
-
Previous Article
Improved SVRG for finite sum structure optimization with application to binary classification
- JIMO Home
- This Issue
-
Next Article
Optimal investment-reinsurance policy with regime switching and value-at-risk constraint
Analytical modeling of laminated composite plates using Kirchhoff circuit and wave digital filters
Department of Information Engineering, Kun Shan University, Taiwan |
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.
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. |
[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. |
[3] |
S. Billbao, Wave and Scattering Methods for Numerical Simulation, New York: Wiley, 2004.
doi: 10.1002/0470870192. |
[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. |
[5] |
T. Bui, M. 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. |
[6] |
R. Byrd, M. 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. |
[7] |
K. Y. Dai, G. R. Liu, K. 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.
|
[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. |
[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. |
[10] |
A. J. M. Ferreira, C. 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. |
[11] |
A. Fettweis,
Wave digital filters: Theory and practice, Proceedings of the IEEE, 74 (1986), 270-327.
doi: 10.1109/PROC.1986.13458. |
[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. |
[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. |
[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. |
[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. |
[16] |
D. Goldberg, Genetic Algorithms in Search, Optimization & Machine Learning. Addison-Wesley, 1989. |
[17] |
S. Han,
A globally convergent method for nonlinear programming, Journal of Optimization Theory and Applications, 22 (1977), 297-309.
doi: 10.1007/BF00932858. |
[18] |
M. Jones, Mechanics of Composite Materials, 2/E. Taylor & Francis, London, 1998.
doi: 10.1201/9781498711067. |
[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. |
[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.
|
[21] |
A. Y. T. Leung, C. Xiao, B. 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. |
[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. |
[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. |
[24] |
K. M. Liew, Y. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[30] |
D. Ngo-Cong, N. Mai-Duy, W. 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. |
[31] |
H. Nguyen-Van, N. 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. |
[32] |
G. Nitsche, Numerische L$\ddot{o}$sung partieller Differentialgleichungen mit hilfe von Wellendigitalfiltern., PhD thesis, Ruhr-Universit$\ddot{a}$t Bochum, 1993. |
[33] |
J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operations Research, Springer Verlag, 1999.
doi: 10.1007/b98874. |
[34] |
O. Ochoa and J. Reddy, Finite Element Analysis of Composite Laminates (Solid Mechanics and Its Applications), Kluwer Acedemic Publisher, The Netherlands, 1992. |
[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.
|
[36] |
J. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2/ed, CRC Press, London, 2004.
![]() |
[37] |
C. Roque, D. Cunha, C. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[45] |
C. H. Tseng, The full parallel architecture of multidimensional wave digital filtering network., TW patent I501149, 21/9/2015-14/10/2033. |
[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. |
[47] |
J. Zhang, J. Yuan, E. Feng, H. 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. |
[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. |
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. |
[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. |
[3] |
S. Billbao, Wave and Scattering Methods for Numerical Simulation, New York: Wiley, 2004.
doi: 10.1002/0470870192. |
[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. |
[5] |
T. Bui, M. 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. |
[6] |
R. Byrd, M. 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. |
[7] |
K. Y. Dai, G. R. Liu, K. 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.
|
[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. |
[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. |
[10] |
A. J. M. Ferreira, C. 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. |
[11] |
A. Fettweis,
Wave digital filters: Theory and practice, Proceedings of the IEEE, 74 (1986), 270-327.
doi: 10.1109/PROC.1986.13458. |
[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. |
[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. |
[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. |
[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. |
[16] |
D. Goldberg, Genetic Algorithms in Search, Optimization & Machine Learning. Addison-Wesley, 1989. |
[17] |
S. Han,
A globally convergent method for nonlinear programming, Journal of Optimization Theory and Applications, 22 (1977), 297-309.
doi: 10.1007/BF00932858. |
[18] |
M. Jones, Mechanics of Composite Materials, 2/E. Taylor & Francis, London, 1998.
doi: 10.1201/9781498711067. |
[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. |
[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.
|
[21] |
A. Y. T. Leung, C. Xiao, B. 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. |
[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. |
[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. |
[24] |
K. M. Liew, Y. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[30] |
D. Ngo-Cong, N. Mai-Duy, W. 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. |
[31] |
H. Nguyen-Van, N. 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. |
[32] |
G. Nitsche, Numerische L$\ddot{o}$sung partieller Differentialgleichungen mit hilfe von Wellendigitalfiltern., PhD thesis, Ruhr-Universit$\ddot{a}$t Bochum, 1993. |
[33] |
J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operations Research, Springer Verlag, 1999.
doi: 10.1007/b98874. |
[34] |
O. Ochoa and J. Reddy, Finite Element Analysis of Composite Laminates (Solid Mechanics and Its Applications), Kluwer Acedemic Publisher, The Netherlands, 1992. |
[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.
|
[36] |
J. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2/ed, CRC Press, London, 2004.
