doi: 10.3934/jimo.2019051

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

Department of Information Engineering, Kun Shan University, Taiwan

Received  April 2018 Revised  January 2019 Published  May 2019

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.

Citation: Chien Hsun Tseng. Analytical modeling of laminated composite plates using Kirchhoff circuit and wave digital filters. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019051
References:
[1]

S. Atluri and S. Zhu, A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Computational Mechanics, 22 (1998), 117-127. doi: 10.1007/s004660050346. Google Scholar

[2]

S. Basu and A. Zerzghi, Multidimensional digital filter approach for numerical solution of a class of PDEs of the propagating wave type, ISCAS '98. Proceedings of the 1998 IEEE International Symposium on Circuits and Systems (Cat. No.98CH36187), 5 (1998), 74–77, Monterey, California, USA. doi: 10.1109/ISCAS.1998.694411. Google Scholar

[3]

S. Billbao, Wave and Scattering Methods for Numerical Simulation, New York: Wiley, 2004. doi: 10.1002/0470870192. Google Scholar

[4]

B. P. Bogert, M. J. R. Healy and J. W. Tukey, The quefrency alanysis of time series for echoes: Cepstrum, pseudo autocovariance, cross-cepstrum and saphe cracking, Proceedings of the Symposium on Time Series Analysis (M. Rosenblatt, Ed) New York: Wiley, 1963, Chapter 15, 209–243.Google Scholar

[5]

T. BuiM. Nguyen and C. Zhang, An efficient meshfree method for vibration analysis of laminated composite plates, Computational Mechanics, 48 (2011), 175-193. doi: 10.1007/s00466-011-0591-8. Google Scholar

[6]

R. ByrdM. Hribar and J. Nocedal, An interior point algorithm for large-scale nonlinear programming, SIAM Journal on Optimization, 9 (1999), 877-900. doi: 10.1137/S1052623497325107. Google Scholar

[7]

K. Y. DaiG. R. LiuK. M. Lim and X. L. Chen, A mesh-free method for static and free vibration analysis of shear deformation laminated composite plates, J. Sound & Vibration, 269 (2004), 633-652. Google Scholar

[8]

Z. Dong, B. Tan, Y. Zhang, J. Yuan, E. Feng, H. Yin and Z. Xiu, Strong stability of an optimal control hybrid system in fed-batch fermentation, International Journal of Biomathematics, 11 (2018), 1850045, 17pp. doi: 10.1142/S1793524518500456. Google Scholar

[9]

A. A. J. Ferreira and G. Fasshauer, Analysis of natural frequencies of composite plates by an RBF-pseudospectral method, Composite Structures, 79 (2007), 202-210. doi: 10.1016/j.compstruct.2005.12.004. Google Scholar

[10]

A. J. M. FerreiraC. M. C. Roque and R. M. N. Jorge, Free vibration analysis of symmetric laminated composite plates by FSDT and radial basis functions., Computer Methods in Applied Mechanics and Engineering, 194 (2005), 4265-4278. doi: 10.1016/j.cma.2004.11.004. Google Scholar

[11]

A. Fettweis, Wave digital filters: Theory and practice, Proceedings of the IEEE, 74 (1986), 270-327. doi: 10.1109/PROC.1986.13458. Google Scholar

[12]

A. Fettweis and G. Nitsche, Numerical integration of partial differential equations using principles of multidimensional wave digital filters, J. VLSI Signal Processing, 3 (1991), 7-24. doi: 10.1007/978-1-4615-4036-6_2. Google Scholar

[13]

A. Fettweis and G. Nitsche, Transformation approach to numerical integrating PDEs by means of WDF principles, Multidimensional Systems and Signal Processing, 2 (1991), 127-159. doi: 10.1007/BF01938221. Google Scholar

[14]

A. Fettweis, Robust numerical integration using wave-digital concepts, Multidim. Syst. Signal Process., 17 (2006), 7-25. doi: 10.1007/s11045-005-6236-3. Google Scholar

