Abstract
This study focuses on the optimal design of an Auxetic Re-entrant Honeycomb (ARH) metamaterial structure aimed at enhancing energy harvesting (ES) performance in doubly-curved sandwich shells (DCSS). To accomplish this, the geometric parameters of the ARH are optimized using a Multi-Objective Genetic Algorithm (MOGA) and considering geometric and physical constraints. Initially, the study provides an accurate calculation for the density of the ARH layer and compares it with previous models. Then, the governing electromechanical coupled equations are derived using Modified First-Order Shear Deformation Theory (MFSDT) and Hamilton’s principle, assuming simple support boundary conditions at all four edges. Utilizing Galerkin’s principle, the frequency response function (FRF) of the specific output power is analyzed. In addition to evaluating the free vibration results against authoritative references, the finite element simulation is also used to validate both the eigenfrequencies and FRFs data derived from the analytical solution. The optimization algorithm aims to maximize the specific power output while minimizing the resonant frequency of the structure. The results from optimizing the ARH core layer demonstrate that the new designs achieve ES capabilities that are 42.96 times greater than those with an isotropic core and 6.92 times greater than those of the conventional ARH. Additionally, the optimized structures show improved stiffness metrics compared to their pre-optimization counterparts. Thus, this design successfully enhances both ES and structural behavior, particularly in terms of stiffness.
Keywords: Multi-objective genetic algorithm (MOGA), Energy harvesting, Doubly-curved sandwich shell (DCSS), ARH metastructure, Specific power
Subject terms: Engineering, Materials science, Mathematics and computing
Introduction
Vibration-based piezoelectric ES has received widespread attention in recent years as one of the most significant approaches in mechanical engineering to energize self-powered sensing networks, monitoring systems, and smart structures. Liang, et al.1 demonstrated that, despite ongoing advancements in the development of piezoelectric materials, the primary limitation in power harvesting stems from structural dynamics. As a result, effective energy conversion is generally limited to narrow regions around the resonant frequencies. Subsequent studies also clarified that this limitation is fundamentally mechanical and is not only related to the design of the electrical circuits, but also involves factors such as stiffness distribution, boundary conditions, and the modal properties of the structure2,3. Recognizing this challenge, Shin, et al.4 demonstrated that the simultaneous activation of multiple vibrational modes can significantly enhance the total energy harvested, particularly when the performance criterion focuses on the amount of energy generated within a specific frequency range. Additionally, nonlinear dynamic effects, whether intentionally introduced through geometric design or arising naturally at higher excitation amplitudes, have been identified as effective methods to increase the operational bandwidth and reduce sensitivity to frequency mismatches5–7. Furthermore, mechanical tuning strategies, such as adjusting the effective inertia and stiffness distribution, have been reported as practical ways to improve resonance-based ES without altering the active material system8–10.
Curvature is a common feature in many real mechanical components, making it important to explore piezoelectric ES from curved plates and shells. Early electromechanical models demonstrated that curvature leads to a coupling of membrane and bending behaviors, fundamentally changing the distribution of strains available for piezoelectric energy conversion11,12. Jha and Ray13 found that in doubly-curved shells, this coupling can enhance strain energy in specific resonant modes, resulting in a higher electrical output compared to flat structures. Numerical and experimental studies on doubly-curved shells have confirmed that the presence of curvature enhances the modal spectrum and leads to stronger resonant responses under harmonic excitation14,15. Furthermore, even moderate curvature can alter the nodal patterns of vibration modes and change the regions of strain concentration. This is particularly important when considering continuous piezoelectric layers on the face sheets16,17. Recent studies have expanded these concepts to include layered and sandwich shells, demonstrating that curvature interacts with layer sequencing and shear effects. This interaction provides a broader design space for vibration analysis, active control, and resonance-based ES18–20. Moreover, significant research has focused on investigating and analyzing the dynamic-vibrational behavior of composite sandwich cylindrical shells under various boundary and environmental conditions21,22. Dong, et al.23 developed a composite corrugated shell structure that offers lightweight advantages and superior mechanical properties for engineering applications such as aerospace. In another study, Dong, et al.24 presented a more accurate and comprehensive analytical model for corrugated cylindrical sandwich shells in real engineering applications, such as engine shells and aircraft fuselages. This model aids in detecting and preventing dynamic behavior issues in components during service. Additionally, the combination of a honeycomb structure with a corrugated structure as a core layer has been explored to determine the vibration behavior of cylindrical sandwich shells under different environmental and thermal conditions25,26.
