Dynamic Topology Optimization of Constrained Damping Plates considering Frequency and Temperature Characteristics Based on an Efficient Strategy

. Te frequency-and temperature-dependent characteristics of viscoelastic materials signifcantly afect the vibration response of the damped composite structures. In this paper, an efcient 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. Te similarity index is defned to distinguish the correlation among diferent design layouts. Two instances demonstrated the validity of the proposed approach. Te fndings indicated that a positive compromise between accuracy and efciency can be achieved, and the computational time can be signifcantly reduced while ensuring the accuracy of the results. Furthermore, it has been discovered that the excitation frequency and temperature signifcantly impact the optimal confguration of damping material. Te efects of layer thicknesses and volume fractions on optimization designs are also further investigated.


Introduction
Tin-walled constructions are frequently utilized as loadbearing 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 efective technique to suppress the level of vibration in structures [1][2][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 identifcation [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 efectiveness, aiming to achieve greater performance in decreasing vibration and abating noise under lightweight constraints.In the feld of dynamic characteristics, fundamental or multiple modal loss factors are frequently employed as objective functions.Zheng et al. [7] investigated the structural design confguration using the genetic algorithm for minimizing the overall vibration energy.Wang et al. [8] proposed an artifcial 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 efcient 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 specifc design confgurations are sensitive to the damping coefcients.Nevertheless, the frequency-and temperature-dependent properties of viscoelastic materials are still rarely considered in the optimization design of viscoelastic structures, and the corresponding difculties and challenges have not been well addressed.
It should be emphasized that the dynamic mechanical properties of viscoelastic materials are afected by the coupling of several factors, the most relevant of which are the temperature and frequency [6,18,19].Te 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 efect 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 efect 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.Tese 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 difculties 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 refected in the efcient optimization design approach which aims to reduce the dynamic response of viscoelastic structural materials comprising the frequency-and temperaturedependent damping materials.Terefore, 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.Tese fndings provide the potential to be a driving factor in the application and popularization of the nonproportional damped systems in the feld of topology optimization.
Te aim of this paper is to study the topology optimization design of viscoelastic composite structures at diferent temperatures and frequencies to minimize the steady-state response.For this purpose, the GHM model was incorporated into the fnite element method to describe the energy dissipation of materials.To handle the topology optimization difculties 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.Specifc attention focuses on the infuence of the fuctuation of viscoelastic material properties with the change of temperature and frequency on the vibration response and optimal confguration of composite structures.
Te 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 specifc procedure for calculating the displacement response of CLD plates under harmonic excitations by combining hybrid expansion and dynamic reduction methods.Te 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 efects of frequency and temperature on the optimal layouts.Besides, the infuence of the layer thicknesses and volume fractions is also further discussed.Conclusions are drawn in section 6.

Numerical Modeling Procedure
Te CLD plates, as a king of common multilayer structures, are extensively applied in many engineering felds.Tis section presents the related theories for the numerical modeling procedure of CLD plates incorporating the GHM model.

Basic Kinetics Relationships.
Figure 1 depicts a schematic illustration of the CLD plate.Te symbol h represents the thickness, while the subscripts p, c, and v, respectively, denote the base plate, constrained layer, and damping layer.Te 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 u v and v v and the shear strains β x and β y are given as 2 Shock and Vibration where u c and v c and u p and v p denote midplane displacements of constrained layer and base plate, respectively.w is the out-of-plane displacement, θ x � zw/zx and θ y � zw/zy denote the partial derivatives of w in x and y directions, and d � h v + (h c + h p )/2 is the midplanes distance.

Finite Element Models.
Te fnite element model is discreted into four-node planar plate element, and the nodal displacement vector is determined by q e � q e 1 q e 2 q e 3 q e 4   T , ( in which Terefore, the displacement at any position within the element can be represented by the node displacement as follows: where N uc , N vc , N up , N vp , N w , N θx , and N θy are the spatial interpolating vectors corresponding to u c , v c , u p , v p , w, θ x , and θ y , respectively.In addition, the spatial interpolating vectors corresponding to u v , v v , β x , and β y can be derived from (1) and (2), given as Ten, the elemental mass and stifness matrices can be obtained according to the energy approach and variational Hamilton principle as follows: where k e sv denotes the shear stifness, D i identifes the elastic matrix, and strain matrices are presented as Te elemental dynamic equation of composite structures is given as m e q .. e + k e q e + k e sv q e � f, where m e � m e p + m e c + m e v and k e � k e p + k e c + k e v .m e and k e represent the composite elemental mass and stifness matrixes, respectively.