[15]

A. Fettweis and K. Meerkötter, On parasitic oscillations in digital filters under looped conditions, IEEE Transactions on Circuits and Systems, 25 (1978), 1060-1066. doi: 10.1109/TCS.1978.1084425. Google Scholar

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D. Goldberg, Genetic Algorithms in Search, Optimization & Machine Learning. Addison-Wesley, 1989.Google Scholar

[17]

S. Han, A globally convergent method for nonlinear programming, Journal of Optimization Theory and Applications, 22 (1977), 297-309. doi: 10.1007/BF00932858. Google Scholar

[18]

M. Jones, Mechanics of Composite Materials, 2/E. Taylor & Francis, London, 1998. doi: 10.1201/9781498711067. Google Scholar

[19]

K. Kant and K. Swaminathan, Estimation of transverse/interlaminar stresses in laminated composites-a selective review and survey of current developments, Composite Structures, 49 (2000), 65-75. doi: 10.1016/S0263-8223(99)00126-9. Google Scholar

[20]

A. Khdeir and L. Librescu, Analysis of symmetric cross-ply elastic plates using a higher-order theory, Part II: buckling and free vibration, Composite Structures, 9 (1988), 259-277. Google Scholar

[21]

A. Y. T. LeungC. XiaoB. Zhu and S. Yuanet, Free vibration of laminated composite plates subjected to in-plane stresses using trapezoidal p-element, Composite Structures, 68 (2005), 167-175. doi: 10.1016/j.compstruct.2004.03.011. Google Scholar

[22]

R. Lewis and V. Torczon, A globally convergent augmented Lagrangian pattern search algorithm for optimization with general constraints and simple bounds, SIAM Journal on Optimization, 12 (2002), 1075-1089. doi: 10.1137/S1052623498339727. Google Scholar

[23]

K. Liew, Solving the vibration of thick symmetric laminates by Reissner/Mindlin plates theory and the p-Ritz method, Journal of Sound and Vibration, 198 (1996), 343-360. doi: 10.1006/jsvi.1996.0574. Google Scholar

[24]

K. M. LiewY. Q. Huang and J. N. Reddy, Vibration analysis of symmetrically laminated plates based on FSDT using the moving least squares differential quadrature method, Computer Methods in Applied Mechanics and Engineering, 192 (2003), 2203-2222. doi: 10.1016/S0045-7825(03)00238-X. Google Scholar

[25]

X. Liu and A. Fettweis, Multidimensional digital filtering by using parallel algorithms based on diagonal processing, Multidimensional Systems and Signal Processing, 1 (1990), 51-66. doi: 10.1007/BF01812206. Google Scholar

[26]

P. Malekzadeh and A. Setoodeh, Large deformation analysis of moderately thick laminated plates on nonlinear elastic foundations by DQM, Composite Structures, 80 (2007), 569-579. doi: 10.1016/j.compstruct.2006.07.004. Google Scholar

[27]

P. Malekzadeh, Differential quadrature large amplitude free vibration analysis of laminated skew plates based on FSDT, Composite Structures, 83 (2008), 189-200. doi: 10.1016/j.compstruct.2007.04.007. Google Scholar

[28]

F. Mohammadi and R. Sedaghati, Nonlinear free vibration analysis of sandwich shell structures with a constrained electrorheological fluid layer, Smart Materials and Structures, 21 (2012), 075035. doi: 10.1088/0964-1726/21/7/075035. Google Scholar

[29]

P. M. Mohite and C. S. Upadhyay, Region-by-region modeling of laminated composite plates, Computers & Structures, 85 (2007), 1808-1827. doi: 10.1016/j.compstruc.2007.04.005. Google Scholar

[30]

D. Ngo-CongN. Mai-DuyW. Karunasena and T. Tran-Cong, Free vibration analysis of laminated composite plates based on FSDT using one-dimensional IRBFN method, Computers & Structures, 89 (2011), 1-13. doi: 10.1016/j.compstruc.2010.07.012. Google Scholar