![]() |
[37] |
C. Roque, D. Cunha, C. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[45] |
C. H. Tseng, The full parallel architecture of multidimensional wave digital filtering network., TW patent I501149, 21/9/2015-14/10/2033. |
[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. |
[47] |
J. Zhang, J. Yuan, E. Feng, H. 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. |
[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. |

















Options for ASA selection | Value | Options for ASA termination | Value |
Number of objective function evaluations | Maxi. number of iterations | ||
Max. change for finite-difference gradients | Tolerance on objective function | ||
Min. change for finite-difference gradients | Tolerance of maxi. constraint | ||
Finite difference type | forward | Maxi. number of SQP iterations | |
Tolerance on SQP constraint violation | |||
Options for IPA selection | Value | Options for IPA termination | Value |
Number of objective function evaluations | Maximum number of iterations | ||
Initial barrier value | Tolerance on objective function | ||
Max. change for finite-difference gradients | Tolerance of maxi. constraint | ||
Min. change for finite-difference gradients | Maxi. number of PCG iterations | 2 | |
Finite difference type | forward | Tolerance of PCG algorithm |
Options for ASA selection | Value | Options for ASA termination | Value |
Number of objective function evaluations | Maxi. number of iterations | ||
Max. change for finite-difference gradients | Tolerance on objective function | ||
Min. change for finite-difference gradients | Tolerance of maxi. constraint | ||
Finite difference type | forward | Maxi. number of SQP iterations | |
Tolerance on SQP constraint violation | |||
Options for IPA selection | Value | Options for IPA termination | Value |
Number of objective function evaluations | Maximum number of iterations | ||
Initial barrier value | Tolerance on objective function | ||
Max. change for finite-difference gradients | Tolerance of maxi. constraint | ||
Min. change for finite-difference gradients | Maxi. number of PCG iterations | 2 | |
Finite difference type | forward | Tolerance of PCG algorithm |
Options for ALGA selection | Value | Options for ALGA termination | Value |
Size of population | Maxi. number of iterations | 100 | |
Probability of crossover | Tolerance on fitness function | ||
Probability of mutation | Tolerance of maxi. constraint | ||
Size of elitism | Tolerance of nonlinear constraint | ||
Initial penalty parameter | Max. stall generations | ||
Penalty update parameter | |||
Options for ALPSA selection | Value | Options for ALPSA termination | Value |
Number of objective function evaluations | Maxi. number of iterations | ||
Initial penalty parameter | Tolerance on function value | ||
Penalty update parameter | Min. tolerance for mesh size | ||
Poll method | Tolerance of nonlinear constraint | ||
