Virtual Modelling Framework-Based Inverse Study for the Mechanical Metamaterials with Material Nonlinearity

Abstract

Mechanical metamaterials have become a critical research focus across various engineering fields. Recent advancements have pushed the development of reprogrammable mechanical metamaterials to achieve adaptive mechanical behaviours against external stimuli. The relevant designs strongly depend on a thorough understanding of the response spectrum of the original structure, where establishing an accurate virtual model is regarded as the most efficient approach to this end up to now. By employing an extended support vector regression (X-SVR), a powerful machine learning algorithm model, this study explores the uncertainty and sensitivity analysis and inverse study of re-entrant honeycombs under quasi-static compressive loads. The proposed framework enables accurate uncertainty quantification, sensitivity analysis, and inverse study, facilitating the related design and optimisation of metastructures when extended to responsive materials. The proposed framework is considered an effective tool for uncertainty quantification and sensitivity analysis, enabling the identification of key parameters affecting mechanical performance. Finally, the inverse study approach leverages X-SVR to swiftly obtain the required structural configurations based on targeted mechanical responses.

1. Introduction

In recent years, mechanical metamaterials have established a strong presence in a group of engineering fields, including material science, structure design, artificial intelligence (AI), biomedicine, etc. [1,2,3,4,5] With specific man-made geometric configurations of the underlying structural layouts, they are able to accomplish exotic and desirable physical characteristics to deal with the targeted task [6,7]. Mechanical metamaterials are typically composed of unit cells with customised structures arranged in specific patterns. Initially, scientists emphasised the unit cell design based on topological optimisation to realise the expected physical properties and attained commendable achievements [8]. The products created through this approach are categorised as traditional mechanical metamaterials [9]. In this domain, a multitude of interesting and valuable functionalities such as negative Poisson’s ratio and high stiffness-to-weight ratio were unveiled to diversify the applications of engineering products [10].

Beyond merely optimising the arrangement and geometric shape of unit cells, recent research attempted to bring mechanical metamaterial into the smart area by customising more related fabrication phases. For instance, reprogrammable mechanical metamaterial is envisioned to become one of the pillar industries in the future engineering market [11]. Its design principle is incorporating responsive materials or living biological cells into the material phase to result in flexible mechanical characteristics when exposed to different external impacts [12]. By the changing shapes of the unit cells through the deformed smart materials, the applications made with reprogrammable mechanical metamaterials are expected to automatically adjust their physical properties to real-time external loads. To this end, before fabricating the metamaterials with advanced techniques, predicting their mechanical behaviours under various impacts, which guides the introduction of smart materials, plays a fundamental role in the whole process.

With the demands of modern engineering design, a series of response spectrums of structure which indicate the structural behaviours against various potential loads during service life play an important role in many projects since they can give a theoretical guide in the design and manufacturing process. It is noticed that a response spectrum is accomplished by computing the structural performance thousands of times. In this case, although numerical simulation has displayed its excellent calculation ability and precision in modern structural analysis, the costs of the enormous calculation times still present a challenge to the utilisation in response spectrums during the process of metamaterial design. Fortunately, the development of data-driven techniques shows compelling potential to compensate for the drawbacks of numerical tools. Employing the numerical model as the sampler, a virtual model for the response spectrum can be established to achieve rapid mapping between the structure parameters and behaviours. Singh et al. [13] employed a physics-informed neural network and an artificial neural network with equivalent hyperparameters to solve a fourth-order differential equation related to solid mechanics problems. Based on the neural network framework, Chen et al. [14] predicted the vibrational behaviours of a 42-storey concrete building in an intense earthquake.

Lately, the Extended Support Vector Regression (X-SVR) has emerged as a powerful tool for virtual model construction, particularly in applications requiring uncertainty quantification and structural reliability analysis across diverse physical problems, including static, dynamic, and elastoplastic systems. Grounded in rigorous mathematical theory, X-SVR formulates a convex optimisation problem, leveraging Quadratic Programming (QP) to identify an optimal hyperplane and kernelised mapping to enhance its processing capacity, effectively capturing both localised and broader structural patterns within the data. Furthermore, Bayesian optimisation is employed for automatic hyperparameter tuning, ensuring an optimal balance between model complexity and generalisation, while cross-validation mitigates the risk of overfitting. Tian et al. [15] established a framework based on the data-driven method to conduct the uncertainty prediction for structures. Combining the X-SVR virtual modelling technique with the phase field method, Liu et al. [16] conducted a study on crack propagation based on dynamic fracture analysis.

