adjoint variable method (AVM) has been developed for the fully coupled acoustic–structure system. Numerical examples have been provided to validate theoptimization technique.

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