Search method/Iteration limit | Bind tolerance | ||
Options for ALGA selection | Value | Options for ALGA termination | Value |
Size of population | Maxi. number of iterations | 100 | |
Probability of crossover | Tolerance on fitness function | ||
Probability of mutation | Tolerance of maxi. constraint | ||
Size of elitism | Tolerance of nonlinear constraint | ||
Initial penalty parameter | Max. stall generations | ||
Penalty update parameter | |||
Options for ALPSA selection | Value | Options for ALPSA termination | Value |
Number of objective function evaluations | Maxi. number of iterations | ||
Initial penalty parameter | Tolerance on function value | ||
Penalty update parameter | Min. tolerance for mesh size | ||
Poll method | Tolerance of nonlinear constraint | ||
Search method/Iteration limit | Bind tolerance | ||
ASA | IPA | |||||||||
Iter. | F-count | Max. constr. | Iter. | F-count | Max. constr. | |||||
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 | -1.225e-15 | 2.37e-6 | 20 | 52 | 11.88414 | 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 | -4.458e-12 | 1.05e-4 | 20 | 52 | 16.34150 | 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 | -1.789e-17 | 1.92e-7 | 23 | 58 | 19.77282 | 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 | -1.351e-15 | 1.28e-6 | 22 | 54 | 22.67716 | 8.846e-9 | 3.960e-7 |
ASA | IPA | |||||||||
Iter. | F-count | Max. constr. | Iter. | F-count | Max. constr. | |||||
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 | -1.225e-15 | 2.37e-6 | 20 | 52 | 11.88414 | 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 | -4.458e-12 | 1.05e-4 | 20 | 52 | 16.34150 | 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 | -1.789e-17 | 1.92e-7 | 23 | 58 | 19.77282 | 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 | -1.351e-15 | 1.28e-6 | 22 | 54 | 22.67716 | 8.846e-9 | 3.960e-7 |
Model( |
CFL no. | Div. step | |||||||
10 | 4.33 | 0.1552 | 1.18 | -0.19 | 49.95 | [-45, 2471] | [-22, 2505] | 108 | |
4.27 | 0.1530 | 1.22 | -0.10 | 51.90 | [-63, 1224] | [-34, 1263] | 182 | ||
4.24 | 0.1522 | 1.23 | -0.06 | 52.67 | [-54,454] | [-37,499] | 331 | ||
4.21 | 0.1507 | 1.26 | 0 | 54.03 | reference | reference | X | ||
4.16 | 0.1492 | 1.28 | 0.07 | 55.43 | [-47, 12] | [-36, 2.2] | X | ||
4.14 | 0.1485 | 1.30 | 0.10 | 56.15 | [-45, 14] | [-39, 4.4] | X | ||
4.08 | 0.1462 | 1.34 | 0.20 | 5C.37 | [-42, 19] | [-40, 8.9] | X | ||
1.68 | 0.0602 | 8.17 | 17.13 | 416.5 | [-30, 32] | [-30, 23] | X | ||
1.05 | 0.0376 | 21.0 | 48.94 | 1089 | [-30, 32] | [-30, 24] | X | ||
20 | 3.15 | 0.1129 | 7.17 | -0.32 | 127.5 | [-44, 2712] | [-18, 2746] | 102 | |
3.11 | 0.1112 | 7.40 | -0.16 | 132.4 | [-47, 1590] | [-30, 1628] | 153 | ||
3.09 | 0.1107 | 7.48 | -0.11 | 134.1 | [-49, 1031] | [-46, 1073] | 210 | ||
3.06 | 0.1096 | 7.64 | 0 | 137.4 | reference | reference | X | ||
3.03 | 0.1085 | 7.79 | 0.11 | 140.8 | [-49, A.7] | [-56, 2.9] | X | ||
3.01 | 0.1080 | 7.88 | 0.16 | 142.5 | [-47, 9.6] | [-32, 5.4] | X | ||
