Optimization of functionally graded plates under Thermo-elastic free vibration using nature-based algorithms

Abstract

This article presents the parametric optimization of the free vibration characteristics of a functionally graded material (FGM) plate exposed to nonlinear thermal loading using the finite element method and nature-based algorithms. The one-dimensional (1D) Fourier heat conduction equation is used to calculate temperature distributions over the thickness of the plate. Using Lagrange’s equation, we get the dynamic equation of motion for the plate. An eight-noded iso-parametric plate element with five degrees of freedom per node is used in the finite element formulation based on first-order shear deformation theory. Rectangular plates have temperature-dependent material characteristics; the thickness of the plates is scaled using a straightforward power law distribution. Here, the investigation of the FGM plate is conducted with two different boundary conditions, such as simple support and fully clamped. Additionally, new hybrid optimization approaches, namely RSM-Composite Desirability Optimization (RSM-CDO), Whale Optimization Algorithm (WOA), Corrected Moth Search Optimization (CMSO), Lichtenberg Algorithm Optimization (LAO), Sunflower Optimization Algorithm (SFO), and Forensic-Based Investigation Algorithm (FBI), are utilised to determine the best optimal solution, and the obtained findings are validated using confirmatory tests.

Keywords

FGM finite element method metaheuristic optimization plate thermal environment

1. Introduction

High temperatures may cause laminated composites to delaminate, separate the adhesive connection, or other concerns that demand the use of functionally graded material (FGM), a kind of composite material with equal strength but none of these drawbacks. Depending on the size of the object, the characteristics of FGM might change.¹ A power law is established to meet the service needs, which alters the material characteristics. Metal-ceramic composites are common functionally graded materials used in thin-walled structures for high thermal gradient applications.² Due to the fact that the FGM structures are also subjected to a high thermal gradient, it is of interest to investigate the free vibration response of the FGM plate modelled by FEM and later on optimise parameters by employing nature-based algorithms. The optimization of FGM plate settings in a nonlinear thermal environment has eluded many studies despite the fact that functional graded (FG) plate free vibration is well-documented.^3,4,5,6 Hao et al.⁷ investigated the nonlinear dynamics of a functionally graded simply supported plate subjected to in-plane and transverse stimulation in a temperature environment using Reddy’s third-order shear deformation theory. Zhang et al.⁸ studied the chaotic dynamics of a FG rectangular plate using Galerkin’s method. Hao et al.⁹ exposed a cantilevered functionally graded plate to transverse stress and nonlinear oscillations in a temperature environment, who focused on resonance circumstances.

From the extensive literature reviews, the authors found that very few investigators have addressed the parametric optimization of the response of FGM structures. As a consequence, an attempt has been made to narrow this gap via the use of the finite element method and nature-based optimization techniques which is the novelty and originality of this paper. Eight nodded, five-degree of freedom plate ele0ments are employed to discretise the surface. With the help of Hamilton’s theory, we may deduce the dynamic governing equations. In this work, authors have tried to reach the global minimum points with the optimum settings in input variables such as volume fraction index (n), the length to thickness ratio (L/h), the length to width ratio (L/w), and the temperature of the ceramic zone (Tc) in both the cases of simple support and fully clamped scenario. Following the determination of the non-dimensional natural frequency, the RSM-Composite Desirability Optimization (RSM-CDO), Whale Optimization Algorithm (WOA), Corrected Moth Search Optimization (CMSO), Lichtenberg Algorithm Optimization (LAO), Sunflower Optimization Algorithm (SFO), and Forensic-Based Investigation Algorithm (FBI) approaches were used to obtain global optimum settings. A confirmatory test was also done to make sure that the global optimal value and the theoretical simulation value were the same.

The structure of this paper is stated as follows: Section 2 elaborates the theoretical formulation with two sub parts such as (a) Kinematics of plate and (b) Finite element method; Section 3 describes the optimization methodology of RSM-CDO; Section 4 represents the results and discussion with four sub parts such as (i) Free Vibration Analysis, (ii) Parametric Optimization Analysis, (iii) Comparative Analysis and (iv) Confirmatory Test; and lastly, conclusions are drawn and summarised in Section 5.

2. Theoretical formulation

All of the plates studied in this study have the following dimensions: L, W, and Thickness (h). Material composition of the plate changes continually along the thickness direction, based on a blend of metals and ceramics (z-direction). Bottom (z = −h/2) and top (z = h/2) surfaces of the plate panel are made of metal and ceramic, respectively. Due to the compositional change of metals and ceramics in the z-direction, it is essential to calculate accurately the material characteristics using a suitable technique. Numerous techniques are used by researchers in the literature, but some of the most often utilised include the self-consistent scheme, the Frohlich and Sack three-phase model, the mean field approach, the Mori-Tanaka methodology, and the Voigt method. The Voigt method is used in this study to determine the effective properties of the plate panel, which include Young’s modulus (E), Poisson’s ratio (υ), and thermal expansion coefficient (α_t), as a function of position and temperature, while mass density (ρ) and thermal conductivity (K) are assumed to be position-dependent only.

The twisted plate’s effective material characteristics (P) are stated as follows:

P (z, T) = P_{m} (T) + [P_{c} (T) - P_{m} (T)] {[\frac{z}{h} + \frac{1}{2}]}^{n}

(1)

where P_m and P_c denote the respective component characteristics of metal and ceramic. n indicates a non-negative constant known as the volume fraction index, which represents the material variation profile along the z-axis.

The volume fractions of the component materials are provided as follows:

V_{m} + V_{c} = 1

(2)

V_{c} = {(\frac{z}{h} + \frac{1}{2})}^{n}

(3)

The temperature-dependent material characteristics of ceramics and metals (E, υ and α_t ) are calculated as

P_{c} (T) = P_{0} (P_{- 1} T^{- 1} + 1 + P_{1} T + P_{2} T^{2} + P_{3} T^{3})

(4)

P_{m} (T) = P_{0} (P_{- 1} T^{- 1} + 1 + P_{1} T + P_{2} T^{2} + P_{3} T^{3})

(5)

where P_i (i=−1, 0, 1, 2, 3) denotes the temperature-dependent coefficients, which vary across metals and ceramics.

The temperature change throughout the thickness of a functional graded plate is not uniform; the temperature distribution over the thickness can be determined by solving the one-dimensional (1D) Fourier heat conduction equation.

- \frac{d}{d z} [K (z) \frac{d T}{d z}] = 0

(6)

where

K (z) = K_{m} + [K_{c} - K_{m}] {[\frac{z}{h} + \frac{1}{2}]}^{n}

(7)

By solving equation (6) with equation (7) and imposing the boundary conditions of T = T_m at z = −h/2 and T = T_c at z = h/2, the temperature distribution throughout the plate’s thickness may be derived as follows:

T (z) = T_{m} + \frac{Δ T}{F} [\begin{array}{l} (\frac{z}{h} + \frac{1}{2}) - \frac{K_{c m}}{(n + 1) K_{m}} {(\frac{z}{h} + \frac{1}{2})}^{n + 1} + \frac{K_{c m}^{2}}{(2 n + 1) K_{m}^{2}} {(\frac{z}{h} + \frac{1}{2})}^{2 n + 1} \\ - \frac{K_{c m}^{3}}{(3 n + 1) K_{m}^{3}} {(\frac{z}{h} + \frac{1}{2})}^{3 n + 1} + \frac{K_{c m}^{4}}{(4 n + 1) K_{m}^{4}} {(\frac{z}{h} + \frac{1}{2})}^{4 n + 1} \\ - \frac{K_{c m}^{5}}{(5 n + 1) K_{m}^{5}} {(\frac{z}{h} + \frac{1}{2})}^{5 n + 1} \end{array}]

(8)

where

F = 1 - \frac{K_{c m}}{(n + 1) K_{m}} + \frac{K_{c m}^{2}}{(2 n + 1) K_{m}^{2}} - \frac{K_{c m}^{3}}{(3 n + 1) K_{m}^{3}} + \frac{K_{c m}^{4}}{(4 n + 1) K_{m}^{4}} - \frac{K_{c m}^{5}}{(5 n + 1) K_{m}^{5}} a n d K_{c m} = K_{c} - K_{m}

(9)

Kinematics of the plate

For the FGM plate based on first-order shear deformation theory, the in-plane (u, v) and transverse displacements (w) are shown as:

u (x, y, z, t) = u_{0} (x, y, t) + z θ_{x} (x, y, t) v (x, y, z, t) = v_{0} (x, y, t) + z θ_{y} (x, y, t) w (x, y, z, t) = w_{0} (x, y, t)

(10)

where u₀, v₀, w₀ are the mid plane translation displacements and θ_x, θ_y are two rotations. The strain-displacement connection for the plate is shown by the equation:

{\begin{array}{c} ε_{x} \\ ε_{y} \\ γ_{x y} \\ γ_{x z} \\ γ_{y z} \end{array}} = {\begin{array}{c} \frac{\partial u}{\partial x} \\ \frac{\partial v}{\partial x} \\ \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \\ θ_{x} + \frac{\partial w}{\partial x} \\ θ_{y} + \frac{\partial w}{\partial y} \end{array}} = {\begin{array}{c} ε_{x}^{0} \\ ε_{y}^{0} \\ γ_{x y}^{0} \\ γ_{x z}^{0} \\ γ_{y z}^{0} \end{array}} + z {\begin{array}{c} κ_{x} \\ κ_{y} \\ κ_{x z} \\ 0 \\ 0 \end{array}}

(11)

where

{\begin{array}{c} ε_{x}^{0} \\ ε_{y}^{0} \\ γ_{x y}^{0} \\ γ_{x z}^{0} \\ γ_{y z}^{0} \end{array}} = {\begin{array}{c} u_{0, x} \\ v_{0, x} \\ u_{0, y} + v_{0, x} \\ u_{0, z} + w_{0, x} \\ v_{0, z} + w_{0, y} \end{array}}

(12)

and

{\begin{array}{c} κ_{x} & κ_{y} & κ_{x y} \begin{array}{c} 0 & 0 \end{array} \end{array}}^{T} = {\begin{array}{c} θ_{x, x} & θ_{y, y} & θ_{x, y} + θ_{y, x} & 0 & 0 \end{array}}^{T}

(13)

The constitutive equations of the plate are as follows:

{\begin{array}{c} σ_{x} \\ σ_{y} \\ τ_{x y} \\ τ_{x z} \\ τ_{y z} \end{array}} = [\begin{array}{c} Q_{11} & Q_{12} & 0 & 0 & 0 \\ Q_{21} & Q_{22} & 0 & 0 & 0 \\ 0 & 0 & Q_{66} & 0 & 0 \\ 0 & 0 & 0 & Q_{44} & 0 \\ 0 & 0 & 0 & 0 & Q_{55} \end{array}] ({\begin{array}{c} ε_{x} \\ ε_{y} \\ γ_{x y} \\ γ_{x z} \\ γ_{y z} \end{array}} - α_{t} Δ T {\begin{array}{c} 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}})

(14)

where ΔT is the temperature increase relative to a reference condition and Q_ij are the stiffnesses due to plane stress reduction.

Integrating the stresses in the plate’s thickness direction yields the following in-plane force, moment, and transverse shear forces:

[\begin{array}{c} N_{x} \\ N_{y} \\ N_{x y} \\ M_{x} \\ M_{y} \\ M_{x y} \\ Q_{x z} \\ Q_{y z} \end{array}] = [\begin{array}{c} A_{11} & A_{12} & A_{13} & B_{11} & B_{12} & B_{13} & 0 & 0 \\ A_{21} & A_{22} & A_{23} & B_{21} & B_{22} & B_{23} & 0 & 0 \\ A_{31} & A_{32} & A_{31} & B_{31} & B_{32} & B_{33} & 0 & 0 \\ B_{11} & B_{11} & B_{11} & D_{11} & D_{12} & D_{13} & 0 & 0 \\ B_{11} & B_{11} & B_{11} & D_{21} & D_{22} & D_{23} & 0 & 0 \\ B_{11} & B_{11} & B_{11} & D_{31} & D_{32} & D_{33} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & S_{44} & S_{45} \\ 0 & 0 & 0 & 0 & 0 & 0 & S_{54} & S_{55} \end{array}] {\begin{array}{c} ε_{x}^{0} \\ ε_{y}^{0} \\ γ_{x y}^{0} \\ κ_{x} \\ κ_{y} \\ κ_{x y} \\ γ_{x z}^{0} \\ γ_{y z}^{0} \end{array}} - [\begin{array}{c} N_{x}^{T} \\ N_{y}^{T} \\ N_{x y}^{T} \\ M_{x}^{T} \\ M_{y}^{T} \\ M_{x y}^{T} \\ 0 \\ 0 \end{array}]

(15)

In abbreviated form, the preceding constitutive relationship may be represented as:

{F} = [D] {\bar{ε}} - {F^{T}}

(16)

As a consequence of thermal loading, the stress and moment resultants (N^T and M^T) are represented as

{\begin{array}{c} N_{x}^{T} \\ N_{y}^{T} \\ N_{x y}^{T} \end{array}} = \int_{- h / 2}^{h / 2} [\begin{array}{c} Q_{11} & Q_{12} & 0 \\ Q_{21} & Q_{22} & 0 \\ 0 & 0 & 0 \end{array}] α_{t} (z, T) Δ T {\begin{array}{c} 1 \\ 1 \\ 0 \end{array}} d z

(17)

{\begin{array}{c} M_{x}^{T} \\ M_{y}^{T} \\ M_{x y}^{T} \end{array}} = \int_{- h / 2}^{h / 2} [\begin{array}{c} Q_{11} & Q_{12} & 0 \\ Q_{21} & Q_{22} & 0 \\ 0 & 0 & 0 \end{array}] α_{t} (z, T) Δ T {\begin{array}{c} 1 \\ 1 \\ 0 \end{array}} z d z

(18)

The stiffnesses of the plates A_ij , B_ij and D_ij are given by

(A_{i j}, B_{i j}, D_{i j}) = \int_{- h / 2}^{h / 2} Q_{i j} (1, z, z^{2}) d z (i, j = 1, 2, 6) (S_{i j}) = k_{s} \int_{- h / 2}^{h / 2} Q_{i j} d z (i, j = 4,5)

(19)

where k_s equals 5/6, is the shear correction factor.

Finite Element method

An eight-noded iso-parametric plate element with five degrees of freedom per node is used to build the finite element model of the plate. The following may be used to represent any point in the plate element’s displacement vector generically:

{δ} = \sum_{i = 1}^{8} [N_{i}] {δ_{e}}_{i}

(20)

where {δ_e} and [N_i] denote the vector of nodal displacement and the matrix of nodal shape functions, respectively.¹⁰

Formula for calculating strain energy of functionally graded plate element:

V_{e 1} = \frac{1}{2} {δ_{e}}^{T} [K_{e}] {δ_{e}}

(21)

where [K_e] is the element’s elastic stiffness matrix and is the nodal displacement vector i.e.

{δ_{e}}^{T} = {\begin{array}{c} u_{i}^{0} & v_{i}^{0} & w_{i}^{0} & θ_{x i} & θ_{y i} \end{array}}_{i = 1,8}^{T}

. The elastic stiffness matrix may be calculated using the conventional finite element method in the following manner:

[K_{e}] = \int_{A} {[B]}^{T} [D] [B] dA

(22)

Where [B] is the strain displacement matrix.

The kinetic energy of the plate element denoted by the formula:¹¹

T_{e} = \frac{1}{2} {{\dot{δ}}_{e}}^{T} [M_{e}] {{\dot{δ}}_{e}}

(23)

The elemental mass matrix, [M_e] , may be deduced as follows:

[M_{e}] = \int_{A} {[N]}^{T} [m] [N] d A

(24)

[M] stands for the matrix of shape functions and [N] stands for a matrix of inertia functions.¹¹

As a result of the initial strains and the subsequent strain energy generated by the temperature change, we may derive the following:

V_{e 0} = \int_{v} {ε_{n}}^{T} {σ_{0 T}} d v

(25)

{ε_n} is the non-linear strain component vector and which can be defined as:

{ε_{n}} = \frac{1}{2} [\begin{array}{c} \frac{\partial u}{\partial x} & \frac{\partial v}{\partial x} & \frac{\partial w}{\partial x} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{\partial u}{\partial y} & \frac{\partial v}{\partial y} & \frac{\partial w}{\partial y} & 0 & 0 & 0 \\ \frac{\partial u}{\partial y} & \frac{\partial v}{\partial y} & \frac{\partial w}{\partial y} & \frac{\partial u}{\partial x} & \frac{\partial v}{\partial x} & \frac{\partial w}{\partial x} & 0 & 0 & 0 \\ \frac{\partial u}{\partial z} & \frac{\partial v}{\partial z} & \frac{\partial w}{\partial z} & 0 & 0 & 0 & \frac{\partial u}{\partial x} & \frac{\partial v}{\partial x} & \frac{\partial w}{\partial x} \\ 0 & 0 & 0 & \frac{\partial u}{\partial z} & \frac{\partial v}{\partial z} & \frac{\partial w}{\partial z} & \frac{\partial u}{\partial y} & \frac{\partial v}{\partial y} & \frac{\partial w}{\partial y} \end{array}] {\begin{array}{c} \frac{\partial u}{\partial x} \\ \frac{\partial v}{\partial x} \\ \frac{\partial w}{\partial x} \\ \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial y} \\ \frac{\partial w}{\partial y} \\ \frac{\partial u}{\partial z} \\ \frac{\partial v}{\partial z} \\ \frac{\partial w}{\partial z} \end{array}}

(26)

It may be stated in this manner after solving the previous equation.