Given the strong dependence of the amount of energy harvested by piezoelectric systems on factors such as geometric configuration, stiffness distribution, and electromechanical arrangement, optimization has become a key approach in designing these ES systems. Biswal, et al.27 demonstrated that classical parametric optimization, which adjusts geometric dimensions, thickness ratios, and boundary conditions, can significantly enhance the harvested power at resonant frequencies for planar harvesters. Following this, He, et al.28 introduced topology optimization techniques to address challenges like charge neutralization and inefficient strain utilization. They showed that optimal material redistribution could substantially improve electromechanical coupling and energy output. Similar findings have been reported in studies focused on multi-degree-of-freedom planar harvesters driven by topology optimization29. As optimization methods advanced, research efforts naturally shifted towards broadband and multi-amplitude objectives. Hu, et al.30 utilized evolutionary and combinatorial strategies to optimize multi-amplitude harvesters, achieving notable improvements in ES within a specific frequency range. Concurrently, new frameworks for broadband topology optimization have been developed, which enhance power generation efficiency across a broader frequency spectrum by selectively activating multiple vibrational modes31. In recent years, optimization frameworks have been enhanced to include shape and geometry-based designs. Peralta, et al.32 applied shape optimization to piezoelectric energy harvesters using isogeometric analysis (IGA), enabling smooth geometric modifications. By optimizing for peak voltage output at a target frequency, they demonstrated improved ES capabilities. Their approach was later refined by integrating IGA with particle swarm optimization, resulting in better convergence and stability for the design process33. Additionally, utilizing parametric analysis alongside shape optimization has demonstrated that unconventional geometries can effectively tune natural frequencies and ES output in a targeted manner34. On the other hand, optimization in layered and composite structures has introduced greater design flexibility. Shukla and Pradyumna35 demonstrated that optimizing composite laminates with variable stiffness can significantly improve ES efficiency and broadband performance, by simultaneously adjusting fiber orientation and structural configuration. In addition to deterministic formulations, multi-objective and robust optimization strategies are increasingly being utilized to address realistic mechanical constraints. These frameworks aim to maximize harvested energy while also limiting stress, mass, and displacement36,37. To reduce the computational cost of repeated finite element analyses, particularly for shell and curved plate models, researchers have explored surrogate model-based and data-driven optimization methods. Jegadeesan, et al.38 utilized an artificial neural network-genetic algorithm (ANN–GA) framework to expedite the optimization process of vibration harvesters. Similarly, Selim, et al.39 demonstrated the effectiveness of machine learning approaches in navigating high-dimensional design spaces. Mhiri, et al.40 proposed an optimization approach for improving piezoelectric ES performance using a neural network surrogate model and a genetic algorithm. In a related effort, topology-driven designs that prioritize manufacturability have been proposed to address challenges in manufacturing and the local polarization of piezoelectric materials41. The optimization of nonlinear harvesters with self-coupling structures has been proposed, focusing on enhancing both bandwidth and output power simultaneously42. Additionally, there have been targeted reports on the modeling and optimization of harvesters for specific frequency ranges. This approach is closely related to formulating goals aimed at maximizing the harvested energy within the frequency range near resonances43.
Mechanical metamaterials offer innovative ways to manipulate vibrational properties and strain transfer beyond traditional geometric and material designs. Among these materials, auxetic structures, which are characterized by a negative Poisson’s ratio, can enhance both transverse and shear strain components due to their unique deformation mechanisms. This makes them particularly effective for piezoelectric energy conversion44. Subsequent research has demonstrated that utilizing auxetic substrates and configurations can significantly improve electromechanical coupling and ES output by concentrating strain under resonant excitation45,46. In sandwich configurations, auxetic re-entrant honeycomb (ARH) cores significantly influence the dynamic stiffness and the shape of vibration modes. This means that the geometry of the unit cell can be adjusted to control resonance and strain distribution47,48. Recent reviews and comparative studies have highlighted that designs based on acoustics hold great promise, particularly for tuning resonance and enhancing the effective bandwidth in vibration ES systems49,50. However, most existing research has concentrated on beams or flat plates, typically using fixed auxetic geometries51. As a result, the combined effects of curvature, sandwich structures, and the systematic tuning of ARH cores on energy conversion performance, especially in doubly-curved shells, have not been thoroughly explored in the mechanical engineering literature.
This study aims to determine the optimal design of an ARH meta-structure to enhance the ES performance of doubly-curved structures. The key point of this research lies in the application of a MOGA that considers the geometric parameters of the ARH, in conjunction with geometric and physical constraints, to improve the ES capabilities of a DCSS. Initially, the study presents an exact calculation for the density of the ARH layer and compares this with previous models. The electromechanical governing coupled equations are then derived using the MFSDT and Hamilton’s principle, assuming simple support boundary conditions on all four edges. By employing the Galerkin principle and simplifying the resulting differential equations, the FRF of specific power output is investigated based on the geometric and physical parameters of the structure. Next, the study identifies parameters with a significant influence on the ES results, which are then selected as optimization variables. The objective functions focus on maximizing specific output power and minimizing the structure’s resonance frequency. Optimal results are generated using the genetic algorithm, leading to an evolutionary improvement in ES capability of the DCSS with an auxetic meta-structure core after optimization. Generally, the primary innovations and achievements of this research, compared to previous studies, are as follows:
Utilization of a lightweight reinforcement ARH core to enhance energy harvesting from a DCSS under harmonic vibration.
Introduction of the accurate relation to calculate the density of the ARH structure.
Implementation of an MOGA that incorporates both physical and geometric constraints to design the optimal configuration of the ARH core layer.
Simultaneous improvement in both energy harvesting and structural stiffness in the optimized ARH designs compared to pre-optimization structures.
Subsequently, the next section details the geometric model and governing equations of the problem. The third section presents the results derived from analytical solutions computed using MATLAB. These results are further validated through numerical solutions implemented in the finite element software COMSOL, alongside comparisons with authoritative articles. It also includes the results of a sensitivity analysis on the selected parameters. In the fourth section, the multi-objective genetic algorithm, as well as the input parameters and constraints of the problem, are described. The optimization results achieved through the genetic algorithm for ES from the doubly-curved sandwich structure are presented, along with a discussion of the best structural design and its evaluation. Finally, the last section concludes the simulations and optimization study while outlining suggestions for future work.
Theoretical formulation
Model description
Consider a double-curvature sandwich shell with a re-entrant auxetic core of 
thickness and piezoelectric surface layers, each having a constant thickness of 
, as depicted in Fig. 1. The shell’s dimensions are a and b, and it has two radii of curvature of 
and 
. The piezoelectric layers are connected in series with an external electric resistance of 
to generate electricity from the vibrational harmonic extended load of 
.
Fig. 1.
Schematic of the system in: (a) 3D view, (b) electrical configuration, (c) unit cell of the ARH.
Assuming the electrodes covering the piezoelectric layers are perfectly conductive, the in-plane terms of electric fields are neglected. Therefore, the non-zero component of the electric field for the considered series connection is formulated as follows52:
 |
1 |
Where 
, and 
represents the Heaviside function.
