Analytical modeling of laminated composite plates using Kirchhoff circuit and wave digital filters

Chien Hsun Tseng

doi:10.3934/jimo.2019051

Article Contents

2020, Volume 16, Issue 5: 2213-2252. Doi: 10.3934/jimo.2019051

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Analytical modeling of laminated composite plates using Kirchhoff circuit and wave digital filters

Chien Hsun Tseng

Department of Information Engineering, Kun Shan University, Taiwan

Received: March 31, 2018

Revised: December 31, 2018

Early access: May 2019

Published: August 2020

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Abstract

A physical-based numerical algorithm using Kirchhoff circuit is detailed for modelling the free vibration of moderate thick symmetrically laminated plates based on the first order shear deformation theory (FSDT). With the help of multidimensional passivity of analog circuit and nonlinear optimization solvers, the philosophy gives rise to a nonlinear programming (NLP) model that can apply further to explore stability characteristics and optimum performance of the resultant multidimensional wave digital filtering network representing the FSDT plate. Various optimization methods exploiting gradient-based and direct search methods are adopted with efficient broad search power to tackle the NLP model. As a result, the necessary Courant-Friedrichs-Levy stability criterion can be fully satisfied at all time with least restriction on the spatially discretized geometry of the scattering problem. With full stability guaranteed, the waveform is analyzed by the power cepstrum for spectra peaks detection, which has led to more accurate estimate of various vibration effects in predicting nature frequencies with different fiber orientations, stacking sequences, stiffness ratios and boundary conditions. These results have shown in excellent agreement with early published works based on the finite element solutions of the high-order shear deformation theory and other well known numerical techniques.

Keywords:

Mathematics Subject Classification: Primary: 35L10; Secondary: 58J45.

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Figure 1. A schematic flow diagram towards modeling a general MDWDF network

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Figure 3. A MDKC representation for a symmetrically laminated composite FSDT plate with free vibration

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Figure 4. A MDWDF network for numerical simulation of the laminated plate system (2.6)-(2.7)

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Figure 2. Geometry of square laminated plates with (a)threelayer cross-ply stacking sequence [0°/90°/0°] (b) four-lay crossply stacking sequence [0°/90°/90°/0°], and (c) five-layer cross-ply stacking sequence [0°/90°/90°/90°/0°]

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Figure 5. Key parameters obtained by the ASA solver for the four-layer cross-ply $ [0^\circ/90^\circ/90^\circ/0^\circ] $ square plate with stiffness ratios $ E_1/E_2 = 10, 20, 30, 40 $. (a1)-(d1): Objective function value ($ \chi $). (a2)-(d2): The maximum constraint violation corresponding to the objective value. (a3)-(d3): The first-order optimality

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Figure 6. MDWDF network robustness indicated by the percentage error distribution ($ E_{L_{SE}}(\%)) $ w.r.t. various scales of CFL number for four stiffness ratios $ E_1/E_2 = 10, 20, 30, 40 $ where model 4 of $ \chi_{min} $ is used as a reference

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Figure 7. MDWDF network stability indicated by the percentage error distribution ($ E_{L_{KE}}(\%)) $ w.r.t. various scales of CFL number for four stiffness ratios $ E_1/E_2 = 10, 20, 30, 40 $

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Figure 8. Vibration waveform and its corresponding power cepstrum for four-layer cross-ply layup $ [0^\circ/90^\circ/90^\circ/0^\circ] $ square laminates with the stiffness ratios $ E_1/E_2 = 10, 20 $

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Figure 9. Vibration waveform and its corresponding power cepstrum for four-layer cross-ply layup $ [0^\circ/90^\circ/90^\circ/0^\circ] $ square laminates with the stiffness ratios $ E_1/E_2 = 30, 40 $

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Figure 10. The first six modes vibration for the four-layer crossply [0°/90°/90°/0°] SS2 square laminate.