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 G ∞ is the equilibrium value of modulus.Each term includes α k ,  ζ k , and  ω k , which are determined via G * v (ω) at temperature T( °C).Te number of mini-oscillator terms r could be adjusted to refect the corresponding damping performance of diferent viscoelastic materials.Te Laplace's transform of ( 9) is Substituting ( 10) into (11) and introducing auxiliary coordinates called dissipation coordinates are defned as follows: Equation ( 9) is converted into the time domain and rewritten as follows: where where Λ v is a diagonal matrix composed of the positive eigenvalues of the shear stifness matrix of viscoelastic materials and R T v is the matrix composed of the corresponding eigenvectors.
Each element can be considered to be composed of fve 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.
Te global matrices are assembled based on the physical coordinates while reserving auxiliary coordinates.Te equation of motion is shown as follows: where Te fnite 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, stifness, and damping matrix.Te dynamic equation obtained by this modeling method is a standard second-order linear system model, which performs well in response and sensitivity analysis.

Solutions of the Governing Equation
As previously stated, dissipative coordinates appear as the augmented state variables, signifcantly increasing the dimension of the governing equation.As a result of this, such a method sufers from the fnite 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.Terefore, model reduction is necessary to reduce the high-order fnite element models to a smaller size for undertaking more efcient dynamic analysis and optimization procedures.

Dynamic Reduction Method.
Te dynamic reduction method pioneered by Leung [30] and afterward Petersmann [31] as a type of hybrid coordinates reduction method is frequently and widely employed.Tis 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.Te displacement subvector of slave DOFs can be obtained as follows: 4 Shock and Vibration Rewrite ( 16) into matrix form as follows: where I 1 and I 2 are the unit matrix; t 1 � − (K ss q ) − 1 K sm q ; and t 2 � − (K ss q ) − 1 [K sm qz K ss qz ] denote the static contribution, and Φ represents the corresponding dynamic contribution.
Terefore, the reduced-space governing equation is showed as follows: where , respectively, denote the reduced mass, stifness, damping matrices, and load vector.Furthermore, please see [32] for the specifc theories and the derivation of corresponding reduction space.

Complex Mode Superposition Method for Higher-Order
Modes Correction.Te dynamic equation in the frequency domain of reduced order for the CLD structure with n R DOFs can be expressed as follows: where the matrix is the dynamic stifness matrix and ω refers to the exciting frequency.
Te complex frequency response function (FRF) matrix can be expressed using complex mode superposition theory, as shown as follows: where where ϕ r is the complex eigenvector that corresponds to the complex eigenvalue λ r .We suppose that the complex eigenvalues are distinct and the number is 2 N R .In practice, only limited low-order modes can be involved.
Equation ( 22) is the modal displacement method (MDM).Te modal truncation error may be generated because some higher-order modes are disregarded.When L ≪ 2N R , 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.
Te K d (ω) can be reformed and the inverse calculated as follows:

Shock and Vibration
where [I N + jωK − 1 (C + jωM)] − 1 can be expanded as follows: Combining ( 24) and ( 25), we get It should be highlighted that Γ k may be obtained specifcally by the iterative approach, and only needs to be calculated once at diferent frequencies due to the frequency independence.
Combining ( 23) and ( 26), we provide the more broader connections between the eigensolutions and system matrices for k � 1, 2, . . ., ∞, 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 h term in power-series is retained for meeting appropriate precision criteria, the estimated displacement vector is shown as follows:

Topology Optimization Model
4.1.Problem Statement.Te 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 specifed locations.Te steady displacement response is obtained by where Y R j and Y I j (j � 1, 2, . . ., N) are the real and imaginary parts of the complex amplitude Y j , respectively.
Te mathematical model is established as follows: where ρ e are the pseudodensity variables, N e denotes the total number of desirable elements, and m is the number of specifed DOFs.Te volume fraction is represented by the symbol f v .ρ min is the lower bound of the design variables to prevent the system matrices from becoming singular, in this case ρ min � 0.001.
In the framework of the polynomial interpolation scheme (PIS), the elemental mass matrices m where m e0 v and m e0 c and k e0 v , k e0 c , and k e0 sv are the mass matrices and stifness matrices when mass density is equal to 1.

