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Open Access 05.01.2025 | Original Article

Elementary-level intrusive coupling of machine learning for efficient mechanical analysis of variable stiffness composite laminates: a spatially-adaptive fidelity-sensitive computational framework

verfasst von: A. Garg, S. Naskar, T. Mukhopadhyay

Erschienen in: Engineering with Computers

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Abstract

Mechanical analysis of the complex configurations of composite laminates can be computationally prohibitive based on accurate higher-order theories, especially when the analyses involve multiple realizations corresponding to different sets of input parameters such as uncertainty quantification, optimization, reliability and sensitivity analysis. Efficient lower-order theories should not be adopted in such situations since the error accumulates with multiple realizations, leading to poor outcomes. We propose an elementary-level coupling of machine learning for efficient, yet accurate mechanical analysis of laminated composites based on finite element simulations coupled with gaussian process regression. The generic parameter space of material properties, mesh size, number of layers, and ply angle in composite laminates are accounted for forming an efficient mapping with the augmentation of lower-order theory-based elementary-level structural matrices. The computationally efficient machine learning models predict the difference in the elements of the stiffness matrix for higher-order zigzag theory (HOZT) and first-order shear deformation theory (FSDT) at the first stage. Based on such machine learning-based difference mapping, we augment the elementary stiffness matrices obtained using FSDT efficiently to the equivalent of HOZT theory without any additional computational expenses (referred to here as augmented FSDT, or aFSDT). However, it is not necessary to augment all the elements in the analysis domain which might otherwise lead to unnecessary computational expenses and loss in accuracy. To achieve an optimal level of computational efficiency and accuracy, we further propose spatially-adaptive fidelity-sensitive coupling of machine learning, only for the elements within the analysis domain where it is necessary to adopt higher-order theories. The selective augmentation strategy essentially brings in a scope of integrating physics-based insights of critical stress resultant distribution into the algorithm based on best theory diagram. Subsequently, the global structural matrices are computed exploiting such adaptive criteria containing a mixed set of elements formed using FSDT and aFSDT, which leads to an accuracy equivalent to HOZT in the mechanical analysis of composite laminates almost at the computational expense of FSDT. The proposed spatially-adaptive fidelity-sensitive scheme ensures optimal performance in terms of computational efficiency by augmenting selective elements while minimizing the loss of accuracy due to the involvement of surrogates. Detailed numerical results are presented for static, dynamic and stability characterization of composite laminates including the demonstration for variable stiffness composite configurations based on the efficient machine learning-assisted elementary-level intrusive computational framework, wherein the notion of engineering judgement is introduced concerning the trade-off between computational efficiency and required level of accuracy.
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Abkürzungen
\({N}^{u}\) and\({N}^{l}\)
Numbers of upper and lower layers
\({u}_{0}, {v}_{0}\) and\({w}_{0}\)
Displacement of any point lying on the mid-plane
\({\xi }_{\left(x\right)}\), \({\xi }_{\left(y\right)}\), \({\psi }_{\left(x\right)}\) and\({\psi }_{\left(y\right)}\)
Higher-order unknowns
\(T\left(z-{z}_{e}^{u}\right)\) and\(T\left(-z+{z}_{f}^{u}\right)\)
Unit functions
\({\varphi }_{\left(xu\right)}^{e}\), \({\varphi }_{\left(yu\right)}^{e}\), \({\chi }_{\left(xl\right)}^{f}\) and\({\chi }_{\left(yl\right)}^{f}\)
Slopes of \(e\)-th and \(f\)-th layer for the upper and lower layers
h
Thickness of plate
\(a\) and \(b\)
Dimesnions of plate along X and Y axes
\(\left\{\bar{\sigma }\right\}\)
Stress vector
\(\left\{\overline{\varepsilon }\right\}\)
Strain vector
\({\left[\overline{Q }\right]}_{k}\)
Transformed rigidity matrix for the \(k\)-th layer
\([B]\)
Strain–displacement relationship in cartesian coordinate
\(\left\{\delta \right\}\)
Nodal displacement vector
\({U}_{s}\)
Strain energy of the plate
\({W}_{ext}\)
Energy due to the external loading
\(\left[{K}_{e}\right]\)
Elemental stiffness matrix
\(\left[{P}_{e}\right]\)
Mechanical load vector
\(T\)
Kinetic energy of plate
\(\dot{U}\left(x\right)\), \(\dot{V}\left(y\right)\) and\(\dot{W}\left(z\right)\)
Derivatives of \(U\left(x\right)\), \(V\left(y\right)\), and \(W\left(w\right)\)
\(\rho\)
Density of the material
\(\left[M\right]\)
Global mass matrix
\(\left[K\right]\)
Stiffness matrix
\(\left\{\overline{\delta }\right\}\)
Nodal variable vector
\(\left\{\frac{{\partial }^{2}\overline{\delta }}{\partial {t}^{2}}\right\}\)
Acceleration vector of the system
\(\left[{K}_{ge}\right]\)
Geometric stiffness matrix

1 Introduction

Laminated composites are extensively employed for composing or erecting structures in several engineering areas such as aerospace, civil, mechanical, automobile, etc., involving macro and nanoscale components [19]. Diverse theories are available for examining the behavior of laminates under different kinds of loadings, ranging from classical laminated theory to layerwise theory (LWT). A brief review of these theories can be consulted from the literature [16, 17, 38, 4346]. First-order shear deformation theories (FSDT) are computationally efficient but cannot predict the behavior of layered structures accurately [32, 49, 52]. Higher-order shear deformation theories (HSDT), a refined version of LWTs, could forecast the nature of laminates accurately but requires high computational costs. Many HSDTs exist in the literature for analyzing laminates concerning different loading conditions [3, 4, 7, 14, 18, 22, 36, 37, 50]. Higher-order zigzag theory (HOZT) is a variant of HSDT that can predict the mechanical behavior of layered structures like laminated composites and sandwich structures accurately in a wide variety of loading conditions. In general, mechanical analysis of the complex configurations of composite laminates can be computationally prohibitive based on accurate higher-order theories, especially when the analyses involve multiple realizations corresponding to different sets of input parameters such as uncertainty quantification, optimization, reliability and sensitivity analysis. Efficient lower-order theories should not be adopted in such situations as an alternative since the error accumulates with multiple realizations, leading to poor overall outcomes. In this paper we aim to propose a spatially adaptive computational framework that intrusively integrates machine learning with elementary-level structural matrices for achieving computational efficiency in the mechanical analysis of composite laminates without losing accuracy.
Integration of different machine learning and surrogate models in the analysis of composite materials and structures following intrusive and non-intrusive schemes has witnessed increasing interest from the scientific community over the last decade [8, 47]. Machine learning enables us to develop an efficient computational mapping between the input variables and the output/ response quantity of interest, thereby allowing an efficient characterization of the system without invoking the actual computational simulation model further. In this context, it must be noted that the machine learning model is formed on the basis of a few algorithmically chosen simulations of the computationally intensive model. Thus, having the actual simulation model (or experimental data) is mandatory, while the involvement of machine learning can be quite helpful in new predictions once it is formed, particularly in analyses where multiple realizations are necessary. There is another predominant path of machine learning model formation by incorporating the governing differential equations directly, commonly referred to as physics-informed machine learning (PIML). This approach aims to infer the mathematical descriptions (e.g., partial differential equations, constitutive laws) that govern the behavior of a system for more accurate and efficient prediction. In contrast, the data-driven surrogate models offer different benefits depending on the context such as not having direct governing equations corresponding to multi-stage analyses for complex systems such as composite laminates. Although PIML integrates physics to improve generalization and accuracy, it often involves solving the governing equations alongside the neural network optimization during the training process. This can make PIML computationally more intensive than surrogate models, especially for complex physics or high-dimensional systems. Implementing PIML can be more challenging, as it requires encoding physical constraints, boundary conditions, and governing equations into the machine learning model. While PIML integrates known physics to improve predictions, it may struggle with problems where the physics is not well understood or where the governing equations are too complex to encode directly. Considering the complexity of the current laminated composite models, will adopt a data-driven surrogate modelling approach in the current work.
Takeuchi and Kosugi [48] were among the first to exploit finite element (FE) along with machine learning techniques. Hashash et al. [23] employed a constitutive model based on a neural network for predicting structural behavior using FE. Koide et al. [28] used support vector regression and neural networks for studying the buckling behavior of laminates. Uncertainty quantification on free vibration analysis of laminated plates was carried out by Dey et al. [10, 11] using neural networks. Stiffened laminates under buckling conditions were studied by Mallela and Upadhyay [31] using neural networks. Oishi and Yagawa [39] investigated the accuracy of machine learning to predict the stiffness matrix. Fuzzy-based frequency analysis of functionally graded plates was carried out by Karsh et al. [27]. Mishra et al. [33] employed multivariate adaptive regression spline and minimax probability machine regression techniques for buckling analysis of skew laminated plates. Petrolo and Carrera (Petrolo and Carrera) carried out free vibration analysis of laminated plates using neural networks. Under the action of hygrothermal conditions, Kallannavar et al. [26] proposed the application of neural networks for the buckling analysis of laminates. With the help of deep learning, Jung et al. [25] evaluated the stiffness matrix for frames. Further, based on deep learning, Jung et al. [24] bypassed the iterative technique employed for predicting distortions and automatically updating the FEs in response to distortions, thus making the model computationally efficient. Kumar et al. [30] carried out a comparative study on various machine learning techniques to determine the natural frequencies of sandwich plates using surrogate models. Garg et al. [13] predicted the natural frequencies of functionally graded nanoplates using Gaussian Process Regression (GPR) surrogate model. To accurately predict the distribution of stresses across the thickness of the laminated composite beams, Garg et al. [20] coupled FSDT with GPR. Random forest-based surrogate was employed by Garg et al. [21] for correcting the FSDT-based behavior of laminated to the Elasticity-based solutions. A comprehensive review of surrogate models on predicting laminate vibration and dynamic behavior was reported by Dey et al. [9] and subsequently, Mukhopadhyay et al. [35] investigated the performance of different kriging model variants in depth.
A careful review concerning the application of machine learning in the domain of composite plates and shells reveals that neural network is the most widely used technique for studying the behavior of laminated composite plates. A significant number of investigations can be traced to deal with the static, dynamic and stability of composite laminates, sandwich structures and functionally graded structures covering both deterministic and stochastic computational frameworks. Most of these investigations follow a non-intrusive approach of exploiting machine learning model where the capability of prediction becomes limited to only a few targeted response parameters. For example, if the machine learning model is formed for a natural frequency of composite laminates in terms of the design input parameters, it will only be able to predict the natural frequency. However, if the machine learning model is integrated at a more elementary level, such as the formation or augmentation of stiffness matrices, various static, dynamic and stability analyses can be performed based on the efficiently obtained stiffness matrix or other structural matrices. In this article, we aim to propose an elementary-level coupling of machine learning for efficient, yet accurate mechanical analysis of laminated composites based on finite element simulations coupled with gaussian process regression. In this context, it may be noted that, while gaussian process based surrogates can lead to a significant level of accuracy in prediction from a reasonably small training dataset [35], it has been explored only in limited instances concerning composite structures where it is necessary to minimize the number of actual expensive simulations for developing the machine learning model.
In this work, the generic parameter space of material properties, mesh size, number of layers, and ply angle in composite laminates will be accounted for forming an efficient machine learning-based mapping with the augmentation of lower-order theory-based elementary-level structural matrices. The computationally efficient machine learning models can predict the difference in the elements of the stiffness matrix for higher-order zigzag theory (HOZT) and first-order shear deformation theory (FSDT) at the first stage. Based on such machine learning-based difference mapping, we will further augment the elementary stiffness matrices obtained using FSDT efficiently to the equivalent of HOZT theory without any additional computational expenses (referred to here as augmented FSDT, or aFSDT). However, it is not necessary to augment all the elements in the analysis domain which might otherwise lead to unnecessary computational expenses and loss in accuracy. To achieve an optimal level of computational efficiency and accuracy, we will develop spatially-adaptive fidelity-sensitive coupling of machine learning, only for the elements within the analysis domain where it is necessary to adopt higher-order theories. Subsequently, the global structural matrices will be computed based on such adaptive criteria containing a mixed set of elements formed using FSDT and aFSDT, leading to an accuracy equivalent to HOZT in the mechanical analysis of composite laminates almost at the computational expense of FSDT. The proposed spatially adaptive fidelity-sensitive scheme will further ensure optimal performance in terms of computational efficiency by augmenting selective elements while minimizing the loss of accuracy due to the involvement of surrogates. Hereafter, this article is organized into four sections, describing the machine learning-based computational framework (with adequate details on the finite element models, lower and higher order plate theories including FSDT and HOZT, gaussian process regression and its elementary-level spatially-adaptive integration), numerical results with related discussions, and concluding remarks.