[31]

H. Nguyen-VanN. Mai-Duy and T. Tran-Cong, Free vibration analysis of laminated plate/shell structures based on FSDT with a stabilized nodal-integrated quadrilateral element, Journal of Sound and Vibration, 313 (2008), 205-223. doi: 10.1016/j.jsv.2007.11.043. Google Scholar

[32]

G. Nitsche, Numerische L$\ddot{o}$sung partieller Differentialgleichungen mit hilfe von Wellendigitalfiltern., PhD thesis, Ruhr-Universit$\ddot{a}$t Bochum, 1993.Google Scholar

[33]

J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operations Research, Springer Verlag, 1999. doi: 10.1007/b98874. Google Scholar

[34]

O. Ochoa and J. Reddy, Finite Element Analysis of Composite Laminates (Solid Mechanics and Its Applications), Kluwer Acedemic Publisher, The Netherlands, 1992.Google Scholar

[35]

M. K. Pandit, Free vibration analysis of laminated composite rectangular plate using finite element method, Journal of Reinforced Plastics & Composites, 26 (2007), 69-80. Google Scholar

[36] J. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2/ed, CRC Press, London, 2004. Google Scholar
[37]

C. RoqueD. CunhaC. Shu and A. A. J. Ferreira, A local radial basis functions-Finite differences technique for the analysis of composite plates, Engineering Analysis with Boundary Elements, 35 (2011), 363-374. doi: 10.1016/j.enganabound.2010.09.012. Google Scholar

[38]

C. H. Tseng, Modelling and visualization of a time-dependent shallow water system using nonlinear Kirchhoff circuit, IEEE Transactions on Circuits and Systems I: Regular Papers, 59 (2012), 1265-1277. doi: 10.1109/TCSI.2011.2173511. Google Scholar

[39]

C. H. Tseng, Numerical stability verification of a two-dimensional time-dependent nonlinear shallow water system using multidimensional wave digital filtering network, Circuits, Systems and Signal Processing, 32 (2013), 299-319. doi: 10.1007/s00034-012-9461-7. Google Scholar

[40]

C. H. Tseng, Analysis of parallel multidimensional wave digital filtering network on IBM cell broadband engine, Journal of Computational Engineering, 2014 (2014), 1-13. doi: 10.1155/2014/793635. Google Scholar

[41]

C. H. Tseng and S. Lawson, Initial and boundary conditions in multidimensional wave digital filter algorithms for plate vibration, IEEE Transactions on Circuits and Systems I: Regular papers, 51 (2004), 1648-1663. doi: 10.1109/TCSI.2004.832796. Google Scholar

[42]

C. H. Tseng, An optimal modeling of multidimensional wave digital filtering network for free vibration analysis of symmetrically laminated composite FSDT plates, Mechanical Systems and Signal Processing, 52/53 (2015), 465-494. doi: 10.1016/j.ymssp.2014.07.001. Google Scholar

[43]

C. H. Tseng, The multidimensional wave digital filtering network for dynamic vibration analysis of laminated composite FSDT plates, TW patent I494783, 1/8/2015-30/3/2034.Google Scholar

[44]

C. H. Tseng, Static bending deflection and free vibration analysis of moderate thick symmetric laminated plates using multidimensional wave digital filters, Mechanical Systems and Signal Processing, 106 (2018), 367-394. doi: 10.1016/j.ymssp.2017.12.044. Google Scholar

[45]

C. H. Tseng, The full parallel architecture of multidimensional wave digital filtering network., TW patent I501149, 21/9/2015-14/10/2033.Google Scholar

[46]

S. Xiang, et al., Natural frequencies of generally laminated composite plates using the Gaussian radial basis function and first-order shear deformation theory. Thin-Walled Structures, 47 (2009), 1265–1271. doi: 10.1016/j.tws.2009.04.002. Google Scholar