As illustrated in Fig. 1, the auxetic unit cell has four independent dimensions of the inclined angle 
, the length of vertical cell rib 
, the length of inclined cell rib 
, and the thickness of the wall t. These dimensions are commonly used to calculate the physical properties of the ARH. Moreover, 
and 
represent horizontal and vertical distances between the ARH ribs, respectively. The effective mechanical properties of the ARH are formulated in terms of dimensionless parameters of 
and 
as follows53:
 |
2 |
Where Em and Gm=Em/2(1+ν) represent the elastic modulus and the shear modulus of the constructing material, respectively. Also, theARH's density is defined in most recent studies as follows [53-55]:
 |
3 |
Where 
represents the density of the constructing material. On the other hand, it can be clearly understood that the relation between 
and 
is derived using the following equation:
 |
4 |
In which 
refers to the surface area of an ARH unit cell, while 
represents the surface area of a rectangle that surrounds the unit cell, with dimensions of 
. The surface areas and dimensions can be calculated based on the geometry of the ARH unit cell as follows:
 |
5 |
Then, Eq. (4) is expanded and mathematically simplified:
 |
6 |
The comparison between Eq. (3) and Eq. (6) reveals a discrepancy of 
, which has significant effects for large values of 
. Consequently, Eq. (3) serves as an approximate relation that provides semi-accurate values primarily for small 
, as noted in one of the initial investigations56. Figure 2 further supports this assertion by comparing the two equations. The results from Eq. (3) are also confirmed by those presented in ref.55. The figure illustrates that as 
increases, the difference between the density calculated by Eq. (3) and the true value given by Eq. (6) becomes more pronounced, especially at larger inclined angles 
. Therefore, for ARH configurations with substantial values of t and 
, Eq. (3) is not appropriate.
Fig. 2.
Comparison of ARH density predictions between present exact formulation and ref.55.
In this study, to account for natural numbers of unit cells in intended dimensions, N unit cells along the 
axis, and M unit cells along the 
axis are assumed. For the square-shaped shell (a=b) the dimensionless parameters of 
and 
are defined:
 |
7 |
Constitutive relations
Deploying the MFSDT, the displacements’ components of each point of the shell 
in the orthogonal curved coordinate 
are presented in terms of the normal displacements of the neutral surface 
and the transversal rotations of neutral surface 
as follows57:
 |
8 |
The linear relation between mechanical strain and displacement fields defined in Eq. (8) is obtained, adopting the modified Sander’s shell assumptions appropriate for moderately-thick shells57:
 |
9 |
Where 
and 
represent normal and shear strains, respectively. To prevent rigid body motion’s zero strains, 
is considered, called the characteristic of Sanders’ Theory.
The linear stress-strain relationships are expressed:
 |
10 |
Where, 
represent stress tensor for the core layer, and 
denotes elastic matrix components in terms of elastic modulus 
and Poisson’s ratio 
. Additionally, for the ARH substrate, the elastic matrix components 
can be defined in terms of its effective mechanical properties reported in Eq. (2) as follows53:
 |
11 |
For the piezoelectric layers, the electro-mechanical constitutive relationships between stress tensor 
, strain tensor 
, electric fields 
, and electric displacements 
are presented:
 |
12 |
Where the elastic 
, the piezoelectric 
, and the dielectric 
coefficients are denoted as follows:
 |
13 |
Also, the resultants of normal forces 
, moments 
, and transverse forces 
are represented:
 |
14 |
In this equation, 
is called the shear correction factor and is valued differently based on the material used.
Coupled electro-mechanical governing equations
Applying Hamilton’s principle, the motion equations for the MFSDT are formulated as follows:
 |
15 |
Where 
for 
represent the mass inertias. Since all edges are assumed to be simply-supported, the boundary conditions are mathematically defined:
 |
16 |
The sixth motion equation is derived from Gauss’s Law, which states that the coupled mechanical strain and the generated electrical displacements are related in an integral form. This is expressed for the electric resistance current58:
 |
17 |
Substituting from Eqs. (1) and (9), then calculating the integral over the piezoelectric surface 
, and finally employing Ohm’s Law, Eq. (17) is extended as follows:
 |
18 |
Where 
represents equivalent piezoelectric layers capacitance, and 
resulted from averaging through the normal direction. In these equations, the coupling between mechanical deformation and electrical output is governed by the piezoelectric constitutive relations, where mechanical strain generates electrical charge as expressed in Eq. (17) through Gauss’s Law, while the resulting electric field conversely contributes to the mechanical response captured in the motion equations (15).
Galerkin method solution
According to the Galerkin’s method to discretize the set of equations (15), the following Fourier series are assumed for the unknown response functions, satisfying the simply-supported boundary conditions expressed in Eq. (16):
 |
19 |
Where 
, 
, 
, 
, 
are modal coordinates.
Assuming the harmonic load of 
, the modal coordinate, as well as the output electrical voltage, are expressed:
 |
20 |
In which, 
, 
, 
, 
, 
, and 
have unknown complex values.
By substituting Eq. (20) into Eq. (19), and then inserting the result into Eq. (15), the equations of motion in the mass matrix 
and stiffness matrix 
form are derived, utilizing mode orthogonality as well:
 |
21 |
The components of the matrices are listed in Appendix A. To prevent instability in the system at the resonant points, the Rayleigh damping as a mechanical damping effect is added as follows:
 |
22 |
In which 
and 
that indicate mass and stiffness proportional coefficients, respectively. For any given damping ratio, the values of the coefficients could be calculated59. Finally, Eq. (21) is rewritten considering the damping effect:
 |
23 |
Then Eq. (18) is reformulated:
 |
24 |
Where 
and 
are expressed as follow:
 |
25 |
Equation (24) gives 
for any given stimulus frequency 
. Ultimately, the obtained 
is used to calculate the magnitude of the power FRF:
 |
26 |
In the present study, the specific power is studied, aiming to simultaneously investigate changes in the electrical power and the core layer’s mass, using the following formulation:
 |
27 |
Results validation and discussion
Comparison study
This section aims to assess the precision of the derived formulations by comparing the current results of natural frequencies with those previously disseminated. All material properties used to do so are represented in Table 1.