2.97 | 0.1063 | 8.11 | 0.33 | 147.6 | [-44, 14] | [-47, 9] | X | ||
1.22 | 0.0438 | 48.7 | 28.50 | 1017 | [-37, 38] | [-26, 23] | X | ||
0.76 | 0.0274 | 125.0 | 81.31 | 2651 | [-32, 40] | [-41, 22] | X | ||
Model( |
CFL no. | Div. step | |||||||
10 | 4.33 | 0.1552 | 1.18 | -0.19 | 49.95 | [-45, 2471] | [-22, 2505] | 108 | |
4.27 | 0.1530 | 1.22 | -0.10 | 51.90 | [-63, 1224] | [-34, 1263] | 182 | ||
4.24 | 0.1522 | 1.23 | -0.06 | 52.67 | [-54,454] | [-37,499] | 331 | ||
4.21 | 0.1507 | 1.26 | 0 | 54.03 | reference | reference | X | ||
4.16 | 0.1492 | 1.28 | 0.07 | 55.43 | [-47, 12] | [-36, 2.2] | X | ||
4.14 | 0.1485 | 1.30 | 0.10 | 56.15 | [-45, 14] | [-39, 4.4] | X | ||
4.08 | 0.1462 | 1.34 | 0.20 | 5C.37 | [-42, 19] | [-40, 8.9] | X | ||
1.68 | 0.0602 | 8.17 | 17.13 | 416.5 | [-30, 32] | [-30, 23] | X | ||
1.05 | 0.0376 | 21.0 | 48.94 | 1089 | [-30, 32] | [-30, 24] | X | ||
20 | 3.15 | 0.1129 | 7.17 | -0.32 | 127.5 | [-44, 2712] | [-18, 2746] | 102 | |
3.11 | 0.1112 | 7.40 | -0.16 | 132.4 | [-47, 1590] | [-30, 1628] | 153 | ||
3.09 | 0.1107 | 7.48 | -0.11 | 134.1 | [-49, 1031] | [-46, 1073] | 210 | ||
3.06 | 0.1096 | 7.64 | 0 | 137.4 | reference | reference | X | ||
3.03 | 0.1085 | 7.79 | 0.11 | 140.8 | [-49, A.7] | [-56, 2.9] | X | ||
3.01 | 0.1080 | 7.88 | 0.16 | 142.5 | [-47, 9.6] | [-32, 5.4] | X | ||
2.97 | 0.1063 | 8.11 | 0.33 | 147.6 | [-44, 14] | [-47, 9] | X | ||
1.22 | 0.0438 | 48.7 | 28.50 | 1017 | [-37, 38] | [-26, 23] | X | ||
0.76 | 0.0274 | 125.0 | 81.31 | 2651 | [-32, 40] | [-41, 22] | X | ||
Model( |
CFL no. | Div. step | |||||||
30 | 2.61 | 0.0933 | 21.68 | -0.45 | 209.1 | [-46, 2761] | [-20, 2810] | 101 | |
2.57 | 0.0919 | 22.37 | -0.23 | 217.0 | [-49, 1692] | [-47, 1743] | 147 | ||
2.55 | 0.0915 | 22.61 | -0.15 | 219.7 | [-51, 1177] | [-37, 1231] | 190 | ||
2.53 | 0.0905 | 23.07 | 0 | 225.0 | reference | reference | X | ||
2.50 | 0.0897 | 23.55 | 0.15 | 230.5 | [-51, 12] | [-47, 5.1] | X | ||
2.49 | 0.0892 | 23.79 | 0.23 | 233.3 | [-49, 15] | [-51, 7.6] | X | ||
2.46 | 0.0879 | 24.49 | 0.46 | 241.5 | [-47, 22] | [-30, 11] | X | ||
1.01 | 0.0362 | 146.2 | 39.8 | 1644 | [-37, 36] | [-30, 24] | X | ||
0.63 | 0.0226 | 374.9 | 113.7 | 4280 | [-34, 38] | [-33, 24] | X | ||
40 | 2.27 | 0.0814 | 48.41 | -0.58 | 291.9 | [-47, 2792] | [-26, 2850] | 100 | |
2.24 | 0.0801 | 49.94 | -0.29 | 302.9 | [-50, 1735] | [-333, 1796] | 144 | ||
2.23 | 0.0798 | 50.46 | -0.19 | 306.6 | [-52, 1238] | [-34, 1302] | 184 | ||
2.20 | 0.0789 | 51.50 | 0 | 314.1 | reference | reference | X | ||
2.18 | 0.0782 | 52.54 | 0.19 | 321.6 | [-52, 10] | [-52, 2.7] | X | ||
2.17 | 0.0778 | 53.10 | 0.29 | 325.4 | [-50, 14] | [-44, 4.7] | X | ||
2.14 | 0.0766 | 54.67 | 0.59 | 336.9 | [-47, 20] | [-36, 8.3] | X | ||
0.88 | 0.0315 | 325.3 | 51.1 | 2280 | [-36, 37] | [-22, 23] | X | ||
0.55 | 0.0197 | 833.7 | 146.1 | 5932 | [-35, 38] | [-30, 24] | X | ||
Model( |
CFL no. | Div. step | |||||||