[47]

J. ZhangJ. YuanE. FengH. Yin and Z. Xiu, Strong stability of a nonlinear multi-stage dynamic system in batch culture of glycerol bioconversion to 1, 3-propanediol, Mathematical Modelling and Analysis, 21 (2016), 159-173. doi: 10.3846/13926292.2016.1142481. Google Scholar

[48]

J. Zhang, J. Yuan, Z. Dong, E. Feng, H. Yin and Z. Xiu, Strong stability of optimal design to dynamic system for the fed-batch culture, International Journal of Biomathematics, 10 (2017), 1750018, 18pp. doi: 10.1142/S1793524517500188. Google Scholar

show all references

References:
[1]

S. Atluri and S. Zhu, A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Computational Mechanics, 22 (1998), 117-127. doi: 10.1007/s004660050346. Google Scholar

[2]

S. Basu and A. Zerzghi, Multidimensional digital filter approach for numerical solution of a class of PDEs of the propagating wave type, ISCAS '98. Proceedings of the 1998 IEEE International Symposium on Circuits and Systems (Cat. No.98CH36187), 5 (1998), 74–77, Monterey, California, USA. doi: 10.1109/ISCAS.1998.694411. Google Scholar

[3]

S. Billbao, Wave and Scattering Methods for Numerical Simulation, New York: Wiley, 2004. doi: 10.1002/0470870192. Google Scholar

[4]

B. P. Bogert, M. J. R. Healy and J. W. Tukey, The quefrency alanysis of time series for echoes: Cepstrum, pseudo autocovariance, cross-cepstrum and saphe cracking, Proceedings of the Symposium on Time Series Analysis (M. Rosenblatt, Ed) New York: Wiley, 1963, Chapter 15, 209–243.Google Scholar

[5]

T. BuiM. Nguyen and C. Zhang, An efficient meshfree method for vibration analysis of laminated composite plates, Computational Mechanics, 48 (2011), 175-193. doi: 10.1007/s00466-011-0591-8. Google Scholar

[6]

R. ByrdM. Hribar and J. Nocedal, An interior point algorithm for large-scale nonlinear programming, SIAM Journal on Optimization, 9 (1999), 877-900. doi: 10.1137/S1052623497325107. Google Scholar

[7]

K. Y. DaiG. R. LiuK. M. Lim and X. L. Chen, A mesh-free method for static and free vibration analysis of shear deformation laminated composite plates, J. Sound & Vibration, 269 (2004), 633-652. Google Scholar

[8]

Z. Dong, B. Tan, Y. Zhang, J. Yuan, E. Feng, H. Yin and Z. Xiu, Strong stability of an optimal control hybrid system in fed-batch fermentation, International Journal of Biomathematics, 11 (2018), 1850045, 17pp. doi: 10.1142/S1793524518500456. Google Scholar

[9]

A. A. J. Ferreira and G. Fasshauer, Analysis of natural frequencies of composite plates by an RBF-pseudospectral method, Composite Structures, 79 (2007), 202-210. doi: 10.1016/j.compstruct.2005.12.004. Google Scholar

[10]

A. J. M. FerreiraC. M. C. Roque and R. M. N. Jorge, Free vibration analysis of symmetric laminated composite plates by FSDT and radial basis functions., Computer Methods in Applied Mechanics and Engineering, 194 (2005), 4265-4278. doi: 10.1016/j.cma.2004.11.004. Google Scholar

[11]

A. Fettweis, Wave digital filters: Theory and practice, Proceedings of the IEEE, 74 (1986), 270-327. doi: 10.1109/PROC.1986.13458. Google Scholar

[12]

A. Fettweis and G. Nitsche, Numerical integration of partial differential equations using principles of multidimensional wave digital filters, J. VLSI Signal Processing, 3 (1991), 7-24. doi: 10.1007/978-1-4615-4036-6_2. Google Scholar