{ε_{n}} = \frac{1}{2} [S_{l}] [ϕ] = \frac{1}{2} [S_{l}] [G] {δ_{e}}

(27)

Derivatives of form functions may be found in [G], a matrix.

The stored strain energy is given by equation (27), where the non-linear strain values are substituted.

V_{e 0} = \frac{1}{2} (\int_{v} {S_{e}}^{T} {[G]}^{T} {[S_{l}]}^{T} {σ_{0 T}} d v)

(28)

But

{[S_{l}]}^{T} {σ_{0 T}} = [N_{σ T}] [G] {δ_{e}}

where [N_σT] is the matrix of in-plane stress resultants caused by thermal loading.¹²

As a result of calculating equation (28), the strain energy stored by thermal load is

V_{e 0} = \frac{1}{2} {δ_{e}}^{T} [K_{G T e}] {δ_{e}}

(29)

Due to thermal loads, [K_GTe] is the geometric stiffness matrix.

[K_{G T e}] = \int_{v o l} {[G]}^{T} [N_{σ T}] [G] d v

(30)

The plate element has a total strain energy capacity of:

V_{e} = V_{e 0} + V_{e 1}

(31)

Lagrange’s equation allows one to obtain the governing equation of motion for the FGM plate, which may be found here.

\frac{d}{d t} [\frac{\partial T_{e}}{\partial {\dot{δ}}_{e}}] - [\frac{\partial T_{e}}{\partial δ_{e}}] + \frac{\partial V_{e}}{\partial δ_{e}} = 0

(32)

The solution to Lagrange’s equation may be found by simply plugging in the appropriate numbers for kinetic energy and strain energy.

[M_{e}] {{\ddot{δ}}_{e}} + ([K_{e}] + [K_{G T e}]) {δ_{e}} = 0

(33)

When all of the matrices and vectors have been put together, the dynamic equation of motion for the FGM plate will look like this:

[M] {\ddot{δ}} + ([K] + [K_{G T}]) {δ} = 0

(34)

The fundamental frequencies may be determined by using the QR iteration approach, which is based on the conventional eigenvalue problem.^13,14

[C] {δ} = λ {δ}

(35)

where

[C] = [K] + [K_{G T}]

(36)

3. Optimization methodology

RSM-composite desirability optimization (RSM-CDO)

The response surface methodology (RSM) is a combination of mathematical and statistical methodologies in which mathematical models are constructed by regression using experimental data. RSM may be thought of as an acronym for “response surface methodology.”^15,16 In this work, the dependent variable, namely the volume fraction index (n), the length to thickness ratio (L/h), the length to width ratio (L/w), and the temperature of the ceramic zone (T_c), is stated as follows.

Natural Frequency (\bar{ω}) = f (n, L / h, L / w, T_{c})

(37)

To maximise the natural Frequency ( $\bar{ω}$ ), it is necessary to construct a second-order polynomial equation that may be written as

\bar{ω} = b_{0} + \sum b_{i} x_{i} + \sum \sum b_{i j} x_{i} x_{j} + \sum b_{i i} {x_{i i}}^{2} + ε

(38)

Where b₀ is the constant term in the regression equation, b_i denotes the linear term, b_ij denotes the interaction term, and b_ii denotes the quadratic term. Two Box-Behnken Designs of RSM models, namely Natural Frequencies of

{\bar{ω}}_{1_{S S}}

and

{\bar{ω}}_{1_{C}}

, were created in this study using MINITAB V17 software and are stated as follows.

{\bar{ω}}_{1_{S S}} = (1.34) + (- 4.75 (n)) + (0.526 (L / h)) + (2.11 (L / w)) + (0.01003 (T_{c})) + (2.175 {(n)}^{2}) + (- 0.01026 {(L / h)}^{2}) + (4.026 {(L / w)}^{2}) + (- 0.000005 {(T_{c})}^{2}) + (0.0023 (n) (L / h)) + (- 2.875 (n) (L / w)) + (0.00055 (n) (T_{c})) + (0.059 (L / h) (L / w)) + (- 0.000779 (L / h) (T_{c})) + (0.00041 (L / w) (T_{c}))

(39)

{\bar{ω}}_{1_{C}} = (9.29) + (- 4.67 (n)) + (0.454 (L / h)) + (- 5.19 (L / w)) + (0.00411 (T_{c})) + (2.260 {(n)}^{2}) + (- 0.01284 {(L / h)}^{2}) + (9.156 {(L / w)}^{2}) + (- 0.000001 {(T_{c})}^{2}) + (- 0.0163 (n) (L / h)) + (- 2.695 (n) (L / w)) + (0.00025 (n) (T_{c})) + (0.1935 (L / h) (L / w)) + (- 0.000408 (L / h) (T_{c})) + (- 0.00070 (L / w) (T_{c}))

(40)

The desire function method, often known as the DF approach, is a strategy that is frequently used in multiple response optimization. With the use of this method, the optimum working circumstances, x, for achieving the sought-after response values may be determined. Each response is then converted to an individual discrete factor (d_i), using a scale factor that ranges from 0 to 1. The values for these functions are specified as the intended response (y_i), the minimum response (y_i), and the maximum response (y_i) attained during the testing. Let the intended minimum, target, and maximum values for response y be denoted respectively by L, U, and T, with the order L > T > U. In the event that a response is of the kind that is wanted, the individual desirability function for that answer is as follows:

d_{i} = {\begin{array}{c} 0 \\ \begin{array}{c} {(\frac{y_{i} - L}{T - L})}^{r} \\ {(\frac{{U - y}_{i}}{U - T})}^{s} \end{array} \\ 0 \end{array} \begin{array}{c} y_{i} < L \\ \begin{array}{c} {L \leq y}_{i} \leq T \\ {T \leq y}_{i} \leq U \end{array} \\ y_{i} > U \end{array}

(41)

with the r and s exponents indicating how important it is to reach the target value. When r and s are both equal to one, the desirability function rises linearly toward T; when r is less than one and s is less than one, the function is convex; and when r is more than one and s is greater than one, the function is concave.

To minimise a response, the individual desirability is defined as:

d_{i} = {\begin{array}{c} 1 \\ {(\frac{{U - y}_{i}}{U - T})}^{s} \\ 0 \end{array} \begin{array}{c} y_{i} < L \\ {T \leq y}_{i} \leq U \\ y_{i} > U \end{array}

(42)

with T indicating a low enough response value.

In contrast, if the response is to be maximised, then the individual desirability is defined as the following:

d_{i} = {\begin{array}{c} 0 \\ {(\frac{y_{i} - L}{T - L})}^{r} \\ 1 \end{array} \begin{array}{c} y_{i} < L \\ {L \leq y}_{i} \leq T \\ y_{i} > T \end{array}

(43)

with T regarded as a sufficiently large value for the response in this instance.

The individual desirability is then averaged, yielding the composite desirability D which is determined by following equation:

D = {(d_{1}, d_{2}, d_{3}, \dots d_{n})}^{\frac{1}{n}} = {(\prod_{i = 1}^{n} d_{n})}^{\frac{1}{n}}

(44)

The greatest feasible value of D is sought to ensure that the reactions occur under optimum circumstances.¹⁶

4. Results and discussions

Free vibration analysis

The mathematical formulas described above are utilised to create an in-house MATLAB algorithm for analysing the FGM plate’s free vibration response under nonlinear temperature loading. In the course of this investigation, we made use of functional graded plates that were composed of ceramic silicon nitride (Si₃N₄) and metal stainless steel (SUS304). Table 1 lists the coefficients of Si₃N₄ and SUS304 that are temperature dependent and temperature independent. To begin, benchmark problems have been solved to validate the code’s capabilities and correctness. Table 2 presents the results of the validation of the existing formulation by means of a FGM plate carried out in a thermal environment. The proposed technique produces result that are extremely similar to the benchmark solutions, indicating that it is capable of calculating the free vibration response of a functionally graded plate that has been subjected to a nonlinear temperature change. Additionally, a convergence analysis is performed with mesh sizes of 2 × 2, 4 × 4, 6 × 6, 8 × 8 and 10 × 10. As can be observed, the result converges at a mesh size of 4 × 4, and the difference between the results produced at mesh sizes 6 × 6 and 8 × 8 is much very smaller. The Mesh Convergence plot is provided in Figure 1, and it can be seen here. As a result, for the sake of optimal computer performance, the whole research uses a mesh size of 6 × 6. For the sake of this study, the boundary conditions of simply supported (SS) and completely clamped (CC) are represented as follows:

A t x = 0, L v = w = θ_{y} = 0 y = 0, b u = w = θ_{x} = 0

(45)

A t x = 0, L u = v = w = θ_{x} = θ_{y} = 0 y = 0, b u = v = w = θ_{x} = θ_{y} = 0

(46)

Table 1.