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Figure 11. MDWDF network optimality w.r.t. various scales of $ \chi $ (model) and different stiffness ratios. (a) $ E_1/E_2 = 10:\bar{\omega}_f(\chi_{min}) = 9.8336 $. (b) $ E_1/E_2 = 20:\bar{\omega}_f(\chi_{min}) = 12.1655 $. (c) $ E_1/E_2 = 30:\bar{\omega}_f(\chi_{min}) = 13.7703 $. (d) $ E_1/E_2 = 40:\bar{\omega}_f(\chi_{min}) = 15.0852 $

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Figure 12. Optimal parameters obtained for the study of network stability and optimality w.r.t the SS2 simply supported cross-ply $ [0^\circ/90^\circ/90^\circ/0^\circ] $ laminated plate with various stiffness ratios and $ a/h = 10 $: (a) Optimal CFL number. (b) Optimal $ \bar{\omega}_f $

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Figure 13. Key parameters obtained by the IPA for the SS2 simply supported four-layer cross-ply $ [0^\circ/90^\circ/90^\circ/0^\circ] $ square plate with stiffness ratios $ E_1/E_2 = 10, 20 $. (a1)-(d1) Objective function value ($ \chi $) at every iteration. (a2)-(d2) The maximum constraint violation corresponding to the objective value. (a3)-(d3) The first-order optimality

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Figure 14. Key parameters obtained by the IPA for the SS2 simply supported four-layer cross-ply $ [0^\circ/90^\circ/90^\circ/0^\circ] $ square plate with stiffness ratios $ E_1/E_2 = 10, 20 $. (a1)-(d1) Objective function value ($ \chi $) at every iteration. (a2)-(d2) The maximum constraint violation corresponding to the objective value. (a3)-(d3) The first-order optimality

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Figure 15. Key parameters obtained by the ALGA for the SS2 simply supported four cross-ply $ [0^\circ/90^\circ/90^\circ/0^\circ] $ square plate with various stiffness ratios $ E_1/E_2 = 10, 20, 30, 40 $. (a1)-(d1) Score histograms at every iteration w.r.t. number of individuals. (a2)-(d2) The corresponding fitness values

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Figure 16. Key parameters obtained by the ALPSA for the SS2 simply supported four-layer cross-ply $ [0^\circ/90^\circ/90^\circ/0^\circ] $ square plate with stiffness ratios $ E_1/E_2 = 10, 20, 30, 40 $. (a1)-(d1) Score histograms at every iteration w.r.t. number of functions evaluated. (a2)-(d2) The corresponding function values

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Figure 17. Feasible comparisons in terms of performance measured by $ E_{\bar{\omega}}(\%) $ among (a) nonlinear optimization solvers within the MDWDF network w.r.t stiffness ratios $ E_1/E_2 = 10, 20, 30, 40 $, (b) the CPU runtime of NLP solvers w.r.t stiffness ratios, and (c) MDWDF networks and GRBF method

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Table 1. Option parameters set for the gradient-based NLP solvers

Options for ASA selection	Value	Options for ASA termination	Value
Number of objective function evaluations	$ 150 $	Maxi. number of iterations	$ 400 $
Max. change for finite-difference gradients	$ 0.1 $	Tolerance on objective function	$ 10^{-8} $
Min. change for finite-difference gradients	$ 10^{-8} $	Tolerance of maxi. constraint	$ 10^{-6} $
Finite difference type	forward	Maxi. number of SQP iterations	$ 40 $
		Tolerance on SQP constraint violation	$ 10^{-6} $
Options for IPA selection	Value	Options for IPA termination	Value
Number of objective function evaluations	$ 150 $	Maximum number of iterations	$ 1000 $
Initial barrier value	$ 0.1 $	Tolerance on objective function	$ 10^{-8} $
Max. change for finite-difference gradients	$ 0.1 $	Tolerance of maxi. constraint	$ 10^{-6} $
Min. change for finite-difference gradients	$ 10^{-8} $	Maxi. number of PCG iterations	2
Finite difference type	forward	Tolerance of PCG algorithm	$ 10^{-10} $