[13]

A. Fettweis and G. Nitsche, Transformation approach to numerical integrating PDEs by means of WDF principles, Multidimensional Systems and Signal Processing, 2 (1991), 127-159. doi: 10.1007/BF01938221. Google Scholar

[14]

A. Fettweis, Robust numerical integration using wave-digital concepts, Multidim. Syst. Signal Process., 17 (2006), 7-25. doi: 10.1007/s11045-005-6236-3. Google Scholar

[15]

A. Fettweis and K. Meerkötter, On parasitic oscillations in digital filters under looped conditions, IEEE Transactions on Circuits and Systems, 25 (1978), 1060-1066. doi: 10.1109/TCS.1978.1084425. Google Scholar

[16]

D. Goldberg, Genetic Algorithms in Search, Optimization & Machine Learning. Addison-Wesley, 1989.Google Scholar

[17]

S. Han, A globally convergent method for nonlinear programming, Journal of Optimization Theory and Applications, 22 (1977), 297-309. doi: 10.1007/BF00932858. Google Scholar

[18]

M. Jones, Mechanics of Composite Materials, 2/E. Taylor & Francis, London, 1998. doi: 10.1201/9781498711067. Google Scholar

[19]

K. Kant and K. Swaminathan, Estimation of transverse/interlaminar stresses in laminated composites-a selective review and survey of current developments, Composite Structures, 49 (2000), 65-75. doi: 10.1016/S0263-8223(99)00126-9. Google Scholar

[20]

A. Khdeir and L. Librescu, Analysis of symmetric cross-ply elastic plates using a higher-order theory, Part II: buckling and free vibration, Composite Structures, 9 (1988), 259-277. Google Scholar

[21]

A. Y. T. LeungC. XiaoB. Zhu and S. Yuanet, Free vibration of laminated composite plates subjected to in-plane stresses using trapezoidal p-element, Composite Structures, 68 (2005), 167-175. doi: 10.1016/j.compstruct.2004.03.011. Google Scholar

[22]

R. Lewis and V. Torczon, A globally convergent augmented Lagrangian pattern search algorithm for optimization with general constraints and simple bounds, SIAM Journal on Optimization, 12 (2002), 1075-1089. doi: 10.1137/S1052623498339727. Google Scholar

[23]

K. Liew, Solving the vibration of thick symmetric laminates by Reissner/Mindlin plates theory and the p-Ritz method, Journal of Sound and Vibration, 198 (1996), 343-360. doi: 10.1006/jsvi.1996.0574. Google Scholar

[24]

K. M. LiewY. Q. Huang and J. N. Reddy, Vibration analysis of symmetrically laminated plates based on FSDT using the moving least squares differential quadrature method, Computer Methods in Applied Mechanics and Engineering, 192 (2003), 2203-2222. doi: 10.1016/S0045-7825(03)00238-X. Google Scholar

[25]

X. Liu and A. Fettweis, Multidimensional digital filtering by using parallel algorithms based on diagonal processing, Multidimensional Systems and Signal Processing, 1 (1990), 51-66. doi: 10.1007/BF01812206. Google Scholar

[26]

P. Malekzadeh and A. Setoodeh, Large deformation analysis of moderately thick laminated plates on nonlinear elastic foundations by DQM, Composite Structures, 80 (2007), 569-579. doi: 10.1016/j.compstruct.2006.07.004. Google Scholar

[27]

P. Malekzadeh, Differential quadrature large amplitude free vibration analysis of laminated skew plates based on FSDT, Composite Structures, 83 (2008), 189-200. doi: 10.1016/j.compstruct.2007.04.007. Google Scholar

[28]

F. Mohammadi and R. Sedaghati, Nonlinear free vibration analysis of sandwich shell structures with a constrained electrorheological fluid layer, Smart Materials and Structures, 21 (2012), 075035. doi: 10.1088/0964-1726/21/7/075035. Google Scholar

