
==== Front
iScience
iScience
iScience
2589-0042
Elsevier

S2589-0042(24)01873-X
10.1016/j.isci.2024.110648
110648
Article
Vibro-acoustic topology optimization for improving the acoustic insulation and mechanical stiffness performance of periodic sandwich structure
Luo Kui 1
Hu Jie jiehu@gzu.edu.cn
125∗
Yao Song 3
Gan Ning 34
Cao Chenfei 1
Xu Jiao 1
Cao Wenkang wkcao@gzu.edu.cn
1∗∗
1 School of Mechanical Engineering, Guizhou University, Guiyang 550025, China
2 Key Laboratory of Modern Manufacturing Technology, Ministry of Education, Guizhou University, Guiyang 550025, China
3 Key Laboratory of Traffic Safety on the Track of Ministry of Education, School of Traffic & Transportation Engineering, Central South University, Changsha 410075, China
4 Frontiers Science Center for Extreme Mechanics and Engineering, Central South University, Changsha 410075, China
∗ Corresponding author jiehu@gzu.edu.cn
∗∗ Corresponding author wkcao@gzu.edu.cn
5 Lead contact

03 8 2024
20 9 2024
03 8 2024
27 9 11064819 3 2024
20 6 2024
30 7 2024
© 2024 The Author(s)
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
Summary

The traditional parameter adjustment design makes it difficult to effectively regulate the acoustic insulation performance of periodic sandwich structures while meeting the lightweight and mechanical stiffness requirements. A dynamic three-field floating projection topology optimization (FPTO) method for periodic structures is proposed to meet the optimization requirements of low-noise and high-stiffness performance of lightweight periodic sandwich structures. The sound transmission loss is taken as the optimization objective, and the lightweight volume and mechanical stiffness performance are taken as the multiple constraints. The results show that a smooth topology configuration with superior sound insulation performance, high stiffness, and a freely customizable number of periodic cores can be obtained via the proposed method. The accuracy and effectiveness of the presented method are verified via 3D printing technology and impedance tube sound insulation experiments, providing an important reference for the optimal design of lightweight composite structures for vibration and noise reduction in transportation equipment.

Graphical abstract

Highlights

• A novel topological design is carried out for lightweight periodic sandwich structures

• New smooth topology results with superior vibro-acoustic performance are achieved

• 3D printing technology and impedance tube sound insulation experiments are conducted

Mechanical engineering; Mechanics; Physics

Subject areas

Mechanical engineering
Mechanics
Physics
Published: August 3, 2024
==== Body
pmcIntroduction

Lightweight sandwich panels with typical periodic forms and acoustic properties are widely used in the floors and other main load-bearing structures of transportation equipment, such as high-speed train floors,1,2 new energy vehicles,3 and airplanes,4 due to their light weight,5 superior acoustic performance,6,7 high specific stiffness/strength,8 and strong energy absorption capacity.9,10 Existing techniques generally use one or a mixture of analytical, numerical, and experimental methods to perform noise and stiffness control studies of periodic sandwich structures.11 Md et al.12 combined analytical and experimental methods to solve the long-term problem of noise radiation in a spacecraft. The authors used different material stacking combinations and considered the optimized sandwich structure of the composition material, layer number, thickness, and sequence. Moreover, the authors also performed acoustic optimization design of spacecraft sandwich material through parameter adjustment. Zhang et al.13 developed a high-speed train floor sound transmission finite element (FE) model. This was combined with experimental methods involving on-site measurements to analyze the high-speed train floor sound transmission characteristics. Through multi-parameter adjustment and optimization, the overall radiated sound intensity of the floor structure was reduced. Wennberg et al.14 proposed a multidimensional, multi-functional parameter optimization method, which could effectively reduce the weight of the car body. The optimization objective was to achieve a lightweight composite car body structure considering multiple constraints such as mechanical strength, stiffness, geometry, and acoustic/thermal attenuation. Most of the above methods use a combination of size/shape optimization and experience to adjust the structural or material parameters to improve the comprehensive performance of the periodic sandwich structure in terms of acoustic insulation and load-bearing stiffness. Nevertheless, those methods have shortcomings, such as high dependence on the initial configuration and difficulty in realizing the lightweight while regulating the acoustic performance of periodic sandwich structure at a limited cost.15,16 Therefore, there is an urgent need to seek innovative optimization methods that offer high design flexibility and the potential to improve performance considerably.

Structural topology optimization, as an art of "digging holes" in the structure,17 can override the size and shape optimization design limits, break the topological connection relationship of the original structure, and achieve optimal performance under the given multi-constraint conditions by changing the structural topology configuration.18 As a highly advantageous tool in the product’s positive conceptual design phase, topology optimization has become one of the most active research hotspots in structural and multi-disciplinary optimization in the past three decades. The mainstream topology optimization methods can be divided into two main categories based on the different ways of defining the design variables. The first concerns the boundary-based methods that take the structural boundaries as the design variable, including the level-set method (LSM),19 moving morphable components (MMC) method,20 and feature-driven optimization (FDO) method.21 The second concerns the element-based methods, which take the elements as the design variables, including the solid isotropic material penalization (SIMP) method,22 the evolutionary structural optimization (ESO) method/bi-directional evolutionary structural optimization (BESO) method, i.e., ESO/BESO,23 and the topology optimization of binary structures (TOBSs) method.24 In recent years, Huang25,26 proposed an element-based three-field floating projection topology optimization (FPTO) method based on a linear material interpolation model. Unlike conventional topology optimization methods, which use material penalty functions to obtain clear topology configurations, this method breaks through the constraints posed by material penalties by adopting a non-punitive form based on implicit floating projection functions. In particular, the method adopts a 0/1 topology design that gradually drives the design variables to be clear, solves the "variable-thickness-sheet" problem25 caused by matching the element-based topology optimization method with the linear material interpolation model, and has outstanding application potential in the optimization of multi-physics coupled design.27 In this research, the three-field FPTO method based on multi-field coupling forward conceptual optimization design is extended to perform the dynamic vibro-acoustic topology optimization design of transportation equipment periodic core structures.

