Abstract
The frequency- and temperature-dependent characteristics of viscoelastic materials significantly affect the vibration response of the damped composite structures. In this paper, an efficient strategy of hybrid expansion combined with dynamic reduction is developed to solve the steady-state response of the frequency- and temperature-dependent viscoelastic structure characterized by nonproportional system, and the sensitivity analysis is carried out based on the adjoint variable method. The similarity index is defined to distinguish the correlation among different design layouts. Two instances demonstrated the validity of the proposed approach. The findings indicated that a positive compromise between accuracy and efficiency can be achieved, and the computational time can be significantly reduced while ensuring the accuracy of the results. Furthermore, it has been discovered that the excitation frequency and temperature significantly impact the optimal configuration of damping material. The effects of layer thicknesses and volume fractions on optimization designs are also further investigated.
1. Introduction
Thin-walled constructions are frequently utilized as load-bearing components in autos, railways, ships, space shuttles, and other engineering structures. Due to the growing desire for lightweight design, these structures are prone to severe vibration and noise concerns when subjected to external dynamic excitations. Passive control treatments, such as incorporating viscoelastic damping materials into structures, were generally regarded as a practical and effective technique to suppress the level of vibration in structures [1–3]. Since the 1960s, constrained layer damping (CLD) structures have been widely used in vibration control of aerospace structures due to their substantially higher energy dissipation on account of shear deformation than free layer damping. Various CLD structures have emerged as hot topics and trends, with key research focusing on dynamic modeling and analysis [3, 4], performance identification [5, 6], position optimization, and viscoelastic material parametric design.
Topology optimization technology has been widely applied in the conceptual design of the optimal layout of damping materials due to its effectiveness, aiming to achieve greater performance in decreasing vibration and abating noise under lightweight constraints. In the field of dynamic characteristics, fundamental or multiple modal loss factors are frequently employed as objective functions. Zheng et al. [7] investigated the structural design configuration using the genetic algorithm for minimizing the overall vibration energy. Wang et al. [8] proposed an artificial density and heuristic methods for optimization design of CLD shells. Kim et al. [9] optimize the distribution of damping materials by maximizing the structural modal loss factor. Xu et al. [10] applied topology design technology to estimate the material distribution in the spindle box of machine tools to increase vibration suppression ability. Recently, Zhang et al. [11] investigated the topology optimization of multiphase viscoelastic microstructures to enhance macroscopic damping performance. For dynamic response optimization, many notable works have been carried out to suppress the structural vibration. Fang et al. [12] adopted the density-based method to reduce the structural dynamic response. Zheng et al. [13] discussed the topological optimization of CLD structures to minimize sound radiation. Takezawa [14] took the complex dynamic compliance as a novel objective function to lessen the resonant response by maximizing the energy dissipation near the resonance. Delissen et al. [15] proposed a constraint function based on enhanced modal truncation to limit the frequency response peak at the resonance using an efficient reduced-order model. Furthermore, Kang et al. [16, 17] adopted the classical Rayleigh damping model to describe the energy dissipation of viscoelastic materials and found that the specific design configurations are sensitive to the damping coefficients. Nevertheless, the frequency- and temperature-dependent properties of viscoelastic materials are still rarely considered in the optimization design of viscoelastic structures, and the corresponding difficulties and challenges have not been well addressed.
It should be emphasized that the dynamic mechanical properties of viscoelastic materials are affected by the coupling of several factors, the most relevant of which are the temperature and frequency [6, 18, 19]. The following lists some recent meaningful work. Li et al. [20] investigated the vibrations of composite structures subjected to partial CLD treatment by using the strain-dependent shear modulus. Oh [21] investigated the damping performance of laminated shells considering the frequency and temperature dependence. Shu et al. [22] established an accurate dynamic equation of CLD structure and analyzed its vibration characteristics using the coupling effect of temperature and frequency. Mokhtari et al. [23] quantitatively analyzed the dynamic response and behavior of viscoelastic layers in CLD cylindrical shells. Sun et al. [24] conducted a thorough examination and discussion of the effect of frequency on the vibration behavior of composite structures. Dai et al. [25] then investigated the damping performance of laminated shells with frequency-dependent properties and derived the analytical expression of dynamic response. These published works all concur with the conclusion that the viscoelastic materials’ frequency and temperature dependence are essential when evaluating the dynamic response and damping behavior of CLD structures.