Sensitivity Analysis.
To perform optimization, the globally convergent method of moving asymptotes (GCMMA) is adopted to obtain the optimized solution.Te AVM is preferred compared to the direct method for topology optimization problems involved in this study Te sensitivity of objective function in (30) is given by where zY j /zρ e can be derived as follows: where L is so-called the column vector with concerned DOF j 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 K d , one obtains where λ can be calculated by following equation: 6

Shock and Vibration
Almost no additional computation is required because of the same FRF whether the excitation load is F or L.
Once the λ is obtained, then the frst-order sensitivity of the displacement response can be calculated as follows: Te frst-order sensitivity of the augmented global matrix, namely, the mass, stifness and damping matrices in (37), can be obtained by the chain rule through interpolation relations in (31) at the element level.Ten, remaining derivatives of the corresponding sensitivity are removed.

Topology Optimization Process.
Te sensitivity fltering scheme based on the image [33] is applied to avoid checkerboard and mesh dependence.Te flter modifes the design sensitivity of a specifc element based on a weighted average of the element sensitivities in a fxed neighborhood, which is purely heuristic but produces results very similar to local gradient-constrained results, requiring little extra CPU time.Te GCMMA algorithm [34] is used to fnd the optimal solution.All cases are performed in the commercial software MATLAB R2018a.Te optimization process will stop until the absolute error of the design objectives satisfes ‖(f new − f old )/f old ‖ < 0.0005.Figure 3 presents the fow diagram of the optimization process.

Numerical Examples
Tis section presents two typical numerical examples with diferent boundary conditions.One is a cantilever plate clamped at the left side, as shown in Figure 4(a).An external force f(t) � Fe iωt is applied at the midpoint of the right side, with F � 10 5 N and ω � 2πf p .Te 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.Te modulus of 242F01 is characterized by the GHM model.Te parameters at 10 °C and 25 °C are obtained through the dynamic mechanical analysis (DMA) and least square method and are given in Table 2.For comparison, material 2 is an artifcial material, characterized by the complex constant model, whose properties are obtained by averaging the modulus of 242F01 in the frequency domain.Tis section frst investigates the validity of response and sensitivity analysis, then verifes 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 infuence of layer thicknesses and volume fractions on optimal layouts of damping material is further discussed.In order to quantitatively characterize the correlation between diferent optimal layouts, the similarity index is defned as follows: where S same is the overlapped area of damping layer between the diferent optimal layouts and S total 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 diferent optimized confgurations is the possible potential limitation.
Although some researchers have shown that solid elements have higher computational accuracy, they then have much lower computational efciency.Terefore, comparisons in the accuracy and efciency 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 40 × 40 threedimensional continuous solid elements.Te 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 signifcantly reduced by 74.8% compared with common commercial software, ANSYS.Te excellent performance of the proposed model in terms of precision and efciency is fully demonstrated.
In order to verify the efectiveness 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.Te 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 f p � 100Hz 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.Te 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 0.073m 2 to 0.011m 2 and converge to the optimal value.Te 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 confgurations 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 signifcantly improves optimization efciency and also provides potential and possibility for topology optimization of large-scale nonproportional damping structures.

A Cantilever
Plate.Tis section focuses on the structure shown in Figure 4(a) to investigate the infuence of the response-solving methods and frequency-and temperaturedependent viscoelastic materials on the optimal confguration and vibration suppression of the CLD plate.