One of the main challenges in the topology optimization of dynamic acoustic-structural coupling structures is the dependency problem of multi-physic field interfaces. More specifically, due to the continuous change of the acoustic and structural field coupling boundaries during the topology optimization iterative process, the conventional segregated finite element (FE) method is unable to solve the problem; thus, there is a need for an explicit description of the multi-physics field coupling boundaries due to the inconsistency between the equations of different materials and the different physical fields. The second is when the mass-to-stiffness ratio is too high, leading to localized modal problems and the need to solve singularities in low-density regions. The mainstream classes of topology optimization methods mentioned above have been gradually utilized to address the above two problems and develop corresponding treatments based on the core characteristics of their respective algorithms. One class concerns boundary-based topology optimization methods, e.g., the level set method. For instance, Shu et al.28,29 applied the level set method to two-dimensional structures. This method takes acoustic pressure minimization as the optimization objective and lightweight volume as the constraint for the optimal design of acoustic-solid coupling structures. The mesh redistribution technique and zero-dimensional level set function are combined to overcome the interfacial dependence problem of vibro-acoustic structures due to the inconsistency between the equations of different materials and physical fields, i.e., the need for an explicit description of the coupling boundary. However, the level set method depends more on the initial topology configuration due to local design variables based on structural boundaries.19,30,31 The other class is the element-based topology optimization method, which can be divided into two types based on continuous or discrete element design variables. One is the element-based topology optimization method with continuous design variables. Yoon et al.32 and Cetin et al.33 used the SIMP method to obtain optimal two-dimensional structures. They combined a hybrid FE format and a two-phase material interpolation model to seamlessly transition the coupling boundary between the acoustic and solid fields in the two-dimensional structure. Nevertheless, an inappropriate material interpolation model can make the low-density region prone to localized modes, preventing the iterative oscillations from converging.32 The second is the element-based topology optimization method based on discrete design variables. For example, Kook34 and Picelli24 used the BESO and TOBS methods, respectively, to optimize the acoustic and eigenfrequency performance of a simple planar structure. Their approaches take advantage of the 0/1 discrete expression of the design variables of the element density in conjunction with the FE format separation, enabling the explicit description of the acoustic-solid field coupling boundary. At the same time, the local mode problem occurring at the weak element is solved by utilizing an improved material interpolation model. Nonetheless, the separated FE format ignores the effect of the acoustic field, resulting in an inaccurate estimation of the structural field sensitivity. Therefore, further adjustments are required to solve the strong coupling problem.33

Another challenge involves matching the topology optimization of acoustic-structural coupling systems to achieve a periodic distribution of topological configuration. Periodic structures are widely used in the structural design of various acoustic vibration scenarios due to their lightweight, superior sound absorption/insulation and vibration isolation performance, and many identical or similar periodic configurations, which can significantly reduce the manufacturing cost and simplify the assembly process. For example, the periodic lightweight sandwich panels, widely used in the floors and other main load-bearing structures of transportation equipment, comprise several core structures with a similar periodicity. Several experts and scholars have attempted to optimize the periodic structure topology configuration from both the micro and macro scales. In terms of the micro-scale, Sigmund et al.35 used an inverse homogenization approach to design periodically distributed microstructural topologies to obtain desirable equivalent macroscopic material properties, followed by integrating macro-micro-design methods.36 Nevertheless, such a solution does not apply to the investigated dynamic vibro-acoustic coupling problem, partially because the dynamic homogenization method is very time-consuming and immature, and further development is required to improve the corresponding algorithmic framework.37 On the other hand, optimizing the macroscopic material properties through designing periodic microstructures involves homogenizing the material properties over the entire design domain. This requires the assumption that the size of the selected single cell is much smaller than the sampling point of the material. However, this does not apply to arbitrary loading boundary conditions at the macroscopic scale, since different boundary conditions significantly affect the structural properties and thus exhibit inhomogeneous characteristics, which cannot be determined by the inverse homogenization method. In terms of the macro-scale, Huang and Chen et al.38,39 used macroscopic periodic structure algorithms to optimize the structural topology with stiffness and frequency as the optimization objectives. However, there are only a few research studies related to the topology optimization of periodic structures for the structural vibro-acoustic coupling effect using this optimization method, since multiple constraints regarding the lightweight and mechanical stiffness need to be satisfied, while effectively regulating the acoustic performance. This point is significant for optimizing periodic structures with vibro-acoustic coupling, which is widely used in practical engineering.

To this end, the periodic lightweight sandwich panels found in transportation equipment are taken as the research object, a three-field FPTO method for periodic structures is proposed, and finally, the noise reduction performance of sound insulation, i.e., the maximization of the mean sound transmission loss (mean-STL), is taken as the optimization objective. Moreover, the lightweight volume and mechanical stiffness are considered multiple constraints for the optimal design of the vibro-acoustic coupling of lightweight periodic structures. The implicit floating projection constraint function and linear material interpolation model are utilized to form a new 0/1 topology shaping mechanism, which is combined with the mixed displacement/pressure (u/p) FE method to perform multi-physics field coupling analysis of the interface dependence and solve the common local mode problems in dynamic topology optimization. To obtain a periodic smooth topology configuration with superior sound insulation performance, higher stiffness, and free customization of the number of cycles, a simple, effective, and easy-to-manufacture macro-scale periodic structure optimization method is also established by averaging the positional sensitivity of each unit cell. The proposed method can satisfy the urgent need for the optimal design of transportation equipment sandwich panels with lightweight, low-noise, and load-bearing stiffness properties.

Results

Simulation and experimental verification

To verify the validity of the mixed u/p FE formulation, an FE model is established (Figure 1B) based on the impedance tube sound insulation test presented in Figure 1A. A background pressure of 1 Pa is applied to the incident acoustic field, and a perfect matching layer is designed at the top of the incident and transmission acoustic fields, to ensure that no reflected sound wave will interfere with the sound pressure distribution of the acoustic field. The specimen was manufactured with white resin (Resin-PLA) via 3D printing (right side of Figure 1B). The specific material parameters are listed in Table 1.Figure 1 Simulation and experiment

(A) Experiment equipment; (B) Solid square cake FE simulation model; (C) Comparison of sound insulation results.