Table 1.
Material properties in the references.
| Property | Aluminum |
|
|
PZT-4 |
|
|---|---|---|---|---|---|
|
2700 | 3800 | 2370 | 7500 | 7500 |
|
70 | 380 | 384.43 | - | - |
|
0.33 | 0.3 | 0.24 | - | - |
|
- | - | - | 139 | 127.2 |
|
- | - | - | 77.8 | 80.2 |
|
- | - | - | 74.3 | 84.7 |
|
- | - | - | 115 | 117.4 |
|
- | - | - | 25.6 | 23 |
|
- | - | - | 30.6 | 23.5 |
|
- | - | - | -5.2 | -6.6 |
|
- | - | - | 15.1 | 23.2 |
|
- | - | - | 12.7 | 17.0 |
|
- | - | - | 6.75 | 15.1 |
|
- | - | - | 5.9 | 12.7 |
Table 2 reports the first (m=n=1) dimensionless natural frequencies 
of both single-curved and doubly-curved shells made of 
, considering various values for the radii of curvatures. The minimal discrepancies observed confirm the high accuracy of this study. These differences can be attributed to the various methodologies employed in each referenced study: as ref.60 applies a new Sinusoidal Shear Deformation Theory, ref.61 uses Third-order Shear Deformation Theory (TSDT), while ref.62 is based on a 3D Elasticity method. Moreover, FSDT and Classical Shell Theory (CST) are utilized in ref.63 and ref.64, respectively.
Table 2.
The first dimensionless natural frequency 
of both single-curved and doubly-curved shells for different radii of curvatures 
.
Single-curved 
|
Doubly-curved 
|
||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
|
|
Present | Re.60 | Re. | Re. |
|
|
Present | Re. | Re. | Re. |
| 0.5 | 0 | 0.2098 | 0.2080 | 0.2169 | 0.2129 | 0.5 | 0.5 | 0.0740 | 0.0695 | 0.0761 | 0.0779 |
| 1 | 0 | 0.2125 | 0.2133 | 0.2329 | 0.2154 | 0.5 | -0.5 | 0.0534 | 0.0546 | 0.0577 | 0.0597 |
Then, the first closed-circuit natural frequency of a DCSS featuring an isotropic core made of 
and outer surface layers of 
is reported for different geometries in Table 3. The current results demonstrate excellent agreement with those in ref.65 which utilizes higher-order shear deformation theory (HSDT), and ref.66 which is based on the four-variable shear deformation refined shell theory.
Table 3.
The first closed-circuit 
natural frequency 
of an isotropic core DCSS with piezoelectric outer surface layers 
.
|
|
|
Present | Re. | Re. |
|---|---|---|---|---|---|
| -0.2 | 0.1 | 0.1 | 821.84 | 823.13 | 823.99 |
| 0.2 | 784.49 | 787.61 | 788.09 | ||
| 0.2 | 0.1 | 1478.3 | 1484.9 | 1483.89 | |
| 0.2 | 1397.6 | 1400.81 | 1398.09 | ||
| 0 | 0.1 | 0.1 | 838.05 | 839.08 | 839.32 |
| 0.2 | 798.7 | 801.49 | 801.46 | ||
| 0.2 | 0.1 | 1494.3 | 1498.81 | 1498.3 | |
| 0.2 | 1407.9 | 1412.69 | 1411.02 | ||
| 0.1 | 0.1 | 0.1 | 852.09 | 853.03 | 853.09 |
| 0.2 | 810.59 | 813.26 | 813.08 | ||
| 0.2 | 0.1 | 1502.6 | 1506.39 | 1506.10 | |
| 0.2 | 1413.09 | 1418.90 | 1417.63 | ||
| 0.2 | 0.1 | 0.1 | 869.78 | 870.67 | 870.65 |
| 0.2 | 825.46 | 828.03 | 827.77 | ||
| 0.2 | 0.1 | 1511.2 | 1514.34 | 1514.24 | |
| 0.2 | 1418.2 | 1425.28 | 1424.28 |
To validate the derived auxetic formulations, Table 4 represents the first closed-circuit natural frequency for a three-layered piezoelectric sandwich plate with ARH core for different values of the thickness of the piezoelectric layers and the auxetic inclined angle. The material properties for Aluminum and 
are sourced from Table 1. The results reported in ref.53 are obtained employing the FSDT, and both density Eq. (3) and Eq. (6) are used to extract current results. The findings indicate that for small values of 
, there is negligible error, confirming the high precision of the present model. However, for 
, Eq. (3) was used in this study as well as ref.53 results in a larger discrepancy with true results, which have been predicted and discussed in the previous section.
Table 4.