2 Spatially-adaptive machine learning-based computational framework

The entire machine-learning-based model formulation and prediction, as presented in this paper, rely on the data generated through the finite element approach discussed in Sects. 2.1 and 2.2. As the finite element approaches and Gaussian process differ significantly based on the user implementation, it is necessary to provide brief details so that the readers can get the context appropriately.

2.1 Finite element formulation based on FSDT and HOZT

Consider a laminated plate consisting of \({N}^{u}\) and \({N}^{l}\) numbers of upper and lower layers, respectively, with respect to the mid-plane of the plate. The plate's surface is located in the X–Y plane, and its thickness runs parallel to the Z-axis. The thickness of the plate as a whole is equal to h units, and its measurements along the X and Y axes are designated by \(a\) and \(b\) respectively (refer to Fig. 1). The in-plane and transverse displacement fields are taken using HOZT as [29]:
$$\begin{aligned}U\left(x\right)={u}_{0}+{z}^{2}{\xi }_{\left(x\right)}+{z}^{3}{\psi }_{\left(x\right)}+\sum_{e=0}^{{N}^{u}}\left(z-{z}_{e}^{u}\right)T\left(z-{z}_{e}^{u}\right){\varphi }_{\left(xu\right)}^{e}+\sum_{f=0}^{{N}^{l}}\left(z-{z}_{f}^{u}\right)T\left(-z+{z}_{f}^{u}\right){\chi }_{\left(xl\right)}^{f}\\ V\left(y\right)={v}_{0}+{z}^{2}{\xi }_{\left(y\right)}+{z}^{3}{\psi }_{\left(y\right)}+\sum_{e=0}^{{N}^{u}}\left(z-{z}_{e}^{u}\right)T\left(z-{z}_{e}^{u}\right){\varphi }_{\left(yu\right)}^{e}+\sum_{f=0}^{{N}^{l}}\left(z-{z}_{f}^{u}\right)T\left(-z+{z}_{f}^{u}\right){\chi }_{\left(yl\right)}^{f}\\ W\left(z\right)={w}_{0}\end{aligned}$$
(1a)
where, \({u}_{0}, {v}_{0}\) and \({w}_{0}\) are the displacement of any point lying on the mid-plane. \({\xi }_{\left(x\right)}\), \({\xi }_{\left(y\right)}\), \({\psi }_{\left(x\right)}\) and \({\psi }_{\left(y\right)}\) are the higher-order unknowns. \(T\left(z-{z}_{e}^{u}\right)\) and \(T\left(-z+{z}_{f}^{u}\right)\) are the unit functions. \({\varphi }_{\left(xu\right)}^{e}\), \({\varphi }_{\left(yu\right)}^{e}\), \({\chi }_{\left(xl\right)}^{f}\) and \({\chi }_{\left(yl\right)}^{f}\) are the slopes of \(e\)-th and \(f\)-th layer for the upper and lower layers, respectively. Using Eq. (1a), FSDT can be obtained by setting \({\xi }_{\left(x\right)}\), \({\xi }_{\left(y\right)}\), \({\psi }_{\left(x\right)}\), \({\psi }_{\left(y\right)}\), \({\varphi }_{\left(xu\right)}^{e}\), \({\varphi }_{\left(yu\right)}^{e}\), \({\chi }_{\left(xl\right)}^{f}\), and \({\chi }_{\left(yl\right)}^{f}\) equal to zero except for \({\varphi }_{\left(xu\right)}^{0}\), \({\varphi }_{\left(yu\right)}^{0}\), \({\chi }_{\left(xl\right)}^{0}\), and \({\chi }_{\left(yl\right)}^{0}\). This leads to the FSDT-based displacement fields of the following form
$$\begin{array}{c}U\left(x\right)={u}_{0}+z{\varphi }_{\left(xu\right)}+z{\varphi }_{\left(xl\right)}\\ V\left(y\right)={v}_{0}+z{\varphi }_{\left(yu\right)}+z{\varphi }_{\left(yl\right)}\\ W\left(z\right)={w}_{0}\end{array}$$
(1b)
The stress–strain relationship for the \(p\)-th layer made up of an orthotropic material can be explained as
$$\left\{ {\bar{\sigma }} \right\} = \left[ {\bar{Q}} \right]_{k} \left\{ {\bar{\varepsilon }} \right\}$$
(2)
where \(\left\{\bar{\sigma }\right\}\), \(\left\{\overline{\varepsilon }\right\}\) and \({\left[\overline{Q }\right]}_{k}\) are the stress vector, the strain vector, and the transformed rigidity matrix for the \(k\)-th layer, respectively. Applying the conditions for \({\sigma }_{xz}^{\pm h/2}={\sigma }_{yz}^{\pm h/2}=0\) and \({\sigma }_{xz}^{i}={\sigma }_{xz}^{i+1}\), \({\sigma }_{yz}^{i}={\sigma }_{yz}^{i+1}\) at the interfaces, and \(U\left(x\right)={u}_{u}\), \(V\left(y\right)={v}_{u}\) at the top of the plate, and \(U\left(x\right)={u}_{l}\), \(V\left(y\right)={v}_{l}\) at the bottom of the plate, the unknowns, \({\xi }_{\left(x\right)}\), \({\xi }_{\left(y\right)}\), \({\psi }_{\left(x\right)}\), \({\psi }_{\left(y\right)}\), \({\varphi }_{\left(xu\right)}^{0}\), \({\varphi }_{\left(xu\right)}^{1}\), \({\varphi }_{\left(xu\right)}^{2}\),…, \({\varphi }_{\left(xu\right)}^{{N}^{u}}\), \({\varphi }_{\left(yu\right)}^{0}\), \({\varphi }_{\left(yu\right)}^{1}\), \({\varphi }_{\left(yu\right)}^{2}\),…, \({\varphi }_{\left(yu\right)}^{{N}^{u}}\), \({\varphi }_{\left(xl\right)}^{0}\), \({\varphi }_{\left(xl\right)}^{1}\), \({\varphi }_{\left(xl\right)}^{2}\),…, \({\varphi }_{\left(xl\right)}^{{N}^{l}}\), \({\varphi }_{\left(yl\right)}^{0}\), \({\varphi }_{\left(yl\right)}^{1}\), \({\varphi }_{\left(yl\right)}^{2}\),…, \({\varphi }_{\left(yl\right)}^{{N}^{l}}\) can be represented as
$$\begin{array}{c}{\xi }_{\left(x\right)}={A}_{\left(x\right)}{\phi }_{\left(x\right)}+{B}_{\left(x\right)}\frac{\partial {w}_{0}}{\partial x}+{C}_{\left(x\right)}{u}_{0}\\ {\xi }_{\left(y\right)}={A}_{\left(y\right)}{\phi }_{\left(y\right)}+{B}_{\left(y\right)}\frac{\partial {w}_{0}}{\partial y}+{C}_{\left(y\right)}{v}_{0}\\ \begin{array}{c}{\psi }_{\left(x\right)}={D}_{\left(x\right)}{\phi }_{\left(x\right)}+{E}_{\left(x\right)}\frac{\partial {w}_{0}}{\partial x}+{F}_{\left(x\right)}{u}_{0}\\ {\psi }_{\left(y\right)}={D}_{\left(y\right)}{\phi }_{\left(y\right)}+{E}_{\left(y\right)}\frac{\partial {w}_{0}}{\partial y}+{F}_{\left(y\right)}{v}_{0}\\ \begin{array}{c}{\varphi }_{\left(xu\right)}^{e}={G}_{\left(x\right)}^{e}{\phi }_{\left(x\right)}+{H}_{\left(x\right)}^{e}\frac{\partial {w}_{0}}{\partial x}+{I}_{\left(x\right)}{u}_{0}\\ {\varphi }_{\left(yu\right)}^{e}={G}_{\left(y\right)}^{e}{\phi }_{\left(y\right)}+{H}_{\left(y\right)}^{e}\frac{\partial {w}_{0}}{\partial y}+{I}_{\left(y\right)}{v}_{0}\\ \begin{array}{c}{\chi }_{\left(xl\right)}^{f}={M}_{\left(x\right)}^{f}{\phi }_{\left(x\right)}+{N}_{\left(x\right)}^{f}\frac{\partial {w}_{0}}{\partial x}+{O}_{\left(x\right)}{u}_{0}\\ {\chi }_{\left(yl\right)}^{f}={M}_{\left(y\right)}^{f}{\phi }_{\left(y\right)}+{N}_{\left(y\right)}^{f}\frac{\partial {w}_{0}}{\partial y}+{O}_{\left(y\right)}{v}_{0}\end{array}\end{array}\end{array}\end{array}$$
(3)
where \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(I\) and \(O\) are constants and \(G\), \(H\), \(M\) and \(N\) are coefficients whose values depend on engineering constants of the concerned layer.
The strain vector can be written in terms of structural deformations in the linear range as
$$\left\{\overline{\varepsilon }\right\}=\left[H\right]\left\{\varepsilon \right\}$$
(4)
In order to tackle the challenges that are involved with the management of the C-1 continuity requirement, derivative terms are presented in the form of independent variables as follows
$$\partial w_{0} /\partial x = w_{1} \,{\text{and}}\,\partial w_{0} /\partial y = w_{2}$$
(5)
A nine-node quadratic element having seven degrees of freedom per node \(\left({u}_{0} {v}_{0} {w}_{0} {\phi }_{\left(x\right)} {\phi }_{\left(y\right)} {w}_{1} {w}_{2}\right)\) is employed. We write the displacement vector at any point as
$$\left\{\delta \right\}=\sum_{i=1}^{9}{N}_{i}{\left\{\delta \right\}}_{i}$$
(6)
where \({\left\{\delta \right\}}_{i}\) is the nodal displacement vector associated with the \(i\)-th node and \({N}_{i}\) is the shape function matrix [15]. Substituting Eq. (6) in Eq. (5) in terms of unknown field variables we get
$$\left\{\varepsilon \right\}=[B]\left\{\delta \right\}$$
(7)
where, \([B]\) is the strain–displacement relationship in cartesian coordinate.
Static bending analysis: For a plate subjected to mechanical load \(q=f\left(x,y\right)\), the total potential energy can be written as the difference between the strain energy \(\left({U}_{s}\right)\) of the plate and the energy due to the external loading \(\left({W}_{ext}\right)\)
$${\Pi }_{e}={U}_{s}-{W}_{ext}$$
(8)
$${U}_{s}=\frac{1}{2}\sum_{k=1}^{n}{\int }_{0}^{a}{\int }_{0}^{b}{\int }_{-\frac{h}{2}}^\frac{h}{2}{\left\{\overline{\varepsilon }\right\}}^{T}{\left[\overline{Q }\right]}_{k}\left\{\overline{\varepsilon }\right\}\text{dxdydz}=\frac{1}{2}{\int }_{0}^{a}{\int }_{0}^{b}\left[{\left\{\varepsilon \right\}}^{T}\left[D\right]\left\{\varepsilon \right\}\right]\text{dxdy}$$
(9)
where \(\left[D\right]=\sum_{k=1}^{n}\int {\left[H\right]}^{T}{\left[\overline{Q }\right]}_{k}\left[H\right]\text{dz}\).
$$W_{ext} = \mathop \smallint \limits_{0}^{a} \mathop \smallint \limits_{0}^{b} wqdxdy\,{\text{or}}\,W_{ext} = \mathop \smallint \limits_{0}^{a} \mathop \smallint \limits_{0}^{b} w\left\{ \delta \right\}^{T} \left[ {N_{c} } \right]^{T} qdxdy$$
(10)
By combining the Eqs. (8)–(10), the total potential energy at the elemental level can be stated as
$${\Pi }_{e}=\frac{1}{2}{\int }_{0}^{a}{\int }_{0}^{b}{\left\{\delta \right\}}^{T}{\left[B\right]}^{T}\left[D\right]\left[B\right]\left\{\delta \right\}\text{dxdy}-\frac{1}{2}{\int }_{0}^{a}{\int }_{0}^{b}{\left\{\delta \right\}}^{T}{\left[B\right]}^{T}{\left[{N}_{c}\right]}^{T}q\text{dxdy}$$
(11)
$${\Pi }_{e}=\frac{1}{2}{\left\{\delta \right\}}^{T}\left[{K}_{e}\right]\left\{\delta \right\}-\frac{1}{2}{\left\{\delta \right\}}^{T}\left[{P}_{e}\right]$$
(12)
where \(\left[{K}_{e}\right]={\int }_{0}^{a}{\int }_{0}^{b}{\left[B\right]}^{T}\left[D\right]\left[B\right]\text{dxdy}+\left[{P}_{p}\right]\) and \(\left[{P}_{e}\right]={\int }_{0}^{a}{\int }_{0}^{b}{\left[{N}_{c}\right]}^{T}q\text{dxdy}\) are the elemental stiffness matrix and mechanical load vector, respectively. The matrix \({\left[{N}_{c}\right]}^{T}\) is the shape function like matrix composed of nodal transverse displacement at the concerned node. The penalty term \(\left[{P}_{p}\right]\) is defined as