Ceramic and metal materials exhibit temperature-dependent and temperature-independent characteristics.

Material	Properties	P ₀	P ₁	P ₂	P ₃
SUS304	E _m	201.04 × 10⁹	3.079 × 10⁻⁴	−6.534 × 10⁻⁷	0
	α _tm	12.330 × 10⁻⁶	8.08 × 10⁻⁶	0	0
	υ _m	0.3262	−2.002 × 10⁻⁴	3.797 × 10⁻⁷	0
	ρ _m	8166	0	0	0
	K _m	12.04	0	0	0
SI₃N₄	E _c	348:43 × 10⁹	−3.070 × 10⁻⁴	2.160 × 10⁻⁷	−8.946 × 10⁻¹¹
	α _tc	5:8723 × 10⁻⁶	9.095 × 10⁻⁶	0	0
	υ _c	0.24	0	0	0
	ρ _c	2370	0	0	0
	K _c	9.19	0	0	0

Note: E (Pa), α_t (1/K), ρ (kg/m³) and K (W/mK), suffix m and c refer to metal and ceramic, respectively.

Table 2.

Natural frequency parameter ( $ϖ = ω (L^{2} / h) \sqrt{ρ_{0} (1 - ν^{2}) / E_{0}}$ ) for simply supported Si₃N₄/SUS304 square plates in thermal environments. ρ₀ and E₀ are the density and modulus of elasticity of metal at 300 K, L/b = 1, L = 8h, ρ_c = 2770 kg/m³, ρ_m = 8166 kg/m³, ν_c = 0.28,ν_m = 0.28, K_c = 9.19 W/m K, K_m = 12.04 W/m K.

		Huang and Shen	Present FEM	Parida and Mohanty
T_c = 300 K T_m = 300 K	Si₃N₄	12.495	12.518	12.587
	0.5	8.675	8.625	9.094
	1.0	7.555	7.552	7.656
	2.0	6.777	6.778	6.78
	SUS304	5.405	5.411	5.445
T_c = 400 K T_m = 300 K	Si₃N₄	12.397	12.308	12.387
	0.5	8.615	8.468	8.615
	1.0	7.474	7.406	7.51
	2.0	6.693	6.638	6.642
	SUS304	5.311	5.273	5.311
T_c = 600 K T_m = 300 K	Si₃N₄	11.984	11.888	11.971
	0.5	8.269	8.131	8.272
	1.0	7.171	7.087	7.186
	2.0	6.398	6.327	6.327
	SUS304	4.971	4.923	4.989

Figure 1.

Mesh Convergence study (L = w, L/h = 20, T_m = 300 K, T_c = 500 K).

Parametric optimization analysis

According to studies that were conducted in the past, meta-heuristic optimization approaches are able to provide more accurate findings than statistical optimization strategies. In light of this, the present investigation makes use of nature-based metaheuristic optimization strategies in order to ascertain the free vibration response of a rectangular functionally graded material (FGM) plate that is constrained by temperature factors. To optimise the total output parameters concurrently, one statistical approach, i.e., RSM-Composite Desirability Optimization (RSM-CDO), and five different nature-based metaheuristic optimization algorithms were used, namely Whale Optimization Algorithm (WOA), Corrected Moth Search Optimization (CMSO), Lichtenberg Algorithm Optimization (LAO), Sunflower Optimization Algorithm (SFO), and Forensic-Based Investigation Algorithm (FBI). The methods described above were run in MATLAB R2018a on an HP 15s-fr2xxx modelled computer system equipped with an 11th Generation Intel (R) Core (TM) i5-1135G7 @ 2.42 GHz, a 64-bit operating system, an x64-based CPU, and the Microsoft Windows 10 Home operating system. The design of experiment (L₂₇) for RSM was performed using the MINITAB V17 software, with the volume fraction index (n), the length to thickness ratio (L/h), the length to width ratio (L/w), and the temperature of the ceramic zone (T_c) as input parameters. The output response was the non-dimensional fundamental frequency ( $\bar{ω} = ω (L^{2} / h) \sqrt{ρ_{0} (1 - ν^{2}) / E_{0}}$ , ρ₀ and E₀ are the density and modulus of elasticity of metal at 300 K). Two kinds of case studies were analysed in this study: (a) the simple supported boundary condition and (b) the fully clamped boundary condition type FGM plate with varying restrictions. Control parameters and their values for basic supported rectangular FGM plates are given in each case study. For both case studies, only four input variables are considered, with the other parameters remaining constant. The experiment was designed using MINITAB Software’s L₂₇ type Box-Behnken RSM design, which is suitable for statistical analysis of four input variables with three levels. This experimental design, namely the L₂₇ type RSM input matrix, will assist researchers in saving time and effort by avoiding the need to do all probability combinations of 64 runs, i.e., 64 = (4_Variables)^3Levels. The usual sequence of experiments was randomised in order to minimise needless residual errors across all tests. An ANOVA was used to determine the most significant parameters across all experiments. Each output response’s residual plot illustrates the effect of residual mistakes within each run. Additionally, for both case studies, a factorial analysis was performed to determine the impact of each input variable on the output of the first natural frequency. Similarly, factorial analyses were performed on all potential interactions between input variables. Additionally, the most important interactions between two input constraint factors were analysed using enhanced visualisations of 3D surface and 2D contour plots. To begin with, the statistical method of RSM-Composite Desirability Optimization (RSM-CDO) was used to determine the output response’s maximum value. The second ordered regression equation from RSM was used to forecast the global optimal settings for the output natural frequency in each of the five advanced metaheuristic optimization methods. We conducted a comparison analysis of all the convergence charts for each optimization method. Finally, a confirmatory test was performed to verify that the result frequency was within the 95% confidence interval and to forecast the percentage error between the anticipated and experimental values of the output natural frequency.

Case study 1 (For simple supported FGM plate)

In this study, the FGM plate was regarded to be a basic supported structure, and the impact of a variety of factors on the free vibration response was investigated. The ratio of length to thickness (L/h), the ratio of length to width (L/w), and the temperature of the ceramic zone (T_c) were all regarded to be input factors, and the first natural frequency was derived as an output response. The volume fraction index (n) was also taken into consideration. Besides these input parameters, all parameters were kept constant throughout the experimentations. The four input constraint variables and their three levels tabulated in Table 3. The parameters of volume fraction index (n), ratio of length to thickness (L/h), ratio of length to width (L/w) and temperature of ceramic zone (T_c) are presented as A, B, C and D respectively. The first natural frequency for the case of simple supported FGM plate was denoted as

{\bar{ω}}_{1_{S S}}

, which was considered as output response for this case study. Here, the single objective optimization problem is considered and the type of problem is of maximization. The run order, standard order, point type and blocks were tabulated in Table 4 which was designed as per L₂₇ type Box-Behnken RSM design. Further, the analysis of variance (ANOVA) test was carried out and the obtained results were given in Table 5 with total number of degree of freedom (DoF) as 26 including residual errors. Further, statistical terms such as adjusted sum of square (Adj. S.S.), adjusted mean of square (Adj. M.S.), F-Values and p-Values were tabulated in Table 5 with corresponding to each variable. In this table, the number of DoF for the model was obtained as 14 which was consisted of linear terms (DoF = 4), square terms (DoF = 4) and 2-way interaction terms (DoF = 6). Similarly, the number of DoF for the error was found as 12 which was summed of Lack-of-Fit (DoF = 10) and Pure Error (DoF = 2). It can be noted that the value of “F-test” and “P-test” was obtained as null (“*”), because of the lowest residual errors in the whole experimentations. The percentage contribution of each variable was calculated from the Adj. S.S. and represented in Table 5. The linear word category with the “L/w” was determined to have the largest percentage contribution, with a value of 54.481%. Similarly, in the case of square and 2-way interaction terms category, the highest value of percentage contribution was obtained as 6.533% and 2.139% respectively. Lastly, it can be observed that the percentage contribution of Lack-of-Fit and Pure Error terms as 0.886 and 0.000 respectively. Here, it can be concluded that the interaction of “n” and “L/w” input variable has significant contribution in the experimentation and hence the 3D interaction surface and 2D contour plot were plotted and represented in Figures 5 and 6 respectively. The results of this part of the experiment show that the maximum natural frequency was achieved when the value of the “n" variable was set as low as possible while the “L/w" variable was set as high as possible.

