Simulation physics-informed deep neural network by adaptive Adam optimization method to perform a comparative study of the system

Abstract

Taking the marvelous advantages of artificial intelligence (AI) in accelerating the procedure of finding a solution to different engineering analyses is the main motivation of this article to establish a non-model-based mechanism on the basics of fully connected deep neural networks (FC-DNN) to analyze the hygro-thermomechanical buckling response of the multiscale hybrid composite MHC doubly curved panel. First, the system's buckling response at its design points is obtained by applying DQM to motion equations developed based upon the refined-form of third-order shear deformation theory (TSDT). Then the obtained information would be transferred to DNN to acquire the regressor system. Finding the optimal values of weights and biases of the DNN is the key factor to provide an AI system with high-accuracy prediction. For this reason, the adaptive Adam optimization approach is chosen due to its phenomenal speed as well as lower computational costs.

Appendix

The variation of Eq. (17) can be achieved as below:

$$ \begin{aligned} \delta \phi & = \frac{1}{2}\iiint\limits_{V} {\sigma_{ij} \delta \varepsilon_{ij} {\text{d}}V} \\ & = \iint\limits_{A} {\left[ \begin{gathered} \left( {\partial_{\alpha } N_{\alpha \alpha }^{*} + \partial_{\beta } N_{\alpha \beta }^{*} + R_{1}^{ - 1} N_{\alpha \gamma } - R_{1}^{ - 1} N_{\alpha \gamma }^{*} } \right)\delta \alpha_{0} \hfill \\ + \left( {\partial_{\beta } N_{\beta \beta }^{*} + \partial_{\alpha } N_{\alpha \beta }^{*} + R_{2}^{ - 1} N_{\beta \gamma } - R_{2}^{ - 1} N_{\beta \gamma }^{*} } \right)\delta \beta_{0} \hfill \\ + \left( {\partial_{\alpha } N_{\alpha \gamma }^{*} + \partial_{\beta } N_{\beta \gamma }^{*} - R_{1}^{ - 1} N_{\alpha \alpha } - R_{2}^{ - 1} N_{\beta \beta } } \right)\delta \gamma_{0} \hfill \\ + \left( {\partial_{\alpha } M_{\alpha \alpha } + \partial_{\beta } M_{\alpha \beta }^{\beta } + R_{1}^{ - 1} M_{\alpha \gamma } - N_{\alpha \gamma }^{*} } \right)\delta \alpha_{1} \hfill \\ + \left( {\partial_{\beta } M_{\beta \beta } + \partial_{\alpha } M_{\alpha \beta }^{\alpha } + R_{2}^{ - 1} M_{\beta \gamma } - N_{\beta \gamma }^{*} } \right)\delta \beta_{1} \hfill \\ \end{gathered} \right]\,}{\text{d}}A \\ \end{aligned} $$

(62)

where

$$ \left\{ {N_{\alpha \alpha }^{*} ,N_{\alpha \alpha } ,M_{\alpha \alpha } } \right\} = \int_{z} {\frac{1}{{\left( {1 + \frac{z}{{R_{1} }}} \right)}}\sigma_{\alpha \alpha } \left\{ {\left( {1 + \frac{z}{{R_{1} }}} \right),1,z} \right\}\left( {1 + \frac{z}{{R_{1} }}} \right)} \left( {1 + \frac{z}{{R_{2} }}} \right){\text{d}}z $$

(63)

$$ \left\{ {N_{\beta \beta }^{*} ,N_{\beta \beta } ,M_{\beta \beta } } \right\} = \int_{z} {\frac{1}{{\left( {1 + \frac{z}{{R_{2} }}} \right)}}\sigma_{\beta \beta } \left\{ {\left( {1 + \frac{z}{{R_{2} }}} \right),1,z} \right\}\left( {1 + \frac{z}{{R_{1} }}} \right)} \left( {1 + \frac{z}{{R_{2} }}} \right){\text{d}}z $$

(64)

$$ \left\{ {N_{\alpha \gamma } ,M_{\alpha \gamma } } \right\} = \int_{z} {\frac{1}{{\left( {1 + \frac{z}{{R_{1} }}} \right)}}\tau_{\alpha \gamma } \left\{ {1,z} \right\}\left( {1 + \frac{z}{{R_{1} }}} \right)} \left( {1 + \frac{z}{{R_{2} }}} \right){\text{d}}z $$

(65)

$$ \left\{ {N_{\beta \gamma } ,M_{\beta \gamma } } \right\} = \int_{z} {\frac{1}{{\left( {1 + \frac{z}{{R_{2} }}} \right)}}\tau_{\beta \gamma } \left\{ {1,z} \right\}\left( {1 + \frac{z}{{R_{1} }}} \right)} \left( {1 + \frac{z}{{R_{2} }}} \right){\text{d}}z $$