$$\left[{P}_{p}\right]={\int }_{0}^{a}{\int }_{0}^{b}\beta \left[{\left\{\frac{\partial w}{\partial x}-{w}_{1}\right\}\left\{\frac{\partial w}{\partial x}-{w}_{1}\right\}}^{T}+\left\{\frac{\partial w}{\partial y}-{w}_{2}\right\}{\left\{\frac{\partial w}{\partial y}-{w}_{2}\right\}}^{T}\right]\text{dxdy}$$
(13)
where \(\beta\) is the penalty multiplier. After minimizing Eq. (12) with respect to \(\left\{\delta \right\}\), we get
$$\left[{K}_{e}\right]\left\{\delta \right\}=\left\{{P}_{e}\right\}$$
(14)
Based on the above equations and subsequent assembling of the elementary-level structural matrices to the global level, static analysis can be carried out including deflection, stresses and strains.
Free vibration analysis: Using Hamilton's principle of the equation of motion we get
$${\int }_{0}^{t}\delta \left(T-U\right)dt=0$$
(15)
where \(\left(T\right)\) is kinetic energy, \(T=\frac{1}{2}\int \rho \left({\dot{U}\left(x\right)}^{2}+{\dot{V}\left(y\right)}^{2}+{\dot{W}\left(z\right)}^{2}\right)\text{dxdy}\) and \(\dot{U}\left(x\right)\), \(\dot{V}\left(y\right)\) and \(\dot{W}\left(z\right)\) are the derivatives of \(U\left(x\right)\), \(V\left(y\right)\), and \(W\left(w\right)\) respectively and \(\rho\) is the density of the material. The potential energy \(\left(U\right)\) of the plate is
$$U=\frac{1}{2}{\int }_{V}{\sigma }_{xx}{\varepsilon }_{xx}+{\sigma }_{yy}{\varepsilon }_{yy}+{\sigma }_{xz}{\varepsilon }_{xz}+{\sigma }_{yz}{\varepsilon }_{yz}+{\sigma }_{xy}{\varepsilon }_{xy}\text{dV}$$
$${\text{or}},\,U = \frac{1}{2}\mathop \sum \limits_{k = 1}^{n} \iint {\left\{ {\overline{\varepsilon }} \right\}^{T} \left[ Q \right]\left\{ {\overline{\varepsilon }} \right\}dxdy}$$
(16)
Equation (16) neglects external work done by external forces and damping. Hamilton's principle leads to the equilibrium equation of a system, which can be stated as
$$\left[M\right]\left\{\frac{{\partial }^{2}\overline{\delta }}{\partial {t}^{2}}\right\}+\left[K\right]\left\{\overline{\delta }\right\}=0$$
(17)
where \(\left[M\right],\left[K\right],\left\{\overline{\delta }\right\}, \left\{\frac{{\partial }^{2}\overline{\delta }}{\partial {t}^{2}}\right\}\) are global mass matrix, stiffness matrix, nodal variable vector, and acceleration vector of the system, respectively. Frequency \(\left(\uplambda \right)\) can be worked out using Eq. (18)
$$\left[K\right]\left\{\overline{\delta }\right\}={\uplambda }^{2}\left[M\right]\left\{\overline{\delta }\right\}$$
(18)
Acceleration at any point within the plate can be written as
$$\left\{\frac{{\partial }^{2}\overline{\delta }}{\partial {t}^{2}}\right\}=\left\{\begin{array}{c}\begin{array}{c}\frac{{\partial }^{2}\overline{U }\left(x\right)}{\partial {t}^{2}}\\ \frac{{\partial }^{2}\overline{V }\left(y\right)}{\partial {t}^{2}}\end{array}\end{array}\right\}=-{\uplambda }^{2}\left\{\begin{array}{c}\begin{array}{c}U\left(x\right)\\ V\left(y\right)\end{array}\end{array}\right\}=-{\uplambda }^{2}\left[\Delta \right]\left\{\delta \right\}$$
(19)
where \(\left[\Delta \right]\) is a matrix having terms of z and some constant values. As the elementary stiffness matrix \(\left[{K}_{e}\right]\) is derived, the elementary mass matrix (and subsequently the global mass matrix by assembling the elementary level matrices) can be worked out following a similar way.
$$\left[M\right]=\sum_{i=1}^{{N}^{(u)}+{N}^{(l)}}\int {\rho }_{i}{\left[N\right]}^{T}{\left[\Delta \right]}^{T}\left[N\right]\left[\Delta \right]\text{dxdy}=\int {\left[N\right]}^{T}\left[L\right]\left[N\right]\text{dxdy}$$
(20)
where \({\rho }_{i}\) is the mss density of the \(i\)-th layer. \(\left[L\right]\) in the above Equation can be written as
$$L=\sum_{i=1}^{{N}^{u}+{N}^{l}}\int {\rho }_{i}{\left[\Delta \right]}^{T}\left[\Delta \right]\text{dx}$$
(21)
Buckling study: For buckling analysis, strain can be written as
$$\left\{\overline{\varepsilon }\right\}={\left\{\overline{\varepsilon }\right\}}_{Linear}+{\left\{\overline{\varepsilon }\right\}}_{Non-linear}$$
(22)
\({\left\{\overline{\varepsilon }\right\}}_{Linear}\) can be determined as per Eq. (5), while for the second term of the above equation.
\({\left\{\overline{\varepsilon }\right\}}_{Non-linear}=\left[\begin{array}{c}{\frac{1}{2}\left(\frac{\partial W(z)}{\partial x}\right)}^{2}+ {\frac{1}{2}\left(\frac{\partial U\left(x\right)}{\partial x}\right)}^{2}+ {\frac{1}{2}\left(\frac{\partial V(y)}{\partial x}\right)}^{2}\\ {\frac{1}{2}\left(\frac{\partial W(z)}{\partial y}\right)}^{2}+ {\frac{1}{2}\left(\frac{\partial U\left(x\right)}{\partial y}\right)}^{2}+ {\frac{1}{2}\left(\frac{\partial V(y)}{\partial y}\right)}^{2}\\ \left(\frac{\partial W(z)}{\partial x}\right)\left(\frac{\partial W(z)}{\partial y}\right)+ \left(\frac{\partial U\left(x\right)}{\partial x}\right)\left(\frac{\partial U\left(x\right)}{\partial y}\right)+ \left(\frac{\partial V(y)}{\partial x}\right)\left(\frac{\partial V(y)}{\partial y}\right)\end{array}\right]\) or
$${\left\{\overline{\varepsilon }\right\}}_{Non-linear}=\frac{1}{2} \left[\begin{array}{cc}\begin{array}{cc}\frac{\partial W(z)}{\partial x}& \begin{array}{cc}0& \frac{\partial U\left(x\right)}{\partial x}\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}0 & \frac{\partial V\left(y\right)}{\partial x}\end{array}& 0\end{array}\\ \begin{array}{cc}0& \begin{array}{cc}\frac{\partial W(z)}{\partial y}& 0\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}\frac{\partial U\left(x\right)}{\partial y}& 0\end{array}& \frac{\partial V\left(y\right)}{\partial y}\end{array}\\ \begin{array}{cc}\frac{\partial W(z)}{\partial y}& \begin{array}{cc}\frac{\partial W(z)}{\partial x}& \frac{\partial U\left(x\right)}{\partial y}\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}\frac{\partial U\left(x\right)}{\partial x}& \frac{\partial V\left(y\right)}{\partial y}\end{array}& \frac{\partial V\left(y\right)}{\partial x}\end{array}\end{array}\right]\left\{\begin{array}{c}\frac{\partial W(z)}{\partial x}\\ \begin{array}{c}\frac{\partial W(z)}{\partial y}\\ \begin{array}{c}\frac{\partial U\left(x\right)}{\partial x}\\ \begin{array}{c}\frac{\partial U\left(x\right)}{\partial y}\\ \begin{array}{c}\frac{\partial V\left(y\right)}{\partial x}\\ \frac{\partial V\left(y\right)}{\partial y}\end{array}\end{array}\end{array}\end{array}\end{array}\right\}=\frac{1}{2}\left[{A}_{G}\right]\left[\Lambda \right]=\frac{1}{2}\left[{H}_{G}\right]\left[B\right]\left\{\delta \right\}$$
The elements contained in \(\left[H\right], \left[{A}_{G}\right]\) and \(\left[{H}_{G}\right]\) are functions of unit step function and thickness coordinate. The geometric stiffness matrix \(\left[{K}_{ge}\right]\) can be derived as
$$\left[{K}_{ge}\right]= \sum_{i=1}^{nu+nl}\iiint {\left[B\right]}^{T}\left[{S}^{k}\right] \left[B\right]\text{dxdydz}$$
(23)
where \(\left[{S}^{k}\right]\) is the in-plane stress components of the \(k\)-th layer that can be written as
$$\left[{S}^{k}\right]= \left[\begin{array}{ccc}\begin{array}{ccc}\begin{array}{c}{\sigma }_{xx}\\ {\tau }_{xy}\\ \begin{array}{c}0\\ 0\\ \begin{array}{c}0\\ 0\end{array}\end{array}\end{array}& \begin{array}{c}{\tau }_{xy}\\ {\sigma }_{yy}\\ \begin{array}{c}0\\ 0\\ \begin{array}{c}0\\ 0\end{array}\end{array}\end{array}& \begin{array}{c}0\\ 0\\ \begin{array}{c}{\sigma }_{xx}\\ {\tau }_{xy}\\ \begin{array}{c}0\\ 0\end{array}\end{array}\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ 0\\ {\tau }_{xy}\end{array}\\ {\sigma }_{yy}\\ \begin{array}{c}0\\ 0\end{array}\end{array}& \begin{array}{c}\begin{array}{cc}0& 0\end{array}\\ \begin{array}{cc}0& 0\end{array}\\ \begin{array}{c}\begin{array}{cc}0& 0\end{array}\\ \begin{array}{cc}0& 0\end{array}\\ \begin{array}{c}\begin{array}{cc}{\sigma }_{xx}& {\tau }_{xy}\end{array}\\ \begin{array}{cc}{\tau }_{xy}& {\sigma }_{yy}\end{array}\end{array}\end{array}\end{array}\end{array}\right]$$
The \({U}_{s}\) for determining the total potential energy of the plate can be determined as per Eq. (9) whereas the \({W}_{ext}\) is calculated as
$${W}_{ext}=\iiint {\left\{\overline{\varepsilon }\right\}}_{Non-linear}^{T}\left[{S}^{k}\right]{\left\{\overline{\varepsilon }\right\}}_{Non-linear}\text{dxdydz}$$
(24)
Thus, the total potential energy of the plate under buckling conditions can be stated as
$${\Pi }_{e}=\frac{1}{2}\iint {\left\{\delta \right\}}^{T}{\left[B\right]}^{T}\left[D\right]\left[B\right]\left\{\delta \right\}\text{dxdy}-\frac{1}{2}\iint {\left\{\delta \right\}}^{T}{\left[B\right]}^{T}\left[G\right]\left[B\right]\left\{\delta \right\}\text{dxdy}=\frac{1}{2}{\left\{\delta \right\}}^{T}\left[{K}_{e}\right]\left\{\delta \right\}-\frac{1}{2}\alpha {\left\{\delta \right\}}^{T}\left[{K}_{ge}\right]\left\{\delta \right\}$$
(25)
where \(\left[ G \right] = \mathop \sum \limits_{k = 1}^{n} \smallint \left[ H \right]^{T} \left[ {S^{k} } \right]\left[ H \right]{\text{dz}}{.}\)
For buckling analysis, in the first step, a static problem is solved as per the governing Eq. (11) to calculate stresses at the Gauss points of different elements for a given loading condition. Finally, these stresses are used to form the matrix \(\left[{S}^{k}\right]\) of the geometric stiffness matrix. Minimizing the Eq. (25) with respect to \(\left\{\delta \right\}\) will yield
$$\left[{K}_{e}\right]\left\{\delta \right\}=\alpha \left[{K}_{ge}\right]\left\{\delta \right\}$$
(26)
where \(\lambda\) is the buckling load factor.
Plate behavior may be calculated by first determining the contribution of each individual element, and then solving the resulting equations after imposing the required boundary conditions. A single array is used to hold the global stiffness matrix when the skyline approach is implemented. For the purpose of carrying out the static investigation, a Gaussian decomposition technique has been designed. The investigation of free vibration and buckling has been carried out with the use of simultaneous iteration approach.