The first closed-circuit 
natural frequency 
of an ARH-core doubly curved sandwich plate with piezoelectric outer surface layers 
.
|
|
|
|
|
|
|
|
|---|---|---|---|---|---|---|---|
| 0.1 | Present Eq. (6) | 1268.5 | 1261 | 1248.2 | 1205.2 | 1069.5 | 612.1 |
| Present Eq. (3) | 1268.4 | 1260.8 | 1247.9 | 1204.5 | 1067 | 605.58 | |
| Re.53 | 1269.23 | 1261.72 | 1248.89 | 1205.6 | 1068.13 | 605.13 | |
| 0.2 | Present Eq. (6) | 1370.5 | 1366.3 | 1359.2 | 1334.6 | 1248.6 | 839.1 |
| Present Eq. (3) | 1370.4 | 1366.2 | 1359 | 1334.2 | 1246.9 | 831.57 | |
| Re.53 | 1371.6 | 1367.63 | 1360.72 | 1336.48 | 1250.22 | 835.63 |
FEM analysis
In addition to the comparison studies conducted in the previous section, an FEM model via COMSOL Multiphysics v6.3 is investigated. To accomplish this, a piezoelectric DCSS 
with an ARH core 
, where the shell outer surface layers and core layer feature PZT-5 H and Aluminum, respectively. The structure was then meshed using 980,581 tetrahedral elements, ensuring that the first six natural frequencies of the model were adequately converged and computationally cost-effective. The results of the eigen frequency study are elucidated in Table 5. Considering the following expression 
, the relative error between the two outcomes is calculated. The discrepancy between the analytical and finite element natural frequencies can be attributed to three main factors. First, the complex geometry of the ARH structure poses challenges for mesh generation, particularly at the angled corners of doubly-curved shells, leading to localized mesh quality degradation that can affect system stiffness through variations in the effective degrees of freedom. Second, the auxetic structure is not symmetric and exhibits direction-dependent properties along the 
and 
axes, reducing simulation accuracy in regions with higher strain where mesh quality is suboptimal. Third, the mechanical properties of the auxetic layer derived from its geometry vary dynamically during the simulation, whereas they are assumed to be constant in the analytical formulation. Therefore, these factors contribute to the observed discrepancy and make the largest error of 5.92% in the third mode justified, confirming both approaches’ correctness and accuracy.
Table 5.
The sixth first natural frequencies and mode shapes of the piezoelectric DCSS with an ARH core.
| Modes | Present (Analytical) | Present (FEM) | RE 
|
Modes shapes | |
|---|---|---|---|---|---|
| Fre. (Hz) | (1,1) | 1050.1 | 1100 | 4.53 |
|
| (1,2) | 1603.6 | 1682.4 | 4.68 |
|
|
| (2,1) | 1618.5 | 1720.4 | 5.92 |
|
|
| (2,2) | 2311 | 2412.5 | 4.21 |
|
|
| (1,3) | 2795.7 | 2845.5 | 1.75 |
|
|
| (3,1) | 2816.2 | 2943.7 | 4.33 |
|
Moreover, the analytical and FEM models are compared in the frequency domain, by stimulating the shell with the external force of 
to extract the specific power FRF for frequencies ranging from 0 to 2000 Hz in 10 Hz increments. With all boundary conditions assumed to be simply supported, the system’s response is analytically modeled using two infinite Fourier series. Incorporating at least five modes in calculations ensures that the system’s outputs are converged. Thus, a maximum of five modes is used for FRF results. Figure 3 compares the specific power FRFs for both a doubly-curved shell with the ARH core and with conventional Aluminum, considering external electric resistance of 
. The methods demonstrate acceptable agreement, indicating their reliability; the rationale for the observed differences between FEM and analytical results was extensively discussed in the previous section. Furthermore, as elucidated in Fig. 3, the ARH-core shell offers 3 times higher specific power and a lower resonance frequency compared to the isotropic-core shell, both of which indicate improved performance in ES.
Fig. 3.
Comparison between specific power FRFs obtained from analytical investigation and FEM in present study.
Parameters sensitivity
This section aims to investigate the effect of various factors on the first peak of output specific power FRFs, following the procedure coded in the FRF_EH function. As depicted in Fig. 4, the function, firstly, calculates the first natural frequency of any given new parameters. Since the first resonance frequency is always equal to or bigger than the first natural frequency, the start frequency point is set with the first natural frequency obtained minus 10 (Hz), making for lower and faster computation. By calculating the specific power for the considered frequency and making a comparison between the current and previous data through the loop, the peak of the FRF is identified and reported, as well as its frequency. During this section, the shell’s geometric parameters are considered as: 
, with material properties of Aluminum and PZT-5 H.
Fig. 4.
The schematic diagram of the FRF_EH function that returns the peak of the specific power FRF and its resonance frequency.
Electric resistance
To explore how electric resistance affects the maximum specific power generated and the resonance frequency, three different ARH-core shells, as shown in Fig. 5, were examined along with an isotropic-core shell. As illustrated in Fig. 6, ARH-core shells consistently generate significantly more power across the range of electric resistance values. Notably, the shell with the AUX-II core produces approximately 8.8 times greater specific power at its optimal resistance (around 10 Ω) compared to the isotropic-core shell at its optimum (about 40 Ω). In addition, the ARH-core shells tend to have lower resonance frequencies. While the AUX-II core delivers the highest specific power, the AUX-III core achieves the lowest resonance frequency. The maximum specific power for ARH-core shells fluctuates with resistance and typically peaks near 10 Ω, whereas the isotropic-core shell follows a distinct pattern with its optimal value at 40 Ω. This difference is attributed to the unique mechanical properties, such as negative Poisson’s ratios, of ARH-core structures compared to conventional shells.
Fig. 5.
The different auxetic structures studied.
Fig. 6.
The comparison between isotropic-core shell and ARH-core shells in: (a) the peak of specific power, and (b) its resonance frequency in terms of external electric resistance.
