An acoustic radiation model of a cavity with a flexible plate treated with constrained layer damping (CLD) is developed by a combination of finite element method (FEM) and boundary element method (BEM). An acoustic topology optimization model is established with the objective of minimizing sound radiation power at specific modal frequency and design variables defined as locations of CLD treatments. The evolutionary structural optimization (ESO) method and genetic algorithm (GA) are employed to search optimal CLD configurations. Sound power sensitivity for CLD/plate is derived to determine search direction in ESO optimization procedure. The optimal CLD layouts for the flexible plate with two different boundary conditions are obtained and analyzed. Computational time, optimal layouts, and minimum sound power obtained using ESO and GA are compared. The results demonstrate effectiveness of the two methods, and ESO is more efficient to obtain deterministic and more practical optimal CLD material layouts for minimizing sound radiation power. The influences of CLD materials thickness and exciting force locations on optimal results obtained using ESO are discussed in detail. It is shown that the optimal rejection ratio varies with thicknesses of CLD materials and distribution of normal velocity of the flexible plate. Variation trend of the optimal rejection ratio is opposite for the two boundary conditions.
Vibration and acoustic radiation control are important issues in many engineering applications, such as aerospace and automotive industries. Constrained layer damping (CLD) treatment has been regarded as an effective means for suppressing vibration of the flexible engineering structures and their acoustic radiation since it was presented by Kerwin [
Zheng et al. [
The above works on CLD optimization mainly focus on vibration suppression of flexible structures. However, the minimization of vibration response cannot warrant a minimum sound power and fewer studies have considered optimal CLD configurations for acoustic problems. Zheng and Cai [
As that in the above literatures, it can be formulated as a topology optimization problem where the structure is designed optimally to achieve vibration or sound radiation reduction. Du and Olhoff [
In this paper, the work is extended based on [
Numerical example is demonstrated and discussed, in which the two different boundary conditions are considered for the CLD/plate in the cavity. The emphasis is a comparison on computational time, optimal CLD layouts, and minimum sound power obtained by implementing GA optimization procedure in [
Acoustic radiation model of a closed cavity is formulated. Figure
Schematic drawing of a closed acoustic cavity.
Finite element method is employed to obtain the normal vibration velocities of CLD/plate. A composite element shown in Figure
Composite element for CLD/plate.
Furthermore, for the damping layer, shape functions can be derived as follows:
Hence, element mass matrices and stiffness matrices can be obtained based on energy method, as follows:
The element stiffness matrix of damping layer corresponding to shear deformation can be derived as
Finally, assembling the element matrices and exciting force, the global motion equation of CLD/plate is formulated as
As a harmonic force is assumed, the vibration displacement and excitation force are reasonably expressed into
According to (
Boundary element method is employed to formulate acoustic radiation power of the closed cavity. The acoustic field composed of an ideal homogeneous air is assumed. The steady state sound pressure in the field satisfies the following equation over inner enveloping surface
To obtain acoustic power radiated into the cavity, sound pressure of point
Furthermore, every subsurface is meshed with
Schematic diagram of boundary element.
Hence, the sound pressure
Finally, the sound power radiated into the cavity induced by vibration of the flexible plate is defined as
For flexible structures, it is not practical to have full coverage with CLD treatments for vibration or acoustic reduction due to larger added mass. Furthermore, the full coverage of CLD configurations for vibration reduction cannot warrant a minimum sound radiation. Hence, an acoustic topology optimization model taking acoustic parameters as objective function is formulated. For the acoustic cavity with CLD/plate shown in Figure
In the model,
In topology optimization, sensitivity analysis is necessary to determine search direction. Hence, the sensitivity of objective function
In order to calculate
So the derivative of displacement
Hence, the velocity sensitivity of CLD/plate is induced as
Furthermore,
Substituting (
Evolutionary structural optimization (ESO) has been widely used to determine optimal shapes for continuum structures since Xie and Steven [
The basic flowchart of the acoustic topology optimization is shown in Figure
Flow chart of acoustic topology optimization using ESO.
Boundary surfaces of acoustic cavity are divided into 12 ×
Geometrical and physical parameters for closed acoustic cavity.
Young’s modulus | Density | Poisson’s ratio | Thickness | |
---|---|---|---|---|
Base plate | | 2800 kg/m3 | 0.3 | 0.8 mm |
Damping layer | 120 MPa | 1200 kg/m3 | 0.495 | 0.1 mm |
Constrained layer | | 2700 kg/m3 | 0.3 | 0.1 mm |
Cavity dimensions | 0.3 m × 0.3 m × 0.3 m | |||
CLD/plate dimensions | 0.3 m × 0.3 m | |||
Fluid domain | Air at 25°C and 1 atm |
Schematic diagram of discretization of boundary surfaces for closed cavity.