Table 1 Material properties of sound insulation test-piece and acoustic field

Material properties	Elastic modulus E/MPa	Density
ρ/kg·m−3	Poisson’s ratio	Shear modulus
G/MPa	Speed of sound/m·s−1	Bulk modulus
K/MPa	
White resin	3200	1030	0.33	120.3	–	313.7	
Air	–	1.293	–	0	343(1 + 0.01i)	1.013 × 10−3	

Figure 1C shows the simulation and repetitive experimental curves of the sound insulation frequency response of a solid circular plate obtained by the mixed u/p and conventional separated FE methods for the vibro-acoustic coupling problem (more detailed information is provided in Table S1; Figure S1). The contours of the sound pressure level distribution in the acoustic field at the lowest sound insulation frequency of 860 Hz can be observed. Moreover, it can be seen that there is only a small gap between the sound insulation curves obtained by the two FE simulation methods. This is because the order of element interpolation is different, and the mixed FE model uses second and first-order elements to interpolate displacement and pressure, respectively. In the separated FE model, the displacement and pressure are interpolated by second-order elements. However, this difference can be eliminated via mesh refinement.32 It can also be observed that the trend of the sound insulation curve obtained by simulation is consistent with the experimental one (including three repeated tests), indicating the effectiveness and accuracy of the proposed FE simulation method. The error between the experimental and simulation curves and the fluctuation of the experimental curve may have been caused by using PVC tape at the contact surface between the sound insulation specimen and the impedance tube. Thus, the error can be attributed to factors including the ideal boundary condition set in the simulation model, the effect of environmental noise, and the 3D printing material heterogeneity.

Original modeling of the periodic sandwich structure

Periodic sandwich structure is widely used in transportation due to its light weight, high specific stiffness/strength, and strong energy absorption capacity. As a typical periodic sandwich structure, the high-speed train (HST) floor base is selected, and the original model for the body-in-white floor of an HST is established in this article. The body-in-white structure and its cross-section are illustrated in Figure 2A, and the cross-section of the body part is presented in Figure 2B. It can be observed that a combination of shell-base plate structures welds the body-in-white. From the longitudinal perspective of the HST body-in-white, the entire body structure is an isotropic cross-sectional configuration. Therefore, a representative small section of the floor center base plate structure is taken first as the research object. The FPTO method is adopted to optimize the sound insulation performance under the multi-constraint conditions of lightweight volume and stiffness. The floor base plate structure is simplified to facilitate the corresponding topology optimization design. In particular, the suspension equipment interface is simplified, retaining the rectangular welding groove parts and the welding groove weld joints. The simplified cross-section of the floor base plate structure and its structural parameters are exhibited in Figure 2C.Figure 2 Periodic sandwich structures used for HST floors

(A) Simplified schematic diagram of the HST; (B) Partial cross-section of the body panels; (C) Simplified conventional floor base plate and its parameters.

Optimized design of vibro-acoustic coupling systems

Several steps must be followed to perform the topology optimization of the core structure of the floor base plate. First, the initial vibro-acoustic coupling topology optimization model presented in Figure 3, which contains the acoustic and structural domains, is established. The structural domain contains the design and non-design domain parts. The non-design domain comprises the upper and lower skins and the welded groove parts of the floor structure, and the design domain is part of the core layer. The top and bottom parts of the sandwich structure constitute the acoustic field. At the bottom part, an incident background sound pressure field with a magnitude of 1 Pa is set. The material properties of the structural and acoustic fields are listed in Table 2. The minimum mesh size is typically set to 1/5 to 1/10 of the wavelength. The trade-off between the minimum floor thickness, computational scale, and the completeness of the final topological configuration is set to a minimum mesh size of 1 mm. In addition, the top and bottom boundaries of the entire acoustic field are made of perfectly matched layers to absorb all incident sound waves and simulate open or infinite acoustic boundaries. In this way, the effect of no reflected sound waves interfering with the sound pressure distribution in the upper half of the acoustic field can be achieved. When viewed along the longitudinal direction of the HST body (Figure 2A), the entire body-in-white is an isotropic tensile structure. Periodic constraints are imposed along the longitudinal direction of the HST to realize isotropic design.Figure 3 Schematic diagram of the floor optimization model and the boundary conditions

Table 2 Material properties of structural and acoustic domains

Material properties	Elastic modulus
E/GPa	Density
ρ/kg·m−3	Poisson’s ratio	Shear modulus G/GPa	Sound velocity
/m·s−1	Bulk modulus K/GPa	
Structural field (aluminum)	70	2700	0.34	26.12	–	53.03	
Acoustic field (air)	–	1.293	–	0	343(1 + 0.01i)	1.013 × 10-3	

The corresponding optimization target excitation frequency, compliance constraint value, and target volume fraction must be determined before the effective optimization design of the vibro-acoustic coupling systems. The original corrugated configuration in Figure 2C is first subjected to vibro-acoustic coupling FE analysis to select the optimization target excitation frequency and compliance constraint value. Moreover, the boundary conditions shown in Figure 3 are adopted. The frequency response curves of sound insulation, i.e., the transmission loss in the frequency range of 20 Hz–700 Hz, are obtained in the final calculation (Figure 4). It can be observed that the minimum sound insulation value occurs at 320 Hz, which is 12.15 dB. Consequently, the optimization target excitation frequency is selected to cover the frequency corresponding to the lowest value. At the same time, multiple specific frequencies are selected as the optimization target calculation frequency band. More specifically, from 290 Hz to 350 Hz, a calculation frequency is taken every 10 Hz. The objective function is calculated as the mean of STL values corresponding to all calculation frequency points. At the same time, the structural dynamic compliance, i.e., the inverse of the load-bearing stiffness, is constrained not to exceed 90% of the original value. Subsequently, considering the selection of the target volume fraction for the optimized design, the original corrugated configuration in Figure 2C is calculated, which accounts for 14.36% of the structural cross-sectional area in the entire design domain. Thus, the target volume fraction for the original lightweight design is 14.00% (noted as Scheme 1).Figure 4 Frequency response curve of STL for the original corrugated configuration

Figures 5A and 5B show the axonometric view of the final smooth design and its X-Y plane front view obtained after optimization via the proposed vibro-acoustic coupling FPTO method. It can be observed that the resulting smooth topological configuration is composed of slender rods, which is similar to the original corrugated floor structure. The difference is that the thickness distribution of the rods optimized by the algorithm is non-uniform, and their positions and topological configurations are changed to obtain the optimal target performance. In addition, considering the manufacturability of actual engineering, the minimum plate thickness must not be less than 1.5 mm. In comparison, the finest point of the resulting configuration is 2.2 mm, which satisfies the manufacturability requirement. It should be noted that the finest point of the configuration is more than two layers of elements. In contrast, the minimum element size is 1 mm. On the other hand, the resulting 3D optimized configuration is an isotropic tensile structure, which verifies that the chosen longitudinal periodic constraints are effective and in line with the structural characteristics of actual HST floor base panels.Figure 5 Smooth design

(A) Axonometric view; (B) X-Y plane front view.