(66)

$$ \left\{ {N_{\alpha \gamma }^{*} ,M_{\alpha \gamma }^{*} } \right\} = \int_{z} {\tau_{\alpha \gamma } \left\{ {1,z} \right\}\left( {1 + \frac{z}{{R_{1} }}} \right)} \left( {1 + \frac{z}{{R_{2} }}} \right){\text{d}}z $$

(67)

$$ \left\{ {N_{\beta \gamma }^{*} ,M_{\beta \gamma }^{*} } \right\} = \int_{z} {\tau_{\beta \gamma } \left\{ {1,z} \right\}\left( {1 + \frac{z}{{R_{1} }}} \right)} \left( {1 + \frac{z}{{R_{2} }}} \right){\text{d}}z $$

(68)

$$ \left\{ {N_{\alpha \gamma }^{*} ,M_{\alpha \gamma }^{*} } \right\} = \int_{z} {\tau_{\alpha \gamma } \left\{ {1,z} \right\}\left( {1 + \frac{z}{{R_{1} }}} \right)} \left( {1 + \frac{z}{{R_{2} }}} \right){\text{d}}z $$

(67)

$$ \left\{ {N_{\alpha \beta }^{\alpha } ,M_{\alpha \beta }^{\alpha } } \right\} = \int_{z} {\frac{1}{{\left( {1 + \frac{z}{{R_{1} }}} \right)}}\tau_{\alpha \beta } \left\{ {\left( {1 + \frac{z}{{R_{2} }}} \right),z} \right\}\left( {1 + \frac{z}{{R_{1} }}} \right)} \left( {1 + \frac{z}{{R_{2} }}} \right){\text{d}}z $$

(68)

$$ \left\{ {N_{\alpha \beta }^{\beta } ,M_{\alpha \beta }^{\beta } } \right\} = \int_{z} {\frac{1}{{\left( {1 + \frac{z}{{R_{2} }}} \right)}}\tau_{\alpha \beta } \left\{ {\left( {1 + \frac{z}{{R_{1} }}} \right),z} \right\}\left( {1 + \frac{z}{{R_{1} }}} \right)} \left( {1 + \frac{z}{{R_{2} }}} \right){\text{d}}z $$

(69)

Furthermore, the first variation of the work done by temperature gradient is obtained as

$$ \delta \Gamma_{1} = - \int {\left( {N_{1}^{T} \frac{{\partial^{2} \delta \gamma_{0} }}{{\partial x^{2} }} + N_{2}^{T} \frac{{\partial^{2} \delta \gamma_{0} }}{{\partial y^{2} }}} \right)} {\text{d}}A, $$

(70)

In addition, the variation of the work done by moisture coefficient is obtained as

$$ \delta \Gamma_{2} = - \int {\left( {N_{1}^{H} \partial_{\alpha }^{2} \gamma_{0} + N_{2}^{H} \partial_{\beta }^{2} \gamma_{0} } \right)} \delta \gamma_{0} {\text{d}}A \, , $$

(71)

In addition, the work exerted by the mechanical forces and its first variation is formulated as

$$ \delta \Gamma_{3} = - \int {\left( {N_{\alpha \alpha }^{0} \partial_{\alpha }^{2} \gamma_{0} + N_{\beta \beta }^{0} \partial_{\beta }^{2} \gamma_{0} } \right)\delta \gamma_{0} } {\text{d}}A \, , $$

(72)

The equilibrium equations for the MHC doubly curved panel is as follows:

$$ \delta \alpha_{0} :\partial_{\alpha } N_{\alpha \alpha }^{*} + \partial_{\beta } N_{\alpha \beta }^{*} + R_{1}^{ - 1} N_{\alpha \gamma } - R_{1}^{ - 1} N_{\alpha \gamma }^{*} = 0, $$

(73)

$$ \delta \beta_{0} :\partial_{\beta } N_{\beta \beta }^{*} + \partial_{\alpha } N_{\alpha \beta }^{*} + R_{2}^{ - 1} N_{\beta \gamma } - R_{2}^{ - 1} N_{\beta \gamma }^{*} = 0, $$