The established virtual model and response spectrum can be applied to assessments, including uncertainty analysis [17], sensitivity analysis [18], and inverse study [19], all of which pave their way to the structural design of metamaterials. The inverse study is an important engineering design method to determine the optimal geometric and material parameters to achieve a target mechanical response. Based on the response of the structure subjected to external impacts, the inverse study can compute the optimal adjustable parameters for the structure rapidly. Then, the related external inputs are transformed into smart materials, facilitating the structure to the optimal parameters. However, with the intricate microstructure of metamaterials with numerous adjustable parameters, the immense complexity and vastness of the design space present significant challenges in inverse study based on traditional experimental, theoretical, and computational approaches; therefore, data-driven techniques have emerged as powerful tools in this area [20,21]. By establishing the virtual model of the relationship between material parameters and the response of structures. Machine learning methods save numerous computational costs and realise a great number of impossible design projects. The established model is also applicable to the uncertainty and sensitivity analysis, making the cooperativity of the whole design process.

Besides attaining a comprehensive understanding of the mechanical attributes of the structure, the uncertainty analysis and sensitivity analysis also give some additional guides to the inverse study for real-life engineering applications [22,23,24,25,26,27,28]. For instance, the uncertainty analysis results, which is the uncertainty probability of the structural response, are valuable to determining the effective design space in reality [29]. The sensitivity analysis results can reduce engineers’ burden by abandoning the parameter adjustability in the region with insignificant sensitivity [30,31,32,33,34,35].

To realise the preparation work of a metamaterial design, this study explores the uncertainty and sensitivity analysis of a re-entrant honeycomb under compressive loads initially; then, the inverse study is applied. Extended supported vector regression (X-SVR) [15], a newly developed machine learning algorithm with powerful data processing capabilities, is first used in the inverse study of mechanical metamaterials. The primary contributions of this work include: (1) the development of an X-SVR-based surrogate model for predicting the stress-strain behaviour of re-entrant honeycombs under quasi-static compression, (2) an in-depth uncertainty and sensitivity analysis to identify key parameters influencing mechanical response, and (3) an inverse study framework leveraging X-SVR to enable efficient metamaterial optimisation.

By integrating these components, this study provides a comprehensive approach to characterising, understanding, and optimising the mechanical performance of re-entrant honeycombs, paving the way for their intelligent application in advanced engineering systems. The mechanical behaviour of the structure mainly involves elastoplastic deformation on a large scale. Thus, the material and the geometrical nonlinearity display great significance in the establishment of the numerical model. There are various models describing the nonlinearity of deformed structures under external loads. For instance, Filippi et al. [36] adopted the Carrera Unified Formulation to simulate the geometric nonlinearities in wave propagation in prestressed structures. In the present work, the $J_2$ flow theory is employed in a numerical model to describe the nonlinearity of the structure during elastoplastic deformation.

The remainder of this paper is organised as follows. Section 2 provides a theoretical background for the elastoplastic behaviours of the re-entrant honeycomb based on the finite element method and the methodology for the X-SVR algorithm. The numerical results are presented in Section 3, including the results of uncertainty analysis, sensitivity analysis, and inverse study. Finally, Section 4 concludes with a summary of key contributions and recommendations for further studies in this domain.

2. Theoretical Formulation of the Virtual Model

2.1. Finite Element Method

Before establishing the data-driving model of the mechanical properties of structures, generating the data is always the first step. The finite element method, due to its powerful computational ability in solving partial differential equations, is extremely popular in engineering fields. In the present study, the finite element method is employed to investigate the elastoplastic behaviours of structures under external impacts. Since it is responsible for generating training and testing samples of the X-SVR-based virtual model, the framework works on multiple sets of inputs. The variable parameters are uniformly distributed in the considered range, where $u_n$ and $l_n$ represent the upper and lower bounds of the range. The variable parameters ${\theta }_{m\times n}$ can be written as follows:

$${\theta }_{m\times n}=\left[\begin{array}{c}0\\ ⋮\\ n-1\end{array}\right]{\displaystyle \frac{u_n-l_n}{n-1}}+{\left[\begin{array}{c}1\\ ⋮\\ 1\end{array}\right]}_{\left(n-1\right)\times 1}{l}_n$$

(1)

where m represents the number of samples, and n denotes the number of parameters.