Evolutionary structural optimization (ESO) method and genetic algorithm (GA) in reference [
All the above model and optimization process are carried out by programming Matlab code and implemented on a work station with two Intel(R) Xeon(R) CPUs (E5-2623 v3 3.00 GHz) and 64.0 GB RAM. The finite element model and boundary element model of the CLD/plate and acoustical cavity are already validated in [
For CLD/plate, a common boundary condition that all the four sides of base structure are clamped is employed as the first BC. Here, another boundary condition, four sides of base plate are clamped and the sides of CLD treatments on the plate edges are proposed as the second BC for CLD/plate.
In optimization process using ESO, evolution parameters
Relationship between sound power and rejection ratio of CLD materials under various filter radius.
The first boundary condition
The second boundary condition
Figures
Optimal CLD layouts obtained using ESO under various rejection ratios for first BC.
Rejection ratio = 25%
Rejection ratio = 50%
Rejection ratio = 75%
Optimal CLD layouts obtained using ESO under different rejection ratios for second BC.
Rejection ratio = 25%
Rejection ratio = 50%
Rejection ratio = 75%
The frequency responses of sound radiation power are also monitored. Figures
Frequency responses of sound power obtained using ESO under various rejection ratios.
The first boundary condition
The second boundary condition
The relationship between sound power and rejection ratios of CLD materials obtained using ESO.
For comparison purposes, a genetic algorithm with integer encoding in [
Schematic diagram of design variables in GA optimization process.
Rejection ratio = 25%
Rejection ratio = 50%
Rejection ratio = 75%
In optimization process, other parameters are the same as that in [
Optimization process is simulated ten times for every above rejection ratio and the best solutions are selected. The corresponding layouts of CLD treatments with the above two BCs for first mode are presented in Figures
Optimal CLD layouts obtained using GA under various rejection ratios for first BC.
Rejection ratio = 25%
Rejection ratio = 50%
Rejection ratio = 75%
Optimal CLD layouts obtained using GA under various rejection ratios for second BC.
Rejection ratio = 25%
Rejection ratio = 50%
Rejection ratio = 75%
Frequency responses of sound power under various rejection ratios obtained using GA.
The first boundary condition
The second boundary condition
The minimum sound power for the two BCs obtained using ESO and GA is listed in Table
Minimum sound power obtained using of ESO and GA (dB).
Rejection ratio | ESO | GA | ||
---|---|---|---|---|
First BC | Second BC | First BC | Second BC | |
25% | 96.51 dB | 99.71 dB | 96.12 dB | 97.18 dB |
50% | 96.67 dB | 98.61 dB | 96.51 dB | 96.37 dB |
75% | 99.21 dB | 98.00 dB | 98.91 dB | 98.00 dB |
The total execution time
Execution time in seconds using ESO.
Rejection ratio | 25% | 50% | 75% | |||
---|---|---|---|---|---|---|
| | | | | | |
First BC | 403 | 44.8 | 739 | 41.0 | 1036 | 38.4 |
Second BC | 355 | 39.4 | 659 | 36.6 | 945 | 35.0 |
The minimum sound power, the execution time
Execution time in seconds using GA for first BC.
Simulation | Rejection ratio = 25% | Rejection ratio = 50% | Rejection ratio = 75% | ||||||
---|---|---|---|---|---|---|---|---|---|
SPL/dB | | | SPL/dB | | | SPL/dB | | | |
| 96.16 dB | 6552 | 384s | 96.67 | 5634 | 421 | 99.03 | 4910 | 327 |
| 96.29 dB | 8392 | 378s | 96.75 | 7998 | 409 | 99.03 | 5989 | 323 |
| 96.27 dB | 6556 | 385s | 96.67 | 10372 | 401 | | 5047 | 330 |
| 96.27 dB | 6502 | 382s | 96.87 | 8377 | 412 | | 4606 | 340 |
| 96.27 dB | 4055 | 338s | 96.63 | 8919 | 414 | 99.14 | 3630 | 338 |
| 96.33 dB | 7208 | 392 | 96.79 | 8022 | 411 | 99.68 | 5938 | 328 |
| 96.25 dB | 6744 | 390 | | 9346 | 413 | 99.02 | 5601 | 328 |
| | 5644 | 398 | 96.64 | 7879 | 404 | 99.03 | 7257 | 324 |
| 96.26 dB | 5643 | 399 | 96.66 | 12473 | 407 | | 6604 | 326 |
| 96.27 dB | 9561 | 376 | 96.60 | 7099 | 408 | 99.03 | 4548 | 325 |
Execution time in seconds using GA for second BC.