30 | 2.61 | 0.0933 | 21.68 | -0.45 | 209.1 | [-46, 2761] | [-20, 2810] | 101 | |
2.57 | 0.0919 | 22.37 | -0.23 | 217.0 | [-49, 1692] | [-47, 1743] | 147 | ||
2.55 | 0.0915 | 22.61 | -0.15 | 219.7 | [-51, 1177] | [-37, 1231] | 190 | ||
2.53 | 0.0905 | 23.07 | 0 | 225.0 | reference | reference | X | ||
2.50 | 0.0897 | 23.55 | 0.15 | 230.5 | [-51, 12] | [-47, 5.1] | X | ||
2.49 | 0.0892 | 23.79 | 0.23 | 233.3 | [-49, 15] | [-51, 7.6] | X | ||
2.46 | 0.0879 | 24.49 | 0.46 | 241.5 | [-47, 22] | [-30, 11] | X | ||
1.01 | 0.0362 | 146.2 | 39.8 | 1644 | [-37, 36] | [-30, 24] | X | ||
0.63 | 0.0226 | 374.9 | 113.7 | 4280 | [-34, 38] | [-33, 24] | X | ||
40 | 2.27 | 0.0814 | 48.41 | -0.58 | 291.9 | [-47, 2792] | [-26, 2850] | 100 | |
2.24 | 0.0801 | 49.94 | -0.29 | 302.9 | [-50, 1735] | [-333, 1796] | 144 | ||
2.23 | 0.0798 | 50.46 | -0.19 | 306.6 | [-52, 1238] | [-34, 1302] | 184 | ||
2.20 | 0.0789 | 51.50 | 0 | 314.1 | reference | reference | X | ||
2.18 | 0.0782 | 52.54 | 0.19 | 321.6 | [-52, 10] | [-52, 2.7] | X | ||
2.17 | 0.0778 | 53.10 | 0.29 | 325.4 | [-50, 14] | [-44, 4.7] | X | ||
2.14 | 0.0766 | 54.67 | 0.59 | 336.9 | [-47, 20] | [-36, 8.3] | X | ||
0.88 | 0.0315 | 325.3 | 51.1 | 2280 | [-36, 37] | [-22, 23] | X | ||
0.55 | 0.0197 | 833.7 | 146.1 | 5932 | [-35, 38] | [-30, 24] | X | ||
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 |
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 |
Mode |
||||||
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 |
||||||
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 |
ALGA | ALPSA | ||||||||
Gen. | F-count | Max. constr. | Iter. | F-count | 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 |
0 | 4 | 501 | 11.8842 |
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 |
0 | 3.333e-8 | |
10 | 5269 | 16.3415 |
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 |
0 | 3 | 437 | 19.7728 |
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 |
0 | 3 | 441 | 22.6772 |
0 | 3.333e-8 |
ALGA | ALPSA | ||||||||
Gen. | F-count | Max. constr. | Iter. | F-count | 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 |
0 | 4 | 501 | 11.8842 |
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 |
0 | 3.333e-8 | |
10 | 5269 | 16.3415 |
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 |
0 | 3 | 437 | 19.7728 |
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 |
0 | 3 | 441 | 22.6772 |
0 | 3.333e-8 |
NLP | ||||||
Method | Algorithm | Para. | ||||
FSDT-MDWDF | ASA | 11.88413667 | 16.34149675 | 19.77282249 | 22.67715857 | |
0.15072042 | 0.10960942 | 0.09058808 | 0.07898617 | |||
CPU time(ms) | 2411.1 | 2425.98 | 2461.76 | 2502.44 | ||
9.83369172 | 12.28105278 | 13.97527101 | 15.21304757 | |||
0.19596347 | 0.82328368 | 0.59941701 | 0.46257394 | |||
IPA | 11.88413707 | 16.34149709 | 19.77282289 | 22.67715898 | ||
0.15072041 | 0.10960942 | 0.09058808 | 0.07898617 | |||
CPU time(ms) | 2265.58 | 1745.8 | 1824.4 | 1972.86 | ||