Table 3.

Control Parameters and their levels for case study 1.

Constraint variables	Notation	Representation	Levels
Constraint variables	Notation	Representation	1	2	3
Volume fraction index	n	A	0	1	2
Ratio of length to thickness	L/h	B	10	15	20
Ratio of length to width	L/w	C	0.5	1.0	1.5
Temperature of ceramic zone	T _c	D	300	500	700

Table 4.

L₂₇ experimental runs with output response for case study 1.

Run order	Std. Order	Pt. Type	Blocks	n	L/h	L/w	T _c	${\bar{ω}}_{1_{S S}}$
1	3	2	1	0	20	1	500	11.1415
2	11	2	1	0	15	1	700	10.5216
3	14	2	1	1	20	0.5	500	3.2514
4	12	2	1	2	15	1	700	5.023
5	6	2	1	1	15	1.5	300	12.6067
6	21	2	1	1	10	1	300	7.6879
7	8	2	1	1	15	1.5	700	10.583
8	24	2	1	1	20	1	700	3.8397
9	23	2	1	1	10	1	700	6.7612
10	9	2	1	0	15	1	300	12.972
11	15	2	1	1	10	1.5	500	11.774
12	18	2	1	2	15	0.5	500	3.5365
13	4	2	1	2	20	1	500	5.4968
14	10	2	1	2	15	1	300	7.0295
15	20	2	1	2	15	1.5	500	10.413
16	19	2	1	0	15	1.5	500	19.7
17	7	2	1	1	15	0.5	700	2.7349
18	2	2	1	2	10	1	500	6.4831
19	26	0	1	1	15	1	500	6.938
20	5	2	1	1	15	0.5	300	4.922
21	1	2	1	0	10	1	500	12.173
22	17	2	1	0	15	0.5	500	7.0742
23	16	2	1	1	20	1.5	500	11.159
24	25	0	1	1	15	1	500	6.938
25	27	0	1	1	15	1	500	6.938
26	22	2	1	1	20	1	300	7.8828
27	13	2	1	1	10	0.5	500	4.458

Table 5.

Analysis of Variance for output response ${\bar{ω}}_{1_{S S}}$ .

Variable	DoF	AdjS.S.	Percentage contribution	AdjM.S.	F-value	p-value
Model	14	382.941	99.114	27.353	95.830	0.000
Linear	4	335.202	86.758	83.800	293.600	0.000
n	1	105.616	27.336	105.616	370.030	0.000
L/h	1	3.593	0.930	3.593	12.590	0.004
L/w	1	210.495	54.481	210.495	737.480	0.000
T _c	1	15.498	4.011	15.498	54.300	0.000
Square	4	36.903	9.551	9.226	32.320	0.000
n ²	1	25.241	6.533	25.241	88.440	0.000
(L/h) ²	1	0.351	0.091	0.351	1.230	0.289
(L/w) ²	1	5.403	1.398	5.403	18.930	0.001
(T _c ) ²	1	0.212	0.055	0.212	0.740	0.405
2-Way interaction	6	10.836	2.805	1.806	6.330	0.003
n(L/h)	1	0.001	0.000	0.001	0.000	0.967
n(L/w)	1	8.264	2.139	8.264	28.950	0.000
n(T _c )	1	0.049	0.013	0.049	0.170	0.685
(L/h) (L/w)	1	0.087	0.023	0.087	0.310	0.590
(L/h) (T _c )	1	2.428	0.628	2.428	8.510	0.013
(L/w) (T _c )	1	0.007	0.002	0.007	0.020	0.881
Error	12	3.425	0.886	0.285	*	*
Lack-of-Fit	10	3.425	0.886	0.343	*	*
Pure error	2	0.000	0.000	0.000	*	*
Total	26	386.366	100.000	—	—	—

The summery of model with the value of R²(R-sq.) terms were tabulated in the Table 6 and the value of model summery (S), actual R² value, adjusted R² value and predicted R² value were found as 0.534, 99.11%, 98.08% and 94.89% respectively. From this table, it can observe that the value of R² was greater than the 95% and hence the most of the p-values in the ANOVA table became 0.000. Usually, p-value less than 0.05 value suggested as the “significant variable”, in other words the influence of these parameters is more as compared to the parameters having p-value greater than 0.05 in the ANOVA test. Again, if the p-value of most of the terms gets 0.000 then the F-value should be analyzed and the highest F-values are more preferable for predicting the significance of input variables in the whole experimentations. From the ANOVA table of Table 5, it can be observed that the highest F-value obtained for the input variable “L/w” which means it has significant influence among all terms whether it was single term or square term or interaction term. So, the ratio of length to width has most significant characteristics and also percentage contribution value found as 54.481% which was also highest among other terms.

Table 6.

Model Summary for output response ${\bar{ω}}_{1_{S S}}$ .

S	R-sq, %	R-sq(adj), %	R-sq(pred), %
0.534250	99.11	98.08	94.89

In the residual plot for ${\bar{ω}}_{1_{S S}}$ i.e. Figure 2, four subplots i.e. (a) Normal probability plot, (b) standardized residuals versus fit values, (c) Histogram plot with respect to residual’s frequencies and (d) standardized residuals versus run order values were combined together using command of MINITAB V17 Software. It can be observed that all the blue coloured points of residuals were placed near the red line of normal probability which indicates the lower percentage of errors obtained during experimentations. Similarly, in the subplot of standardized residuals versus fit values it can be predicted that there was no any specific pattern or bunch in any part of the figure, hence it can be concluded that there was no relationship among the residual errors. Furtherly, in the histogram subplot, it is clearly seen that the bars of this figure. were placed in a normal distribution curve and so the experimental results can be fitted to 95% confidence interval. Last but not least, the subplot of standardised residuals versus run order values demonstrated the fluctuations that took place in each run order. It is easy to deduce from this graph that the lowest value of natural frequency took place at run order 12 and that the highest value took place at run order 16 respectively.

Figure 2.

Residual plots of ${\bar{ω}}_{1_{S S}}$ .

The single as well as interaction terms related factorial analysis of ${\bar{ω}}_{1_{S S}}$ was carried out among all four considered input variables and plotted in Figures 3 and 4 respectively. In the single type factorial analysis, it can be observed that the mean of natural frequency was decreased with respect to increasing value of “n”, “L/h” and “T_c”, and was increased only with respect to increasing value of “L/w”. Similarly, it can be can predicted the increasing and decreasing order of the value of mean of natural frequency in the plot of interaction terms related factorial plot which is figured in Figure 4. As the interaction of the terms i.e. “n” and “L/w” have highest percentage contribution among all interactions, thus from Figure 4, it is clear that the interaction of "(n) (L/w)" has a negative impact on the natural frequency mean value, as shown by the fact that this value has shrunk but increasing the value of ${\bar{ω}}_{1_{S S}}$ in case of “(L/w) (n)”. In this way, the influence of other interaction terms can be observed with respect to output response of natural frequency mean value i.e. ${\bar{ω}}_{1_{S S}}$ .

Figure 3.

Factorial plot of singular terms for ${\bar{ω}}_{1_{S S}}$ .

Figure 4.

Factorial plot of interaction terms for ${\bar{ω}}_{1_{S S}}$ .

From the Figures 5 and 6, it can be observed that the lowest interval of “n” variable i.e. <0.0, 0.05> and highest interval of “L/w” variable i.e. <1.25, 1.50> contributed the highest natural frequency value i.e. above 17.50 in the whole experimentation. In both the figures, the middle level values of “L/h” and “T_c” were kept as holding values or constant. In the Figure 6, “Blue” coloured portions depicts lowest value and “Green” coloured portions showed the highest values of the mean of output response of natural frequency i.e. ${\bar{ω}}_{1_{S S}}$ .

Figure 5.

3D interaction surface plot between “n” and “L/w” for ${\bar{ω}}_{1_{S S}}$ .

Figure 6.

2D interaction contour plot between “n” and “L/w” for ${\bar{ω}}_{1_{S S}}$ .