In terms of the elastoplastic deformation of the structures, the incremental displacement strain relationship has the following form:

$$\Delta {\epsilon }_i=\Delta {\epsilon }_{e,i}+\Delta {\epsilon }_{p,i}+\Delta {\epsilon }_{g,i}=\left(B_i+B_{g,i}\right)\Delta u_i$$

(2)

where i represents the ith parameter set ($i=1,2,\dots ,n$); $\Delta {\epsilon }_{e,i}$, $\Delta {\epsilon }_{p,i}$, and $\Delta {\epsilon }_{g,i}$ denote the elastic, plastic, and higher-order strain increments, respectively; $B_i$ and $B_{g,i}$ represent the material and geometric nonlinear strain-displacement matrix, individually; $\Delta u_i$ is the incremental displacement of the structure.

The incremental plastic component is derived from the following:

$$\Delta {\epsilon }_{p,i}={\displaystyle \frac{\partial F_i}{\partial {\sigma }_i}}\Delta {\alpha }_i$$

(3)

where $F_i$ is the plastic yield function, dominated by the current stress state $\sigma$ and hardening coefficient $H$, can be expressed as follows, based on the von Mises yielding criteria:

$$F_i\left({\sigma }_i,H_i\left({\epsilon }_{p,i}\right)\right)=\sqrt{{\displaystyle \frac{3}{2}}{J}_{2,i}}-{\sigma }_{f,i}\left({\epsilon }_{p,i}\right)=0$$

(4)

in which $J_{2,i}$ is the second deviatoric stress invariant, and ${\sigma }_{f,i}$ denotes the yield stress condition depending on the plastic strain ${\epsilon }_{p,i}$.

Based on the constitutive equation, the incremental stress is written as the following form with the consideration of the material nonlinearity:

$$\Delta {\sigma }_i=D_{ep,i}\Delta {\epsilon }_i$$

(5)

where

$$D_{ep,i}=D_i-D_i\left({\displaystyle \frac{\partial F_i}{\partial {\sigma }_i}}\right){\left({\displaystyle \frac{\partial F_i}{\partial {\sigma }_i}}\right)}^T{D}_i{\left[{\left({\displaystyle \frac{\partial F_i}{\partial {\sigma }_i}}\right)}^T{D}_i\left({\displaystyle \frac{\partial F_i}{\partial {\sigma }_i}}\right)-{\displaystyle \frac{1}{\Delta {\alpha }_i}}{\displaystyle \frac{\partial F_i}{\partial {\sigma }_i}}d{H}_i\right]}^{-1}$$

(6)

in which $D_i$ is the stress-strain relationship matrix of the material in the elastic phase.

By substituting Equation (6), Equation (5) can be rewritten as follows:

$$\Delta {\sigma }_i=\left(D_{ep,i}{B}_i+D_i{B}_{g,i}\right)\Delta u_i$$

(7)

Based on the Hamilton’s principle, the relationship between the displacement and strain in the structure can be expressed as follows:

$${{\displaystyle \int }}_V\delta {{\epsilon }_i}^T{\sigma }_{i}dV-{{\displaystyle \int }}_V\delta {u_i}^T{\tau }_{i}dS+{{\displaystyle \int }}_V\delta {u_i}^T{\rho }_i{\ddot{u}}_{i}dV=0$$

(8)

where ${\rho }_i$ is the density of the structure; ${\ddot{u}}_i$ indicates the virtual acceleration vector; ${\tau }_i$ represents the external loads on the surface of the structure. According to the incremental strategy, Equation (8) can be rewritten as follows by incorporating Equation (7) and finite element shape function $\Phi$.

$$\left({{\displaystyle \int }}_V{B_i}^T{D}_{ep,i}{B}_{i}dV+{{\displaystyle \int }}_V{B}_{g,i}^T{D}_{ep,i}{B}_{g,i}dV\right)\Delta u={{\displaystyle \int }}_s{{\Phi }_i}^T{\tau }_{i}dS+{{\displaystyle \int }}_V\left(B_i+B_{g,i}\right){\sigma }_{i}dV-{{\displaystyle \int }}_V{{\Phi }_i}^T{\rho }_i{\Phi }_{i}dV{\ddot{u}}_i=0$$