2.2 Gaussian process based machine learning model

Gaussian Process (GP) is a supervised machine learning algorithm that takes a probabilistic, non-parametric and Bayesian approach to the regression model. The fact that the GP regression (GPR) model can readily interpolate the observations, that it functions well with small datasets, and that it allows the user to specify the kernel function in accordance with their preferences are the primary advantages of this model. The Bayesian methodology involves performing a study of the probability distribution on each of the attainable values. Means may be assessed as functions, together with the variances that reflect the confidence levels for the anticipated value [54]. Because the GPR model is based on probabilities, it is possible to make an accurate prediction of the outcome by applying a previously trained instance of the GPR model to the data in question. The general form of regression function using the multivariate Gaussian can be modeled as [12]
$$P\left( {{\mathbf{f}}|{\mathbf{X}}} \right) = \mathcal{N}\left( {{\mathbf{f}}\left| {{\varvec{\mu}},{\mathbf{K}}} \right.} \right)$$
(27)
where \(\mathbf{X}=\left[{x}_{1}, {x}_{2},\dots ,{x}_{n}\right]\) represents the studied data point, the joint distribution of the random variables \(\mathbf{f}=\left[f\left({\text{x}}_{1}\right),f\left({\text{x}}_{2}\right),\dots ,f\left({\text{x}}_{n}\right)\right]\) is Gaussian, \({\varvec{\mu}}=[m\left({\text{x}}_{1}\right),m\left({\text{x}}_{2}\right),\dots ,m\left({\text{x}}_{n}\right)]\) denotes GP mean function and \({\mathbf{K}} = k\left( {x, x^{\prime } } \right)\) is the covariance function, where \(k\) represents the kernel function. \(\mathbf{K}\) governs the shape or smoothness of the distribution function. For data, which is normalized to zero mean, the mean function automatically turns zero \(\left(m\left(\text{x}\right)=0\right)\) having no observation. If \(\left\{f\left(\text{x}\right),\text{x}\in {\mathbb{R}}^{d}\right\}\) is GP, then
$$E\left(f\left(\text{x}\right)\right)=m(\text{x})$$
(28)
$$Cov\left[ {f\left( x \right),f\left( {x^{\prime } } \right)} \right] = E\left[ {\left\{ {f\left( x \right) - m\left( x \right)} \right\}\left\{ {f\left( {x^{\prime } } \right) - m\left( {x^{\prime } } \right)} \right\}} \right] = k\left( {x, x^{\prime } } \right)$$
(29)
A mean function, \(\mathbf{f}\), is estimated with the assistance of the data points that have been provided, and the same function is used for the purpose of forecasting the value at a new \({\mathbf{X}}_{\boldsymbol{*}}\) point that has not been observed before as \(\mathbf{f}\left({\mathbf{X}}_{\boldsymbol{*}}\right)\). The joint distribution of \(\mathbf{f}\) and \({\mathbf{f}}_{\boldsymbol{*}}\) is written as
$$\left[\begin{array}{c}\mathbf{f}\\ {\mathbf{f}}_{\boldsymbol{*}}\end{array}\right]\sim \mathcal{N}\left(\left[\begin{array}{c}m\left(\mathbf{X}\right)\\ m\left({\mathbf{X}}_{\boldsymbol{*}}\right)\end{array}\right],\left[\begin{array}{cc}\mathbf{K}& {\mathbf{K}}_{\boldsymbol{*}}\\ {\mathbf{K}}_{*}^{T}& {\mathbf{K}}_{\boldsymbol{*}\boldsymbol{*}}\end{array}\right]\right)$$
(30)
where \(\mathbf{K}=K\left(\mathbf{X},\mathbf{X}\right)\), \({\mathbf{K}}_{\boldsymbol{*}}=K\left(\mathbf{X},{\mathbf{X}}_{\boldsymbol{*}}\right)\), \({\mathbf{K}}_{\boldsymbol{*}\boldsymbol{*}}=K\left({\mathbf{X}}_{\boldsymbol{*}},{\mathbf{X}}_{\boldsymbol{*}}\right)\) and \(\left[m\left(\mathbf{X}\right), m\left({\mathbf{X}}_{\boldsymbol{*}}\right)\right]=0\). The joint form of the probability distribution is written as \(P\left( {{\mathbf{f}},{\mathbf{f}}_{\user2{*}} \left| {{\mathbf{X}},{\mathbf{X}}_{\user2{*}} } \right.} \right),\) and it is applied across both \(\mathbf{f}\) and \({\mathbf{f}}_{\boldsymbol{*}}\). Nevertheless, the distribution \(P\left( {{\mathbf{f}}_{{\mathbf{*}}} \left| {{\mathbf{f}},{\mathbf{X}},{\mathbf{X}}_{{\mathbf{*}}} } \right.} \right)\) over \({\mathbf{f}}_{\boldsymbol{*}}\) is the only one required for regression. The process of conversion is explained in Appendix A, giving output as
$${\mathbf{f}},{\mathbf{f}}_{\user2{*}} |{\mathbf{X}},{\mathbf{X}}_{\user2{*}} \sim\mathcal{N}\left( {{\mathbf{K}}_{*}^{T} {\mathbf{K}} {\mathbf{f}}, {\mathbf{K}}_{{\user2{**}}} - {\mathbf{K}}_{*}^{T} {\mathbf{K}}^{ - 1} \user2{ }{\mathbf{K}}_{\user2{*}} } \right)$$
(31)
There is some noise in the approximated function, and as a result, the operation exists as y = f(x) + c. In the event that there is such a thing as additive identically distributed and independent Gaussian noise with a variance \({\sigma }_{n}^{2}\) the prior on the noisy observations becomes \(Cov\left(y\right)=\mathbf{K}+{\sigma }_{n}^{2}I\) in this scenario. It may be described as incorporating the noise, the anticipated value, and the function values at new testing sites as
$$\left( {\begin{array}{*{20}c} {\mathbf{y}} \\ {{\mathbf{f}}_{\user2{*}} } \\ \end{array} } \right)\sim \mathcal{N}\left( {0,\left[ {\begin{array}{*{20}c} {{\mathbf{K}} + \sigma_{n}^{2} I} & {{\mathbf{K}}_{\user2{*}} } \\ {{\mathbf{K}}_{*}^{T} } & {{\mathbf{K}}_{{\user2{**}}} } \\ \end{array} } \right]} \right)$$
(32)
Using conditional distribution, GPR predictive Eq. is obtained as
$${\overline{\mathbf{f}}}_{{\mathbf{*}}} \left| {{\mathbf{X}},{\mathbf{y}},{\mathbf{X}}_{*} } \right. \sim \mathcal{N}\left( {{\overline{\mathbf{f}}}_{*} ,{\text{cov}} \left( {{\mathbf{f}}_{*} } \right)} \right)$$
(33)
where
$${\overline{\mathbf{f}}}_{*} \triangleq \mathbb{E}\left[ {{\overline{\mathbf{f}}}_{{\mathbf{*}}} \left| {{\mathbf{X}},{\mathbf{y}},{\mathbf{X}}_{{\mathbf{*}}} } \right.} \right] = {\mathbf{K}}_{*}^{T} \left[ {{\mathbf{K}} + \sigma_{n}^{2} I} \right]^{ - 1} {\mathbf{y}}$$
(34)
$${\text{cov}} \left( {{\mathbf{f}}_{{\mathbf{*}}} } \right) = {\mathbf{K}}_{**} - {\mathbf{K}}_{*}^{T} \left[ {{\mathbf{K}} + \sigma_{n}^{2} I} \right]^{ - 1} {\mathbf{K}}_{*}$$
(35)
It can be seen that \(\text{cov}\left({\mathbf{f}}_{\boldsymbol{*}}\right)\), i.e., variance depends only on the inputs \(\mathbf{X}\) and \({\mathbf{X}}_{\boldsymbol{*}}\), but not on the output \(\mathbf{y}\). This property is exhibited by Gaussian distribution.

