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Application of level set method to optimize the topology of free damping treatments on plates is investigated. The objective function is defined as a combination of several desired modal loss factors solved by the finite element-modal strain energy method. The finite element model for the composite plate is described as combining the level set function. A clamped rectangle composite plate is numerically and experimentally analyzed. The optimized results for a single modal show that the proposed method has the possibility of nucleation of new holes inside the material domain, and the final design is insensitive to initial designs. The damping treatments are guided towards the areas with high modal strain energy. For the multimodal case, the optimized result matches the normalized modal strain energy of the base plate, which would provide a simple implementation way for industrial application. Experimental results show good agreements with the proposed method. The experimental results are in good agreement with the optimization results. It is very promising to see that the optimized result for each modal has almost the same damping effect as that of the full coverage case, and the result for multimodal gets moderate damping at each modal.

In the automotive industry, the door, roof, dash, floor, and cab back panels of automobiles are always treated with damping materials to reduce the structure-borne noise. The effectiveness of damping treatments depends upon design parameters such as damping material types, locations, and size of the treatment. As the weight often plays a key role in the performance and cost in the industry, it is essential to get the optimal distribution of the damping treatments with a limited coverage rate.

Traditionally, experimental techniques conduct laser vibro-meter type tests on the BIW structure or full-vehicle prototypes excited by a shaker at each suspension to generate velocity contours as optimal damping treatment configuration, which is prohibitively time consuming and expensive. Conventional CAE methodology employs the modal strain energy contour based on finite element modeling to determine the distribution of the damping treatment. The method is easy to implement and cost-effective, while the optimal results are not very accurate as the effectiveness of the damping treatment was not taken into account in the FE modeling, and the optimal configuration is determined by the engineer’s selection by trial-and-error based on the strain energy contour [

During the recent years, extensive efforts have been exerted to optimally design damping treatments for vibration and noise reduction, resulting in a large number of studies in the field. Lall et al. [

All of these studies mentioned above which performed shape or size optimization are at the level of macroscopic design, using a macroscopic definition of geometry given by, for example, dimensions or boundaries. One of the limitations of these conventional shape or size optimizations by boundary variations is that the topology of the structure is fixed during the iterative design process. A very interesting idea was proposed by Alvelid [

Since there is a limited type for the PCLD products on the market, the main problem is to determine the shape, location, and sizing of the PCLD patch with certain specification of materials and dimensions. Its very nature is to formulate this optimization problem as a topological design optimization. The idea of using topology optimization to design continuum structures was first introduced about twenty years ago by Bendsøe and Kikuchi [

Unfortunately, the above topology optimization methods tend to suffer from numerical instability problems such as mesh dependency, checkerboard patterns, and gray scales. A different approach is used in level set-based structural optimization methods that have been proposed as a new type of structural optimization. Sethian and Wiegmann [

Ansari et al. [

The parameterized level set can well compensate for the shortcomings of numerical calculation difficulties [

The main goal of this paper is to investigate the use of topology optimization based on a level set-based parameterization method to optimize free damping treatments with partial coverage to improve the damping characteristics of vibrating plates. The material of the paper is organized as follows. In Section

Figure

Damping treatments with a shape of

The objective of the paper is to find the optimal shape of the viscoelastic layer

A linear and viscoelastic, frequency-independent, complex constant modulus is supposed to describe the properties of the viscoelastic materials as follows:

In addition, the constraint should be considered to limit the consumption of damping material treatments, and the volume fraction is limited here. Then, the mathematical formulation based on the level set method of the optimization problem is defined as

When _{1} are the thickness of the base layer and the viscoelastic layer, respectively.

The neutral axis of the composite layer.