Figure 6 shows the iteration history of the floor base panel structure for each optimization objective function, i.e., STL, dynamic compliance constraint, and structural topology of smooth design versus β (more detailed information is provided in Table S2). The inset in Figure 6 presents the smooth design for element density values larger than 0.5. It can be observed that the maximum STL occurs approximately at iteration 61. However, this design does not satisfy the smooth design criterion defined by Equation 28 and is undesirable. Thus, stricter 0/1 constraints using a larger β should be adopted. After 127 iterations, the iterative convergence of the optimization process stops when β = 8. The original mean value of the optimization objective increases from J = 52.68 dB to J = 53.71 dB, which is an improvement of 1.03 dB, indicating that the proposed method improves the STL of the base plate and achieves a better optimization result. The volume constraint is always satisfied throughout the iteration process; the structural dynamic compliance constraint first slowly decreases from the original value of 1 to the constraint value of 0.9, and finally remains unchanged until the end of the iteration process, reflecting the effectiveness of the constraints of the proposed method. The optimized objective function of the mean-STL of the resulting smooth design solution is 53.48 dB (corresponding to the blue mark at the top of Figure 6), which is only 0.43% lower than that of the element-based design. The small difference can be attributed to the adoption of the linear material interpolation model (Equation 12), which can accurately describe the properties of the intermediate density elements at the coupling boundary, as well as to the convergence criterion (Equation 28), which reflects the accuracy and stability of the proposed method. Moreover, compared to the traditional element-based topology optimization (not conducive to forming virtual element boundaries), the smooth design obtained using the proposed method has clear and smooth boundaries. It can be directly exported as stereolithography files (.stl) for secondary design or 3D printing and fabrication, reflecting improved engineering practicability.Figure 6 Iteration history of the objective function J, dynamic compliance constraint values, structural topology of the smooth design, and β during the optimization process (blue “+” marks denote the objective function of the smooth design)

Figure 7 compares the sound pressure level contours of the original corrugated configuration and the smooth design obtained after optimization at the excitation frequency of 320 Hz. As can be observed, the incident acoustic wave in the background field of the original configuration (Figure 7A) is somewhat attenuated by the attenuation of the original configuration; however, its value is still higher and further optimization is required. Figure 7B exhibits the sound pressure level contours corresponding to the smooth design obtained after optimization. It can be seen that the acoustic domains in the upper half are lighter in color, indicating that the STL corresponding to the obtained smooth design solution is better than that of the original configuration, visually reflecting the effectiveness of the proposed optimization method.Figure 7 Sound pressure level (dB) distribution (f = 320 Hz)

(A) Corrugated configuration; (B) Smooth design.

Figure 8 shows the frequency response curves of the STL and dynamic compliance for the original corrugated floor structure with a smooth design in the frequency range of 20–700 Hz. It can be observed that, in the optimization target calculation frequency band, i.e., from 290 Hz to 350 Hz, the topological configuration obtained using the proposed method, which uses fewer materials (14.00%), effectively improves the acoustic insulation at around 320 Hz compared to the original corrugated floor configuration (14.36%), which relies on the experience of the designer and exhibits a sound insulation trough phenomenon. The corresponding STL value at 320 Hz is improved from 12.15 dB to 54.03 dB. The sound insulation of the entire floor base panel is effectively improved over a wider frequency band other than the optimized target band, demonstrating the proposed method’s superiority. Moreover, according to Figure 8, the dynamic compliance of the structure in the optimized frequency range is significantly reduced from 4.16e−6 Nm to 5.41e−10 Nm at 320 Hz. In other words, the structural load-bearing stiffness has been improved, verifying the necessity of the topology optimization design and the effectiveness of the proposed topology optimization method.Figure 8 Frequency response comparison of STL (dB) and dynamic compliance between the original corrugated structure and Scheme 1

Comparison

Subsequently, the effect of the optimized target volume fraction on the optimization results was further investigated by keeping the original structural and material parameters, boundary conditions, dynamic compliance constraints, and optimized filter radius unchanged, and decreasing only the optimized target volume fraction. This corresponds to a reduction in the weight of the structure in order to achieve the lightweight requirement. Therefore, the optimized target volume fraction is set to 10.00% (Scheme 2), which implies a weight reduction of 30.36% compared to the original corrugated configuration (14.36%). After optimization, the axonometric and X-Y plane front views of the smooth design are obtained (Figures 9A and 9B). Compared to the optimization result of Scheme 1 (14.00%; Figure 5), the two topological configurations are similar. The differences are that the rod structure is locally finer, the layout is compact, the distribution of rods is aligned to the axis of symmetry, and the thinnest part of the smooth design is 2.1 mm, meeting the manufacturability requirements.Figure 9 Smooth design

(A) Axonometric view; (B) X-Y plane front view.

Considering the periodic distribution of the corrugated reinforcement structures in the original configuration and the manufacturability requirements, periodic constraints are added to the above topology optimization scheme for comparative design to assess the effect of periodic constraints on the optimization results (Figure 10). Based on the number of corrugated cycles in the original configuration, the periodic number of the unit cell is X × Y = 5 × 1. The volume fraction is 14.00% (Scheme 3) and 10.00% (Scheme 4). At the time, the rest of the conditions are kept unchanged. The smooth design obtained after optimization is presented in Figures 10A and 10B. It can be observed that the periodic constraints are used to generate a regular periodic structure, which is more similar to that in the original configuration, with the only difference being that the resulting configuration is symmetrical along the Y-direction. Processing and manufacturing are more convenient than Scheme 1 due to its periodic symmetric topology, and it also provides a variety of multi-solution references for engineering design with better performance.Figure 10 Smooth designs

(A) Scheme 3: Axonometric view; (B) Scheme 3: X-Y plane front view; (C) Scheme 4: Axonometric view; (D) Scheme 4: X-Y plane front view.