Figure 7 shows the results obtained from RSM-CDO statistical approach in which the maximum value of natural frequency was obtained as 19.8504 with optimal settings of “n”, “L/h”, “L/w” and “T_c” as 0.0, 18.2240, 1.50 and 300.0 respectively. In this graph, the red line mark depicts the optimal value in each input parameters. Further, the Figure 8 shows optimum result of natural frequency of simple supported FGM plate from the Whale Optimization Algorithm (WOA) where the value of optimum parameters is obtained as “n” = 0.000, “L/h” = 18.558, “L/w” = 1.500 and “T_c” = 300.000 with global optimum value as 19.840 for the natural frequency. Similarly, Figure 9 describes the results of Corrected Moth Search Optimization (CMSO) with maximum value of natural frequency as 19.451 with input parameters settings as 0.007, 13.922, 1.499 and 327.752 values of “n”, “L/h”, “L/w” and “T_c” respectively. Further, Lichtenberg Algorithm Optimization (LAO) approach was applied to the data set of RSM to find out the global maximum value of natural frequency and the exact value of 19.303 was obtained from the simulation of MATLAB code and the obtained optimum parametric settings as “n” = 0.000, “L/h” = 12.067, “L/w” = 1.500 and “T_c” = 352.314, and the convergence plot of LAO is plotted in the Figure 10. The Sunflower Optimization Algorithm (SFO) was practised to data set of RSM design of experiment of simple supported FGM plate and the convergence plot of SFO is shown in Figure 11 having global optimum value as 18.116 which is the second lowest value among all approaches. In the SFO methodology, the optimum setting for this maximization problem of natural frequency was obtained at value of 0.099, 12.438, 1.481 and 382.776 with corresponding to “n”, “L/h”, “L/w” and “T_c” input variables. Lastly, recent optimization technique such as Forensic-Based Investigation Algorithm (FBI) approach was worked out to obtain the global maximum point in the space value of natural frequency using the RSM designed dataset and the final convergence plot is plotted in Figure 12. The value of optimum parameters is obtained as “n” = 0.144, “L/h” = 10.654, “L/w” = 1.294 and “T_c” = 357.427 with global optimum value as 15.084 for the natural frequency from the FBI optimization approach. In all five metaheuristic optimization approaches, the value of population size and number of iterations were kept constant and respective values were 1000 and 100.

Figure 7.

Optimum results obtained by RSM-CDO for ${\bar{ω}}_{1_{S S}}$ .

Figure 8.

Convergence plot obtained by WOA for ${\bar{ω}}_{1_{S S}}$ .

Figure 9.

Convergence plot obtained by CMSO for ${\bar{ω}}_{1_{S S}}$ .

Figure 10.

Convergence plot obtained by LAO for ${\bar{ω}}_{1_{S S}}$ .

Figure 11.

Convergence plot obtained by SFO for ${\bar{ω}}_{1_{S S}}$ .

Figure 12.

Convergence plot obtained by FBI for ${\bar{ω}}_{1_{S S}}$ .

Case study 2 (for fully clamped typed FGM plate).

In this particular investigation, a clamped kind of FGM plate was used, and the influence of a number of different parameters on the free vibration response was investigated. As input parameters, the volume fraction index (n), the length to thickness ratio (L/h), the length to width ratio (L/w), and the temperature of the ceramic zone (T_c) were employed, and the first non-dimensional natural frequency was the output response. Apart from these input parameters, all other variables remained constant throughout the experiments. Table 7 summarises the four input constraint variables and their three levels. For the completely clamped kind of FGM rectangular plate, the variables designated by the letters A, B, C, and D, are the volume fraction index (n), the length to thickness ratio (L/h), the length to width ratio (L/w), and the temperature of the ceramic zone (T_c), respectively. There is only parameter i.e. volume fraction index in case of fully clamped as compared to simple supported such as “n = [0.5, 1.0, 1.5]” keeping other parameters levels kept same as previous case. The volume fraction index (n) was considered differently in both cases because, in the simply supported and fully clamped cases, the reactive forces are different. There is a moment involved in a clamped case, which necessitates a different type of strength than in a simply supported case. Thus, we have increased the value of “n" by 0.5 for the Fully Clamped Scenario. In general, it is observed that FGM plates with clamed boundary conditions are found to be stiffer than those of simply supported cases. For instance, for a clamped type FGM plate, the first natural frequency was designated as

{\bar{ω}}_{1_{C}}

, which was used as the output response in this case study.

Table 7.

Control Parameters and their levels for case study 2.

Constraints	Notation	Representation	Levels
Constraints	Notation	Representation	1	2	3
Volume fraction index	n	A	0.5	1.0	1.5
Ratio of length to thickness	L/h	B	10	15	20
Ratio of length to width	L/w	C	0.5	1.0	1.5
Temperature of ceramic zone	T _c	D	300	500	700

The single objective optimization issue is addressed in this section, as is the maximisation problem. Table 8 contains information about the run order, standard order, point type, and blocks. It was developed using the L₂₇ type Box-Behnken RSM design. In addition, an analysis of variance (ANOVA) test was carried out, and the findings are shown in Table 9. The overall degree of freedom (DoF), which takes into account any residual errors, was calculated to be 26. Additionally, Table 9 contains statistical terminology such as adjusted sum of squares (Adj. S.S.), adjusted mean of squares (Adj. M.S.), F-Values, and p-Values for each variable. The number of degrees of freedom (DoF) for the model was determined to be 14 in this table, which included linear terms (DoF = 4), square terms (DoF = 4), and two-way interaction terms (DoF = 6). Similarly, the number of degrees of freedom (DoF) for the mistake was determined to be 12, which was calculated by adding the degrees of freedom for Lack-of-Fit (DoF = 10) and Pure Error (DoF = 2). It should be noted that the values for the “F-test” and “P-test” were set to null ("*") due to the lowest residual errors across all experiments. The percentage contribution of each variable to the Adj. S.S. was computed and is shown in Table 9. The greatest percentage contribution of the “L/w" linear term category was determined to be 87.087%. Similarly, the greatest percentage contribution values were found in the square and two-way interaction terms categories, at 4.820% and 0.313%, respectively. Finally, the percentage contributions of the Lack-of-Fit and Pure Error components are 0.143 and 0.000, respectively. As a result of the interaction between the input variables “n" and “L/w,” a 3D interaction surface and a 2D contour plot were produced and shown in Figures 16 and 17, respectively. It can be seen here that the lowest value of the “n" variable and the greatest value of the “L/w" variable resulted in the highest natural frequency throughout the whole experiment.

Table 8.

L₂₇ experimental runs with output response for case study 2.

Run order	Std. Order	Pt. Type	Blocks	n	L/h	L/w	T _c	${\bar{ω}}_{1_{C}}$
1	3	2	1	0.5	20	1	500	14.9993
2	11	2	1	0.5	15	1	700	14.3196
3	14	2	1	1	20	0.5	500	8.7521
4	12	2	1	1.5	15	1	700	11.5415
5	6	2	1	1	15	1.5	300	22.665
6	21	2	1	1	10	1	300	13.1543
7	8	2	1	1	15	1.5	700	21.1207
8	24	2	1	1	20	1	700	11.7307
9	23	2	1	1	10	1	700	12.3888
10	9	2	1	0.5	15	1	300	15.8167
11	15	2	1	1	10	1.5	500	20.4934
12	18	2	1	1.5	15	0.5	500	8.3876
13	4	2	1	1.5	20	1	500	12.1365
14	10	2	1	1.5	15	1	300	12.9397
15	20	2	1	1.5	15	1.5	500	20.4659
16	19	2	1	0.5	15	1.5	500	25.109
17	7	2	1	1	15	0.5	700	8.3213
18	2	2	1	1.5	10	1	500	11.9356
19	26	0	1	1	15	1	500	13.1807
20	5	2	1	1	15	0.5	300	9.5837
21	1	2	1	0.5	10	1	500	14.635
22	17	2	1	0.5	15	0.5	500	10.3358
23	16	2	1	1	20	1.5	500	22.27312
24	25	0	1	1	15	1	500	13.1807
25	27	0	1	1	15	1	500	13.1807
26	22	2	1	1	20	1	300	14.1278
27	13	2	1	1	10	0.5	500	8.90744

Table 9.

Analysis of Variance for output response ${\bar{ω}}_{1_{C}}$ .