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Table 2. Option parameters set for the direct search NLP solvers

Options for ALGA selection	Value	Options for ALGA termination	Value
Size of population	$ 30 $	Maxi. number of iterations	100
Probability of crossover	$ 0.85 $	Tolerance on fitness function	$ 10^{-8} $
Probability of mutation	$ 1 $	Tolerance of maxi. constraint	$ 10^{-6} $
Size of elitism	$ 2 $	Tolerance of nonlinear constraint	$ 10^{-6} $
Initial penalty parameter	$ 30 $	Max. stall generations	$ 50 $
Penalty update parameter	$ 50 $
Options for ALPSA selection	Value	Options for ALPSA termination	Value
Number of objective function evaluations	$ 500 $	Maxi. number of iterations	$ 100 $
Initial penalty parameter	$ 30 $	Tolerance on function value	$ 10^{-8} $
Penalty update parameter	$ 100 $	Min. tolerance for mesh size	$ 10^{-6} $
Poll method	$ a^\ast $PM	Tolerance of nonlinear constraint	$ 10^{-6} $
Search method/Iteration limit	$ b^\ast $SM/$ 30 $	Bind tolerance	$ 10^{-6} $
$a^\ast$PM: GPS positive basis 2N; $b^\ast$SM: Latin hypercube search

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Table 3. The optimum value of $\chi_{min}$ and its corresponding optimization processes performed by the gradient-based methods (ASA and IPA) with the algorithm termination options listed in Table 1 for the four-layer cross-ply $[0^\circ/90^\circ/90^\circ/0^\circ]$ laminated square plates with $E_1/E_2=10,20,30,40$

	ASA					IPA
$E_1/E_2$	Iter.	F-count	$\chi$	Max. constr.	$1^{st}$ opt.	Iter.	F-count	$\chi$	Max. constr.	$1^{st}$ opt.
10	0	2	2.53311	0.001187	Infeasible	0	2	3.523114	0.5000	Infeasible
	1	4	29.1438	-0.0007862	3.86	1	5	3.239829	0.4950	0.2833
	2	6	16.995	-0.0002634	2.22	5	16	2.558054	2.486e-2	0.1092
	3	8	12.6526	-4.356e-5	0.531	9	25	5.851129	5.661e-3	3.331e-2
	4	10	11.9075	-1.283e-6	6.59e-2	13	33	12.04055	1.342e-1	1.866
	5	12	11.8842	-1.256e-9	2.19e-3	17	41	11.88414	3.321e-6	3.957e-5
	6	14	11.8841$^\ast$	-1.225e-15	2.37e-6	20	52	11.88414$^\ast$	2.196e-13	0
20	0	2	2.53311	0.002611	Infeasible	0	2	3.523114	0.5000	Infeasible
	1	4	53.9773	-0.005372	27.4	1	5	3.243753	0.4950	0.2794
	2	6	29.4623	-0.00122	4.2	5	16	2.574543	4.123e-2	9.694e-2
	3	8	19.2631	-2.112e-4	1.59	9	25	9.093263	2.175e-3	1.320e-2
	4	10	16.5631	-1.48e-5	0.184	13	33	16.41998	8.452e-2	1.516
	5	12	16.343	-9.832e-8	1.55e-2	17	41	16.34150	2.417e-6	3.959e-5
	6	14	16.3415$^\ast$	-4.458e-12	1.05e-4	20	52	16.34150$^\ast$	2.654e-13	0
30	0	2	2.53311	0.004084	Infeasible	0	2	3.523114	0.5000	Infeasible
	1	4	78.4373	-0.01117	8.75	1	5	3.247713	0.4950	0.2754
	2	6	41.7109	-0.002616	5.44	5	16	2.589383	5.592e-2	8.980e-2
	3	8	25.542	-5.07e-4	2.63	9	26	11.41252	3.059	8.636
	4	10	20.4244	-5.079e-5	0.486	13	36	21.52822	1.733	8.5
	5	12	19.7832	-7.972e-7	3.76e-2	17	44	19.77675	4.03e-3	1.526e-2
	6	14	19.7728	-2.093e-10	6.29e-4	20	50	19.77282	2.0e-6	3.960e-5
	7	16	19.7728$^\ast$	-1.789e-17	1.92e-7	23	58	19.77282$^\ast$	7.136e-9	2.069e-14
40	0	2	2.53311	0.005573	Infeasible	0	2	3.523114	0.5000	Infeasible
	1	4	102.773	-0.01904	20.1	1	5	3.251616	0.4950	0.2715
	2	6	53.8883	-0.004527	5.94	5	16	2.603136	6.948e-2	8.390e-2
	3	8	31.7156	-9.314e-4	3.55	9	26	13.44820	3.121	10.49
	4	10	23.9651	-1.138e-4	0.874	13	36	24.51400	1.666	9.295
	5	12	22.7118	-2.976e-6	6.42e-2	17	44	22.68110	4.026e-3	1.542e-2
	6	14	22.6772	-2.266e-9	1.87e-3	20	50	22.67716	1.754e-6	3.960e-5
	7	16	22.6772$^\ast$	-1.351e-15	1.28e-6	22	54	22.67716$^\ast$	8.846e-9	3.960e-7