Auxetic geometric parameters
The four-dimensional graphs presented in Fig. 7 analyze how ARH geometric parameters affect the ES outputs, with the z-axis representing the maximum generated specific power and the color bar as the fourth dimension indicating the resonance frequency. The electric resistance of 
is considered, which is optimal for the ARH-core shells in the proposed configuration, as discussed previously. The analysis reveals that increasing M and N results in a decrease in specific power; specifically, the lowest values of M and N (both set at 5) yield about four times the specific power compared to configurations with M and N at 15. Additionally, the resonance frequency decreases with an increase in M or a decrease in N. In analyzing this observation, it can be stated that increasing the number of unit cells ( M and N) increases the elastic modulus of the layer and consequently increases the structural stiffness. This increase in stiffness reduces the strain transferred from the ARH core layer to the piezoelectric layers, thereby reducing the output power. Additionally, as the density of the core layer rises due to an increase in the number of ARH cells, there is a further decline in specific power. However, directly analyzing the changes in resonance frequency is complicated by the simultaneous variations in stiffness and mass, as well as the indirect relationships of these parameters with the respective values.
Fig. 7.
The influence of ARH parameters: (a) M and 
; (b) 
and 
, on both the peak of specific power (Z-axis) and the resonant frequency (color-bar).
It is also observed that the thicker t is, the less specific power is output. This reduction occurs sharply because the 
thickness has a considerable effect on the ARH density that leads to a dramatic change in specific power. An increase in the value of the auxetic inclined angle 
for constant
slightly dwindles the specific power. It is worth noting that the effect of t and 
on the resonance frequency does not follow a linear pattern. For example, when 
, the resonance frequency decreases from 
to 
as 
increases from 
to 
. Conversely, the resonance frequency remains relatively unchanged for any values of 
when 
. Overall, the unexpected effects of geometric ARH parameters on ES outputs resulting from indirect and sometimes opposing effects on physical properties highlight the necessity for optimization investigations to identify the most effective ARH structure that maximizes specific power while minimizing stimulus frequency.
MOGA optimization
This section aims to optimize the ARH-core shell’s energy conversion performance, utilizing the MOGA to simultaneously maximize the peak of specific output power while minimizing resonant frequency. For the purpose of this analysis, the system is modeled as a DCSS composed of Aluminum and PZT-5 H, characterized by the dimensions of 
. The optimization problem is defined as follows:
Objective functions
As mentioned earlier, the MOGA is employed to achieve two goals:
 |
28 |
Where 1<M,N<50 and 
are integers and 
changes with the increments of 
. To address significant numerical order disparities between power 
and frequency 
, both objectives are normalized to comparable ranges between 0 and 1 as follows:
 |
29 |
Where 
, and 
. As the MOGA just able to minimize the objectives, there are defined as follows:
 |
30 |
Constraints
Geometric constraints
Selecting inappropriate sets of geometric ARH parameters can result in excessively small or even zero horizontal 
and vertical 
) distances between the ARH ribs shown previously in Fig. 1. To avoid the algorithm from generating and analyzing these non-ARH structures, which can lead to inconsistencies in geometry and inaccurate properties, the following two geometric constraints are applied:
 |
31 |
By applying some geometric and mathematical calculations, these constraints are formulated as follows:
 |
32 |
Stiffness constraint
To ensure practical manufacturability and structural integrity, a stiffness constraint based on the in-plane elastic matrix determinant was implemented. According to Eq. (10) for the FSDT, the in-plane stiffness matrix is denoted as:
 |
33 |
The determinant of this matrix 
represents the material’s resistance to in-plane deformation while accounting for Poisson coupling effects67,68. The normalized stiffness metric 
is defined as:
 |
34 |
Where 
represents the value for the solid reference material, Aluminum. To represent relative stiffness thresholds from 
to 
of the solid material, four constraint levels are investigated as follows:
 |
35 |
Algorithm implementation
The parameters reported in Table 6 are selected to perform optimization, using MATLAB’s “gamultiobj” function. The algorithm employs tournament selection, intermediate crossover, and adaptive feasible mutation operators. According to the flowchart demonstrated in Fig. 8, the multi-objective genetic algorithm optimizes the auxetic energy harvester by evolving populations of design variables 
. For each individual, the FRF_EH function is called to compute the peak power and its resonance frequency, while constraints ensure geometric feasibility and minimum stiffness. The algorithm iteratively selects, crossovers, and mutates solutions across generations, creating Pareto-optimal fronts that reveal power-frequency trade-offs. This process is repeated for four stiffness thresholds 
in order to systematically analyze how structural integrity requirements impact harvesting performance.
Table 6.
MOGA’s parameters.
| Parameter | Value | Justification |
|---|---|---|
| Population size | 80 | 20×number of variables |
| Maximum generation | 200 | Ensure convergence |
| Pareto fraction | 0.35 | Maintain solution diversity |
| Crossover fraction | 0.8 | Balance exploration/exploitation |
| Function tolerance | 0.0001 | Convergence criterion |
| Constraint tolerance | 0 | Strict constraint enforcement |
Fig. 8.
The integrated MOGA-FRF flowchart.
Results
Monitoring the convergence of the MOGA, Fig. 9 illustrates the evolution of the average Pareto spread over generations for the representative case of 
. The average Pareto spread is a metric that quantifies the diversity of non-dominated solutions across generations. The figure demonstrates that the average Pareto spread decreased rapidly from an initial value of 1.0 to approximately 0.1 after 85 generations, subsequently stabilizing with only minor fluctuations. This indicates that the algorithm effectively converges to a stable Pareto front within 85 to 108 generations. Furthermore, this behavior is consistent across all other stiffness constraint levels, highlighting the MOGA’s robust performance.
Fig. 9.
Average Pareto spread versus generation number during MOGA optimization for 
.