9.83369205 | 12.28105303 | 13.97527129 | 15.21304784 | |||
0.19596011 | 0.82328166 | 0.59941904 | 0.46257571 | |||
ALGA | 11.88413869 | 16.34149995 | 19.77282581 | 22.67716160 | ||
0.15072040 | 0.10960941 | 0.09058807 | 0.07898617 | |||
CPU time(ms) | 6262.2 | 6137.32 | 4223.16 | 4194.09 | ||
9.83370380 | 12.28105800 | 13.97527405 | 15.21304960 | |||
0.1958408 | 0.82324156 | 0.59943894 | 0.46258734 | |||
ALPSA | 11.88415178 | 16.34150453 | 19.77282756 | 22.67716223 | ||
0.15072013 | 0.10960929 | 0.09058806 | 0.07898616 | |||
CPU time(ms) | 2924.57 | 2781.15 | 2593.6 | 2857.36 | ||
9.83370423 | 12.28105862 | 13.97527458 | 15.21305002 | |||
0.19583654 | 0.82323649 | 0.59944269 | 0.46259008 | |||
FSDT-GRBF [46] | 9.539 | 11.977 | 13.716 | 15.059 | ||
3.18684664 | 3.27868852 | 1.26691621 | 0.55471175 | |||
FSDT-EFG [7] | 9.670 | 12.115 | 13.799 | 15.068 | ||
1.85730234 | 1.00506618 | 0.66945004 | 0.49527834 | |||
FSDT-FEM [36] | 9.841 | 12.138 | 13.864 | 15.107 | ||
0.12179031 | 0.16342539 | 0.20155485 | 0.23773360 | |||
TSDT-EFG [7] | 9.842 | 12.138 | 14.154 | 15.145 | ||
0.11164112 | 0.16342539 | 1.90757270 | 0.01320742 | |||
TSDT-FEM [36] | 9.853 | 12.238 | 13.892 | 15.143 | ||
NLP | ||||||
Method | Algorithm | Para. | ||||
FSDT-MDWDF | ASA | 11.88413667 | 16.34149675 | 19.77282249 | 22.67715857 | |
0.15072042 | 0.10960942 | 0.09058808 | 0.07898617 | |||
CPU time(ms) | 2411.1 | 2425.98 | 2461.76 | 2502.44 | ||
9.83369172 | 12.28105278 | 13.97527101 | 15.21304757 | |||
0.19596347 | 0.82328368 | 0.59941701 | 0.46257394 | |||
IPA | 11.88413707 | 16.34149709 | 19.77282289 | 22.67715898 | ||
0.15072041 | 0.10960942 | 0.09058808 | 0.07898617 | |||
CPU time(ms) | 2265.58 | 1745.8 | 1824.4 | 1972.86 | ||
9.83369205 | 12.28105303 | 13.97527129 | 15.21304784 | |||
0.19596011 | 0.82328166 | 0.59941904 | 0.46257571 | |||
ALGA | 11.88413869 | 16.34149995 | 19.77282581 | 22.67716160 | ||
0.15072040 | 0.10960941 | 0.09058807 | 0.07898617 | |||
CPU time(ms) | 6262.2 | 6137.32 | 4223.16 | 4194.09 | ||
9.83370380 | 12.28105800 | 13.97527405 | 15.21304960 | |||
0.1958408 | 0.82324156 | 0.59943894 | 0.46258734 | |||
ALPSA | 11.88415178 | 16.34150453 | 19.77282756 | 22.67716223 | ||
0.15072013 | 0.10960929 | 0.09058806 | 0.07898616 | |||
CPU time(ms) | 2924.57 | 2781.15 | 2593.6 | 2857.36 | ||
9.83370423 | 12.28105862 | 13.97527458 | 15.21305002 | |||
0.19583654 | 0.82323649 | 0.59944269 | 0.46259008 | |||
FSDT-GRBF [46] | 9.539 | 11.977 | 13.716 | 15.059 | ||
3.18684664 | 3.27868852 | 1.26691621 | 0.55471175 | |||
FSDT-EFG [7] | 9.670 | 12.115 | 13.799 | 15.068 | ||
1.85730234 | 1.00506618 | 0.66945004 | 0.49527834 | |||
FSDT-FEM [36] | 9.841 | 12.138 | 13.864 | 15.107 | ||
0.12179031 | 0.16342539 | 0.20155485 | 0.23773360 | |||
TSDT-EFG [7] | 9.842 | 12.138 | 14.154 | 15.145 | ||
0.11164112 | 0.16342539 | 1.90757270 | 0.01320742 | |||
TSDT-FEM [36] | 9.853 | 12.238 | 13.892 | 15.143 | ||
NLP | ||||||