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Table 4. Key results measured by the first $ 800 $ temporal steps for the study of network robustness and optimality based on the ASA solver w.r.t the four-layer cross-ply layup $ [0^\circ/90^\circ/90^\circ/0^\circ] $ laminated square plate with $ E_1/E_2 = 10, 20 $

$ E_1/E_2 $	Model($ \chi $)	$ T_t(ms) $	CFL no.	$ \bar{L}_1 $	$ \bar{L}_4 $	$ \bar{L}_5 $	$ [E_{L_{KE}}(\%)] $	$ [E_{L_{SE}}(\%)] $	Div. step
10	$ 1(\chi_{min}-3.0\%) $	4.33	0.1552	1.18	-0.19	49.95	[-45, 2471]	[-22, 2505]	108
	$ 2(\chi_{min}-1.5\%) $	4.27	0.1530	1.22	-0.10	51.90	[-63, 1224]	[-34, 1263]	182
	$ 3(\chi_{min}-1.0\%) $	4.24	0.1522	1.23	-0.06	52.67	[-54,454]	[-37,499]	331
	$ 4(\chi_{min}) $	4.21	0.1507	1.26	0	54.03	reference	reference	X
	$ 5(\chi_{min}+1.0\%) $	4.16	0.1492	1.28	0.07	55.43	[-47, 12]	[-36, 2.2]	X
	$ 6(\chi_{min}+1.5\%) $	4.14	0.1485	1.30	0.10	56.15	[-45, 14]	[-39, 4.4]	X
	$ 7(\chi_{min}+3.0\%) $	4.08	0.1462	1.34	0.20	5C.37	[-42, 19]	[-40, 8.9]	X
	$ 8(\chi_{min}+1.5\mbox{x}) $	1.68	0.0602	8.17	17.13	416.5	[-30, 32]	[-30, 23]	X
	$ 9(\chi_{min}+3.0\mbox{x}) $	1.05	0.0376	21.0	48.94	1089	[-30, 32]	[-30, 24]	X
20	$ 1(\chi_{min}-3.0\%) $	3.15	0.1129	7.17	-0.32	127.5	[-44, 2712]	[-18, 2746]	102
	$ 2(\chi_{min}-1.5\%) $	3.11	0.1112	7.40	-0.16	132.4	[-47, 1590]	[-30, 1628]	153
	$ 3(\chi_{min}-1.0\%) $	3.09	0.1107	7.48	-0.11	134.1	[-49, 1031]	[-46, 1073]	210
	$ 4(\chi_{min}) $	3.06	0.1096	7.64	0	137.4	reference	reference	X
	$ 5(\chi_{min}+1.0\%) $	3.03	0.1085	7.79	0.11	140.8	[-49, A.7]	[-56, 2.9]	X
	$ 6(\chi_{min}+1.5\%) $	3.01	0.1080	7.88	0.16	142.5	[-47, 9.6]	[-32, 5.4]	X
	$ 7(\chi_{min}+3.0\%) $	2.97	0.1063	8.11	0.33	147.6	[-44, 14]	[-47, 9]	X
	$ 8(\chi_{min}+1.5\mbox{x}) $	1.22	0.0438	48.7	28.50	1017	[-37, 38]	[-26, 23]	X
	$ 9(\chi_{min}+3.0\mbox{x}) $	0.76	0.0274	125.0	81.31	2651	[-32, 40]	[-41, 22]	X
$\bar{L}_j=L_j\times10^4, j=1, 4, 5.$ $E_1/E_2=10:\chi_{min}=11.8841$, $E_1/E_2=20:\chi_{min}=16.3415$.