The Pareto fronts for all twenty-eight solutions and the actual values of the objective functions, which are specific power and the resonant frequency of the ARH designs, are demonstrated in Fig. 10, from the parametric stiffness constraint study. The actual values of the objectives reveal that the designs with maximum specific power outputs are independent of resonant frequency, maintaining a value around 1060 (Hz). From the Pareto fronts as well as the trade-off between stiffness metric 
and both specific power and resonant frequency depicted in Fig. 11, designs cluster in distinct regions: high-stiffness solutions 
achieve moderate specific power at higher frequencies, while low-stiffness solutions 
attain superior specific power across a wider frequency range. This suggests that the designs optimized primarily for power tend toward higher frequencies, while those with lower frequencies sacrifice substantial power output.
Fig. 10.
Pareto Fronts and objectives’ actual values for all levels of stiffness constraints.
Fig. 11.
The stiffness metric 
of the Pareto-optimal solutions versus: (a) specific power; (b) resonant frequency.
The first seven of twenty-eight Pareto-optimal solutions from the parametric stiffness constraint study are reported in Table 7, confirming the discussed fundamental trade-off between energy conversion and structural integrity. As 
is decreased from
to 
, the maximum achievable specific power surges dramatically from 
to 
, representing a remarkable 5.5-fold improvement. However, this enhancement in power is accompanied by a substantial reduction in stiffness ratio, from 
to 
. This also indicates that auxetic configurations characterized by minimal unit cells in one direction (N=1), thin walls 
, and near-vertical ribs 
effectively optimize specific power output. These configurations, moreover, achieve frequencies above 1060 Hz, indicating that maximum power extraction occurs at higher resonant frequencies, contrary to our objective of frequency minimization. This suggests a fundamental trade-off: designs optimized primarily for power tend toward higher frequencies, while those with lower frequencies sacrifice substantial power output.
Table 7.
The initial seven out of twenty-eight Pareto-optimal solutions derived from the study on parametric stiffness constraints.
| De. |
|
|
|
|
|
|
|
|
|---|---|---|---|---|---|---|---|---|
|
1 | 21 | 1 | 0.6 | 1 | 0.004417 | 1060 |
|
| 2 | 23 | 2 | 0.7 | 12 | 0.003159 | 1048 |
|
|
| 3 | 26 | 2 | 1.7 | 60 | 0.000983 | 979 |
|
|
| 4 | 33 | 2 | 0.5 | 11 | 0.003232 | 1050 |
|
|
| 5 | 48 | 1 | 1.0 | 15 | 0.001138 | 983 |
|
|
| 6 | 45 | 1 | 1.1 | 27 | 0.001093 | 980 |
|
|
| 7 | 31 | 3 | 1.5 | 63 | 0.000873 | 965 |
|
|
|
1 | 29 | 1 | 0.2 | 1 | 0.010182 | 1060 |
|
| 2 | 35 | 7 | 0.9 | 69 | 0.000835 | 935 |
|
|
| 3 | 33 | 6 | 0.8 | 67 | 0.001090 | 961 |
|
|
| 4 | 42 | 3 | 0.2 | 6 | 0.006524 | 1052 |
|
|
| 5 | 30 | 3 | 0.3 | 8 | 0.005717 | 1050 |
|
|
| 6 | 32 | 7 | 0.8 | 66 | 0.001030 | 955 |
|
|
| 7 | 41 | 2 | 0.2 | 5 | 0.006984 | 1054 |
|
|
|
1 | 34 | 1 | 0.1 | 1 | 0.017705 | 1064 |
|
| 2 | 22 | 4 | 0.9 | 70 | 0.001329 | 967 |
|
|
| 3 | 19 | 4 | 0.9 | 69 | 0.001443 | 972 |
|
|
| 4 | 22 | 4 | 0.6 | 63 | 0.002286 | 1011 |
|
|
| 5 | 20 | 4 | 1.2 | 71 | 0.001023 | 943 |
|
|
| 6 | 23 | 5 | 0.7 | 67 | 0.001574 | 981 |
|
|
| 7 | 29 | 1 | 0.2 | 2 | 0.010176 | 1058 |
|
|
|
1 | 24 | 1 | 0.1 | 1 | 0.024547 | 1067 |
|
| 2 | 39 | 6 | 0.6 | 76 | 0.001032 | 936 |
|
|
| 3 | 39 | 5 | 0.9 | 76 | 0.000808 | 921 |
|
|
| 4 | 43 | 2 | 0.1 | 4 | 0.013480 | 1060 |
|
|
| 5 | 29 | 1 | 0.2 | 3 | 0.010235 | 1058 |
|
|
| 6 | 36 | 6 | 0.5 | 76 | 0.001255 | 950 |
|
|
| 7 | 32 | 6 | 0.3 | 67 | 0.002822 | 1017 |
|
Design selection depends critically on application requirements, providing designers with limitations, such as minimum structural stiffness required, stimulus frequency range, or even manufacturing restrictions. To illustrate these trade-offs, consider three representative designs from the case that engineering consideration applies 
. Aiming to extract the maximum power output, Design.1 
provides 
at 1060 (Hz), which is significantly higher power than the low-frequency Design.2 
, while maintaining frequency near the optimal range. For applications prioritizing low-frequency operation, though, Design.2 at 935 Hz offers the best frequency-power compromise, reducing frequency by 125 (Hz) compared to the Design.1. Successfully balancing both objectives, Design.3 
provides 
at 
; a 
reduction from the Desing.1 while maintaining reasonable power, offering moderate power with reduced frequency.