As far as the authors know, despite the fact that dynamic topology optimization of CLD structures has been extensively and thoroughly researched, few research reports address the difficulties of optimizing the design of viscoelastic structures. Recently, Zhang et al. [26] performed the topology optimization study for maximizing the modal loss factor of the CLD structure considering temperature and frequency dependence by means of a parametric level set method based on the generalized Maxwell model. However, their work did not involve attenuating the structural response. As a result, the originality of this study is primarily reflected in the efficient optimization design approach which aims to reduce the dynamic response of viscoelastic structural materials comprising the frequency- and temperature-dependent damping materials. Therefore, the application scope of Golla–Hughes–McTavish (GHM) model [27, 28] is extended to solve the optimal layout problem of the CLD plate. However, this composite structure is characterized by nonproportional damping system owing to the nonuniform distribution of viscoelastic materials. Recently, Li et al. [29] proposed an N-space approach to correct the higher-order responses using the established relationship between the eigensolutions and the system matrix. These findings provide the potential to be a driving factor in the application and popularization of the nonproportional damped systems in the field of topology optimization.
The aim of this paper is to study the topology optimization design of viscoelastic composite structures at different temperatures and frequencies to minimize the steady-state response. For this purpose, the GHM model was incorporated into the finite element method to describe the energy dissipation of materials. To handle the topology optimization difficulties characterized by nonproportional damping of the CLD plate resulting from this, an improved approach integrating hybrid expansion and dynamic reduction methods is proposed and developed, and the sensitivity analysis of adjoint schemes is performed. Specific attention focuses on the influence of the fluctuation of viscoelastic material properties with the change of temperature and frequency on the vibration response and optimal configuration of composite structures.
The remainder of the paper was structured as follows. Section 2 established the augmented vibration equations incorporating the GHM model, taking into account the frequency- and temperature-dependent properties of viscoelastic materials. Section 3 described the specific procedure for calculating the displacement response of CLD plates under harmonic excitations by combining hybrid expansion and dynamic reduction methods. The formulations of topology optimization of CLD plate and sensitivity analysis based on the adjoint variable method (AVM) are reviewed in Section 4. In section 5, several numerical examples are implemented to validate the proposed optimization method and evaluated the effects of frequency and temperature on the optimal layouts. Besides, the influence of the layer thicknesses and volume fractions is also further discussed. Conclusions are drawn in section 6.
2. Numerical Modeling Procedure
The CLD plates, as a king of common multilayer structures, are extensively applied in many engineering fields. This section presents the related theories for the numerical modeling procedure of CLD plates incorporating the GHM model.
2.1. Basic Kinetics Relationships
Figure 1 depicts a schematic illustration of the CLD plate. The symbol represents the thickness, while the subscripts , , and , respectively, denote the base plate, constrained layer, and damping layer. The coordinate system is set on the neutral plane of the CLD plates, and the position of the neutral axis can be obtained by balancing the forces in the out-of-plane direction.
Considering the dynamic deformations, as shown in Figure 1, the midplane displacements and and the shear strains and are given aswhere and and and denote midplane displacements of constrained layer and base plate, respectively. is the out-of-plane displacement, and denote the partial derivatives of in and directions, and is the midplanes distance.
2.2. Finite Element Models
The finite element model is discreted into four-node planar plate element, and the nodal displacement vector is determined byin which
Therefore, the displacement at any position within the element can be represented by the node displacement as follows:where , , , , , , and are the spatial interpolating vectors corresponding to , , , , , , and , respectively.
In addition, the spatial interpolating vectors corresponding to , , , and can be derived from (1) and (2), given as
Then, the elemental mass and stiffness matrices can be obtained according to the energy approach and variational Hamilton principle as follows:where denotes the shear stiffness, identifies the elastic matrix, and strain matrices are presented as
The elemental dynamic equation of composite structures is given aswhere and . and represent the composite elemental mass and stiffness matrixes, respectively.