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Table 5. Key results measured by the first $ 800 $ temporal steps for the study of network robustness and optimality based on the ASA solver w.r.t the four-layer cross-ply layup $ [0^\circ/90^\circ/90^\circ/0^\circ] $ laminated square plate with $ E_1/E_2 = 30, 40. $

$ E_1/E_2 $	Model($ \chi $)	$ T_t(ms) $	CFL no.	$ \bar{L}_1 $	$ \bar{L}_4 $	$ \bar{L}_5 $	$ [E_{L_{KE}}(\%)] $	$ [E_{L_{SE}}(\%)] $	Div. step
30	$ 1(\chi_{min}-3.0\%) $	2.61	0.0933	21.68	-0.45	209.1	[-46, 2761]	[-20, 2810]	101
	$ 2(\chi_{min}-1.5\%) $	2.57	0.0919	22.37	-0.23	217.0	[-49, 1692]	[-47, 1743]	147
	$ 3(\chi_{min}-1.0\%) $	2.55	0.0915	22.61	-0.15	219.7	[-51, 1177]	[-37, 1231]	190
	$ 4(\chi_{min}) $	2.53	0.0905	23.07	0	225.0	reference	reference	X
	$ 5(\chi_{min}+1.0\%) $	2.50	0.0897	23.55	0.15	230.5	[-51, 12]	[-47, 5.1]	X
	$ 6(\chi_{min}+1.5\%) $	2.49	0.0892	23.79	0.23	233.3	[-49, 15]	[-51, 7.6]	X
	$ 7(\chi_{min}+3.0\%) $	2.46	0.0879	24.49	0.46	241.5	[-47, 22]	[-30, 11]	X
	$ 8(\chi_{min}+1.5\mbox{x}) $	1.01	0.0362	146.2	39.8	1644	[-37, 36]	[-30, 24]	X
	$ 9(\chi_{min}+3.0\mbox{x}) $	0.63	0.0226	374.9	113.7	4280	[-34, 38]	[-33, 24]	X
40	$ 1(\chi_{min}-3.0\%) $	2.27	0.0814	48.41	-0.58	291.9	[-47, 2792]	[-26, 2850]	100
	$ 2(\chi_{min}-1.5\%) $	2.24	0.0801	49.94	-0.29	302.9	[-50, 1735]	[-333, 1796]	144
	$ 3(\chi_{min}-1.0\%) $	2.23	0.0798	50.46	-0.19	306.6	[-52, 1238]	[-34, 1302]	184
	$ 4(\chi_{min}) $	2.20	0.0789	51.50	0	314.1	reference	reference	X
	$ 5(\chi_{min}+1.0\%) $	2.18	0.0782	52.54	0.19	321.6	[-52, 10]	[-52, 2.7]	X
	$ 6(\chi_{min}+1.5\%) $	2.17	0.0778	53.10	0.29	325.4	[-50, 14]	[-44, 4.7]	X
	$ 7(\chi_{min}+3.0\%) $	2.14	0.0766	54.67	0.59	336.9	[-47, 20]	[-36, 8.3]	X
	$ 8(\chi_{min}+1.5\mbox{x}) $	0.88	0.0315	325.3	51.1	2280	[-36, 37]	[-22, 23]	X
	$ 9(\chi_{min}+3.0\mbox{x}) $	0.55	0.0197	833.7	146.1	5932	[-35, 38]	[-30, 24]	X
$\bar{L}_j=L_j\times10^4, j=1, 4, 5.$ $E_1/E_2=30:\chi_{min}=19.7728$, $E_1/E_2=40:\chi_{min}=22.6772$.