2.3 Adaptive elementary-level framework

In most the articles available in the literature, machine learning techniques are exploited for determining the behavior of laminated composite structures following a non-intrusive framework. In such cases the surrogate based computational mapping is developed for global responses like natural frequency, buckling or deflection as a function of the input parameters. Thus, multiple surrogate models need to be built to explore different responses and corresponding number of expensive finite element simulations are required. In this article we aim to integrate the machine learning model at the elementary level, so that same structural matrices can be utilized for various types of analyses. For example, a stiffness matrix is useful in static deflection, vibration and buckling analyses. We will augment the elementary level stiffness matrices obtained using FSDT to that of HOZT employing gaussian process based machine learning models. A spatially-adaptive scheme will be used based on the best theory diagram [5, 42] for choosing the elements optimally that need to be augmented. A detailed description about the proposed machine learning assisted computational methodology is presented following a diagrammatic form in Fig. 2.
Preparing dataset for training GPR: For building a GPR model, the plate is assumed to be made up of ten layers. Each layer can have any ply orientation. The variables adopted during the dataset preparation are material properties, thickness, and ply-angle of each layer, along with the dimension of the elements. Thus, the proposed model will be able to capture a generic behavior of the plate at the element level. Sobol sequence [34] is used to determine the training set based on a quasi-random distribution. The Sobol sequence will have upper bound and lower bound values equal to one and zero, respectively. The upper and lower bound values for the variables used are reported in Table 1. Using a linear interpolation, the actual values of Table 1 are bridged with the normalized domain of Sobol sequence so that they can further be used for machine learning formation.
Table 1
Upper and lower bound values taken for determining the material properties using Sobol sequences (for element length along X- and Y-axis, the values given by Sobol are used)
 
\({E}_{11}\) (GPa)
\({E}_{22}={E}_{33}\) (GPa)
\({G}_{12}={G}_{13}\) (GPa)
\({G}_{23}\) (GPa)
\({\nu }_{12}={\nu }_{13}\)
\({\nu }_{23}\)
\(\theta^\circ\)
Lower bound value
20
1
0.5
0.4
0.22
0.25
− 90
Upper bound value
300
10.5
8
7
0.28
0.5
90
Training GPR model: The difference between the elements of the stiffness matrix of HOZT and FSDT is worked out for different sets of input parameters and is given as output value to be predicted by the trained GPR model. Each GPR model will give the corresponding value for the element of the stiffness matrix (in the form of a difference between HOZT and FSDT based stiffness matrix), by using which we proposed to develop the augmented FSDT (aFSDT) based elementary stiffness matrix. Note that such an aFSDT based elementary stiffness matrix is equivalent to that of HOZT in terms of accuracy while obtained almost at the computational cost of FSDT. A diagrammatic representation of the upper triangular matrix showing the difference in the elements of the stiffness matrix (for HOZT and FSDT) for the first twenty-one columns is presented in Fig. 3. It can be noticed that many of the matrix elements are repeated, and some of them are zero. Hence, a much lower number of independent GPR models is required in the analysis than the number of elements in the elementary stiffness matrix. The surrogate model formation becomes multidimensional in terms of output in the current context. It may be noted here that separate simulations are not needed to obtain different elements of the stiffness matrix. We consider that the computational cost for surrogate model formation, once all the training data are generated, is negligible compared to that required for even a single training data generation based on the actual simulation model. Thus, the upfront cost primarily lies in generating the training data, not in forming the model itself. The obtained dataset is split randomly into three parts for training, validation, and testing of the surrogate GPR algorithm. From the dataset, 80% of the data is used for training the algorithm whereas the remaining 10% each for validating and testing the accuracy of the surrogate. This random process is repeated multiple times to ensure the accuracy of prediction.
It is worth noting here that most of the predominant data-driven machine learning-based approaches (a simulation-based model as considered here because of the complexity of composite structures) are limited to the trained dataset. However, considering that we have incorporated a large range of influencing input parameters and the machine learning is integrated at the elementary stiffness level, the prediction capability here is quite significant. The capability of the proposed approach can be further improved by expanding the input dataset, albeit a similar approach can be followed. The current work does not include boundary conditions of composite laminates in the machine-learning model because of the fact that the machine learning model is incorporated at the stiffness level, and the incorporation of boundary conditions in finite element analysis comes after the machine learning prediction or augmentation of the stiffness elements. In this context, it can be noted that the computational savings of the proposed machine learning based analyses should not be judged based on a single prediction, but the powerful capability in terms of carrying out simulations where multiple (thousands of simulations) realizations are necessary such as optimization, detailed parametric study, uncertainty quantification, reliability analysis etc. In such cases, the computational efficiency will be tremendously high.
Implementation of GPR model in a spatially adaptive framework: The built ML models are used to determine the difference in elements of the stiffness matrix between HOZT and FSDT based formulation for an unseen problem. The predicted differences are added to the corresponding elements of the FSDT based stiffness matrix for the same problem to augment it to aFSDT based elements. This effectively upgrades the elementary stiffness matrix from a lower theory (FSDT) to an advanced theory (HOZT) without extra computational effort. At this stage, we identify which elements are necessary to be upgraded based on their susceptibility to complex nonlinear deformation. In other words, to achieve computational efficiency, the elements in the plate are identified where implementing FSDT does not affect the plate's behavior to a large extent, compared to the HOZT based predicted behavior for the same elements. We exploit the concept of best theory diagram (BTD) for this purpose. For different placements of FSDT and higher-order FEs in plate, the configuration which gives minimum percentage error is adopted. By analyzing these BTDs in detail, it is observed that for all edges clamped or simply supported boundary conditions, distributing FSDT elements in the corner region along with higher-order elements at the mid-span portion yields good results. For free edge, lower-order elements should be present at the mid-edge portion. In order to determine the effective distribution of lower and higher-order elements, such criteria based on spatial distribution of critical stress resultants are adopted here based on a detailed study of the percentage difference in the quantities of interest for various prospective placements of FSDT and aFSDT elements. When the percentage error is plotted against the number of higher-order elements, a crescent-moon-shaped curve is obtained (refer to Fig. 2).
The spatially-adaptive reconstruction based on BTD is a key aspect of our proposed computational framework. The strategy is designed to selectively augment the lower-order theory (FSDT) to higher-order theory (HOZT-equivalent aFSDT) elements based on the local requirements of accuracy and computational efficiency. The decision to augment elements is made by analyzing the behavior of the structure under different conditions, considering areas where higher-order theory provides significant improvements. The BTD approach is not intended to be static; it is adaptable based on the structural characteristics and loading conditions. The diagram serves as a visual tool to map regions where adopting higher-order elements is necessary, allowing for efficient placement of augmented elements. A general trend can be noticed from the figures concerning the best theory diagram (as presented later in Figs. 5, 8 and 10) that for SSSS boundary conditions the elements away from the four corners are more critical for augmenting to higher-order theory. In the case of CCFF boundary condition, elements away from the central portion of the free edges are more critical. These trends are true for all three kinds of analyses presented in this paper concerning, static, vibration and buckling. A comparative insight into the FSDT and HOZT leads to identifying such critical elements (i.e. the zone of critical elements) that need to be augmented from FSDT to HOZT. Note that HOZT is based on the assumption that the displacement field (both transverse shear deformation and transverse normal deformation) through the thickness of the plate is zigzag-shaped, which means that the displacement does not vary smoothly from one layer to the next but instead exhibits abrupt changes in slope at the interfaces between layers. This accurately captures the kinematic behavior of multilayered plates where stiffnesses between layers differ significantly. On the contrary, FSDT assumes continuous displacement fields and approximates them insufficiently, leading to inaccuracies in thick or layered plates. We note from the trend discussed above that the zones with higher bending stress resultant become more critical for augmenting from FSDT to HOZT. Since obtaining the spatial distribution of such bending stress resultant (even accurate calculations are not necessary and just a qualitative idea is sufficient) for a simple plate-like structure (consideration of the actual layered structure is also not necessary) under different boundary and loading conditions is rather a trivial exercise, the powerful and generic capability of the algorithm for selectively and optimally augmenting lower to higher order theories can be readily realized. Thus, the proposed selective augmentation strategy essentially brings in a scope of integrating physics-based insights of bending stress resultant distribution into the algorithm.
In the next step, the optimally identified (under the constraint of the required level of accuracy and computational efficiency) set of FSDT and aFSDT based elementary stiffness matrices are assembled to form the global stiffness matrix and subsequently used to analyze the plate's bending, free vibration, and buckling behavior. Since the GPR model is implemented at an intermediate step during analysis, the present methodology is generic in terms of its ability to carry out different such analyses. The proposed idea of selective augmentation of the FSDT-based elements in a spatially adaptive framework is quite fundamental in nature and it can further be implemented to characterize plates with spatially varying system properties such as spatially varying material properties or degraded stiffness due to damage or defects. Moreover, since we propose to incorporate the element size as a machine learning parameter, it is possible to have different number of elements in the analysis domain with spatially varying effective material attributes, damage parameters and the most optimum distribution of elements with different orders of theory in complex geometries.