Variable	DoF	AdjS.S.	Percentage contribution	AdjM.S.	F-value	p-value
Model	14	578.947	99.857	41.353	598.120	0.000
Linear	4	538.412	92.865	134.603	1946.850	0.000
n	1	26.429	4.558	26.429	382.260	0.000
L/h	1	0.523	0.090	0.523	7.560	0.018
L/w	1	504.911	87.087	504.911	7302.870	0.000
T _c	1	6.548	1.129	6.548	94.710	0.000
Square	4	37.089	6.397	9.272	134.110	0.000
n ²	1	1.702	0.294	1.702	24.620	0.000
(L/h) ²	1	0.550	0.095	0.550	7.950	0.015
(L/w) ²	1	27.944	4.820	27.944	404.180	0.000
(T _c ) ²	1	0.013	0.002	0.013	0.190	0.674
2-Way interaction	6	3.446	0.594	0.574	8.310	0.001
n(L/h)	1	0.007	0.001	0.007	0.100	0.761
n(L/w)	1	1.816	0.313	1.816	26.260	0.000
n(T _c )	1	0.002	0.000	0.002	0.040	0.854
(L/h) (L/w)	1	0.936	0.161	0.936	13.540	0.003
(L/h) (T _c )	1	0.666	0.115	0.666	9.630	0.009
(L/w) (T _c )	1	0.020	0.003	0.020	0.290	0.602
Error	12	0.830	0.143	0.069	*	*
Lack-of-Fit	10	0.830	0.143	0.083	*	*
Pure error	2	0.000	0.000	0.000	*	*
Total	26	579.777	100.000	—	—	—

The summery of the model along with the R² (R-sq.) values were tabulated in Table 10, and the values of the model summery (S), the actual R²value, the adjusted R² value, and the predicted R²value were determined to be 0.262,942, 99.86%, 99.69%, and 99.18%, respectively. Due to the fact that the value of R² was more than 95%, the vast majority of the p-values in the ANOVA table became 0.000. This is something that can be observed in this table. In statistical parlance, variables are said to be “significant” if they have p-values that are lower than 0.05, which suggests that certain aspects of the ANOVA test have a bigger influence than those that have p-values that are higher than 0.05. Again, if the p-values for the majority of the terms are 0.000, the F-value should be evaluated, with the greatest F-values preferred for forecasting the importance of input variables across all experiments. As can be seen from the ANOVA table in Table 9, the greatest F-value was obtained for the input variable “L/w,” indicating that it has a substantial effect on all factors, whether single, square, or interaction. Thus, the ratio of length to breadth has the most important features and also possesses the greatest percentage contribution value of 87.087% among other variables.

Table 10.

Model Summary for output response ${\bar{ω}}_{1_{C}}$ .

S	R-sq, %	R-sq(adj), %	R-sq(pred), %
0.262942	99.86	99.69	99.18

The residual plot for ${\bar{ω}}_{1_{C}}$ was created by combining four subplots: (a) normal probability plot; (b) standardised residuals versus fit values; (c) histogram plot with regard to residual frequencies; and (d) standardised residuals versus run order values which is plotted in Figure 13. As can be seen, all of the residuals' blue points are located around the red line of normal probability, indicating a reduced proportion of mistakes produced throughout experiments. Similarly, in the subplot of standardised residuals against fit values, it is possible to deduce that there was no distinct pattern or clump in any area of the image, indicating that there was no connection between the residual errors. Additionally, the histogram subplot demonstrates that the bars in this Fig. were put inside a normal distribution curve, allowing the experimental findings to be fitted to a 95% confidence interval. Finally, the subplot of standardised residuals against run order values demonstrated the oscillations that occurred in each run order, and from this graph, it is easy to deduce that the natural frequency values were lowest and greatest at run orders 11 and 16, respectively.

Figure 13.

Residual plots of ${\bar{ω}}_{1_{C}}$ .

The factorial analysis of ${\bar{ω}}_{1_{C}}$ was performed on both the single and interaction terms, and the results are shown in Figures 14 and 15. The single type factorial analysis revealed that the mean of natural frequency dropped as the values of “n” and “T_c” rose, but increased when the value of “L/h” and “L/w” grew. Similarly, the rising and decreasing order of the value of the mean of natural frequency may be anticipated from the plot of interaction terms linked factorial plot shown in Figure 15. As the interaction between the terms “n” and “L/w” has the highest percentage contribution of all interactions, it can be seen in Figure 16 that the mean value of natural frequency decreases when the interaction “(n) (L/w)” occurs, while the value of ${\bar{ω}}_{1_{C}}$ ” increases when the interaction “(L/w) (n)” occurs. In this manner, the effect of additional interaction factors on the output response of natural frequency mean value, i.e., ${\bar{ω}}_{1_{C}}$ , may be seen.

Figure 14.

Factorial plot of singular terms for ${\bar{ω}}_{1_{C}}$ .

Figure 15.

Factorial plot of interaction terms for ${\bar{ω}}_{1_{C}}$ .

Figure 16.

3D interaction surface plot between “n” and “L/w” for ${\bar{ω}}_{1_{C}}$ .

As shown in Figure 16 and Figure 17, the lowest interval of the “n” variable, i.e. <0.0, 0.05>, and the largest interval of the “L/w” variable, i.e. <1.25, 1.50>, provided the greatest natural frequency value, i.e. more than 22.50, throughout the whole experimentation. In both figures, the holding values or constants for the intermediate level values of “L/h = 15” and “T_c = 500” were used. The blue part of Figure 17 represents the lowest value, while the green portion represents the greatest value of the mean of the output response, i.e., ${\bar{ω}}_{1_{C}}$ .

Figure 17.

2D interaction contour plot between “n” and “L/w” for ${\bar{ω}}_{1_{C}}$ .

Figure 18 illustrates the results of the RSM-CDO statistical method, which yielded a maximum natural frequency of 26.284 with optimum values for “n”, “L/h”, “L/w”, and “T_c” of 0.5, 20.0, 1.50, and 300.0, respectively. The red line indicates the optimum value for each of the input parameters in this graph. The optimal parameters for the natural frequency of a simple supported FGM plate using the Whale Optimization Algorithm (WOA) are “n” = 0.5, “L/h” = 20.0, “L/w” = 1.5, and “Tc” = 300.0, with the global optimum value for the natural frequency being 26.318. Additionally, Figure 19 highlights the optimal result for the natural frequency of a simple supported FGM plate using the WOA. Similarly, Figure 20 illustrates the results of Corrected Moth Search Optimization (CMSO) with a maximum natural frequency of 25.081 and input parameters of 0.565, 15.800, 1.499, and 334.689 for “n”, “L/h”, “L/w”, and “T_c”, respectively. Additionally, the Lichtenberg Algorithm Optimization (LAO) approach was applied to the RSM data set in order to determine the global maximum value of natural frequency, and the precise value of 25.505 was obtained from the simulation of the MATLAB code, along with the optimum parametric settings of “n” = 0.500, “L/h” = 16.947, “L/w” = 1.500, and “T_c” = 387.040, as shown in Figure 21. The Sunflower Optimization Algorithm (SFO) was applied to the data set of RSM design of experiments using a basic supported FGM plate, and the SFO convergence plot is displayed in Figure 22, with a global optimal value of 25.848, which is the second lowest value among all methods. The optimal configuration for this natural frequency maximisation issue was found using the SFO technique with values of 0.533, 18.962, 1.500, and 325.997 for the input variables “n”, “L/h”, “L/w”, and “T_c”. Finally, using the RSM-designed dataset, a new optimization method called Forensic-Based Investigation Algorithm (FBI) was used to determine the global maximum point in the space value of natural frequency, and the final convergence plot is shown in Figure 23. The optimal parameters are defined as “n” = 0.759, “L/h” = 15.160, “L/w” = 1.426, and “T_c” = 388.350, with the global optimum value for the natural frequency set at 22.009 using the FBI optimization method. In each of the five metaheuristic optimization methods, the population size and number of iterations were maintained constant at 1000 and 100, respectively.

Figure 18.

Optimum results obtained by RSM-CDO for ${\bar{ω}}_{1_{C}}$ .

Figure 19.

Convergence plot obtained by WOA for ${\bar{ω}}_{1_{C}}$ .

Figure 20.

Convergence plot obtained by CMSO for ${\bar{ω}}_{1_{C}}$ .

Figure 21.

Convergence plot obtained by LAO for ${\bar{ω}}_{1_{C}}$ .

Figure 22.

Convergence plot obtained by SFO for ${\bar{ω}}_{1_{C}}$ .

Figure 23.

Convergence plot obtained by FBI for ${\bar{ω}}_{1_{C}}$ .

Comparative analysis

Comparative bar chart of percentage contribution each terms for ${\bar{ω}}_{1_{S S}}$ and ${\bar{ω}}_{1_{C}}$ is plotted in Figure 24 on the basis of ANOVA test in each cases. In this figure, it can be clearly seen that the variable of “L/w” has most influential characteristics on first non-dimensional natural frequency in both simple supported and clamped typed FGM plate.

Figure 24.

Comparative bar chart of percentage contribution each terms for ${\bar{ω}}_{1_{S S}}$ and ${\bar{ω}}_{1_{C}}$ .

A comprehensive evaluation of all of the suggested optimization strategies has been carried out, and the findings of this evaluation are summarised in Table 11. The maximum global maximum optimal values and parametric optimal values are plotted in Figures 25 and 26 where green coloured bars shows the value of resulted maximum value of output response of natural frequency. Out of all six optimization approaches it is found that the maximum value of

{\bar{ω}}_{1_{S S}}

and

{\bar{ω}}_{1_{C}}

occurred in RSM-Composite Desirability Optimization (RSM-CDO) and Whale Optimization Algorithm (WOA) approaches respectively.