The composite layer can be meshed with the rectangular nonconforming plate-bending element. Based on the classical plate and finite element theory, the displacement field of the element can be written as follows [

Then, the mass matrix and the stiffness matrix of the composite element can be derived from the variational principle as follows [

Then, the global mass matrix and the stiffness matrix of the composite structure in equation (

In this paper, the Lagrangian formulation is applied by means of the Lagrangian multiplier

When the derivative of equation (

The sensitivity of the stiffness matrix about

The numerical solution form of the level set equation can be written as

Therefore, the sensitivity of the objective function about shape, that is, the velocity field of the element can be expressed as

Please note that

The MQ function is proposed and used as a common form of radial basis function, which had been performing well in function approximation, curve fitting, partial differential equation solving, etc. MQ functions can be expressed as

MQ function and its partial derivatives about

Interpolating the level set function with

In order to guarantee the uniqueness of the solution of the level set function, the interpolation coefficients must satisfy the following constraints:

Equations (

Since the matrix

Then, the level set function (

The fast marching method and the partial differential method are commonly used in the reinitialization of the level set function. The purpose is to ensure that the level set function is a symbol distance function at least near the boundary, that is,

Level set function reinitialization.

There is a linear relationship between the level set function

Compared with the common reinitialization method, the method used in this paper was simple and would not hinder the ability of autonomous opening and can fully retain the boundary information and only change the relative size of the value.

After the level set function was interpolated with the MQ radial basis function, the numerical calculation was performed. Substituting formula (

For this time-varying interpolation problem, in order to ensure that the interpolation coefficients can be solved without being limited to the positive definition of the condition of the MQ function, a boundary constraint needs to be introduced.

So far, the time-dependent H-J PDEs were discrete into a set of coupled ordinary differential equations. Equation (

The set of coupled nonlinear ordinary differential equations can be numerically solved by the first-order forward Euler method, and the approximate solution can be expressed as

At the same time, the

Combined with the reinitialization method, the iterative equation (

Among them,

In addition, since the time step and the number of interpolation nodes of the radial basis function are determined by the requirements of the optimization convergence rather than the need to solve the H-J equation, the time and space discretization requirements of the CFL condition can be relaxed. Using equation (

In the calculation of equation (

When the volume fraction of the initial design is not equal to the volume fraction of the constraint, the volume constraint can be relaxed during the initial iteration, expressed as follows:

The flow chart of the damping layer parameterization level set optimization algorithm based on the modal loss factor is shown in Figure

Given the initial design of the structure and meshing, initialize the level set function and radial basis function.

Run a finite element analysis to calculate the modal loss factor of the current structure.

Determine whether the number of iterations exceeds the specified minimum number of iterations, determine whether the current volume fraction reaches the constraint condition, and determine whether the difference between the current modal loss factor and the modal loss factor of the first five calculations is sufficiently small. When all three are satisfied, the loop ends, otherwise, the following steps are continued.

Update the Lagrange multiplier according to the current volume fraction and related parameters, and calculate the evolution velocity field and update the level set function.

Update the radial basis function interpolation coefficients and return to step (2).

Flow chart of the parameterized level set method for optimizing the modal loss factor.

The dimensions of the rectangle base plate is

The evolution of the damping layer for the 5th modal with four holes as the initial design is shown in Figure

Evolution history of the optimized design of the damping layer for the 5th modal with four holes as the initial design. (a) Step 1. (b) Step 15. (c) Step 25. (d) Step 81.

Evolution history of the level set surface of the damping layer for the 5th modal with four holes as the initial design. (a) Step 1. (b) Step 15. (c) Step 25. (d) Step 81.

Convergence curves of the modal loss factor (a) and volume fraction (b) of the damping layer for the 5th modal with four holes as the initial design (

Figure

Evolution history of the optimized design of the damping layer for the 5th modal with one hole as the initial design. (a) Step 1. (b) Step 14. (c) Step 24. (d) Step 76.

Evolution history of the level set surface of the damping layer for the 5th modal with one hole as the initial design. (a) Step 1. (b) Step 12. (c) Step 25. (d) Step 75.