Simulation | Rejection ratio = 25% | Rejection ratio = 50% | Rejection ratio = 75% | ||||||
---|---|---|---|---|---|---|---|---|---|
SPL/dB | | | SPL/dB | | | SPL/dB | | | |
| 97.181 | 4962 | 327 | 96.60 | 8758 | 357 | | 5194 | 321 |
| 92.227 | 5587 | 330 | 96.58 | 7267 | 358 | 98.62 | 1253 | 328 |
| 97.181 | 4091 | 334 | 96.72 | 5729 | 345 | 98.20 | 4924 | 322 |
| | 3074 | 329 | 96.72 | 5467 | 338 | 98.16 | 10096 | 302 |
| | 5914 | 332 | 96.72 | 5588 | 340 | 98.27 | 5545 | 322 |
| | 4133 | 337 | 96.47 | 11385 | 346 | | 7820 | 304 |
| 97.181 | 7638 | 330 | | 9298 | 344 | 98.16 | 3809 | 310 |
| 97.180 | 4375 | 332 | 96.58 | 7024 | 351 | | 4658 | 308 |
| 97.316 | 4701 | 325 | 96.58 | 7537 | 351 | 98.51 | 3476 | 310 |
| | 6372 | 321 | 96.61 | 5812 | 341 | 98.44 | 3777 | 309 |
Further, a consistency ratio is defined to describe the relativity of optimal CLD treatments layouts obtained using ESO and GA, that is, a ratio of total number of the same locations of CLD treatments to total number of CLD treatments under a rejection ratio, seen in Table
Consistency ratios of optimal CLD layouts using ESO and GA.
Rejection ratio | 25% | 50% | 75% |
---|---|---|---|
First BC | 0.93 | 0.83 | 0.78 |
Second BC | 0.67 | 0.61 | 1.00 |
Integrating the above comparison and analysis, it can be concluded that the ESO is more efficient to obtain deterministic and more practical optimal CLD material layouts for minimizing sound radiation power compared to GA.
In topology optimization study for CLD structures, the positions of CLD materials are usually considered as design variables while a specific thickness of CLD materials is defined, such as that in [
Influence for sound power with rejection ratios under various
The first boundary condition
The second boundary condition
Influence for sound power with rejection ratio under various
The first boundary condition
The second boundary condition
The influence of exciting force locations on optimal CLD materials layouts is also discussed. Figure
Locations of exciting force shown at finite element partition of 1/4 CLD/plate.
Optimal layouts of CLD treatments under exciting force at point 1 for first BC.
Rejection ratio = 25%
Rejection ratio = 50%
Rejection ratio = 75%
Optimal layouts of CLD treatments under exciting force at point 1 for second BC.
Rejection ratio = 25%
Rejection ratio = 50%
Rejection ratio = 75%
The vibration normal velocity in the full coverage plate with CLD materials for first BC.
The force is applied at point C
The force is applied at point 1
The vibration normal velocity in the full coverage plate with CLD materials for second BC.
The force is applied at point C
The force is applied at point 1
The best rejection ratios with varied locations of exciting force are also monitored, given in Table
The best rejection ratios with varied locations of exciting force.
Point 1 | Point 2 | Point 3 | Point 4 | Point C | |
---|---|---|---|---|---|
First BC | 27.8% | 33.3% | 38.9% | 30.6% | 41.7% |
Second BC | 55.6% | 52.8% | 61.1% | 66.7% | 63.9% |
Finite element method and acoustic boundary element method are combined to predict sound radiation power of a closed acoustic cavity with a flexible surface treated with constrained layer damping (CLD). A topology optimization model is established with the objective of minimizing sound radiation power at the specified frequency. The evolutionary structural optimization (ESO) method and genetic algorithm (GA) with integer coding are employed to search optimal layouts of CLD materials bonded on the flexible plate in the cavity. The sound power sensitivity is formulated to determine the search direction in ESO optimization process.
A new boundary condition is defined for CLD/plate in the cavity besides a common boundary condition. Optimal layouts of CLD material with different rejections ratios are obtained and analyzed for the first mode using ESO and GA. It shows the two methods are feasible to solve the optimization problem. The computational time, the optimal layouts, and the minimum sound radiation power obtained using ESO and GA are compared. The results demonstrate that the ESO is more efficient to obtain deterministic and more practical optimal CLD material layouts for minimizing sound radiation power.
The optimal results obtained using ESO are discussed in detail. There is an optimal rejection ratio of CLD materials which makes the sound power reduction highest. It varies with variation of thickness of constrained layer material and viscoelastic material. The variation trend is opposite for the two boundary conditions. The locations of exciting force have also effect on the optimal layouts of CLD materials. The best rejection ratios of CLD materials lie in a small range but are different for the two boundary conditions.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (no. 51275126).