[29]

P. M. Mohite and C. S. Upadhyay, Region-by-region modeling of laminated composite plates, Computers & Structures, 85 (2007), 1808-1827. doi: 10.1016/j.compstruc.2007.04.005. Google Scholar

[30]

D. Ngo-CongN. Mai-DuyW. Karunasena and T. Tran-Cong, Free vibration analysis of laminated composite plates based on FSDT using one-dimensional IRBFN method, Computers & Structures, 89 (2011), 1-13. doi: 10.1016/j.compstruc.2010.07.012. Google Scholar

[31]

H. Nguyen-VanN. Mai-Duy and T. Tran-Cong, Free vibration analysis of laminated plate/shell structures based on FSDT with a stabilized nodal-integrated quadrilateral element, Journal of Sound and Vibration, 313 (2008), 205-223. doi: 10.1016/j.jsv.2007.11.043. Google Scholar

[32]

G. Nitsche, Numerische L$\ddot{o}$sung partieller Differentialgleichungen mit hilfe von Wellendigitalfiltern., PhD thesis, Ruhr-Universit$\ddot{a}$t Bochum, 1993.Google Scholar

[33]

J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operations Research, Springer Verlag, 1999. doi: 10.1007/b98874. Google Scholar

[34]

O. Ochoa and J. Reddy, Finite Element Analysis of Composite Laminates (Solid Mechanics and Its Applications), Kluwer Acedemic Publisher, The Netherlands, 1992.Google Scholar

[35]

M. K. Pandit, Free vibration analysis of laminated composite rectangular plate using finite element method, Journal of Reinforced Plastics & Composites, 26 (2007), 69-80. Google Scholar

[36] J. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2/ed, CRC Press, London, 2004. Google Scholar
[37]

C. RoqueD. CunhaC. Shu and A. A. J. Ferreira, A local radial basis functions-Finite differences technique for the analysis of composite plates, Engineering Analysis with Boundary Elements, 35 (2011), 363-374. doi: 10.1016/j.enganabound.2010.09.012. Google Scholar

[38]

C. H. Tseng, Modelling and visualization of a time-dependent shallow water system using nonlinear Kirchhoff circuit, IEEE Transactions on Circuits and Systems I: Regular Papers, 59 (2012), 1265-1277. doi: 10.1109/TCSI.2011.2173511. Google Scholar

[39]

C. H. Tseng, Numerical stability verification of a two-dimensional time-dependent nonlinear shallow water system using multidimensional wave digital filtering network, Circuits, Systems and Signal Processing, 32 (2013), 299-319. doi: 10.1007/s00034-012-9461-7. Google Scholar

[40]

C. H. Tseng, Analysis of parallel multidimensional wave digital filtering network on IBM cell broadband engine, Journal of Computational Engineering, 2014 (2014), 1-13. doi: 10.1155/2014/793635. Google Scholar

[41]

C. H. Tseng and S. Lawson, Initial and boundary conditions in multidimensional wave digital filter algorithms for plate vibration, IEEE Transactions on Circuits and Systems I: Regular papers, 51 (2004), 1648-1663. doi: 10.1109/TCSI.2004.832796. Google Scholar

[42]

C. H. Tseng, An optimal modeling of multidimensional wave digital filtering network for free vibration analysis of symmetrically laminated composite FSDT plates, Mechanical Systems and Signal Processing, 52/53 (2015), 465-494. doi: 10.1016/j.ymssp.2014.07.001. Google Scholar

[43]

C. H. Tseng, The multidimensional wave digital filtering network for dynamic vibration analysis of laminated composite FSDT plates, TW patent I494783, 1/8/2015-30/3/2034.Google Scholar

[44]

C. H. Tseng, Static bending deflection and free vibration analysis of moderate thick symmetric laminated plates using multidimensional wave digital filters, Mechanical Systems and Signal Processing, 106 (2018), 367-394. doi: 10.1016/j.ymssp.2017.12.044. Google Scholar