(9)

where the left-side two terms are designated as elastoplastic and geometric stiffness matrix $K_{ep,i}$ and $K_{g,i}$ as follows:

$$\begin{array}{c}{K}_{ep,i}={{\displaystyle \int }}_V{B_i}^T{D}_{ep,i}BdV\\ K_{g,i}={{\displaystyle \int }}_V{B}_{g,i}^T{D}_i{B}_{g,i}dV\end{array}$$

(10)

The external load vector, internal load vector and mass matrix $F_{ex,i}$, $R_{in,i}$, and $M_i$ are adopted to describe the right-side three terms in the equilibrium.

$$\begin{array}{c}{F}_{ex,i}={{\displaystyle \int }}_S{{\Phi }_i}^T{\tau }_{i}dS\\ R_{in,i}={{\displaystyle \int }}_V{\left(B_i+B_{g,i}\right)}^T{\sigma }_{i}dV\\ M_i={{\displaystyle \int }}_V{{\Phi }_i}^T{\rho }_i {\Phi }_{i}dV\end{array}$$

(11)

2.2. Extended Support Vector Regression

In the present research, the X-SVR, a novel machine learning algorithm, is considered the tool to realise the comprehensive study of mechanical behaviours of structures under external loads because of its outstanding accuracy and efficiency displayed in previous work. The Quadratic ε-insensitive loss function plays a crucial role in improving the training stability and performance of X-SVR (Extended Support Vector Regression) by reducing sensitivity to small errors and enhancing robustness. Additionally, a mapping function, ζ(x), is introduced to effectively manage complex and large-scale training data, ensuring better adaptability to diverse data distributions. Furthermore, kernelised mapping is applied to $x_i$, enabling efficient processing by transforming the input data into a higher-dimensional space, which improves model accuracy while maintaining computational efficiency. Together, these enhancements contribute to a more powerful and reliable X-SVR model.

$$x_i=\left[x_{i,1},x_{i,2},x_{i,3},x_{i,4},\cdots ,x_{i,n}\right]\to \widehat{w}\left(x_i\right)=\left[\begin{array}{c}\zeta {(x_1)}^T\zeta \left(x_i\right)\\ \zeta {(x_2)}^T\zeta \left(x_i\right)\\ ⋮\\ \zeta {(x_n)}^T\zeta \left(x_i\right)\end{array}\right]=W\left(x_i\right)$$

(12)

where $W\left(x_i\right)$ and n indicate the kernelised mapping function and the number of training samples, respectively.

The governing formulation based on the kernelised X-SVR technique is shown as follows:

$$\underset{p_x,q_x,\lambda ,\theta ,\widehat{\theta }}{min}:{\displaystyle \frac{{\delta }_1}{2}}\left({\parallel p_x\parallel }_2^2+{\parallel q_x\parallel }_2^2\right)+{\delta }_2{e}_j^T\left(p_x+q_x\right)+{\displaystyle \frac{z}{2}}({\theta }^T\theta +{\widehat{\theta }}^T\widehat{\theta })$$

(13)

subjected to

$$s.t.\{\begin{array}{c}{W}_{train}\left(p_x-q_x\right)-\lambda e_j-y_{train}\le \omega e_j+\theta \\ y_{train}-W_{train}\left(p_x-q_x\right)+\lambda e_j\le \omega e_j+\widehat{\theta }\\ p_x,q_x,\theta ,\widehat{\theta }\ge 0^j\end{array}$$

(14)

where ${\delta }_1$ and ${\delta }_2$ are the coefficients to adjust for the feature selection; x donates the mapping procedure; both $p_x$ and $q_x$ are larger than 0 and is expressed as the normal of the hyperplane; $\theta$ and $\widehat{\theta }$ are the slack variables for allowing some excess deviations; $\lambda$ is considered as the bias parameter; $e_j$ is a unit vector; and $\omega$ is introduced for the acceptable deviation in predicting function and training data.

By leveraging the Lagrange procedure with KKT conditions, the nonlinear regression function of X-SVR can be written as follows:

$$\widehat{f}\left(x\right)={(p_x-q_x)}^T\widehat{w}\left(x\right)-{\widehat{e}}_x^T{\widehat{K}}_x{\zeta }_x^{*}$$

(15)

where ${\zeta }_x^{*}$ is assumed as the solution for the dual formulation of the Equation (13).