2.3. Incorporation of Viscoelastic Materials
For the CLD structures comprising viscoelastic material, equation (9) cannot capture their frequency-dependence. Hence, Golla–Hughes and McTavish [27, 28] introduced the GHM model to describe the energy dissipation of viscoelastic structures and expressed it in the following form:where is the equilibrium value of modulus. Each term includes , , and , which are determined via at temperature . The number of mini-oscillator terms could be adjusted to reflect the corresponding damping performance of different viscoelastic materials.
The Laplace’s transform of (9) is
Substituting (10) into (11) and introducing auxiliary coordinates called dissipation coordinates are defined as follows:
Equation (9) is converted into the time domain and rewritten as follows:wherewhere is a diagonal matrix composed of the positive eigenvalues of the shear stiffness matrix of viscoelastic materials and is the matrix composed of the corresponding eigenvectors.
Each element can be considered to be composed of five nodes, four of which are physical and the other one is virtual. Each physical node has seven DOFs, and the introduced auxiliary coordinates are regarded as the DOFs of the corresponding virtual nodes, which are jointly determined by the number of positive eigenvalues and the orders of the mini-oscillators terms. Figure 2 presents the schematic diagram of the element nodes and assembly process of the global matrices.
The global matrices are assembled based on the physical coordinates while reserving auxiliary coordinates. The equation of motion is shown as follows:where , , and , respectively, denote the global mass, stiffness, and damping matrices. and represent the displacement vector and external load vector. Here, indicates the full DOFs consisting of physical and dissipative coordinates.
The finite element method (FEM) provides a carrier for the fusion of the GHM model, so that the properties of viscoelastic materials can be introduced into the equation of motion of the structure in the form of the element mass, stiffness, and damping matrix. The dynamic equation obtained by this modeling method is a standard second-order linear system model, which performs well in response and sensitivity analysis.
3. Solutions of the Governing Equation
As previously stated, dissipative coordinates appear as the augmented state variables, significantly increasing the dimension of the governing equation. As a result of this, such a method suffers from the finite element dimension system being extended twice or more, necessitating expensive computing costs. Meanwhile, owing to the nonuniform design layouts of damping material, the composite structure represents the nonproportional damping behavior. Solving the governing equation directly is a computationally expensive task. Therefore, model reduction is necessary to reduce the high-order finite element models to a smaller size for undertaking more efficient dynamic analysis and optimization procedures.
3.1. Dynamic Reduction Method
The dynamic reduction method pioneered by Leung [30] and afterward Petersmann [31] as a type of hybrid coordinates reduction method is frequently and widely employed. This method employs the modal synthesis in conjunction with the dynamic reduction, which is analogous to the substructuring technique. According to the master and slave DOFs, the structural displacement vector can be separated into two subvectors. The displacement subvector of slave DOFs can be obtained as follows:
Rewrite (16) into matrix form as follows:where is the transformation matrix, ; and are the unit matrix; ; and denote the static contribution, and represents the corresponding dynamic contribution.
Therefore, the reduced-space governing equation is showed as follows:where , , , and , respectively, denote the reduced mass, stiffness, damping matrices, and load vector. Furthermore, please see [32] for the specific theories and the derivation of corresponding reduction space.
3.2. Complex Mode Superposition Method for Higher-Order Modes Correction
The dynamic equation in the frequency domain of reduced order for the CLD structure with DOFs can be expressed as follows:where the matrix is the dynamic stiffness matrix and refers to the exciting frequency.
The complex frequency response function (FRF) matrix can be expressed using complex mode superposition theory, as shown as follows:wherewhere is the complex eigenvector that corresponds to the complex eigenvalue . We suppose that the complex eigenvalues are distinct and the number is 2 . In practice, only limited low-order modes can be involved.
Equation (22) is the modal displacement method (MDM). The modal truncation error may be generated because some higher-order modes are disregarded. When , the calculation results are hardly credible.