STL’s corresponding frequency response curves are plotted and compared in Figure 11 to compare the advantages and disadvantages of the above four optimization schemes. It can be observed that the optimization objective function, i.e., the STL, of the four optimization schemes in the optimized frequency range is better than that of the original corrugated configuration. Among them, the corresponding STL values at 320 Hz in descending order are 54.03 dB (Scheme 1), 52.39 dB (Scheme 2), 50.91 dB (Scheme 3), and 47.82 dB (Scheme 4). In particular, Scheme 1 has the best sound insulation performance, Scheme 4 has the simplest structure and high manufacturability, and Schemes 2 and 3 possess both advantages. Furthermore, according to the above comparison of values and topological configurations, the value of STL of the resulting optimized configuration increases with increasing volume fraction. Although imposing periodic constraints decreases the STL performance of the acoustic vibration structure, an optimized result of more homogeneous distribution and easier fabrication is obtained. In addition, by comparing the sound pressure level distribution contours of Schemes 2, 3, and 4 in Figure 12, it can be seen that Scheme 2 has the lightest color in the upper half of the acoustic domain, indicating the best acoustic insulation performance. Regarding Schemes 3 and 4, which correspond to the imposition of the periodic constraints, the optimized performance of the acoustic insulation is slightly worse. However, a more homogeneous distribution of the structural material and the acoustic field is obtained.Figure 11 Frequency response comparison of STL (dB) between the original corrugated configuration and four optimization schemes

Figure 12 Sound pressure level (dB) distributions of different schemes, f = 320 Hz

(A) Scheme 2; (B) Scheme 3; (C) Scheme 4.

Finally, this research investigated the effect of the number of periods on the STL performance of the core structures of the floor base panels. Based on the original periodic number of X × Y = 5 × 1, three different periodic numbers of X × Y = 2 × 1, 4 × 1, and 6 × 1 are included in the optimization design, which is denoted as Schemes 5, 6, and 7, respectively. The optimization target volume fraction is set not to exceed 10.00% of the design domain, while the other conditions are kept unchanged. The final obtained smooth designs corresponding to different periodic numbers are exhibited in Figure 13. It can be observed that a regular periodic structure similar to the corrugated base plate structure in the original configuration is produced after applying the periodic constraints. Moreover, the generated topological configuration is more uniformly distributed as the number of periods increases. This also indicates that different periodic configurations can be obtained by applying various periodic constraints, providing a multi-solution reference with superior performance and diverse forms for engineering design.Figure 13 Smooth designs under different unit cell numbers with a volume fraction of 10.00%

(A) Schemes 5: X×Y = 2 × 1 Axonometric view; (B) X-Y plane front view; (C) Schemes 6: X×Y = 4 × 1, Axonometric view; (D) X-Y plane front view; (E) Schemes 7: X×Y = 6 × 1, Axonometric view; (F) X-Y plane front view.

Figure 14 compares the frequency response curves of the STL values for the original corrugated configuration and those corresponding to different numbers of periods in the frequency range of 20 Hz–700 Hz. It can be observed that, in the target optimization frequency range of 290 Hz–350 Hz, the objective function STL values of the obtained optimization schemes with different numbers of periods are better than those of the original corrugated design configuration. At the same time, although a portion of the acoustic insulation effect is slightly sacrificed with the increase of the number of periods, an optimized scheme with more uniform distribution and easier fabrication capability of the topological configurations is obtained under the premise of better performance than that of the original corrugated configuration, which is more practical for engineering applications.Figure 14 Frequency response comparison of STL (dB) between the original corrugated configuration and optimization schemes with different numbers of periods

As depicted in Figure 15, the resulting smooth design can be directly output as a stereolithography file (.stl) through format conversion. Three new topological configurations of 1:3 isotropic scaling of the floor base plate are obtained using an aluminum alloy powder (AISI0MG) in conjunction with 3D printing technology. The above results reveal that the proposed vibro-acoustic coupling FPTO method can generate an optimized configuration with better acoustic and mechanical performance and is easily integrated with CAD/CAE software. Lastly, the structural configuration can be conveniently fabricated via 3D printing demonstrating the proposed method’s effectiveness and the feasibility of practical engineering applications.For a detailed demonstration of the proposed method, Figure 16 shows the difference between the two boundary conditions. Figure 17 illustrates the periodic expression principle of the method used. Figure 18 illustrates the process of generating smooth configurations by the proposed method. Figure 19 shows the entire computational flow of the proposed method, illustrating in detail the joint calls between MATLAB and COMSOL. The details can be referred to the STAR Methods section.Figure 15 The 1:3 physical object with a volume fraction of 10.00%

(A) X×Y = 1 × 1; (B) X×Y = 2 × 1; (C) X×Y = 6 × 1.

Figure 16 Common boundary conditions

(A) Segregated FE formulation; (B) Mixed u/p formulation.

Figure 17 Schematic diagram of the 2D sandwich periodic structure

Figure 18 Schematic diagram of a smooth representation of the optimized topology

Figure 19 Algorithm flowchart of the FPTO method for vibro-acoustic coupling problems

Discussion

(1) The periodic sandwich structures are taken as the research object. Moreover, a multi-constraint dynamic three-field FPTO method for periodic structures that involve vibro-acoustic coupling systems has been proposed. The topology optimization design of the lightweight periodic vibro-acoustic structure is based on the optimization objective of maximizing the STL and the use of lightweight volume and stiffness as multi-constraints. It also effectively solves the local mode problems that may occur in the low-density regions, avoids the interfacial dependence problem of the multi-physical field coupling in the topology optimization process, and obtains a smooth topology configuration with stable performance and clear boundaries, which can be directly used for manufacturing via 3D printing.

(2) The proposed optimization method can generate various novel periodic smooth topological configurations with competitive comprehensive properties such as lightweight, superior sound insulation performance, high stiffness, and a freely customizable number of periodic cores. Consequently, the periodic number of the sandwich structures can be regulated and different periodic configurations can be obtained, providing a multi-solution reference with better performance and diverse forms for engineering design. Compared to the traditional corrugated floor core structure of HSTs, the mean-STL value corresponding to an optimization frequency range of 290 Hz–350 Hz is increased from 52.68 dB to 53.71 dB. Moreover, the minimum STL value under the single frequency of 320 Hz is increased from 12.15 dB to 54.03 dB. Simultaneously, the dynamic compliance of the optimized structure decreases from 4.16 e−6 Nm to 5.41 e−10 Nm, indicating a favorable increase in the structural load-bearing stiffness.