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 diferent methods.Te results calculated by the direct frequency response method (DRFM) and fnite diference method (FDM) are regarded as the exact results of the displacement response and sensitivity, respectively.Te detailed error comparisons are   8 Shock and Vibration 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 (q 50, namely, i, ii) and CHEM (h � 2, 3, namely, iii, iv).As expected, the CHEM results agree well with the exact results, especially when h is equal to 3. Te errors of the vibration amplitudes A I and A II are only 1.01% and 0.78%, respectively, and the sensitivity diferences of the three specifed elements are less than 2%, about 0.92%, 1.02%, and 1.33%, respectively.However, there is a signifcant 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., q � 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-efective measure worth popularizing.
Table 4 lists the computing times of the diferent methods for responses and sensitivities analysis.It can be observed that CHEM, both for response and sensitivity analysis, shows remarkable advantages in computational efciency.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-of between computational accuracy and efciency.
For the cantilever plate structure shown in Figure 4(a), through modal analysis, Figure 8 presents the mode shapes, and the frst 6 natural frequencies are obtained as 58 Hz, 97 Hz, 142 Hz, 195 Hz, 242 Hz, and 304 Hz, respectively.As well known, damping efects are mostly observed around each natural frequency.Terefore, the excitation frequency      Shock and Vibration 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 diferent frequencies (i.e., f p � 100 Hz, 200 Hz, and 300 Hz) and also lists the optimization details.Te blue and green color in optimal material confgurations 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.Te reason is that higher excitation frequencies are prone to excite higher-order modes with more localized characteristics.Tis can also be confrmed 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 confgurations 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 confgurations despite the diference in calculation accuracy.However, there are also signifcant diferences in convergence and clarity.Tis simulation results show that CHEM works well while CMDM converges poorly when large-scale problems are considered, as evidenced by the fuctuation of the initial stage and the oscillation of the later stage in the iterative curves.It can be concluded that highprecision response and sensitivity analysis methods are more conducive to improving the convergence of topology optimization and obtaining the expected results.Te 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.Tis 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 diferent frequencies, respectively.Te performance on computational efciency 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   Shock and Vibration iteration, which greatly reduces the surge of computational time caused by complex modal analysis and multiple iterations.

Frequency-and Temperature-Dependent Properties.
Te frequency-dependent properties of viscoelastic material at 10 °C and 25 °C are considered.To design an artifcial modulus at room temperature as the comparison group, the averaged real and imaginary part of modulus at 25 °C are obtained, given by G v � 8.4(1 + 1.72j)MPa.Figure 9 presents the frequency-dependent and averaged shear moduli of 242F01 for 10 °C and 25 °C.Obviously, the stifness and damping performance of 242F01 exhibit signifcant changes with frequencies and temperatures varying.
Te optimal layouts and iteration curves induced by the complex constant model and GHM model are presented in Table 6.Te 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 signifcant differences among the design layouts of damping materials.Te reason comes from the fact that the frequency and temperature dependence of viscoelastic materials can infuence the stifness 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 � 25 °C.However,  due to the discrepancy between the averaged and actual moduli at diferent frequencies, the similarity of the SI-II group at f p � 300Hz was 0.702, which was signifcantly lower than that of 0.862 at f p � 200Hz.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).Tis 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 f p � 300Hz are significantly lower than that of f p � 200Hz.Tis attributes to signifcant changes in the optimized confguration at higher frequencies due to the involvement of more local modes, even if temperature makes a limited diference.

A Plate Clamped by Four Sides.
To further validate the accuracy and efciency of the proposed CHEM-based approach and to compare the efect 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.Te optimization objectives are to minimize the sum of squares of vibration amplitudes at specifc points of concern as indicated by the red dots in Figure 5.

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 frst 6 natural frequencies are obtained as 128 Hz, 197 Hz, 289 Hz, 392 Hz, 428 Hz, and 511 Hz, respectively.Terefore, the selection strategy of frequency points is based on the natural frequencies, where the values are 200 Hz, 400 Hz, and 500 Hz, respectively.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 diferences 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.Te similarity indices between the optimal layouts obtained are relatively high, with values of 0.92, 0.83, and 0.81, respectively.Te 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 confguration based on CMDM, with obvious oscillation and fuctuations existing in curves, presented unclear and ambiguous results, and a complete 0-1 distribution cannot be formed, especially the optimal layout corresponding to f p � 400Hz.Te 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 efciency, which can be reduced by 66.65%, 70.49%, and 69.73% CPU time when compared with CMDM.