According to the method proposed by Li et al. [29], (20) is rewritten as the matrix form, and expanded the inverse term using the Neumann series as follows:where
It should be noted that (23) can achieve the expected accuracy as long as all corresponding modes whose resonance frequency is within the concerned frequency interval are retained. It can be further proved that the series expansion in (23) is convergent.
The can be reformed and the inverse calculated as follows:where can be expanded as follows:
Combining (24) and (25), we get
It should be highlighted that may be obtained specifically by the iterative approach, and only needs to be calculated once at different frequencies due to the frequency independence.
Combining (23) and (26), we provide the more broader connections between the eigensolutions and system matrices for , as follows:
Hence, a novel mode superposition method called the hybrid expansion method (HEM) for calculating the displacement response can be derived. Assuming that the term in power-series is retained for meeting appropriate precision criteria, the estimated displacement vector is shown as follows:
4. Topology Optimization Model
4.1. Problem Statement
The motive of the research is to seek for the design layouts of a given amount of damping material to minimize the vibration amplitude at the specified locations. The steady displacement response is obtained bywhere and are the real and imaginary parts of the complex amplitude , respectively.
The mathematical model is established as follows:where are the pseudodensity variables, denotes the total number of desirable elements, and is the number of specified DOFs. The volume fraction is represented by the symbol . is the lower bound of the design variables to prevent the system matrices from becoming singular, in this case .
In the framework of the polynomial interpolation scheme (PIS), the elemental mass matrices and and stiffness matrices , , and are expressed bywhere and and , , and are the mass matrices and stiffness matrices when mass density is equal to 1.
4.2. Sensitivity Analysis
To perform optimization, the globally convergent method of moving asymptotes (GCMMA) is adopted to obtain the optimized solution. The AVM is preferred compared to the direct method for topology optimization problems involved in this study
The sensitivity of objective function in (30) is given bywhere can be derived as follows:where is so-called the column vector with concerned DOF being 1.
Take the derivative of both sides of (30) with respect to the design variables as
Substituting (34) into (33) and taking into account the symmetry of , one obtainswhere can be calculated by following equation:
Almost no additional computation is required because of the same FRF whether the excitation load is or .
Once the is obtained, then the first-order sensitivity of the displacement response can be calculated as follows:
The first-order sensitivity of the augmented global matrix, namely, the mass, stiffness and damping matrices in (37), can be obtained by the chain rule through interpolation relations in (31) at the element level. Then, remaining derivatives of the corresponding sensitivity are removed.
4.3. Topology Optimization Process
The sensitivity filtering scheme based on the image [33] is applied to avoid checkerboard and mesh dependence. The filter modifies the design sensitivity of a specific element based on a weighted average of the element sensitivities in a fixed neighborhood, which is purely heuristic but produces results very similar to local gradient-constrained results, requiring little extra CPU time. The GCMMA algorithm [34] is used to find the optimal solution. All cases are performed in the commercial software MATLAB R2018a. The optimization process will stop until the absolute error of the design objectives satisfies . Figure 3 presents the flow diagram of the optimization process.
5. Numerical Examples
This section presents two typical numerical examples with different boundary conditions. One is a cantilever plate clamped at the left side, as shown in Figure 4(a). An external force is applied at the midpoint of the right side, with and . The other is the structure clamped by four sides shown in Figure 4(b), and the same external force is applied at the center of the plate. For viscoelastic composite structures, the material 1 called 242F01, manufactured by 3M, is employed as the damping layer, which exhibits frequency and temperature dependence. Table 1 presents the material properties for the viscoelastic plate. The modulus of 242F01 is characterized by the GHM model. The parameters at and are obtained through the dynamic mechanical analysis (DMA) and least square method and are given in Table 2. For comparison, material 2 is an artificial material, characterized by the complex constant model, whose properties are obtained by averaging the modulus of 242F01 in the frequency domain. This section first investigates the validity of response and sensitivity analysis, then verifies the validity of introducing the dynamic reduction method into topology optimization. Next, the performance of the proposed optimization methods are compared in detail. Finally, the impact of the viscoelastic material’s frequency- and temperature-dependent properties on optimal layouts is emphatically analyzed. Furthermore, the influence of layer thicknesses and volume fractions on optimal layouts of damping material is further discussed. In order to quantitatively characterize the correlation between different optimal layouts, the similarity index is defined as follows:where is the overlapped area of damping layer between the different optimal layouts and is the total area of damping layer. However, since the relative density is not the absolute 0-1 distribution, the calculation accuracy of the overlap area between different optimized configurations is the possible potential limitation.