[45]

C. H. Tseng, The full parallel architecture of multidimensional wave digital filtering network., TW patent I501149, 21/9/2015-14/10/2033.Google Scholar

[46]

S. Xiang, et al., Natural frequencies of generally laminated composite plates using the Gaussian radial basis function and first-order shear deformation theory. Thin-Walled Structures, 47 (2009), 1265–1271. doi: 10.1016/j.tws.2009.04.002. Google Scholar

[47]

J. ZhangJ. YuanE. FengH. Yin and Z. Xiu, Strong stability of a nonlinear multi-stage dynamic system in batch culture of glycerol bioconversion to 1, 3-propanediol, Mathematical Modelling and Analysis, 21 (2016), 159-173. doi: 10.3846/13926292.2016.1142481. Google Scholar

[48]

J. Zhang, J. Yuan, Z. Dong, E. Feng, H. Yin and Z. Xiu, Strong stability of optimal design to dynamic system for the fed-batch culture, International Journal of Biomathematics, 10 (2017), 1750018, 18pp. doi: 10.1142/S1793524517500188. Google Scholar

Figure 1.  A schematic flow diagram towards modeling a general MDWDF network
Figure 3.  A MDKC representation for a symmetrically laminated composite FSDT plate with free vibration
Figure 4.  A MDWDF network for numerical simulation of the laminated plate system (2.6)-(2.7)
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°]
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 $). (a2)-(d2): The maximum constraint violation corresponding to the objective value. (a3)-(d3): The first-order optimality
Figure 6.  MDWDF network robustness indicated by the percentage error distribution ($ 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
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 $
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 $
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 $
Figure 10.  The first six modes vibration for the four-layer crossply [0°/90°/90°/0°] SS2 square laminate.
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 $
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 $
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
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
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
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
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
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} $
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} $
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
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
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 $E_1/E_2=10,20,30,40$
ASAIPA
$E_1/E_2$Iter.F-count$\chi$Max. constr.$1^{st}$ opt.Iter.F-count$\chi$Max. constr.$1^{st}$ opt.
10022.533110.001187Infeasible023.5231140.5000Infeasible
1429.1438-0.00078623.86153.2398290.49500.2833
2616.995-0.00026342.225162.5580542.486e-20.1092
3812.6526-4.356e-50.5319255.8511295.661e-33.331e-2
41011.9075-1.283e-66.59e-2133312.040551.342e-11.866
51211.8842-1.256e-92.19e-3174111.884143.321e-63.957e-5
61411.8841$^\ast$-1.225e-152.37e-6205211.88414$^\ast$2.196e-130
20022.533110.002611Infeasible023.5231140.5000Infeasible
1453.9773-0.00537227.4153.2437530.49500.2794
2629.4623-0.001224.25162.5745434.123e-29.694e-2
3819.2631-2.112e-41.599259.0932632.175e-31.320e-2
41016.5631-1.48e-50.184133316.419988.452e-21.516
51216.343-9.832e-81.55e-2174116.341502.417e-63.959e-5
61416.3415$^\ast$-4.458e-121.05e-4205216.34150$^\ast$2.654e-130
30022.533110.004084Infeasible023.5231140.5000Infeasible
1478.4373-0.011178.75153.2477130.49500.2754
2641.7109-0.0026165.445162.5893835.592e-28.980e-2
3825.542-5.07e-42.6392611.412523.0598.636
41020.4244-5.079e-50.486133621.528221.7338.5