Overall, the results show that the three-field FPTO method based on the linear material interpolation model has a wide application range in solving vibro-acoustic coupling problems and can provide a reference for the optimal design of lightweight composite structures of HSTs and other transportation equipment for vibration and noise reduction.

Limitations of the study

This article aims to optimize the acoustic insulation and mechanical stiffness performance of periodic sandwich structures using the vibro-acoustic topology optimization method. To enhance the robustness and practicality of this optimization approach, the following work will be further conducted in the future: (a) Extending the proposed dynamic three-field FPTO method to other design problems, such as efficiently computing large-size structures and nonlinear dynamic optimization. (b) Conducting further testing of the full-scale 1:1 model using real aluminum material in an experimental system comprising an anechoic chamber and a reverberation chamber.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER	
Software and algorithms	
	
MATLAB 2015a	MathWorks	https://ww2.mathworks.cn/products/matlab.html	
OriginPro 9.1	OriginLab	https://www.originlab.com/	
COMSOL 5.4	COMSOL	https://cn.comsol.com/product-download	
FPTO method	Paper and code	https://doi.org/10.1016/j.advengsoft.2020.102942	

Resource availability

Lead contact

Further information and requests for resources should be directed to and will be fulfilled by the Lead Contact, Jie Hu (jiehu@gzu.edu.cn).

Materials availability

This study did not generate new unique reagents.

Data and code availability

This paper does not report original code.

All data reported in this paper will be shared by the lead contact upon request.

Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.

Experimental model and study participant details

See the simulation and experimental verification part in the results section.

Method details

Mixed u/p FE format for vibro-acoustic coupled systems

The propagation of acoustic waves in the acoustic field is characterized by the acoustic pressure (p), i.e., the Helmholtz equation can be adopted for the acoustic field. On the other hand, when the acoustic wave enters the structural field, the wave propagation is characterized by the structural displacement (u), i.e., the linear elastodynamic equation can be adopted for the structural field. The coupling between the acoustic and structural fields can generally be described by the separated FE scheme illustrated in Figure 16A. However, to overcome the multi-field coupling interface dependence problem, i.e., the need for predefining the coupling boundaries, the mixed FE scheme shown in Figure 16B can be used to represent the acoustic and structural fields uniformly. The principal idea is to introduce an additional sound pressure design variable into the structural field. The typical linear elastodynamic equation describing the propagation of time-varying-simple harmonics in elastic media can be expressed by neglecting the effect of volume force as follows:(Equation 1) ∇Tσ+ω2ρu=0

where σ is the stress vector, ρ is the density of the structure, u is the displacement vector, and ω is the radial frequency of the wave. Then, based on the bulk modulus K and the shear modulus G, the constitutive relation for the solid structure can be described as(Equation 2) σ=Kεvδ+2Gε′

where δ is the Kronecker delta function, and εv and ε′ are the volumetric strain and volume partial strain, respectively.

If the sound pressure variable p is added to the structural domain as an additional variable, the relationship between the sound pressure p and the volumetric strain εv is:(Equation 3) p=−Kεv

By substituting Equation 3 into Equation 2, the constitutive equation of the mixed u/p formulation can be expressed as:(Equation 4) σ=−pδ+2Gε′

The shear modulus of the acoustic medium in the acoustic field is 0 (G = 0). Considering that and substituting Equation 4 into Equation 1, Equations 1 and 3 can be further written as:(Equation 5) ∇Tp−ω2ρu=0

(Equation 6) pK+∇·u=0

By substituting Equation 6 into Equation 5, the standard form of the Helmholtz equation, i.e., the governing equation of the acoustic field, can be obtained as:(Equation 7) ∇·(1ρ∇p)+ω2Kp=0

Equation 7 shows that the wave propagation in the acoustic and structural fields can be characterized uniformly by changing the shear modulus G and bulk modulus K in Equations 4 and 6, respectively. The changes of the shear modulus G and bulk modulus K can be realized by the material interpolation model in topology optimization; thus, it is no longer necessary to explicitly define the coupling boundary between the acoustic and structural fields before optimization.

According to the Galerkin’s method, the weak form of the mixed u/p FE scheme is:(Equation 8) [Kuu−ω2Muu−Cup−CupT−Kpp][uˆpˆ]=[fufp]

where Kuu and Kpp are the stiffness matrices for the displacement and pressure variables, respectively, Muu is the mass matrix, and Cup is the coupling matrix. Equation 8 can be briefly written as:(Equation 9) ZU=f

where Z is the system matrix, U={uˆ,pˆ}T is the state vector, and f is the load vector.

Figure 16 shows the common boundary conditions for solving the vibro-acoustic coupling problem using the mixed u/p FE format. It should be noted that the boundary conditions of the structural field in the mixed u/p FE scheme are consistent with those in the traditional separated FE scheme. The difference is that the boundary conditions of the acoustic field represented by the original Helmholtz equation must be changed accordingly. The FE analysis and calculation of the vibro-acoustic coupling problem are performed with the new boundary conditions,32 as depicted in Figure 16.

FPTO method for vibro-acoustic coupling problems

Topological optimization problem and linear material interpolation model

The optimization objective of this paper is to maximize the STL of the periodic sandwich structure under the coupling effect of the acoustic and structural fields by identifying the optimal periodic layout of the materials in the design domain, while satisfying the lightweight and stiffness constraints. The 2D sandwich structure depicted in Figure 17 is taken as an example to fully illustrate the concept of structural periodic distributing design in a given design domain. The entire design domain is divided into m = m1 × m2 identical unit cells, where m1 and m2 represent the numbers of unit cells along the X and Y directions, respectively.