Spline kernels are widely used in function estimation and non-parametric regression tasks due to their ability to capture local data trends effectively using polynomial basis functions. Among them, the B-spline kernel is particularly valued for its compact support and numerical stability, making it a reliable choice in practical applications. Additionally, T-spline functions have gained attention for their smooth local refinement capabilities, especially in high-dimensional polynomial degree settings [16]. Building on these advancements, a novel T-spline polynomial kernel has been introduced, specifically designed to enhance the performance of the proposed kernelised X-SVR, providing improved adaptability and precision in regression modelling. The T-spline polynomial kernel function $T_{sp}\left({\zeta }_i,{\zeta }_j\right)$ can be defined as follows:

$$T_{sp}\left({\zeta }_i,{\zeta }_j\right)={\displaystyle \frac{S_i{d}_i{B}_{2n+1}\left({\zeta }_i-{\zeta }_j\right)}{{{\displaystyle \sum }}_{j=1}^m{d}_j{B}_{2n+1}\left({\zeta }_i-{\zeta }_j\right)}}$$

(16)

and

$$B_n\left({\zeta }_i\right)={\displaystyle \sum }_{r=0}^{i+1}{\displaystyle \frac{{(-1)}^r}{i!}}\cdot {\displaystyle \frac{\left(i+1\right)!}{r!\left(i+1-r\right)!}}\cdot {({\zeta }_i+{\displaystyle \frac{i+1}{2}}-r)}_{max}$$

(17)

In the process of X-SVR calculation, the statistical measures of R-square ($R^2$), root mean square error (RMSE), and relative error (RE), are utilised to quantitively validate the eligibility of the predicted results from the derived virtual model, where

$$R^2=1-{\displaystyle \frac{{{\displaystyle \sum }}_{i=1}^j{({\widehat{f}}_{s}i-y_i)}^2}{{{\displaystyle \sum }}_{i=1}^j{(\overline{y}-y_i)}^2}}$$

(18)

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{j}}{\displaystyle \sum }_{i=1}^j{({\widehat{f}}_{s}i-y_i)}^2}$$

(19)

$$\mathrm{RE}={\displaystyle \frac{{\widehat{f}}_{s}i-y_i}{y_i}}\times 100\%,i=1,\dots ,j$$

(20)

in which ${\widehat{f}}_s$, y, and $\overline{y}$ denote the predicted result, the benchmark result and the mean of the benchmark results, respectively; j denotes the number of sampling points.

3. Numerical Investigations

In this section, based on the virtual model established by X-SVR, a series of numerical experiments are conducted to reveal the mechanical behaviours of the concerned auxetic honeycomb under axial compressive stress. It is assumed that the structure is subjected to a quasi-static axial compressive load along the y-direction. The geometrical dimensions of the studied structures are displayed in Figure 1. As shown in the diagram, the walls of the unit cell of the auxetic structure have a thickness of 0.8 mm. The lengths of the walls are 8 mm and 4.5 mm, respectively, and the angle between them is 50°. In terms of the whole honeycomb, the length, width, and thickness of the structure are 80 mm, 64 mm, and 30 mm, respectively.

In the present section, before leveraging the X-SVR to construct the virtual model, a necessary comparison is performed to validate the computation results through finite element methods. After that, the accuracy of the data-driven model is tested in the uncertainty analysis again. The uncertainty study framework can provide a guide for the relevant design in practical engineering as well. The sensitivity study results are displayed and discussed to reveal the underlying law of the physical properties of the structure under compressive loads. Finally, a series of inverse studies are exhibited to realise the customised structure with targeted mechanical properties. It is noticed that all the numerical experiments are carried out by focusing on four parameters of the unit cells, which are elastic modulus $E$, Poisson’s ratio $\upsilon$, density $\rho$, and yield stress ${\sigma }_y$ of the used materials.

Before performing the numerical analysis, it is necessary to conduct a comparison between the results from the present model and some benchmark data by Alomarah et al. [36], who performed compression tests on the concerned re-entrant honeycomb structure using a Zwick Roell (Zwick/Z010, Ulm, Germany) material testing machine. In their experiment, a vertical compressive load was applied at a quasi-static speed, with the specimen positioned between two platens, one fixed and the other moving downward at 0.066 m/s. Displacement and reaction force data were recorded via sensors and subsequently analysed. The experimentally obtained stress-strain response was reported and further validated through numerical simulations in their work. The comparison highlights the accuracy and reliability of their simulation model in replicating the mechanical behaviour observed in the benchmark study.