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Although some researchers have shown that solid elements have higher computational accuracy, they then have much lower computational efficiency. Therefore, comparisons in the accuracy and efficiency of the calculated results between the proposed model and common commercial software ANSYS are conducted. For the numerical model in ANSYS, the structure is discrete into three-dimensional continuous solid elements. The results are in line with expectations. Under the premise of less than 3.42% error of calculation accuracy, the CPU time of the proposed model is significantly reduced by 74.8% compared with common commercial software, ANSYS. The excellent performance of the proposed model in terms of precision and efficiency is fully demonstrated.
In order to verify the effectiveness of the proposed strategy combining modal polycondensation and hybrid expansion for solving frequency- and temperature-dependent optimization design problems, the comparison of optimal design, as shown in Figure 4(a) between full-order and reduced-order models is carried out in this section. The volume fraction of the damping material is set at 50%, and the initial layout is uniform distribution with a mass density of 0.5. An excitation frequency of is considered. Figure 5 presents the iterative process of the optimal layouts and objective functions for the full-order and reduced-order models in the topology optimization. The performance comparison details during the optimization process are listed in Table 3.
Obviously, it can be observed from the iteration histories shown in Figure 5 that all design objectives decrease from to and converge to the optimal value. The reduced-order and full-order models are highly consistent in the initial and converged objective values, especially the similarity index of the two, which is as high as 99%, and nearly identical optimal configurations are obtained. Furthermore, in terms of computing time, the full-order model takes 976.9 s, while the reduced-order model only requires 213.3 s, and the acceleration ratio reaches 78.17%. It proved that the proposed strategy is feasible, which significantly improves optimization efficiency and also provides potential and possibility for topology optimization of large-scale nonproportional damping structures.
5.1. A Cantilever Plate
This section focuses on the structure shown in Figure 4(a) to investigate the influence of the response-solving methods and frequency- and temperature-dependent viscoelastic materials on the optimal configuration and vibration suppression of the CLD plate.
5.1.1. Response-Solving Methods
Here, the design domain is discretized into a total of 400 elements as shown in Figure 6. Figures 7(a) and 7(b) present the comparisons of the displacement amplitudes (points I and II) and the sensitivities (elements numbered 30, 210, and 370, denoted by blue elements in Figure 6) obtained with different methods. The results calculated by the direct frequency response method (DRFM) and finite difference method (FDM) are regarded as the exact results of the displacement response and sensitivity, respectively. The detailed error comparisons are depicted in the form of the line chart, as shown in Figure 7(c). For the sake of the narrative, the combination of MDM and HEM with the dynamic reduction method is, respectively, abbreviated CMDM ( = 30, 50, namely, i, ii) and CHEM ( = 2, 3, namely, iii, iv). As expected, the CHEM results agree well with the exact results, especially when is equal to 3. The errors of the vibration amplitudes and are only 1.01% and 0.78%, respectively, and the sensitivity differences of the three specified elements are less than 2%, about 0.92%, 1.02%, and 1.33%, respectively.
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However, there is a significant gap in the performance of calculation accuracy for the traditional method (i.e., CMDM). Although the vibration amplitudes retain a reasonable accuracy of around 20% when the limited low-order modes are involved (e.g., = 30), the sensitivity analysis error has increased dramatically to over 30%. A negative phenomenon worthy of attention is that when the number of low-order modes involved reaches 50, however the accuracy of sensitivity analysis cannot be guaranteed, indicating that excessive increase in the number of low-order modes is not a cost-effective measure worth popularizing.