51219.7832-7.972e-73.76e-2174419.776754.03e-31.526e-2
61419.7728-2.093e-106.29e-4205019.772822.0e-63.960e-5
71619.7728$^\ast$-1.789e-171.92e-7235819.77282$^\ast$7.136e-92.069e-14
40022.533110.005573Infeasible023.5231140.5000Infeasible
14102.773-0.0190420.1153.2516160.49500.2715
2653.8883-0.0045275.945162.6031366.948e-28.390e-2
3831.7156-9.314e-43.5592613.448203.12110.49
41023.9651-1.138e-40.874133624.514001.6669.295
51222.7118-2.976e-66.42e-2174422.681104.026e-31.542e-2
61422.6772-2.266e-91.87e-3205022.677161.754e-63.960e-5
71622.6772$^\ast$-1.351e-151.28e-6225422.67716$^\ast$8.846e-93.960e-7
ASAIPA
$E_1/E_2$Iter.F-count$\chi$Max. constr.$1^{st}$ opt.Iter.F-count$\chi$Max. constr.$1^{st}$ opt.
10022.533110.001187Infeasible023.5231140.5000Infeasible
1429.1438-0.00078623.86153.2398290.49500.2833
2616.995-0.00026342.225162.5580542.486e-20.1092
3812.6526-4.356e-50.5319255.8511295.661e-33.331e-2
41011.9075-1.283e-66.59e-2133312.040551.342e-11.866
51211.8842-1.256e-92.19e-3174111.884143.321e-63.957e-5
61411.8841$^\ast$-1.225e-152.37e-6205211.88414$^\ast$2.196e-130
20022.533110.002611Infeasible023.5231140.5000Infeasible
1453.9773-0.00537227.4153.2437530.49500.2794
2629.4623-0.001224.25162.5745434.123e-29.694e-2
3819.2631-2.112e-41.599259.0932632.175e-31.320e-2
41016.5631-1.48e-50.184133316.419988.452e-21.516
51216.343-9.832e-81.55e-2174116.341502.417e-63.959e-5
61416.3415$^\ast$-4.458e-121.05e-4205216.34150$^\ast$2.654e-130
30022.533110.004084Infeasible023.5231140.5000Infeasible
1478.4373-0.011178.75153.2477130.49500.2754
2641.7109-0.0026165.445162.5893835.592e-28.980e-2
3825.542-5.07e-42.6392611.412523.0598.636
41020.4244-5.079e-50.486133621.528221.7338.5
51219.7832-7.972e-73.76e-2174419.776754.03e-31.526e-2
61419.7728-2.093e-106.29e-4205019.772822.0e-63.960e-5
71619.7728$^\ast$-1.789e-171.92e-7235819.77282$^\ast$7.136e-92.069e-14
40022.533110.005573Infeasible023.5231140.5000Infeasible
14102.773-0.0190420.1153.2516160.49500.2715
2653.8883-0.0045275.945162.6031366.948e-28.390e-2
3831.7156-9.314e-43.5592613.448203.12110.49
41023.9651-1.138e-40.874133624.514001.6669.295
51222.7118-2.976e-66.42e-2174422.681104.026e-31.542e-2
61422.6772-2.266e-91.87e-3205022.677161.754e-63.960e-5
71622.6772$^\ast$-1.351e-151.28e-6225422.67716$^\ast$8.846e-93.960e-7
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
$\bar{L}_j=L_j\times10^4, j=1, 4, 5.$ $E_1/E_2=10:\chi_{min}=11.8841$, $E_1/E_2=20:\chi_{min}=16.3415$.
$ 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
$\bar{L}_j=L_j\times10^4, j=1, 4, 5.$ $E_1/E_2=10:\chi_{min}=11.8841$, $E_1/E_2=20:\chi_{min}=16.3415$.
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
$\bar{L}_j=L_j\times10^4, j=1, 4, 5.$ $E_1/E_2=30:\chi_{min}=19.7728$, $E_1/E_2=40:\chi_{min}=22.6772$.
$ 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
$\bar{L}_j=L_j\times10^4, j=1, 4, 5.$ $E_1/E_2=30:\chi_{min}=19.7728$, $E_1/E_2=40:\chi_{min}=22.6772$.
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.
$ 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.
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
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
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
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
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.
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.
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.
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.
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
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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