3 Results and discussion

In this section, we will first present validation concerning the finite element models and subsequently machine learning-based predictions. Detailed numerical results will be presented thereafter for static, dynamic and stability characterization of composite laminates including the demonstration of spatially variable stiffness configurations based on the efficient machine learning-assisted elementary-level fidelity-sensitive intrusive computational framework. Following are the relations used to convert the dimensional quantities into respective non-dimensional forms: non-dimensional deflection: \(\overline{w }=\frac{100{E}_{T}{h}^{3}w}{{a}^{4}q}\), non-dimensional stresses: \({\bar{\sigma }}_{xx}=\frac{{h}^{2}{\sigma }_{xx}}{{a}^{2}q}\), \(\left({\bar{\sigma }}_{xz},{\bar{\sigma }}_{yz}\right)=\frac{h\left({\sigma }_{xx},{\sigma }_{yz}\right)}{aq}\), non-dimensional natural frequency: \(\overline{\uplambda }=\frac{\uplambda {a}^{2}}{h}\sqrt{\frac{\rho }{{E}_{T}}}\), non-dimensional critical buckling load: \(\overline{\alpha }=\frac{\alpha {a}^{2}}{{E}_{T}{h}^{2}}\). Engineering constants used during the present study are \({E}_{1}=25{E}_{2}\), \({E}_{2}={E}_{3}=1E6\), \({G}_{12}={G}_{13}=0.5{E}_{2}\), \({G}_{23}=0.2{E}_{2}\), \({\nu }_{12}={\nu }_{13}={\nu }_{23}=0.25\), \(\rho =100\) (for bending and free vibration study, referred to as MAT-A). For buckling study, following are the values of engineering constants employed: \({E}_{1}=40{E}_{2}\), \({E}_{2}={E}_{3}=1E6\), \({G}_{12}={G}_{13}=0.6{E}_{2}\), \({G}_{23}=0.5{E}_{2}\), \({\nu }_{12}={\nu }_{13}={\nu }_{23}=0.25\) (referred to as MAT-B). For 4-layered laminated composite plates, the material properties used are: \({E}_{1}=181 \text{GPa}\), \({E}_{2}={E}_{3}=10.3 \text{GPa}\), \({G}_{12}={G}_{13}=7.17 \text{GPa}\), \({G}_{23}=2.87 \text{GPa}\), \({\nu }_{12}={\nu }_{13}=0.28\), \({\nu }_{23}=0.33\), \(\rho =1578 \text{kg}/{\text{m}}^{3}\) (referred to as MAT-C). Note that the proposed machine learning-based approach can deal with a wide range of material properties. We have considered these three sets (MAT-A, MAT-B and MAT-C) of material properties depending on the validation literature available and also to demonstrate the capability of the GP model in terms of its ability to deal with different material properties.
For validating the proposed model and determining the converged mesh size, the plate is analyzed for different mesh sizes, and the results are compared with those available in the literature. Results for bending, buckling, and free vibration are reported in Tables 2 and 3, respectively. A 3-layered (0°/90°/0°) all-around simply supported laminated composite plate is considered for bending, vibration and buckling analysis having all layers of equal thickness. The results converge at a mesh size of 10 × 10. The same is adopted for subsequent studies unless otherwise mentioned. The present results agree well with 3D Elasticity results published by Pagano [43] for bending and HOZT based results published by Chalak et al. [6, 7] for free vibration and buckling analysis of laminated composite plates. Note that the present HOZT results are closer to the 3D elasticity solutions compared to FSDT predictions (i.e. HOZT obtains more accurate results).
Table 2
Non-dimensional displacement and stresses for simply supported square-shaped 3-layered laminated composite plate (0°/90°/0°) subjected to sinusoidal loading (Considered material properties: MAT-A, \(\% {\text{Error}} = \left(( {{\text{Converged present results}} - 3{\text{D Elasticity results}}} \right)/(3{\text{D Elasticity results))}}\)
\(a/h\)
Source
\(\overline{w }\)
\({\bar{\sigma }}_{xx}\)
\({\bar{\sigma }}_{xz}\)
100
Present FSDT (4 × 4)
0.4398
0.5591
0.4103
Present FSDT (8 × 8)
0.4367
0.5389
0.3892
Present FSDT (10 × 10)
0.4356
0.5330
0.3861
Present FSDT (12 × 12)
0.4356
0.5329
0.3859
Present HOZT (4 × 4)
0.4379
0.5582
0.4225
Present HOZT (8 × 8)
0.4353
0.5443
0.3952
Present HOZT (10 × 10)
0.4349
0.5404
0.3919
Present HOZT (12 × 12)
0.4349
0.5402
0.3916
3D elasticity [40]
0.4347
0.5392
0.3947
% error (FSDT)
0.00207
− 0.01168
− 0.02229
% error (HOZT)
0.00046
0.001855
− 0.00785
10
Present FSDT
0.7164
0.5831
0.3467
Present HOZT
0.7479
0.5920
0.3510
3D elasticity [40]
0.7530
0.5908
0.3573
% error (FSDT)
− 0.04861
− 0.01303
− 0.02967
% error (HOZT)
− 0.00677
0.002031
− 0.01763
Table 3
Non-dimensional natural frequency and critical buckling load for 3-layered (0°/90°/0°) simply supported laminated composite plates having \(\left(a/h=10\right)\), (Considered material properties: MAT-A for free vibration study, MAT-B for buckling study)
Source
\(\overline{\lambda }\)
\(\overline{\alpha }\)
Present FSDT
18.3241
10.2228
Present HOZT
16.7089
10.1436
[6, 7]
16.7165
10.1485
After the convergence and validation study, the difference between the stiffness matrix elements derived using HOZT and FSDT is worked out for different sets of inputs obtained using the Sobol sequence, as discussed in the preceding section. At first GPR model is trained for the different number of data sets for each element of the stiffness matrix. The scatter plot (agreement line diagram) between the actual and the predicted values of the elements of the stiffness matrix is investigated. The number of data sets for which most of the values lie on or very near the 45° line is adopted. Figure 3 shows the typical agreement line diagram for three elements of the stiffness matrix. For the data set of 500, most of the values lie on/near the agreement line. Therefore, the same training sample size has been used to train the GPR model.
Figure 4 shows the variation of non-dimensional stresses across the thickness for a 3-layered square-shaped laminated composite plate with all edges simply supported \((a/h=10)\). The stresses are obtained for the plate having all elements FE-HOZT. The variation of stresses is also plotted using the upgradation of elements of the stiffness matrix from FSDT to HOZT exploiting a trained GPR model (i.e. using aFSDT elements). The stresses predicted by both methodologies are in good agreement compared with the 3D Elasticity-based solutions given by Pagano [40]. This shows the capability of aFSDT in predicting HOZT equivalent results, which in turn are further compared with 3D elasticity results to ascertain their level of accuracy. In Fig. 4, note that we have not used the spatially-adaptive criteria for augmenting FSDT elements to HOZT elements selectively. We will introduce this algorithm in the following paragraphs.
We adopt the algorithm of selectively augmenting the elements to aFSDT following spatially-adaptive criteria in the results reported in Fig. 5, wherein the variation of percentage error with respect to pure HOZT analysis \(\left( {{\% \text {Error}} = \left( {{\text{HOZT results}} - {\text{Present results}}} \right) \times 100/{\text{HOZT results}}} \right)\) for 3-layered (0°/90°/0°) all-around simply supported laminated composite plate \(\left(a/h=10\right)\) is shown considering different numbers of FSDT and aFSDT elements (refer to the discussions on BTD, presented in the preceding section). With an increase in FSDT elements, the % error increases for all three types of studies (bending, free vibration and buckling). Based on such diagrams, an optimal number of aFSDT elements can be decided along with their spatial distribution, as shown in the insets. Figure 6 shows the % difference in the time taken for the analysis, wherein it becomes evident that GP-based models with adaptive criteria require significantly less time as compared to solely FE-based models. In this context, the computational efficiency gain and avoidance of machine learning-based approximation becomes evident from the number of HOZT elements used in a simulation. Depending on the type of analysis, when 60% higher order elements are retained, ~ 28–40% computational efficiency can be achieved with a significant level of accuracy based on the fidelity-sensitive scheme. Incorporation of machine learning further improves the efficiency in the range of ~ 20–40% depending on the type of analysis. Note that these are remarkable efficiency gains for a single prediction without compromising the accuracy (the quantification of which is thoroughly investigated in Figs. 5 and 6) after having some upfront cost for training the data, which is necessary for any simulation-based data-driven machine learning approaches. In this context, it is worth mentioning that (1) the computational efficiency of the proposed approach (due to both the fidelity-sensitive approach and incorporation of machine learning) will be significantly higher when a more complex problem is considered with higher number of elements and more refined meshes, (2) the above quantification is presented for a single simulation and the impact of the proposed machine learning based fidelity sensitive scheme would be extraordinarily remarkable when thousands of parametric simulations are necessary such as optimization, uncertainty quantification and reliability analysis, (3) based on the proposed fidelity-sensitive approach it is possible to include an unprecedented notion of engineering judgment to ensure a balance between computational efficiency and necessary accuracy depending on the application.
Figure 7 exhibits the variation of transverse shear stress \(\left({\bar{\sigma }}_{yz}\right)\) across the thickness for 4-layered square-shaped, all-around simply supported laminated composite plates (considering different laminate configurations: 0°/0°/0°/0°, − 30°/30°/− 30°/30°, − 45°/45°/− 45°/45°) obtained using GP based stiffness matrix for different number of aFSDT and FSDT based elements. As the number of aFSDT elements (i.e. HOZT-equivalent element) decreases, the curve deviates more from the solely aFSDT based analysis (i.e. as the count of FSDT elements increases, transverse shear stresses are determined with less accuracy). The behavior of the plate studied using GP based stiffness matrix can predict transverse shear stress-free conditions at the top and bottom surfaces of the plate and the continuity at interfaces for all the configurations.
Non-dimensional displacement and stresses in the form of % error with respect to HOZT plate for 4-layered unsymmetric laminated composite plate (− 30°/30°/− 30°/30°) are shown in Fig. 8a and b for SSSS (all-around simply supported) and CCFF (clamped edges parallel to Y-axis and free edges parallel to X-axis).
boundary conditions considering different configurations of distribution of FSDT and HOZT (or aFSDT) elements \((a/h=10)\). Similar results for free vibration and buckling analysis are reported in Figs. 9 and 10, respectively. Values reported in black are for FE-based solutions (i.e. FSDT and HOZT distributions), whereas those in red are for the values obtained using the GP-based stiffness matrix (i.e. FSDT and aFSDT distributions). With an increase in the number of FSDT elements, the % error also increases. It has been observed that the regions away from the corners are critical for refinement (SSSS boundary condition). The elements lying near these areas should be made of HOZT or aFSDT. In CCFF boundary conditions, the central portion of the free edge can be analyzed with good accuracy using lower-order elements. The results are in accordance with those available in the literature [41]. For optimal placement of lower and higher order elements depending on the allowed level of computational intensiveness, the results show that boundary conditions should be kept in mind while carrying out refinements in the mesh and distributing higher and lower order elements. Depending on the spatial distribution of the extent of critical stress resultants, the distribution of higher and lower order elements can be decided for a given proportion of FSDT and HOZT (or aFSDT) elements. The results further confirm that the involvement of GP-based machine learning does not dilute the accuracy of results, while improving the computational efficiency significantly.
Figure 11 exhibits the variation of non-dimensional displacement, natural frequency, and critical buckling load for multilayered cross-ply laminated composite plates made up of a different number of layers. The stiffness matrix for the plate is derived using different proportions of FSDT and aFSDT elements. Material properties used are the same as those used for 4-layered unsymmetric plates. It can be seen that except for 2-layered laminates, no appreciable difference appears in the values for non-dimensional displacement predicted using different proportions of mesh distribution. However, for natural frequency and buckling load, the proportion of aFSDT and FSDT is more sensitive for ensuring the accuracy of results. Here, by providing effective material properties for different layers according to requirements, we can use the constructed GP models to analyze laminates with different numbers of layers as shown in the results. It is not required to form separate GP models for different configurations with varying number of layers. Figure 12 shows the percentage error in the predicted displacement, natural frequency and critical buckling load for different distributions of FSDT and aFSDT elements with respect to a solely HOZT-based analysis. It becomes evident that the error increases with the increase in the number of FSDT elements, albeit depending on the laminate configuration. The analyses presented in Figs. 11 and 12 give a clear understanding of the effect of spatially-adaptive element distributions on the static, dynamic and stability behavior of composite laminates with different numbers of layers.