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Table 6. The cepstrum analysis for four-layer cross-ply layups $ [0^\circ/90^\circ/90^\circ/0^\circ] $ square laminates

$ E_1/E_2 $	$ T_{C_p}(ms) $	$ T_{FP}(ms) $	$ \omega_f(rad/s) $	$ \bar{\omega}_f(rad/s) $	$ E_{\bar{\omega}}(\%) $
10	1211.6	2414.9	2.60175	9.83369172	0.195963
20	985.2	1964.3	3.24926	12.28105278	0.823283
30	852.1	1699.3	3.69751	13.97527101	0.599417
40	782.7	1561.0	4.02499	15.21304757	0.462573
$E_{\bar{\omega}}(\%)$ uses the TSDT-FEM results [30] as the reference.

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Table 7. Nondimensionalized nature frequencies based on the ASA solver for the first six modes of the four-layer cross-ply layup $ [0^\circ/90^\circ/90^\circ/0^\circ] $ square laminates

	Mode$ (\alpha, \beta) $
$ E_1/E_2 $	$ 1(1, 1) $	$ 2(1, 2) $	$ 3(2, 1) $	$ 4(2, 2) $	$ 5(1, 3) $	$ 6(2, 3) $
10	9.83369172	18.81513016	27.66930905	33.20317086	34.20932756	44.79792894
20	12.28105278	22.82830988	32.74947408	38.42388791	40.00837813	51.06332472
30	13.97527101	25.11064738	35.17371580	41.92581303	45.15087558	53.36012568
40	15.21304757	2A.79312856	36.38796514	44.14277738	48.08409679	59.83798712

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Table 9. Optimization process performed by the direct search methods (ALGA and ALPSA) with the algorithm termination options for the four-layer cross-ply scheme $ [0^\circ/90^\circ/90^\circ/0^\circ] $ and various types of stiffness ratio $ E_1/E_2 = 10, 20, 30, 40 $

	ALGA				ALPSA
$ E_1/E_2 $	Gen.	F-count	$ \chi $	Max. constr.	Iter.	F-count	$ \chi $	Max. constr.	Mesh size
10	1	1066	C.28321	1.678e-4	0	1	3.79967	2.088e-4	1
	3	3146	2.53318	1.187e-3	1	94	5.37203	2.596e-4	1.074e-3
	5	5226	2.53311	1.187e-3	2	343	14.091	0	3.333e-4
	7	7306	2.53311	1.187e-3	3	451	11.8842	0	3.333e-6
	10	10516	11.8842$ ^\ast $	0	4	501	11.8842$ ^\ast $	0	3.333e-8
20	1	530	2.53311	2.611e-3	0	1	3.79967	2.108e-3	1
	3	1570	2.53311	2.611e-3	1	61	24.7257	0	3.333e-4
	5	2610	2.53311	2.611e-3	2	262	16.3415	0	3.333e-6
	7	3668	1A.4907	0	3	433	16.3415$ ^\ast $	0	3.333e-8
	10	5269	16.3415$ ^\ast $	0
30	0	0	2.53311	Infeasible	0	1	2.53311	4.084e-3	1
	1	1072	24.6611	0	1	25	43.6032	0	3.333e-4
	2	2138	19.7728	0	2	266	19.7728	0	3.333e-6
	3	3190	19.7728$ ^\ast $	0	3	437	19.7728$ ^\ast $	0	3.333e-8
40	0	0	2.53311	Infeasible	0	1	2.53311	5.573e-3	1
	1	1072	2C.3387	0	1	29	48.9484	0	3.333e-4
	2	2138	22.6772	0	2	258	22.6772	0	3.333e-6
	3	3190	22.6772$ ^\ast $	0	3	441	22.6772$ ^\ast $	0	3.333e-8