Table 4 lists the computing times of the different methods for responses and sensitivities analysis. It can be observed that CHEM, both for response and sensitivity analysis, shows remarkable advantages in computational efficiency. As previously stated, this is because of the fact that the proposed method can achieve expected accuracy as long as it contains all modes with eigenfrequencies within the excitation frequency range, without worrying about the trade-off between computational accuracy and efficiency.
For the cantilever plate structure shown in Figure 4(a), through modal analysis, Figure 8 presents the mode shapes, and the first 6 natural frequencies are obtained as 58 Hz, 97 Hz, 142 Hz, 195 Hz, 242 Hz, and 304 Hz, respectively. As well known, damping effects are mostly observed around each natural frequency. Therefore, the excitation frequency points are selected around the natural frequency, whose values are 100 Hz, 200 Hz, and 300 Hz, respectively. Table 5 depicts the optimal layouts and iteration curves of topology optimization at three different frequencies (i.e., 100 Hz, 200 Hz, and 300 Hz) and also lists the optimization details. The blue and green color in optimal material configurations represent the area covered by the damping layer and without the damping layer, respectively. On the one hand, the damping materials obtained by topological optimization are mainly distributed in the region with larger modal strain energy determined by adjacent mode shapes. On the other hand, the distribution of damping materials tends to be complicated and decentralized with the increase of the excitation frequency. The reason is that higher excitation frequencies are prone to excite higher-order modes with more localized characteristics. This can also be confirmed by the obtained vibration shape from the lower-order to the higher-order modes, showing a consistent variation trend. It can be seen that the optimal configurations based on the two methods have a high degree of similarity, with the values of 0.79, 0.89, and 0.84, indicating that both methods can obtain reasonable configurations despite the difference in calculation accuracy.
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However, there are also significant differences in convergence and clarity. This simulation results show that CHEM works well while CMDM converges poorly when large-scale problems are considered, as evidenced by the fluctuation of the initial stage and the oscillation of the later stage in the iterative curves. It can be concluded that high-precision response and sensitivity analysis methods are more conducive to improving the convergence of topology optimization and obtaining the expected results. The reasons come from the two aspects. On the one hand, CMDM, as an approximate method, inevitably introduces truncation errors especially for sensitivity analysis, which may cause deviations or even errors in the optimization direction. On the other hand, large errors will be generated away from the resonant frequency, and the essence of dynamic response optimization is to drive the resonant frequency away from the excitation frequency to reduce the vibration level, which further reduces the accuracy of response and sensitivity analysis. This may explain why low-precision analysis methods are by far not entirely satisfactory for large-scale optimization design problems.
According to the details in Table 5, CHEM can reduce 70.53%, 65.63%, and 65.69% CPU time for cases of different frequencies, respectively. The performance on computational efficiency is due to the fact that the CHEM-based method only needs to consider the low-order modes within the excitation frequency rather than enough low-order modes, the order of which can be adaptively determined according to the varied structural eigenfrequencies in each iteration, which greatly reduces the surge of computational time caused by complex modal analysis and multiple iterations.
5.1.2. Frequency- and Temperature-Dependent Properties
The frequency-dependent properties of viscoelastic material at and are considered. To design an artificial modulus at room temperature as the comparison group, the averaged real and imaginary part of modulus at are obtained, given by . Figure 9 presents the frequency-dependent and averaged shear moduli of 242F01 for and . Obviously, the stiffness and damping performance of 242F01 exhibit significant changes with frequencies and temperatures varying.