We have considered the size of individual elements as an input parameter in forming the GP models that allows us to change the size efficiently from coarse to fine meshes. This will be particularly useful for spatially varying structural and material properties depending on the nature of distribution (variable stiffness composites). Further, the spatially-adaptive scheme of distributing finite elements of different orders of theory can be implemented more optimally depending on the complexity of the system (such as shell geometries, panels with cutouts etc. that can be investigated in future studies). In the following paragraphs, we demonstrate the advantage of the proposed spatially-adaptive elementary-level machine learning approach in two different aspects: (a) controlling the mesh size efficiently without losing the accuracy, (b) efficient mechanical characterization of variable stiffness composite laminates.
After carrying out a detailed study on predicting the behavior of laminated composite plates with GP-derived stiffness matrix for different material properties, thickness schemes, and ply-angles in the preceding numerical results, an attempt has been made to predict mechanical behavior of the plate by replacing coarser meshes with finer mesh sizes by exploiting GP. To demonstrate the capability of the proposed computational framework in efficiently changing the mesh size, we have replaced the stiffness matrix obtained using 2 × 2 mesh size with the stiffness matrix derived based on finer mesh sizes through GP. Material properties used during the study are \({E}_{1}=276 \text{GPa}\), \({E}_{2}={E}_{3}={G}_{12}={G}_{13}=\) \({G}_{23}=6.9 \text{GPa}\), \({\nu }_{12}={\nu }_{13}={\nu }_{23}=0.25\), \(\rho =681.8 \text{kg}/{\text{m}}^{3}\). The lamination scheme adopted for the plate is 0°/90°/0°, with all layers of equal thickness. The plate is assumed to be square-shaped with \(a/h=10\) having all edges clamped (CCCC). Figure 13, 14 and 15 show that a significant level of accuracy can be achieved for static, vibration and buckling analyses with finer mesh sizes, albeit in a much more computationally efficient way compared to the direct finite element approach. Note that even though we have demonstrated the concept of varying mesh size based on machine learning through uniform meshes here, it will be quite effective in the case of adaptive spatially varying meshes. Further, it is possible to assign adaptive lower and higher-order theories in such spatially varying meshes for computationally efficient analyses. Thus the current framework lays the foundation of a new research direction for the near future.
Here we further show the capability of the proposed computational framework for variable stiffness composites. The aFSDT model is employed to predict static, free vibration and buckling behavior of variable stiffness laminated composite (VSLC) plates accurately. The VSLC plate is made up of curvilinear fibers, in which the fiber orientation changes along the in-plane direction in a regular fashion. In each layer of the plate, the fiber angle can be different, thus, exhibiting a different stiffness compared to the adjacent layers. For the \(k\)-th layer, the orientation of the fiber is defined as: \({\varpi }^{k}\left(x\right)=\frac{2\left({T}_{1}^{k}-{T}_{0}^{k}\right)}{a}\left|x\right|+{T}_{0}^{k}\), where \({T}_{0}\) is the orientation of the fiber at (0, 0) and \({T}_{1}\) is the fiber angle at the edge of the layer with respect to the X-axis as shown in Fig. 16. Different types of VSLC plates are studied in this section using the proposed aFSDT (i.e. equivalent HOZT) approach for bending, free vibration and buckling analyses in order to demonstrate the effectiveness of the current computational framework.
The results for non-dimensional central deflection for a square-shaped VSLC plate having all edges clamped are reported in Table 4\((a/h=100)\). The material properties of the plate are assumed as \({E}_{11}=173\) GPa, \({E}_{22}=7.2\) GPa, \({G}_{12}={G}_{13}={G}_{23}=3.76\) GPa, \(\nu =0.29\). The present results for static analysis are compared with the third-order shear deformation-based results published by Akhavan et al. [2] and the layerwise model of Yazdani et al. [55], wherein good agreements are found considering the same material properties as the respective literature. VSLC plates with lamination scheme as [< 0°, 45° > , < − 45°, − 60° > , < 0°, 45° >] are analyzed for the free vibration analysis, as reported in Table 5. The present results are found to be in good agreement when compared with those published by Akhavan and Ribeiro [1] and Tornabene et al. [51] using HSDT and Ritz method-based LWT by Vescovini and Dozio [53]. Table 6 shows the non-dimensional critical buckling load for square-shaped simply supported VSLC plates [± < 90°, 0° >]symmetric with \(a/h=50\) having material properties \({E}_{11}=150\) GPa, \({E}_{22}=9.08\) GPa, \({G}_{12}={G}_{13}={G}_{23}=5.29\) GPa, \(\nu =0.32\), \(\rho =1500\) kg/m3. Present results are compared with the results obtained by Vescovini and Dozio [53] using Ritz method, wherein a good agreement can be found. Having the VSLC model validated using the proposed aFSDT based framework, we further present new results here to demonstrate the efficacy of the current computational approach.
Table 4
Non-dimensional central deflection for clamped square plate [< 45°, 0° > , < 135°, 90° > , < 135°, 0° > , < 45°, 0° >] subjected to sinusoidal loading \((a/h=100)\) having material properties \({E}_{11}=173\) GPa, \({E}_{22}=7.2\) GPa, \({G}_{12}={G}_{13}={G}_{23}=3.76\) GPa, \(\nu =0.29\)
Source
\(\overline{w }=\frac{100w{E}_{22}h}{q{a}^{2}}\)
Present HOZT
1.031
Present aFSDT
1.032
Yazdani et al. [55]
1.035
Akhavan et al. [2]
1.04
Table 5
Non-dimensional natural frequency for clamped square plate [< 0°, 45° > , < − 45°, − 60° > , < 0°, 45° >] with \(a/h=100\), having material properties \({E}_{11}=173\) GPa, \({E}_{22}=7.2\) GPa, \({G}_{12}={G}_{13}={G}_{23}=3.76\) GPa, \(\nu =0.29\), \(\rho =1540\) kg/m3
Source
\(\overline{\lambda }=\lambda \frac{{a}^{2}}{h}\sqrt{\frac{\rho }{{E}_{22}}}\)
Present HOZT
17.7204
Present aFSDT
17.7114
Akhavan and Ribeiro [1]
17.8361
Tornabene et al. [51]
17.7153
Vescovini and Dozio [53]
17.7105
Table 6
Non-dimensional critical buckling load for simply supported square plate [± < 90°, 0° >]symmetric with \(a/h=50\), having material properties \({E}_{11}=150\) GPa, \({E}_{22}=9.08\) GPa, \({G}_{12}={G}_{13}={G}_{23}=5.29\) GPa, \(\nu =0.32\), \(\rho =1500\) kg/m3
Source
\(\overline{\alpha }=\frac{{\alpha a}^{2}}{{E}_{11}{h}^{3}}\)
Present HOZT
2.0815
Present aFSDT
2.0807
Vescovini and Dozio [53]
2.0898
Figure 17 shows the variation of non-dimensional central deflection for VSLC plate with simply supported edges having \(h/a = 0.10,\, b/a = 0.50\) and layup sequence as [< − T0, T1 > , < T0, − T1 > , < − T0, T1 > , < T0, − T1 > , < − T0, T1 >]. The material properties adopted are as: \({E}_{11}/{E}_{22}=40\), \({G}_{12}={G}_{13}={G}_{23}=0.5{E}_{22}\), \(\nu =0.25\). The plate is subjected to bisinusoidal loading over its entire surface and the solutions are obtained using the proposed aFSDT based computational framework. The results for different values of T0 and T1 are reported in Fig. 17, wherein the dependence of non-dimensional central deflection on the values of tow angles of the fiber is clearly demonstrated. With the increasing value of T0 for smaller values of T1 (Till 50°), the non-dimensional central deflection of the plate increases. For large values of T1 (beyond 50°), the value for non-dimensional central deflection of the plate decreases with an increase in the value of T0. Figure 18 shows the variation of non- dimensional natural frequency \(\left(\overline{\uplambda }=\uplambda a\sqrt{\frac{\rho }{{E}_{T}}}\right)\) for [\(\mp\)< T0,T1 >]symmetric plates \(\left( {h/a = 0.10, \,b/a = 0.50} \right)\) having SSCF edges and material properties as \({E}_{11}/{E}_{22}=40\), \({G}_{12}={G}_{13}=0.6{E}_{22}\), \({G}_{23}=0.5{E}_{22}\), \(\nu =0.29\) considering different values of T0 and T1. The maximum value for the non-dimensional natural frequency has been observed when the value for T0 reaches 90°. Figure 19 shows the influence of the boundary conditions on the buckling behavior of a three-layered VSLC plate. The plate is having geometric properties as \(h/a = 0.10,\, b/a = 1\) and material properties as \({E}_{11}=173\) GPa, \({E}_{22}=7.2\) GPa, \({G}_{12}={G}_{13}={G}_{23}=3.76\) GPa, \(\nu =0.29\). It can be observed that the maximum value for the critical buckling load is observed for CCCC plate and the minimum for SFSF plate. Note that we have presented the validation and new results for VSLC plates here considering different material properties, boundary conditions and laminate configurations to show the effectiveness of the current computational framework for a wide range of applications. Further, since we have adopted the aFSDT based approach, the current results are equivalent to HOZT solutions in terms of accuracy, albeit with a significant level of computational advantage.
Remarks:
We explain here the novelty and contribution of the current paper in light of available literature (primarily our earlier works in the same field to establish the sequential development in this field). Extrusive applications of Gaussian process based machine learning schemes have been widely adopted in the literature concerning the mechanical analyses of laminated composites and functionally graded structures [8, 20, 35]. [13] focuses mainly on FGM stiffness matrix components and subsequently predicting the natural frequencies. The current paper concerning intrusive coupling of Gaussian process deals with composite laminates and deals with free vibration and stability of the structure. In this context, we would like to emphasize that the applicability of the current computational framework is shown for more complicated variable stiffness composite configurations with curved fibers (refer to Fig. 16). Such analysis involving machine learning, while being critical and attractive to industry and academia, is scarce to find in the literature.
[13] involves only a single-order theory, and the elements of the stiffness matrices are predicted using a machine learning algorithm. The current paper is fundamentally different from this aspect as it involves multiple orders of the theory (such as first-order shear deformation theory and higher-order zigzag theory). We propose to map these two orders of theory through a newly proposed machine learning based augmented FSDT. The current multi-fidelity approach is more efficient and accurate.
[13] does not involve any aspect of spatially-adaptive and fidelity-sensitive computational schemes. One of the major contributions of the current paper is to propose a spatially-adaptive and fidelity-sensitive computational framework involving the concept of best theory diagram (refer to Fig. 2). We note that it is not necessary to augment all the elements of the stiffness matrix in the analysis domain which might otherwise lead to unnecessary computational expenses and loss in accuracy. To achieve an optimal level of computational efficiency and accuracy, we further propose spatially-adaptive fidelity-sensitive coupling of machine learning, only for the elements within the analysis domain where it is necessary to adopt higher-order theories. Subsequently, the global structural matrices are computed based on such adaptive criteria containing a mixed set of elements formed using FSDT and aFSDT, which leads to an accuracy equivalent to HOZT in the mechanical analysis of composite laminates almost at the computational expense of FSDT. The proposed spatially-adaptive fidelity-sensitive scheme ensures optimal performance in terms of computational efficiency by augmenting selective elements based on physics-based insights, while minimizing the loss of accuracy due to the involvement of surrogates.
In the current work, we have considered nine-node quadratic elements for demonstrating the proposed computational framework, but the elementary-level intrusive coupling approach of machine learning based on a spatially-adaptive fidelity-sensitive computational framework is generic in nature. It can be extended to any element type, materials or different structures following a similar approach. The current computational framework can further be exploited for extending it to repetitive iterations like optimization and the stochastic regimes, for achieving computational efficiency. It is a standard approach in scientific literature to present results based on a few specific element types, materials or structures that can further be adopted in a more generic sense. Thus it can be noted that the scope of applicability of the developed computational framework is quite broad.