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Table 8. Numerical accuracy involving four different NLP solvers for the four-layer cross-ply laminated SS2 square plates $ [0^\circ/90^\circ/90^\circ/0^\circ] $

	NLP		$ E_1/E_2 $
Method	Algorithm	Para.	$ 10 $	$ 20 $	$ 30 $	$ 40 $
FSDT-MDWDF	ASA	$ \chi_{min} $	11.88413667	16.34149675	19.77282249	22.67715857
		$ CFL_{max} $	0.15072042	0.10960942	0.09058808	0.07898617
		CPU time(ms)	2411.1	2425.98	2461.76	2502.44
		$ \bar{\omega}_f $	9.83369172	12.28105278	13.97527101	15.21304757
		$ E_{\bar{\omega}}(\%) $	0.19596347	0.82328368	0.59941701	0.46257394
	IPA	$ \chi_{min} $	11.88413707	16.34149709	19.77282289	22.67715898
		$ CFL_{max} $	0.15072041	0.10960942	0.09058808	0.07898617
		CPU time(ms)	2265.58	1745.8	1824.4	1972.86
		$ \bar{\omega}_f $	9.83369205	12.28105303	13.97527129	15.21304784
		$ E_{\bar{\omega}}(\%) $	0.19596011	0.82328166	0.59941904	0.46257571
	ALGA	$ \chi_{min} $	11.88413869	16.34149995	19.77282581	22.67716160
		$ CFL_{max} $	0.15072040	0.10960941	0.09058807	0.07898617
		CPU time(ms)	6262.2	6137.32	4223.16	4194.09
		$ \bar{\omega}_f $	9.83370380	12.28105800	13.97527405	15.21304960
		$ E_{\bar{\omega}}(\%) $	0.1958408	0.82324156	0.59943894	0.46258734
	ALPSA	$ \chi_{min} $	11.88415178	16.34150453	19.77282756	22.67716223
		$ CFL_{max} $	0.15072013	0.10960929	0.09058806	0.07898616
		CPU time(ms)	2924.57	2781.15	2593.6	2857.36
		$ \bar{\omega}_f $	9.83370423	12.28105862	13.97527458	15.21305002
		$ E_{\bar{\omega}}(\%) $	0.19583654	0.82323649	0.59944269	0.46259008
FSDT-GRBF [46]		$ \bar{\omega}_f $	9.539	11.977	13.716	15.059
		$ E_{\bar{\omega}}(\%) $	3.18684664	3.27868852	1.26691621	0.55471175
FSDT-EFG [7]		$ \bar{\omega}_f $	9.670	12.115	13.799	15.068
		$ E_{\bar{\omega}}(\%) $	1.85730234	1.00506618	0.66945004	0.49527834
FSDT-FEM [36]		$ \bar{\omega}_f $	9.841	12.138	13.864	15.107
		$ E_{\bar{\omega}}(\%) $	0.12179031	0.16342539	0.20155485	0.23773360
TSDT-EFG [7]		$ \bar{\omega}_f $	9.842	12.138	14.154	15.145
		$ E_{\bar{\omega}}(\%) $	0.11164112	0.16342539	1.90757270	0.01320742
TSDT-FEM [36]		$ \bar{\omega}_f $	9.853	12.238	13.892	15.143
$E_{\bar{\omega}}(\%)$ uses the TSDT-FEM results [36] as the reference.