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The optimal layouts and iteration curves induced by the complex constant model and GHM model are presented in Table 6. The quantitative details are also listed. Since the cantilever plate structure has not changed, the excitation frequencies are still selected as 100 Hz, 200 Hz, and 300 Hz. In general, the optimal layouts obtained all exhibit remarkable convergence. However, there are significant differences among the design layouts of damping materials. The reason comes from the fact that the frequency and temperature dependence of viscoelastic materials can influence the stiffness distribution and damping behavior of CLD structures. Generally speaking, the similarity index of SI-II in the corresponding group is the highest, which is because the complex constant model adopts the average value of the complex modulus of the GHM model at T = . However, due to the discrepancy between the averaged and actual moduli at different frequencies, the similarity of the SI-II group at was 0.702, which was significantly lower than that of 0.862 at . Furthermore, most of the similarities in the SI-I and SI–III groups were at a low level, with two being lower than 0.5 (0.429, 0.441) and two being in the range of 0.5–0.7 (0.579, 0.591, 0.687). This phenomenon is clearly caused by changes in temperature and frequency caused by the viscoelastic material properties of the great change. In addition, it can be observed that the similarity indices corresponding to are significantly lower than that of . This attributes to significant changes in the optimized configuration at higher frequencies due to the involvement of more local modes, even if temperature makes a limited difference.
5.2. A Plate Clamped by Four Sides
To further validate the accuracy and efficiency of the proposed CHEM-based approach and to compare the effect of frequency- and temperature-dependent properties of damping material on optimal layouts and vibration suppression, the CLD structure is considered under another boundary condition, as shown in Figure 4(b), which is clamped by four sides, representing more “rigid” boundary conditions. The optimization objectives are to minimize the sum of squares of vibration amplitudes at specific points of concern as indicated by the red dots in Figure 5.
5.2.1. Response-Solving Methods
For the plate structure clamped by four sides shown in Figure 4(b), through modal analysis, Figure 10 gives the mode shapes, and the first 6 natural frequencies are obtained as 128 Hz, 197 Hz, 289 Hz, 392 Hz, 428 Hz, and 511 Hz, respectively. Therefore, the selection strategy of frequency points is based on the natural frequencies, where the values are 200 Hz, 400 Hz, and 500 Hz, respectively.
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Table 7 depicts the optimal layouts and iteration curves and also presents the performance results of topology optimization. A similar phenomenon can be observed that the optimization results obtained by the two methods present obvious differences while maintaining similarities. In general, the damping materials are concentrated in the region of the larger modal strain energy in the corresponding modes, and the distribution tends to be local and dispersed with the increase in excitation frequency. The similarity indices between the optimal layouts obtained are relatively high, with values of 0.92, 0.83, and 0.81, respectively. The CHEM-based optimization method can obtain clear optimal layouts due to its high accuracy, and the strong convergence is also demonstrated by iterative curves. However, the detailed characteristics of optimization configuration based on CMDM, with obvious oscillation and fluctuations existing in curves, presented unclear and ambiguous results, and a complete 0-1 distribution cannot be formed, especially the optimal layout corresponding to . The reason is also the cumulative error in response and sensitivity analysis due to modal truncation, particularly for large-scale structures. Equally important, topology optimization based on CHEM also has clear advantages in computing efficiency, which can be reduced by 66.65%, 70.49%, and 69.73% CPU time when compared with CMDM.
5.2.2. Frequency- and Temperature-Dependent Properties
The optimal layouts and quantitative similarity indices induced by complex constant modulus and frequency-dependent modulus are presented in Table 8. The specific temperature points are chosen as and , and the excitation frequency remain the same as in the previous section, i.e., 200 Hz, 400 Hz, and 500 Hz.
The optimal layouts of CLD structures differ significantly, which can be attributed to the great changes in mechanical properties of viscoelastic materials at different temperatures. Although taking the average value to approximate the frequency-dependent properties of damping material at room temperature obtains a relatively properly optimized configuration, differences that cannot be ignored still exist objectively. For complex structures and conditions, the discrepancies in the optimized configuration may be magnified and result in unexpected influences. One also notes that at higher frequencies, such as 500 Hz, the similarity indices for different optimized configurations are significantly lower than those at lower frequencies. This is because higher frequency excites higher-order eigenmodes with more local characteristics, which are more sensitive to the stiffness and damping performance of viscoelastic materials, leading to a more differentiated optimal configuration. It is reasonable to infer that the similarity of the optimized configuration will further decrease with increasing temperature differences, even obtaining uncontrollable erroneous results.