With the optimization objective function being the maximization of the mean-STL of the sandwich structure under a specific excitation frequency, the following optimization problem is established:(Equation 10a) Min.:J(x,w)=−1z∑STL(x,w)|g∈[ωg1,ωg2]

(Equation 10b) S.t.:ZU=f

(Equation 10c) Vf=mVi≤V0

(Equation 10d) Cf=Ci/Ciini≤ϕ

(Equation 10e) ∑j=1nxi,jvi,j−Vi=0

(Equation 10f) ∑j=1nEi,jui,jTki,jui,j−Ci=0

(Equation 10g) x1,j=x2,j=x3,j=⋯=xm,j

(Equation 10h) {xi,j=1,xi,j∈Ωsoliddomainxi,j=0,xi,j∈Ωacousticdomain0<xi,j<1,xi,j∈Γcouplingboundaries

where J represents the optimization objective function defined based on the design variable x, w is the characteristic excitation frequency, STL is the sound transmission loss that measures the sound insulation performance of the designed structure in dB, g is the optimized frequency, z is the number of frequency points, and can be calculated as z=(ωg2−ωg1)/q. ωg1 and ωg2 are the start and cutoff frequencies of the optimized frequency band. The frequency band is split into uniformly distributed frequency intervals that are spaced at q Hz intervals (In this paper, q = 10 Hz has been used). Vf and V0 represent the optimization target volume fraction and its constraint value within the design domain respectively, Vi represents the volume fraction of the i-th unit cell, vi,j represents the volume fraction of the j-th element in the i-th unit cell, and Cf denotes the optimization target compliance fraction, which is the ratio of dynamic compliance Ci to the initial dynamic compliance Ciini. It should be noted that dynamic compliance is the reciprocal of dynamic stiffness. The compliance fraction ϕ is the ratio between Ci and Ciini, ui,j denotes the displacement vector of the j-th element in the i-th unit cell, ki,j denotes the stiffness matrix of the j-th element in the i-th unit cell, and E(xi,j) denotes Young’s modulus of the j-th element in the i-th unit cell. Equation 10g represents the periodic constraint; each unit cell has the same design variable value corresponding to the same position. In addition, x = xi,j{(i = 1,2,3,⋯,m)(j = 1,2,3,⋯,n)} is the unit design variable, where n represents the total number of elements in each unit cell, xi,j = 1 indicates that the j-th element in the i-th unit cell is composed entirely of solid materials, xi,j = 0 indicates that it is composed entirely of acoustic media, e.g., air, and 0<xi,j<1 indicates the coupling boundary between the acoustic and structural domains. The STL at the excitation frequency g∈[ωg1,ωg2]40 can be calculated as:(Equation 11) STL=10log10t(x,w)|g∈[ωg1,ωg2]

where t(x,w)=Ein/Eout is the sound power transmission coefficient, Ein=(1/2ρairc)∫A|pin|2dS and Eout=(1/2ρairc)∫A|pout|2dS are the sound power values at the inlet and outlet boundaries of the structure, respectively, c is the speed of sound, and ρair is the density of air.

Based on the linear material interpolation model, the bulk modulus K, shear modulus G, and equivalent density ρsof each element according to its design variables xi,j can be interpolated.(Equation 12) {K(xi,j)=Ksxi,j+Ka(1−xi,j)G(xi,j)=Gsxi,jρ(xi,j)=ρsxi,j+ρa(1−xi,j)

where the subscripts ‘s’ and ‘α’ denote the solid materials and acoustic media, respectively. The material interpolation model in Equation 12 establishes a unified representation model that simultaneously expresses air and solid materials through a single control equation. Therefore, the coupled boundary is transformed into an internal boundary in one control equation and the problem of explicitly defining the coupling boundaries between different governing equations is avoided. In addition, the linear material interpolation model has the same mass-to-stiffness ratio for all design variables, avoiding the local mode problems caused by extremely high mass-to-stiffness ratios in low-density regions in traditional SIMP models.41,42

Sensitivity analysis

The sensitivity of the optimization objective to the design variables and the constraint function must be calculated to solve the minimization problem in Equation 9 through the gradient optimization method. Therefore, the sensitivity of the STL to the design variable xi,j can be expressed as:(Equation 13) ∂J(x,w)∂xi,j=−1z∑∂STL(x,w)∂xi,j=−1z∑∂(10log10t(x,w))∂xi,j=−1z∑10log10et(x,w)∂t(x,w)∂xi,j

The differentiation of the sound power transmission coefficient with respect to the periodic design variables can be further derived as follows:(Equation 14) ∂t(x,w)∂xi,j=∂Pin∂xi,j1Pout−Pin(Pout)2∂Pout∂xi,j

where ∂Pin/∂xi,j and ∂Pout/∂xi,j are the partial derivatives of the sound power to the design variables. Similarly, ∂Pin/∂xi,j can be expressed as:(Equation 15) ∂Pin∂xi,j=1ρairc∫A|pin|∂|pout|∂xi,jdS

where ∂|pin,out|/∂xi,j can be calculated through the partial derivative of the design variables by the weak form matrix equation of Equation 8:(Equation 16a) ∂[Z]∂xi,j{up}+[Z]{∂u∂xi,j∂p∂xi,j}={00}

(Equation 16b) ∂p∂xi,j=[Z]−1[−∂Z∂xi,j]{p}

where ∂Z/∂xi,j is the gradient of the coefficient matrix with respect to the design variables, which can be calculated by combining Equations 1, 2, 3, 4, 5, 6, 7, and 8 with the material interpolation model in Equation 11:(17) ∂Z∂xi,j=[(∂Kuu∂xi,j−ω2∂Muu∂xi,j)∂Cup∂xi,j−ω2∂CupT∂xi,j−∂Kpp∂xi,j]=[AA00∫ΩKs−KaK2(xi,j)NpTNpdΩ]

where AA=∫Ω2Gs(∂Nu)TD∂NudΩ−ω2∫Ω(ρs−ρa)NuTNudΩ.

In summary, the sensitivity of the objective function to the design variables αi,j=∂J(x,w)/∂xi,j can be calculated by combining Equations 13, 14, 15, 16, and 17. Considering the role of periodic constraints, the entire structure can be iteratively updated by calculating the sensitivity in a single representative unit cell. Since the information of its sensitivity at the same position between different cells is consistent, the sensitivity αj of a single representative unit cell can be expressed as:(Equation 18) αj=∑i=1mαi,j

Similarly, the sensitivity of the volume fraction and the compliance fraction to the design variables in a single representative unit cell can be respectively expressed as follows:(Equation 19a) Vˆj=∑i=1mVˆi,j=∑i=1m∂Vf∂xi,j

(Equation 19b) Cˆj=∑i=1mCˆi,j=∑i=1m∂Cf∂xi,j=∑i=1m∂E(xi,j)∂xi,jui,jTki,jui,j=3(1−2υ)∑i=1m∂K(xi,j)∂xi,jui,jTki,jui,j

The FPTO algorithm defines three variables,27 i.e., the design field x¯i,j, the filtering field x˘i,j, and the physical field xi,j. Therefore, the sensitivities of the objective and constraint functions with respect to x¯i,j can be expressed in terms of the chain derivation rule as follows:(Equation 20) ∂J(x)∂x¯i,j=∂J(x)∂xi,j∂xi,j∂x˘i,j∂x˘i,j∂x¯i,j

(Equation 21) ∂Vf∂x¯i,j=∂Vf∂xi,j∂xi,j∂x˘i,j∂x˘i,j∂x¯i,j

(Equation 22) ∂Cf∂x¯i,j=∂Cf∂xi,j∂xi,j∂x˘i,j∂x˘i,j∂x¯i,j

where ∂x˘i,j∂x¯i,j and ∂xi,j∂x˜i,j can be directly derived from the filtering and projection schemes.