The present work explores the finite element software Abaqus2020 to establish the numerical model. In Figure 2a, the results are compared with the two benchmarks from Alomarah et al. It is apparent that the differences between the numerical and the experimental results are in an acceptable range. The stress-strain results reveal an initially nearly linear increase in the curve, followed by a prolonged plateau region. Subsequently, the slope increases dramatically, accompanied by notable fluctuations along the way. It is interesting to note that the behaviour of the auxetic structure resembles that of an ideal plastic material, starting with an elastic stage, followed by a plastic regime, and ultimately losing stiffness. To gain a deeper understanding of the physical properties of the structure, this study focuses on three specific indicators within the results: the yield stress of the auxetic structure ${\sigma }_{ay}$, the Poisson’s ratio ${\nu }_a$, and the absorbed energy in the plastic stage $E{A}_p$, the expressions of the three indicators are illustrated as follows:

Observing Figure 2a again, by looking through the deformed shape of the structure shown in Figure 2b–f, the auxetic structure undergoes a short linear deformation phase followed by a plateau regime, resembling the stress-strain behaviour of plastic materials. The transition between these two phases is analogous to the yield point of the structure, indicating a loss of primary load-bearing capacity beyond this stage. The stress corresponding to this yield point is defined as the yield stress ${\sigma }_{ay}$ of the auxetic structure.

The negative Poisson’s ratio ${\nu }_a$, a key feature of auxetic honeycombs, makes them well-suited for various advanced engineering applications, including protective gear and smart materials. This property significantly influences the performance of auxetic structures and is therefore vital for conducting elastoplastic analyses related to these materials.

The energy absorption ability in the plastic stage is an important indicator to assess the load dissipation capability of structure deformation and the safety of the adjacent objects or individuals. In the present study, the energy absorption ability $E{A}_p$ is defined as the area of the stress–strain curve in the plastic regime. The beginning of this stage is the yield point, and the end is designated as the moment that the slope increases drastically and the stress is equal to the yield stress ${\sigma }_{ay}$.

In addition to the numerical model, the X-SVR virtual model is validated in the current section as well, where its accuracy is compared with the virtual models based on neural Networks and Gaussian processes.

Table 1. Comparison of the results ${\sigma }_{ay}$ of the re-entrant structure based on X-SVR, Neural network, and Gaussian processes.

		X-SVR	Neural Network	Gaussian Processes	Numerical Results
Case 1	Results	0.592507	0.593272	0.592305	0.592516
	RE (%)	0.0015	0.1276	0.0356
Case 2	Results	0.590369	0.590586	0.590562	0.590247
	RE (%)	0.0207	0.0574	0.0533
Case 3	Results	0.594512	0.594243	0.594135	0.594518
	RE (%)	0.0010	0.0462	0.0645

displays the ${\sigma }_{ay}$ of three groups of variable parameter values within the prescribed range derived from X-SVR, neural networks and Gaussian processes, respectively. The results based on the numerical model are also presented as the benchmark. The employed Neural network consists of three fully connected linear layers with dimensions (4,10), (10,20), and (20,1), respectively. As shown in Table 1, the results of the X-SVR virtual model exhibit a higher accuracy in the prediction of ${\sigma }_{ay}$ based on the considered variable parameters, proving its validation in this research. The parameter values for each case are specified as follows: Case 1: $E=1342.51\mathrm{MPa}$, $\upsilon =0.3212$, $\rho =8791.86 {\mathrm{kg}/\mathrm{m}}^3$, ${\sigma }_y=28.90\mathrm{MPa}$; Case 2: $E=1281.26\mathrm{MPa}$, $\upsilon =0.2834$, $\rho =8995.21 {\mathrm{kg}/\mathrm{m}}^3$, ${\sigma }_y=40.97\mathrm{MPa}$; Case 3: $E=1328.90\mathrm{MPa}$, $\upsilon =0.3543$, $\rho =9401.90 {\mathrm{kg}/\mathrm{m}}^3$, ${\sigma }_y=25.65\mathrm{MPa}$.

3.1. Uncertainty Analysis

With the aid of the data-driven method [37,38,39,40,41,42,43], the current section gives an insight into the uncertainty studies of the mechanical properties of the auxetic structures. There are four variable parameters considered as the uncertain inputs of the proposed system, including the elastic modulus $E$, Poisson’s ratio $\upsilon$, density $\rho$, and yield stress ${\sigma }_y$, the relevant information of which is described in

Table 2. The probability distributions of the variational material properties.