5.2.3. Excitation of the Frequency Bands
This section further discusses the optimal configuration of damping materials under frequency-band excitation. The temperature is and , respectively, and the scheme of the excitation frequency band corresponds to the natural frequency of the structure, which is set to , , and , respectively. Table 9 shows the optimal configuration of the damping material excited by the frequency band at different temperatures. Firstly, the distribution of damping materials at different temperatures is quite different, and the similarity indices are only 0.79, 0.74, and 0.66, respectively. Secondly, the configuration of damping materials obtained by frequency band and discrete excitation is inconsistent at specific temperature points, which is also the result of the synergy of discrete frequency points in the frequency band.
5.2.4. Discussion of Layer Thicknesses and Volume Fractions
Layer thickness and volume fraction have been given a lot of attention as two main parameters affecting the configuration of damping materials. At different temperatures, without changing the mass of the CLD material, i.e., the total mass of the damping layer and the constrained layer is constant, the similarity indices between the optimal layout induced by averaged and frequency-dependent shear moduli for different thicknesses of the layer are discussed in detail.
Table 10 lists the six groups with different thicknesses of damping and constrained layers. Figure 11 presents the similarity variation trend of optimal configurations of averaged modulus as well as frequency- and temperature-dependent moduli obtained by changing the thickness of the constrained layer and damping layer. It can be observed that the optimal layouts obtained by the GHM model and averaged complex modulus at has a high similarity, while the similarities of other cases are at a low level, and the similarity gradually decreases with the increase in excitation frequency. In addition, it is concluded that the similarity of the optimal configuration at different temperatures is negatively correlated with the thickness of the damping layer at all three excitation frequencies. However, the above studies are qualitative analyses rather than quantitative description. Moreover, the conclusions taken from the results are restricted to the small set of thickness pairs herein considered. In fact, as Sher and Moreira [35] point out, the effect of the thickness of both layers, constraining and damping layers, is quite more complex and does not follow a monotonic behavior. Therefore, future work will be devoted to further exploring the effects of the thickness of each layer.
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Fixing the thickness of damping and constrained layers, we consider the influence of the different volume fractions, with = 0.3, 0.4, 0.5, 0.6, 0.7, and 0.8, respectively. Figure 12 illustrates the comparison results. Obviously, higher excitation frequency corresponds to lower similarities. Furthermore, the similarity indices at different frequencies show a roughly close increasing trend with the increase in volume fraction of the damping material. As a result, it is particularly necessary to consider the frequency and temperature dependence of damping materials for topology optimization of viscoelastic structures under the requirement of lower volume fractions.
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6. Conclusions
This paper investigates topology optimization of CLD structures comprising frequency- and temperature-dependent viscoelastic material under harmonic excitations. The structure is characterized by a nonproportional damping model. A novel approach combining hybrid expansion and dynamic reduction methods, which can deal with nonproportional damping, is proposed and developed to solve the topology optimization problem. The objective functions are chosen as the sum of the squared vibration amplitudes of the concerned points. The sensitivity analysis is implemented by using the adjoint variable method. The similarity index is suggested to quantitatively distinguish the different optimal layouts of damping materials. The conclusions can be drawn as follows:(1)The involved optimization design problem is essentially a highly nonconvex issue with multiple local optima. Two typical examples demonstrate that the proposed approach not only obtains clear and convergent optimal layouts of damping material but also significantly reduces computational time. The massive improvement in efficiency becomes more meaningful for composite structures with larger DOFs.(2)The frequency and temperature dependence of viscoelastic materials remarkably affect the optimized configuration of CLD structures. The reason is that the change in temperature and frequency will cause great fluctuations in the mechanical properties and energy dissipation of the viscoelastic material, which mainly decides the dynamic optimization results.(3)In the future research, taking into account the fluctuation of material properties caused by the frequency and temperature, it is quite promising to extend the proposed approach to other types of viscoelastic structures rather than just CLD structures and to carry out more refined topology optimization design when subjected to broadband random excitations or even wide-range temperatures.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
Fan Wu is contributed equally.
Acknowledgments
The authors gratefully acknowledge the financial supports from the National Natural Science Foundation of China under Grants U2241274, 12072288, and 12272319.