Updating strategies and implicit floating projection constraints

Based on the obtained objective function as well as the sensitivity of the constraint function, the design field variables are updated via the MMA optimizer. Upper and lower boundary constraints are imposed on the design variables as follows:(Equation 23) x¯i,j={0,x¯i,j≤0x¯i,j,other1,x¯i,j≥1

The design variables are filtered to eliminate the mesh dependency as well as the checkerboard problem in the optimization process as:(Equation 24a) x˘i,j=∑w(rij)x¯i,j∑w(rij)

where rij is the distance between the j-th element in the i-th unit cell and other elements, and w(rij) is the weight factor defined as:(Equation 24b) w(ri,j)={rmin−rij,0,rij≤rminrij>rmin

where rmin is the filtration radius. Subsequently, to eliminate the intermediate density elements, the filtered field variables are projected based on the Heaviside function:(Equation 25) xi,jk=tanh(β1·η)+tanh[β1·(x˘i,j−η)]tanh(β1·η)+tanh[β1·(1−η)]

where η = 0.5 and β1>0 are set as the factors to control the steepness of the Heaviside function. Then, the moving limit δ, and xi,jk for the next FE analysis should be set in advance to ensure that xmin≤xi,jk−1(1−δ)≤xi,jk≤xi,jk−1(1+δ)≤1. In all numerical examples in this paper, δ = 0.02 has been used.

A linear material interpolation model requires an engine-driven mechanism to push the 0/1 representation of the physical field variables. The FPTO27 method proposes implicit filter constraints and floating projection constraints to enforce length scale control as well as 0/1 constraints on the design variables. The implicit filter and implicit floating projection constraints are applied to the design variable field as follows:(Equation 26) x¯i,jk=tanh(β·thk)+tanh[β·(x˘i,j−thk)]tanh(β·thk)+tanh[β·(1−thk)]

where β>0 controls the steepness of the Heaviside function, which controls the extent to which the design variables tend to be distributed in 0/1. Initially, a sufficiently small positive number e.g., 10-6, is set. The thk, which can be calculated by dichotomization, is a floating threshold that can ensure that the total volume of the design variables before and after projection remains consistent, i.e., ∑x¯i,jk=∑x¯i,j. After updating the design variables, x¯i,jk={x¯1,jk,x¯2,jk,…,x¯i,jk,…x¯N,jk} is used for the subsequent FE analysis. Next, the moving limit xmin≤x¯i,jk−1(1−δ)≤x¯i,jk≤x¯i,jk−1(1+δ)≤1 is imposed on the design variables.

Subsequently, once the convergence criterion in Equation 27 is satisfied, Δβ is increased each time; here, Δβ = 1. The specific convergence criterion is expressed as:(Equation 27) Max(x¯i,jk−x¯i,jk−1)≤0.01

Post-processing of the element-based solutions obtained at each β-convergence is performed to achieve a smoothed characterization of the optimization results by ensuring that the total volume of the smoothed design solution (Figure 18B) is equal to the total volume of the element-based design solution (Figure 18A). At the same time, to evaluate the sound insulation performance of the smoothed extracted configuration, the smoothed design solution depicted in Figure 18B needs to be projected back to the original mesh presented in Figure 18C. Furthermore, an additional FE analysis needs to be performed to compute the objective function value of the smoothed design solution J (viter) and compare its difference with that of the objective function value of the element-based design solution J(xiter).(Equation 28) τ=|J(viter)−J(xiter)J(xiter)|≤0.01

Once the convergence criterion in Equation 28 has been satisfied, the smooth design is considered equivalent to the initial element-based design solution; at that point, the value of β is no longer increased, and the iterative process ends.

Algorithmic process

The entire algorithmic flow of the vibro-acoustic coupling periodic sandwich structure optimization using the proposed FPTO method is summarized in Figure 19. The entire process is written as the main optimization algorithm in MATLAB and combined with the commercial software COMSOL Multiphysics 5.4 for FE analysis to improve computational efficiency.

Quantification and statistical analysis

All experimental data obtained were materialized into topological optimization configurations via resin-based 3D printing technology, while the impedance tube test system was integrated with transfer function analysis to derive the necessary measurements. All simulation data were generated using the software MATLAB and COMSOL. Multiple sets of repetitive experiments were conducted to ensure the reliability of the results. For detailed data, please refer to supplemental information.

Supplemental information

Document S1. Figure S1

Table S1. Repeated experimental data, related to Figures 1C and S1

The table compiles data from five replicate experiments, all shown in Figure S1, with only the first three sets displayed in Figure 1C.

Table S2. Optimize iterative process data, related to Figure 6

The table presents evolutionary values of the objective function J, dynamic compliance constraint values, and iterative history values of β.

Acknowledgments

This work was supported by the 10.13039/501100001809 National Natural Science Foundation of China (No. 52305249 , 52305099 ), the Innovation team of Guizhou Province (CXTD2022-009 ), the Guizhou Provincial Basic Research Program (Natural Science) (ZK [2024] general084), the contract of the scientific research project of Guizhou University for introducing talents (2021 general 66), and the Open Fund Project of Key Laboratory of Advanced Manufacturing Technology of the Ministry of Education (GZUAMT2022KF-06 ).

Author contributions

Kui Luo and Jie Hu developed the overarching research goals and edited the original article. Song Yao and Ning Gan analyzed the measured data and guided the method. Chenfei Cao edited the article and conducted the literature review. Jiao Xu conducted the literature review and edited the article. Wenkang Cao provided the project resource.

Declaration of interests

The authors declare no competing interests.

Supplemental information can be found online at https://doi.org/10.1016/j.isci.2024.110648.
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