Table 11.

Comparison of optimization approaches in terms of optimum value and their optimum settings.

Natural frequency	Sl. No.	Optimization	Global optimum Value	n	L/h	L/w	T _c
${\bar{ω}}_{1_{S S}}$	1	RSM-composite desirability optimization (RSM-CDO)	19.850	0.000	18.224	1.500	300.000
	2	Whale optimization algorithm (WOA)	19.840	0.000	18.558	1.500	300.000
	3	Corrected Moth Search optimization (CMSO)	19.451	0.007	13.922	1.499	327.752
	4	Lichtenberg algorithm optimization (LAO)	19.303	0.000	12.067	1.500	352.314
	5	Sunflower optimization algorithm (SFO)	18.116	0.099	12.438	1.481	382.776
	6	Forensic-based investigation algorithm (FBI)	15.084	0.144	10.654	1.294	357.427
${\bar{ω}}_{1_{C}}$	1	RSM-composite desirability optimization (RSM-CDO)	26.284	0.500	20.000	1.500	300.000
	2	Whale optimization algorithm (WOA)	26.318	0.500	20.000	1.500	300.000
	3	Corrected Moth Search optimization (CMSO)	25.081	0.565	15.800	1.499	334.689
	4	Lichtenberg algorithm optimization (LAO)	25.505	0.500	16.947	1.500	387.040
	5	Sunflower optimization algorithm (SFO)	25.848	0.533	18.962	1.500	325.997
	6	Forensic-based investigation algorithm (FBI)	22.009	0.759	15.160	1.426	388.350

Figure 25.

Comparative analysis of all proposed optimization techniques with maximization of natural frequency ( ${\bar{ω}}_{1_{S S}}$ ) in case study 1 i.e. for simple supported FGM plate.

Figure 26.

Comparative analysis of all proposed optimization techniques with maximization of natural frequency ( ${\bar{ω}}_{1_{C}}$ ) in case study 2 i.e. for clamped typed FGM plate.

Confirmatory test

To validate the predicted value acquired using the various methods, confirmatory experiments are used to validate the anticipated output response. An equation (47) was used to calculate the expected output response of natural frequency.¹⁷

U_{E} = U_{m} + \sum_{i = 1}^{n} (Ù_{i} - U_{m})

(47)

where U_m denotes the final level’s total mean response value [69]. The error percentages for the values obtained using the RSM-Composite Desirability Optimization (RSM-CDO) approach and an in-house MATLAB code for predicting natural frequencies for simple supported and clamped type FGM are 5.736% and 1.713%, respectively, indicating the precision of the predicted values and configuration, which is tabulated in the Table 12. At a 95% confidence interval, all outcome responses of natural frequency were significant. The percentage of error was determined through equation (48) and which is mentioned below:

P e r c e n t a g e E r r o r = | \frac{(P r e d i c t e d V a l u e - E x p e r i m e n t a l V a l u e)}{E x p e r i m e n t a l V a l u e} | \times 100

(48)

Table 12.

Analysis of confirmatory test results of both case studies.

Natural frequency	Optimization method used for prediction	Optimal setting				Predicted value ${({\bar{ω}}_{1})}_{P r e d}$	Experimental value ${({\bar{ω}}_{1})}_{Exp}$	95% confidence interval	95% prediction interval	Percentage error, %
Natural frequency	Optimization method used for prediction	n	L/h	L/w	T _c	Predicted value ${({\bar{ω}}_{1})}_{P r e d}$	Experimental value ${({\bar{ω}}_{1})}_{Exp}$	95% confidence interval	95% prediction interval	Percentage error, %
( ${\bar{ω}}_{1_{S S}}$ )	RSM-CDO approach	0	18.5859	1.5	300	19.852	21.06	(18.303, 21.401)	(17.914, 21.789)	5.736
( ${\bar{ω}}_{1_{C}}$ )	RSM-CDO approach	0.5	20	1.5	300	26.284	26.742	(25.409, 27.159)	(25.238, 27.330)	1.713

5. Conclusions

In the current study, the free vibration response of a functionally graded plate that has been exposed to nonlinear temperature change is investigated. The finite element method is used to solve the equation of motion due to first-order shear deformation. Furthermore, new advanced hybrid optimization approaches are applied to determine the best solution for two boundary conditions of FGM plates. In addition, a confirmatory test was carried out in order to validate the findings that were obtained within the confines of a confidence interval of 95%. The research is conducted in a parametric manner, and the study’s main findings are summarised here:

• The value of the fundamental frequency is lowered when there is an increase in the temperature of the ceramic side.

• The parameter of length to width ratio gave the maximum percentage of contribution in the entire analysis, and hence this parameter has a significant contribution to the whole experimentation in both cases for obtaining the global best value of non-dimensional natural frequency.

• The RSM-Composite Desirability Optimization (RSM-CDO) and Whale Optimization Algorithm (WOA) approaches show the best results for obtaining global best value among six hybrid optimization approaches for both simple supported and fully clamped FGM plates.

• Lastly, the percentage error in the confirmatory test was also shown to be minimal in the case of the statistical approach of the RSM-CDO methodology with experimental value at optimum setting.

➢ Highlights

➢ The present paper examines the free vibration response of a functionally graded plate subjected to nonlinear temperature change.

➢ The concept is based on first-order shear deformation theory. The finite element method is used to find the solution to the equation of motion.

➢ Further, new advanced hybrid optimization approaches namely RSM-Composite Desirability Optimization (RSM-CDO), Whale Optimization Algorithm (WOA), Corrected Moth Search Optimization (CMSO), Lichtenberg Algorithm Optimization (LAO), Sunflower Optimization Algorithm (SFO) and Forensic-Based Investigation Algorithm (FBI) are applied to determine best solution for two boundary conditions of FGM plates.

➢ Also, the confirmatory test was conducted to verify the obtained results within the range of 95% confidence interval.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Ethical approval

None of the writers of this article have conducted any experiments using humans or animals.

Informed consent

Each person who took part in the study voluntarily gave their informed permission.

ORCID iD

Dilip Kumar Bagal

Appendix

(A.1)

E l e m e n t a l K i n e t i c E n e r g y, T_{e} = \frac{1}{2} {{\dot{δ}}_{e}}^{T} [M_{e}] {{\dot{δ}}_{e}} + {δ_{e}}^{T} {F_{c e}}

(A.2)

E l e m e n t a l S t r a i n E n e r g y, V_{e} = V_{e 0} + V_{e 1}

(A.3)

w h e r e V_{e 0} = \frac{1}{2} {δ_{e}}^{T} ([K_{G T e}] + [K_{G R e}]) {δ_{e}}

(A.4)

V_{e 1} = \frac{1}{2} {δ_{e}}^{T} [K_{e}] {δ_{e}}

(A.5)

T h u s, \begin{array}{l} V_{e} = \frac{1}{2} {δ_{e}}^{T} ([K_{G T e}] + [K_{G Re}]) {δ_{e}} + \frac{1}{2} {δ_{e}}^{T} [K_{e}] {δ_{e}} \\ = \frac{1}{2} {δ_{e}}^{T} ([K_{e}] + [K_{G T e}] + [K_{G Re}]) {δ_{e}} \end{array}

(A.6)

\frac{\partial V_{e}}{\partial δ_{e}} = ([K_{e}] + [K_{G T e}] + [K_{G Re}]) {δ_{e}}

(A.7)

\frac{\partial T_{e}}{\partial δ_{e}} = {F_{Ω e}}

(A.8)

\frac{\partial T_{e}}{\partial {\dot{δ}}_{e}} = [M_{e}] {{\dot{δ}}_{e}}

(A.9)

\frac{d}{d t} (\frac{\partial T_{e}}{\partial {\dot{δ}}_{e}}) = [M_{e}] {{\ddot{δ}}_{e}}

(A.10)

L a g r a n g e ’ s e q u a t i o n, \frac{d}{d t} [\frac{\partial T_{e}}{\partial {\dot{δ}}_{e}}] - [\frac{\partial T_{e}}{\partial δ_{e}}] + \frac{\partial V_{e}}{\partial δ_{e}} = 0

Now putting the values in the above equation, we have (A.11)

[M_{e}] {{\ddot{δ}}_{e}} - {F_{c e}} + (([K_{e}] + [K_{G T e}]) + [K_{G Re}]) {δ_{e}} = 0

(A.12)

O r, [M_{e}] {{\ddot{δ}}_{e}} + (([K_{e}] + [K_{G T e}]) + [K_{G Re}]) {δ_{e}} = {F_{c e}}

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