Optimum performance appraisal
To assess the enhanced ES performance of the optimal ARH designs resulting from the MOGA, four different configurations as the core of the sandwich shell are considered. Table 8 outlines the properties of these systems, where the “Isotropic” configuration features a standard aluminum core, while “ARH-pre” represents a shell with a conventional ARH core. The configurations “ARH-De.1” and “ARH-De.2” utilize cores based on the optimal designs found in the previous section, which provide the highest specific power output and the lowest resonant frequency, respectively, within the category of 
. Moreover, the performance of each configuration in the frequency domain is analytically analyzed in the range of 800 to 1500 
and is validated against FEM results, as shown in Fig. 12. It is important to note that to obtain FRFs, a load resistance 
of 
is considered for the Isotropic configuration, and 
for the ARH-core configurations, as previously discussed that these values yield the highest generated power. Both methods’ results show acceptable agreements, since the cogent discrepancy between results is priorly addressed. The findings also confirm that the optimized designs significantly enhance ES. Specifically, ARH-De.1 produces 6.92 times and 42.96 times greater specific power peaks compared to the ARH-pre and Isotropic systems, respectively. Although ARH-De.1 has a higher resonant frequency than ARH-pre, it generates more power at the resonant frequency of the ARH-pre configuration. On the other hand, the ARH-De.2 offers the resonant frequency of 
, a decline of 
and 
compared to the ARH-pre and the Isotropic systems, respectively. While the peak specific power for ARH-De.2 is slightly lower than that of ARH-pre, it still generates 4.2 times more specific power than the Isotropic configuration. Finally, both optimized ARH designs exhibit higher stiffness metrics compared to ARH-pre. It is worth noting that the conventional density formulation overestimates density for designs with large 
and t, as observed in ARH-De.2 with the error of 24%. Consequently, the MOGA may be misled toward suboptimal designs, as the specific power objective is calculated using inaccurate density values.
Table 8.
Geometric and physical properties of configurations.
| Property | Isotropic | ARH-pre | ARH-De.1 | ARH-De.2 | |
|---|---|---|---|---|---|
| Geometric | M | - | 10 | 29 | 35 |
| N | - | 10 | 1 | 7 | |
| t (mm) | - | 1 | 0.2 | 0.9 | |
| θ | - |
|
|
|
|
| Physical |
 Eq. (6) |
2700 | 919.55 | 166.86 | 1804.3 |
 Eq. (3) |
2700 | 991.72 | 167.5 | 2242.9 | |
|
70 | 0.5204 | 3.031 | 23.718 | |
|
70 | 2.2719 | 2.02 | 0.1845 | |
|
0.3 | -0.425 | -0.359 | -5.092 | |
|
0.3 | -1.801 | -0.043 | -0.0723 | |
| \varepsilon | 1 |
|
|
|
|
|
0.000237 | 0.001472 | 0.010182 | 0.000835 | |
|
1366 | 1050 | 1060 | 935 |
Fig. 12.
Comparison between specific power FRFs obtained from analytical investigation and FEM for different cases, including Isotropic, ARH-pre, ARH-De.1, and ARH-De.2.
Conclusion
This study mainly aims to determine the optimal design of an ARH structure to enhance the ES capabilities in double-curved sandwich structures. The optimization process utilizes a combination of a MOGA, geometric and physical constraints, and the ARH parameters (parameters with significant impact on the results) as optimization variables. To facilitate this, the coupled electromechanical governing equations using MFSDT and Hamilton’s principle were extracted and simplified, assuming simple support boundary conditions at all four edges. Employing Galerkin’s principle, the FRF for the specific output power was then derived based on the geometric and physical parameters of the structure. The optimization focuses on two main objective functions: maximizing the specific output power and minimizing the resonant frequency of the structure. In the results section, we first extracted the first six natural frequencies and their corresponding shape modes from the free vibration solution of the structures, using both semi-analytical solutions in MATLAB software and the simulated model in COMSOL software. The comparison between the two approaches revealed a maximum error of 5.92%, which indicates an acceptable level of accuracy. Additionally, after analyzing how various parameters affect the ES performance of the structure and implementing the optimization algorithm on the initial design, the key results are presented as follows:
Increasing the values of M and N results in a decrease in specific power. In particular, configurations with the lowest values of M and N (both set to 5) produce approximately four times the specific power compared to configurations where M and N are both set to 15. Additionally, the resonant frequency of the structure decreases as M increases or as N decreases.
The effect of
and 
on the resonance frequency does not follow a linear pattern; additionally, increasing the inclined angle 
, while keeping t constant slightly decreases the specific power.The optimized designs greatly enhance energy harvesting. Specifically, the ARH-De.1 design generates 6.92 times more peak specific power than the ARH-pre system and 42.96 times more than isotropic systems. Similarly, the ARH-De.2 design delivers 4.2 times more specific power compared to the isotropic configuration.
The resonant frequency of the ARH-De.2 design (937 Hz) is 115 Hz lower than the ARH-pre design and 431 Hz lower than the isotropic design.
Both the optimized designs ARH-De.1 and ARH-De.2 exhibit greater stiffness metrics compared to the ARH-pre design.
This study demonstrates that utilizing optimized ARH structures as the core layer in sandwich structures can improve both ES capabilities and structural stiffness by adjusting the geometric and structural parameters of the ARH. Therefore, future research will focus on enhancing ES conditions in sandwich shells by investigating composite structures that incorporate advanced ARH metamaterials. The present study assumes simply-supported boundary conditions and harmonic excitation; extending the framework to other boundary conditions and arbitrary loading profiles remains an important direction for future work.
Appendix A
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Author contributions
**R. Ansarian**: Conceptualization, Software, Validation, Visualization, Data curation, Formal analysis, Methodology, Writing—original draft. **M. Motavasselolhagh**: Supervision, Investigation, Validation, Visualization, Data curation, Methodology, Writing and Resources. **R. Talebitooti**: Project administration, Writing—review & editing. **D. Younesian**: review & editing.
Data availability
The datasets used or analyzed in this study are available from the corresponding author only upon reasonable request, as the data also form part of an ongoing study.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Associated Data
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Data Availability Statement
The datasets used or analyzed in this study are available from the corresponding author only upon reasonable request, as the data also form part of an ongoing study.