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Table 10. Numerical accuracy of MDWDF networks for the three-layer cross-ply $ [0^\circ/90^\circ/0^\circ] $ laminated plate

	NLP		$ E_1/E_2 $
Method	Algorithm	Para.	$ 10 $	$ 20 $	$ 30 $	$ 40 $
FSDT-MDWDF	ASA	$ \chi_{min} $	12.91506498	17.80436624	21.46965556	24.53187819
		$ CFL_{max} $	0.14077503	0.10211645	0.08468317	0.07411249
		$ \bar{\omega}_f $	9.75894349	12.09708139	13.65712101	14.70474902
		$ E_{\bar{\omega}}(\%) $	0.35793863	0.37405733	0.59012307	0.41481091
	IPA	$ \chi_{min} $	12.91506538	17.80436635	21.46965596	24.53187859
		$ CFL_{max} $	0.14077502	0.10211645	0.08468318	0.07411249
		$ \bar{\omega}_f $	9.75894379	12.09708146	13.65712126	14.70474926
		$ E_{\bar{\omega}}(\%) $	0.35793557	0.00374057	0.59012491	0.41480929
	ALGA	$ \chi_{min} $	12.91506688	17.80436836	21.46965818	24.53190153
		$ CFL_{max} $	0.14077501	0.10211643	0.08468316	0.0741124239
		$ \bar{\omega}_f $	9.75894493	12.09708282	13.65712267	14.70476301
		$ E_{\bar{\omega}}(\%) $	0.35792393	0.37406920	0.59013530	0.41471617
	ALPSA	$ \chi_{min} $	12.91825286	17.81031318	21.46968482	24.53190155
		$ CFL_{max} $	0.14074029	0.10208235	0.08468306	0.0741124238
		$ \bar{\omega}_f $	9.76135232	12.10112200	13.65713962	14.70476302
		$ E_{\bar{\omega}}(\%) $	0.33334367	0.00407583	0.00590260	0.41471610
HSDT-FEM [20]		$ \bar{\omega} $	9.794	12.052	13.577	14.766
$E_{\bar{\omega}_f}(\%)$ uses the HSDT-FEM results [20] as the reference.

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Table 11. Numerical accuracy of the MDWDF network for the five-layer cross-ply $ [0^\circ/90^\circ/90^\circ/90^\circ/0^\circ] $ laminated plate $ (\bar{\omega}_f = (\omega_fb^2/\pi^2\sqrt{\rho h/D_0}, \kappa = \pi^2/12, a/h = 10, a/b = 1) $

NLP		$ E_1/E_2 $
Algorithm	Para.	$ 10 $	$ 20 $	$ 30 $	$ 40 $
ASA	$ \chi_{min} $	11.12198459	15.31530671	18.56722410	21.43934366
	$ CFL_{max} $	0.15957798	0.11588562	0.09558908	0.08375115
	$ \bar{\omega}_f $	9.53036311	11.97168174	13.53757952	5.46066961
IPA	$ \chi_{min} $	11.12198460	15.31530709	18.56722348	21.32430073
	$ CFL_{max} $	0.15957798	0.11588562	0.09558908	0.08323011
	$ \bar{\omega}_f $	9.53036311	11.97168204	13.69778710	15.02075228
ALGA	$ \chi_{min} $	11.12198680	15.31531111	18.56722310	21.32430035
	$ CFL_{max} $	0.15957795	0.11588559	0.09558908	0.08323011
	$ \bar{\omega}_f $	9.53036500	11.97168518	13.69778682	15.02075202
ALPSA	$ \chi_{min} $	11.12199160	15.31530987	18.56722509	21.32430176
	$ CFL_{max} $	0.15957788	0.11588560	0.09558907	0.08323010
	$ \bar{\omega}_f $	9.53036911	11.97168421	13.69778829	15.02075301

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