4 Summary and conclusion

In this article we have proposed an intrusive elementary-level coupling of GP-based machine learning in the finite element computation for analyzing laminated composite plates with the same level of accuracy as higher-order theories, but at the computational expense of lower-order theories. Based on the novel spatially-adaptive fidelity-sensitive computational framework, detailed numerical results are presented for static, dynamic and stability characterization of composite laminates including the demonstration for variable stiffness composite configurations.
Mechanical analysis of the complex configurations of composite laminates with different degrees of mesh refinements and variable stiffness configurations can be computationally prohibitive based on accurate higher-order theories, especially when the analyses involve multiple realizations corresponding to different sets of input parameters such as uncertainty quantification, optimization, reliability and sensitivity analysis. Efficient lower-order theories should not be adopted in such situations since the error accumulates with multiple realizations, leading to poor outcomes. We have proposed a spatially-adaptive fidelity-sensitive intrusive coupling of GP with the finite element framework of composite laminates to deal with the dichotomy of accuracy and computational efficiency. The generic parameter space of material properties, mesh size, number of layers, and ply angle in composite laminates are accounted here for forming an efficient mapping with the augmentation of lower-order theory-based elementary-level structural matrices. Based on the machine learning-based difference mapping between lower and higher order theories, we augment the elementary stiffness matrices obtained using FSDT efficiently to the equivalent of HOZT theory (referred to here as augmented FSDT, or aFSDT) without any additional computational expenses. To achieve an optimal level of computational efficiency and accuracy, we have further proposed spatially-adaptive fidelity-sensitive coupling of machine learning, only for the elements within the analysis domain where it is necessary to adopt higher-order theories. Subsequently, the global structural matrices are computed based on such adaptive criteria containing a mixed set of elements formed using FSDT and aFSDT, which leads to an accuracy equivalent to HOZT in the mechanical analysis of composite laminates almost at the computational expense of FSDT. The proposed spatially-adaptive fidelity-sensitive scheme ensures optimal performance in terms of computational efficiency and the desired level of accuracy by augmenting selective elements in a pre-decided proportion, while minimizing the loss of accuracy due to the involvement of surrogates.
A two-fold validation has been presented in this paper to ensure accuracy at the elemental level and predict the global mechanical behavior concerning static, dynamic and stability analysis. We have presented the validation and new results for conventional and variable stiffness laminated composite plates here considering different material properties, boundary conditions and laminate configurations to show the effectiveness of the current simulation framework for a wide range of structural applications. The proposed spatially adaptive computational approach is generic in nature and it can be extended to study various other structural forms including shells and graded configurations along with exploiting it further for extending the computation to stochastic regimes.

Acknowledgements

TM and SN would like to acknowledge the financial support received from the University of Southampton during this research work.

Declarations

Conflict of interest

The authors declare no competing interests.
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Anhänge

Appendix A

The Marginal and Conditional Distributions of MVN theorem: suppose \(X=\left({x}_{1},{x}_{2}\right)\) is a joint Gaussian with parameters
$$\mu = \left[ {\begin{array}{*{20}c} {\mu_{1} } \\ {\mu_{2} } \\ \end{array} } \right],\,\Sigma = \left[ {\begin{array}{*{20}c} {\Sigma_{11} } & {\Sigma_{12} } \\ {\Sigma_{21} } & {\Sigma_{22} } \\ \end{array} } \right],\, \wedge = \Sigma^{ - 1} = \left[ {\begin{array}{*{20}c} { \wedge_{11} } & { \wedge_{12} } \\ { \wedge_{21} } & { \wedge_{22} } \\ \end{array} } \right],$$
Then the marginals are given by
$$\begin{array}{*{20}c} {p\left( {x_{1} } \right) = \mathcal{N}\left( {x_{1} \left| {\mu_{1} \Sigma_{11} } \right.} \right)} \\ {p\left( {x_{2} } \right) = \mathcal{N}\left( {x_{2} \left| {\mu_{2} \Sigma_{22} } \right.} \right)} \\ \end{array} ,$$
and the posterior conditional is given by
$$p\left( {x_{1} \left| {x_{2} } \right.} \right) = \mathcal{N}\left( {x_{1} \left| {\mu_{12} ,\Sigma_{12} } \right.} \right)$$
$$\mu_{12} = \mu_{1} + \Sigma_{12} \Sigma_{22}^{ - 1} \left( {x_{2} - \mu_{2} } \right) = \mu_{1} - \wedge_{11}^{ - 1} \wedge_{12} \left( {x_{2} - \mu_{2} } \right) = \Sigma_{12} \left( { \wedge_{11} \mu_{1} - \wedge_{12} } \right)\left( {x_{2} - \mu_{2} } \right)$$
$$\Sigma_{12} = \Sigma_{11} - \Sigma_{12} \Sigma_{22}^{ - 1} \Sigma_{21} = \wedge_{11}^{ - 1}$$
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Metadaten
Titel
Elementary-level intrusive coupling of machine learning for efficient mechanical analysis of variable stiffness composite laminates: a spatially-adaptive fidelity-sensitive computational framework
verfasst von
A. Garg
S. Naskar
T. Mukhopadhyay
Publikationsdatum
05.01.2025
Verlag
Springer London
Erschienen in
Engineering with Computers
Print ISSN: 0177-0667
Elektronische ISSN: 1435-5663
DOI
https://doi.org/10.1007/s00366-024-02082-z