Frequency-and Temperature-Dependent Properties.
Te optimal layouts and quantitative similarity indices induced by complex constant modulus and frequencydependent modulus are presented in Table 8.Te specifc temperature points are chosen as 10 °C and 25 °C, and the     Te optimal layouts of CLD structures difer signifcantly, which can be attributed to the great changes in mechanical properties of viscoelastic materials at diferent temperatures.Although taking the average value to approximate the frequency-dependent properties of damping material at room temperature obtains a relatively properly optimized confguration, diferences that cannot be ignored still exist objectively.For complex structures and conditions, the discrepancies in the optimized confguration may be magnifed and result in unexpected infuences.One also notes that at higher frequencies, such as 500 Hz, the similarity indices for diferent optimized confgurations are signifcantly lower than those at lower frequencies.Tis is because higher frequency excites higher-order eigenmodes with more local characteristics, which are more sensitive to the stifness and damping performance of viscoelastic materials, leading to a more diferentiated optimal confguration.It is reasonable to infer that the similarity of the optimized confguration will further decrease with increasing temperature diferences, even obtaining uncontrollable erroneous results.9 shows the optimal confguration of the damping material excited by the frequency band at diferent temperatures.Firstly, the distribution of damping materials at diferent temperatures is quite different, and the similarity indices are only 0.79, 0.74, and 0.66, respectively.Secondly, the confguration of damping materials obtained by frequency band and discrete excitation is inconsistent at specifc temperature points, which is also the result of the synergy of discrete frequency points in the frequency band.Layer thickness and volume fraction have been given a lot of attention as two main parameters afecting the confguration of damping materials.At diferent 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 diferent thicknesses of damping and constrained layers.Figure 11 presents the similarity variation trend of optimal confgurations of averaged modulus as well as frequency-and temperaturedependent 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 25 °C 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 confguration at diferent 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 efect of the thickness of both layers, constraining and damping layers, is quite more complex and does not follow a monotonic behavior.Terefore, future work will be devoted to further exploring the efects of the thickness of each layer.

Excitation of the Frequency
Fixing the thickness of damping and constrained layers, we consider the infuence of the diferent volume fractions, with f v � 0.3, 0.4, 0.5, 0.6, 0.7, and 0.8, respectively.16 Shock and Vibration illustrates the comparison results.Obviously, higher excitation frequency corresponds to lower similarities.Furthermore, the similarity indices at diferent 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.

Conclusions
Tis paper investigates topology optimization of CLD structures comprising frequency-and temperaturedependent viscoelastic material under harmonic excitations.Te 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.Te objective functions are chosen as the sum of the squared vibration amplitudes of the concerned points.Te sensitivity analysis is implemented by using the adjoint variable method.Te similarity index is suggested to quantitatively distinguish the diferent optimal layouts of damping materials.Te conclusions can be drawn as follows: (1) Te 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 signifcantly reduces computational time.Te massive improvement in efciency becomes more meaningful for composite structures with larger DOFs.(2) Te frequency and temperature dependence of viscoelastic materials remarkably afect the optimized confguration of CLD structures.Te reason is that the change in temperature and frequency will cause great fuctuations 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 fuctuation 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 refned topology optimization design when subjected to broadband random excitations or even wide-range temperatures.

Figure 1 :
Figure 1: Schematic diagram of the CLD plate.
respectively, denote the global mass, stifness, and damping matrices.x and F ∈ R N G ×1 represent the displacement vector and external load vector.Here, N G indicates the full DOFs consisting of physical and dissipative coordinates.

Figure 2 :
Figure 2: Schematic diagram of element nodes and assembly process.

Figure 5 :
Figure 5: Iterative process of full-order and reduced-order models.

Figure 6 :
Figure 6: Finite element mesh and element numbers.

Figure 7 :
Figure 7: Comparison of response and sensitivity analysis for diferent methods.(a) Displacement amplitudes.(b) Displacement sensitivities.(c) Analysis errors.
Bands.Tis section further discusses the optimal confguration of damping materials under frequency-band excitation.Te temperature is 10 °C and 25 °C, respectively, and the scheme of the excitation frequency band corresponds to the natural frequency of the structure, which is set to [0 − 200]Hz, [0 − 400]Hz, and [0 − 500]Hz, respectively.Table

Table 1 :
Materials and geometric parameters of the CLD plate.

Table 2 :
GHM parameters at diferent temperatures of the 242F01.

Table 3 :
Performance comparison of topology optimization.

Table 4 :
Comparison of computing time for responses and sensitivities.

Table 5 :
Te topology optimization results of the cantilever plate based on diferent solution methods.

Table 6 :
Te topology optimization results of the cantilever plate based on diferent damping models.

Table 7 :
Te topology optimization results of the plate clamped by four sides based on diferent solution methods.

Table 8 :
Te topology optimization results of the plate clamped by four sides based on diferent damping models.

Table 10 :
Diferent thicknesses of damping and constrained layers.