Variational Material Properties	Distribution Type	Mean	Standard Deviation	Interval
$E\left(\mathrm{MPa}\right)$	Uniform	1300	6.5	[1267.5, 1332.5]
$\upsilon$	Normal	0.33	0.0264	[0.23, 0.43]
$\rho \left({\mathrm{kg}/\mathrm{m}}^3\right)$	Normal	1000	60	[810, 1190]
${\sigma }_y\left(\mathrm{MPa}\right)$	Lognormal	3.6376	0.1819	[20, 70]

. The sources of uncertain information on parameters are mainly categorised into two groups: experimental work and published literature. The former usually requires enormous costs to reveal a full stochastic profile of some parameters with convincing precision. Regarding the latter, it is normal that some targeted uncertain parameters show less information in existing papers. The present section focuses on displaying the utilisation of the X-SVR virtual model for the uncertainty analysis, and the precision of the uncertainty data has no relation with the performance of the framework. Thus, a compromise is reached in this study; the mean value of the uncertain parameters is determined according to the experimental work conducted by Alomarah et al. [36], but the distribution type, standard deviation, and interval are prescribed within a rational scale by referring to the published work facing the similar cases [16]. Additionally, in order for a vivid illustration, the probability density functions (PDFs) and cumulative density functions (CDFs) of the four uncertain parameters are displayed in Figure 3. From the figure, it is clear that the elastic modulus E is uniformly distributed between the bounds. On the contrary, the other three factors, within their considered range, exhibit a peak probability, which generally decreases as they move away from the peak.

Based on the stochastic range of the four uncertain parameters, the authors leveraged MATLAB 2024b to generate the 10,000 parameter group samples based on uniform distribution. The uncertainty numerical model of the compressive test was accomplished based on the software Abaqus2020. Then, a series of stress-strain results of the auxetic structures in the test are obtained, and the three targeted indicators (${\sigma }_{ay}$, ${\nu }_a$, $E{A}_p$) in these results are extracted as the foundation of the X-SVR virtual models. It is noticed that each indicator has its own model individually. After that, a second sampling is conducted for these parameters by including their uncertain distributions. The derived samples are input into the virtual model to acquire the stochastic profiles of the three indicators finally, which are exhibited in Figure 4, Figure 5 and Figure 6.

Table 3. The $R^2$ and RMSE of estimated indicators.

Indicator	${\mathit{\sigma }}_{\mathit{a}\mathit{y}}$	${\mathit{\nu }}_{\mathit{a}}$	EA
$R^2$	0.9875	0.9990	0.9779
$\mathrm{RMSE}$	$1.7036\times {10}^{-6}$	$6.1789\times {10}^{-5}$	$9.1370\times {10}^{-8}$

delineates the statistical measures of $R^2$ and RMSE of the stochastic analysis. From the table, it can be concluded that the analysis results show a high accuracy. Meanwhile, the effectiveness of the virtual model is demonstrated. The validation of the data-driven model can also be observed by the good agreement between the results from the model and Monte Carlo simulation (MCS) in Figure 4, Figure 5 and Figure 6.

When paying attention to the stochastic results of the three indicators, the PDF of ${\sigma }_{ay}$ shows two peaks, which are 0.589 and 0.593 MPa, respectively. It is worth mentioning that a slight decrease can be observed between the two peaks. In terms of the stochastic profiles of the other factors, only one peak is displayed in each figure. With such uncertain parameters, the negative Poisson’s ratio has a high probability of being larger, since the peak appears nearly the smallest value in the range. The peak of the $E{A}_p$ stands at the middle of the range. However, from its PDF, it can be inferred that the $E{A}_p$ has a higher probability, but insignificantly, of being smaller with this set of uncertain inputs.

3.2. Sensitivity Analysis

This section provides a sensitivity analysis to explore the influence of the four variable parameters on the three indicators. The sensitivity analysis focuses on the trends of the performance of the structure with the variation of these parameters; therefore, the uncertainty information of the parameters can be ignored. The process is similar to that of the uncertainty analysis, excluding sampling the variables according to their stochastic profiles in the second time. Adversely, the sensitivity analysis results are derived by inputting the variables following a uniform distribution within their ranges. Since it is impossible to describe the coupling influences of the four parameters in a 3-dimensional figure, the authors divided them into two groups: the first group is elastic modulus $E$ and Poisson’s ratio $\nu$, and the second group is density $\rho$ and yield stress ${\sigma }_y$. The influences of the four parameters are studied in these two groups.