The objective function and volume constrain have certain oscillations from step 10 to step 30 during the optimization procedure as shown in Figure

Convergence curves of the modal loss factor (a) and volume fraction (b) of the damping layer for the 5th modal with one hole as the initial design (

Convergence curves of the modal loss factor (a) and volume fraction (b) of the damping layer for the 5th modal with one hole as the initial design (

Convergence curves of the modal loss factor (a) and volume fraction (b) of the damping layer for the 5th modal with one hole as the initial design (

Optimized design of the damping layer for the 5th modal with one hole as the initial design (

Figure

Optimized results for the first 6 modals: (a) 1st modal, (b) 2nd modal, (c) 3rd modal, (d) 4th modal, (e) 5th modal, and (f) 6th modal.

Strain energy contours for the first 6 modals: (a) 1st modal, (b) 2nd modal, (c) 3rd modal, (d) 4th modal, (e) 5th modal, and (f) 6th modal.

For multimodal application using the proposed level set method, the 1st and 6th modals are under consideration in this section. Here, _{1} = _{6} = 0.5. Figure

(a) Optimized result for the 1st and 6th modals, (b) corresponding level set surface of Figure

For the conventional CAE method [

For multimodal,

Figure

In the conventional FE modal, orthonormal modes are often used. That is,

Then,

The images of the plates with different damping distributions under test are shown in Figure

The images of the plates with different damping distributions.

Experimental setup.

The modal frequency and modal damping under test are shown in Table

Modal frequency and damping ratio of test plates.

1st modal | 6th modal | |||
---|---|---|---|---|

Frequency (Hz) | Damping (%) | Frequency (Hz) | Damping (%) | |

Bared | 71.75 | 0.17 | 325.31 | 0.68 |

Shape (a) | 86.87 | 3.81 | 364.23 | 1.28 |

Shape (b) | 77.45 | 2.90 | 359.36 | 1.5 |

Shape (c) | 83.47 | 3.77 | 362.83 | 1.39 |

Shape (d) | 87.02 | 3.84 | 372.79 | 1.58 |

Frequency response functions between the exciting point (0.2 m, 0.14 m) and response point (0.12 m, 0.08 m) are shown in Figure

Frequency response functions between the exciting point (0.2 m, 0.14 m) and response point (0.12 m, 0.08 m): (a) Finite element method and (b) experimental result.

Topology optimization of free damping treatments on plates has been presented in the paper using the parameterized level set method. The objective function is defined as a combination of several desired modal loss factors solved by the FE-MSE method. A clamped rectangle plate has been applied to demonstrate the validation of the proposed approach. The optimized results for a single modal show that the proposed method has the possibility of nucleation of new holes inside the material domain, and the final design is insensitive to initial designs. The damping treatments are guided towards the areas with high modal strain energy. For multimodal case, the optimized result matches the normalized modal strain energy of the base plate, which would provide a simple implementation way for industrial application. Experimental results show good agreements with the proposed method. The experimental results show good agreement with the optimization results. It is very promising to see that the optimized result for each modal has almost the same damping effect as that of the full-coverage case, and the result for multimodal gets moderate damping at each modal for suppressing the plate vibration with 50% of weight saved.

The data used to support the findings of this study are included within the article. The topology optimization results in this paper have been obtained using the software Matlab. The codes have been uploaded as the

The authors declare that they have no conflicts of interest.

This research was financially supported by the Natural Science Foundation of Guangxi (nos. 2018GXNSFAA281276 and 2018GXNSFBA281012), Key Laboratory Project of Guangxi Manufacturing System and Advanced Manufacturing Technology (17-259-05-010Z), Basic Competence Promotion Project for Young and Middle-aged Teachers in Guangxi in 2018 (2018KY0205), Innovation-Driven Development Special Fund Project of Guangxi (Guike AA18242033), and Liuzhou Science Research and Planning Development Project (nos. 2019AD10203 and 2018AA20301).

MATLAB codes of the proposed topology optimization method.