(74)

$$ \begin{aligned} &\delta \gamma_{0} :\partial_{\alpha } N_{\alpha \gamma }^{*} + \partial_{\beta } N_{\beta \gamma }^{*} - R_{1}^{ - 1} N_{\alpha \alpha } - R_{2}^{ - 1} N_{\beta \beta } \\ &\quad- N_{1}^{T} \partial_{\alpha }^{2} \gamma_{0} - N_{2}^{T} \partial_{\beta }^{2} \gamma_{0} - N_{1}^{H} \partial_{\alpha }^{2} \gamma_{0} \\ & \quad- N_{2}^{H} \partial_{\beta }^{2} \gamma_{0} - N_{\alpha \alpha }^{0} \partial_{\alpha }^{2} \gamma_{0} - N_{\beta \beta }^{0} \partial_{\beta }^{2} \gamma_{0} = 0 \end{aligned} $$

(75)

$$ \delta \alpha_{1} :\partial_{\alpha } M_{\beta \beta } + \partial_{\beta } M_{\alpha \beta }^{\beta } + R_{1}^{ - 1} M_{\alpha \gamma } - N_{\alpha \gamma }^{*} = 0 $$

(76)

$$ \delta \beta_{1} :\partial_{\beta } M_{\beta \beta } + \partial_{\alpha } M_{\alpha \beta }^{\alpha } + R_{2}^{ - 1} M_{\beta \gamma } - N_{\beta \gamma }^{*} = 0 $$

(77)

Subsequent relations denote the boundary conditions of the system

$$ \begin{gathered} \begin{array}{*{20}c} {\delta \alpha_{0} = 0\begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {\rm or} & {} \\ \end{array} } \\ \end{array} N_{\alpha \alpha }^{*} } \\ \end{array} n_{\alpha } + N_{\alpha \beta }^{y} n_{\beta } = 0 \, , \hfill \\ \begin{array}{*{20}c} {\delta \beta_{0} = 0\begin{array}{*{20}c} {} & {\rm or} & {} \\ \end{array} N_{\alpha \beta }^{\alpha } } \\ \end{array} n_{\alpha } + N_{\beta \beta }^{*} n_{\alpha } = 0 \, , \hfill \\ \begin{array}{*{20}c} {\delta \gamma_{0} = 0\begin{array}{*{20}c} {} & {\rm or} & {} \\ \end{array} N_{\alpha \gamma } } \\ \end{array} n_{\alpha } + N_{\beta \gamma } n_{\beta } + N_{1}^{T} \partial_{\alpha } \gamma_{0} n_{\alpha } + N_{2}^{T} \partial_{\beta } \gamma_{0} n_{\beta } \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + N_{1}^{H} \partial_{\alpha } \gamma_{0} n_{\alpha } + N_{2}^{H} \partial_{\beta } \gamma_{0} n_{\beta } + N_{\alpha \alpha }^{0} \partial_{\alpha } \gamma_{0} n_{\alpha } + N_{yy}^{0} \partial_{\beta } \gamma_{0} n_{y} = 0 \, , \hfill \\ \end{gathered} $$

(78)

$$ \begin{gathered} \begin{array}{*{20}c} {\delta \alpha_{1} = 0\begin{array}{*{20}c} {} & {{\text{or}}} & {} \\ \end{array} M_{\alpha \alpha } } \\ \end{array} n_{\alpha } + M_{\alpha \beta }^{y} n_{\beta } = 0 \, , \hfill \\ \begin{array}{*{20}c} {\delta \beta_{1} = 0\begin{array}{*{20}c} {} & {{\text{or}}} & {} \\ \end{array} M_{\alpha \beta }^{\alpha } } \\ \end{array} n_{\beta } + M_{\beta \beta } n_{\alpha } = 0 \, , \hfill \\ \end{gathered} $$

(79)

According to Eqs. (78), (79), clamped (C) and simply supported (S) boundaries are provided as

1.1 Clamped edge

$$ \begin{array}{*{20}c} {\left\{ \begin{gathered} \alpha = 0{\text{ or a}} \hfill \\ \beta = 0{\text{ or b}} \hfill \\ \end{gathered} \right.} & {\mathop{\longrightarrow}\limits^{}{}\alpha_{0} = \beta_{0} = \gamma_{0} = \alpha_{1} = \beta_{1} ,} \\ \end{array} $$

(80)

1.2 Simply edge

$$ \left\{ {\begin{array}{*{20}c} {\alpha {\text{ = 0 or a}}} & {\mathop{\longrightarrow}\limits^{}{}\left\{ {N_{\alpha \alpha }^{*} = M_{\alpha \alpha } = 0} \right.} \\ {\beta {\text{ = 0 or b}}} & {\mathop{\longrightarrow}\limits^{{}}\left\{ {N_{\beta \beta }^{*} = M_{\beta \beta } = 0} \right.} \\ \end{array} } \right. \, {.} $$

(81)