Figure 7 illustrates the effects of the considered parameters on the yield stress ${\sigma }_{ay}$ of the structure, where Figure 7a displays the results of elastic modulus $E$ and Poisson’s ratio $\nu$, and Figure 7b is that of the density $\rho$ and yield stress ${\sigma }_y$. As shown in Figure 7a, the influence between the ${\sigma }_{yr}$ and the two factors is almost in a linear relationship. It may be inferred that there are few mutual impacts of the two factors. As the elastic modulus or Poisson’s ratio increases, the yield stress of the structure continuously climbs. With respect to Figure 7b, the material yield stress ${\sigma }_y$ shows similar effects on the ${\sigma }_{yr}$. However, it seems that the effects of material density are insignificant.

Figure 8 exhibits the relationship between the negative Poisson ratio ${\nu }_a$ and the four factors. From the arc shape of the results, the sensitivity of the ${\nu }_a$ becomes feeble when the values of the four factors approach the middle of their considered ranges. However, the effectiveness of the negative Poisson ratio tends to be more significant when they arrive at the middle values.

The influences of the four factors on the energy absorption ability are elucidated in Figure 9. It is clear that more energy can be absorbed in the plastic stage with the increase in the Poisson’s ratio of the used material. However, this influence will change if the elastic modulus is larger than 1336 MPa. For the relationship between elastic modulus and $E{A}_p$, as the material Poisson’s ratio increases, the $E{A}_p$ with a larger elastic modulus generally grows, but that with a smaller elastic modulus continuously drops. The influence of ${\sigma }_y$ is positive to energy absorption ability, but its intensity continuously decreases as ${\sigma }_y$ grows. As for the material density, a middle value can turn out a maximum energy absorption capacity in the plastic stage.

3.3. Inverse Study

Some results of the inverse study are performed in this section. By using the mapping relationship established by the X-SVR-based virtual model reversely, one can capture the values of the variable parameters for the targeted mechanical properties. As shown in Equation (15), the X-SVR virtual model provides a nonlinear regression function for the indicators regarding the variable parameters. Thus, the inverse study can be accomplished by calculating the inverse function of Equation (15) as follows:

$$x={\widehat{w}}^{-1}\left\{{[{(p_x-q_x)}^T]}^{-1}\left[\widehat{f}\left(x\right)+{\widehat{e}}_x^T{\widehat{K}}_x{\zeta }_x^{*}\right]\right\}$$

(21)

The author realised the inverse function of the X-SVR virtual model through MATLAB 2020b. By specifying the target performance of the structure, one can obtain all corresponding parameter values within the design space. To demonstrate the effectiveness of the inverse study based on the proposed model, the authors apply the model to the three considered indicators and show each indicator with two examples in Figure 10, Figure 11 and Figure 12 based on radar plots. It is noticed that the targeted indicator is illustrated in the northwest of each figure, and those not mentioned are considered as the default value.

4. Conclusions

This study presents an X-SVR-based framework for analysing and optimising the mechanical behaviour of re-entrant honeycombs under quasi-static compressive loads. By integrating machine learning techniques with elastoplastic analysis, the proposed approach enables efficient uncertainty quantification, sensitivity analysis, and inverse study, thereby overcoming computational challenges associated with traditional methods.

The results demonstrate that X-SVR serves as a powerful surrogate model capable of accurately predicting the stress–strain response of auxetic structures while significantly reducing computational costs. The uncertainty analysis reveals the probabilistic nature of mechanical responses, highlighting key factors contributing to variability. The sensitivity analysis identifies dominant parameters influencing structural behaviour, providing valuable insights for design optimisation. Furthermore, the inverse study methodology successfully determines optimal configurations that achieve targeted mechanical properties, paving the way for adaptive and reprogrammable metamaterials. Meanwhile, the method proposed can be extended to other metamaterials or sandwich structures [44,45].

The findings of this research contribute to advancing the application of data-driven techniques in metamaterial design, offering a scalable and interpretable framework for engineering applications. Future work will explore the extension of this methodology to dynamic loading conditions and real-time adaptive metamaterials, further enhancing its applicability in advanced engineering systems.