Method | Algorithm | Para. | ||||
FSDT-MDWDF | ASA | 12.91506498 | 17.80436624 | 21.46965556 | 24.53187819 | |
0.14077503 | 0.10211645 | 0.08468317 | 0.07411249 | |||
9.75894349 | 12.09708139 | 13.65712101 | 14.70474902 | |||
0.35793863 | 0.37405733 | 0.59012307 | 0.41481091 | |||
IPA | 12.91506538 | 17.80436635 | 21.46965596 | 24.53187859 | ||
0.14077502 | 0.10211645 | 0.08468318 | 0.07411249 | |||
9.75894379 | 12.09708146 | 13.65712126 | 14.70474926 | |||
0.35793557 | 0.00374057 | 0.59012491 | 0.41480929 | |||
ALGA | 12.91506688 | 17.80436836 | 21.46965818 | 24.53190153 | ||
0.14077501 | 0.10211643 | 0.08468316 | 0.0741124239 | |||
9.75894493 | 12.09708282 | 13.65712267 | 14.70476301 | |||
0.35792393 | 0.37406920 | 0.59013530 | 0.41471617 | |||
ALPSA | 12.91825286 | 17.81031318 | 21.46968482 | 24.53190155 | ||
0.14074029 | 0.10208235 | 0.08468306 | 0.0741124238 | |||
9.76135232 | 12.10112200 | 13.65713962 | 14.70476302 | |||
0.33334367 | 0.00407583 | 0.00590260 | 0.41471610 | |||
HSDT-FEM [20] | 9.794 | 12.052 | 13.577 | 14.766 | ||
NLP | ||||||
Method | Algorithm | Para. | ||||
FSDT-MDWDF | ASA | 12.91506498 | 17.80436624 | 21.46965556 | 24.53187819 | |
0.14077503 | 0.10211645 | 0.08468317 | 0.07411249 | |||
9.75894349 | 12.09708139 | 13.65712101 | 14.70474902 | |||
0.35793863 | 0.37405733 | 0.59012307 | 0.41481091 | |||
IPA | 12.91506538 | 17.80436635 | 21.46965596 | 24.53187859 | ||
0.14077502 | 0.10211645 | 0.08468318 | 0.07411249 | |||
9.75894379 | 12.09708146 | 13.65712126 | 14.70474926 | |||
0.35793557 | 0.00374057 | 0.59012491 | 0.41480929 | |||
ALGA | 12.91506688 | 17.80436836 | 21.46965818 | 24.53190153 | ||
0.14077501 | 0.10211643 | 0.08468316 | 0.0741124239 | |||
9.75894493 | 12.09708282 | 13.65712267 | 14.70476301 | |||
0.35792393 | 0.37406920 | 0.59013530 | 0.41471617 | |||
ALPSA | 12.91825286 | 17.81031318 | 21.46968482 | 24.53190155 | ||
0.14074029 | 0.10208235 | 0.08468306 | 0.0741124238 | |||
9.76135232 | 12.10112200 | 13.65713962 | 14.70476302 | |||
0.33334367 | 0.00407583 | 0.00590260 | 0.41471610 | |||
HSDT-FEM [20] | 9.794 | 12.052 | 13.577 | 14.766 | ||
NLP | |||||
Algorithm | Para. | ||||
ASA | 11.12198459 | 15.31530671 | 18.56722410 | 21.43934366 | |
0.15957798 | 0.11588562 | 0.09558908 | 0.08375115 | ||
9.53036311 | 11.97168174 | 13.53757952 | 5.46066961 | ||
IPA | 11.12198460 | 15.31530709 | 18.56722348 | 21.32430073 | |
0.15957798 | 0.11588562 | 0.09558908 | 0.08323011 | ||
9.53036311 | 11.97168204 | 13.69778710 | 15.02075228 | ||
ALGA | 11.12198680 | 15.31531111 | 18.56722310 | 21.32430035 | |
0.15957795 | 0.11588559 | 0.09558908 | 0.08323011 | ||
9.53036500 | 11.97168518 | 13.69778682 | 15.02075202 | ||
ALPSA | 11.12199160 | 15.31530987 | 18.56722509 | 21.32430176 | |
0.15957788 | 0.11588560 | 0.09558907 | 0.08323010 | ||
9.53036911 | 11.97168421 | 13.69778829 | 15.02075301 |
NLP | |||||
Algorithm | Para. | ||||
ASA | 11.12198459 | 15.31530671 | 18.56722410 | 21.43934366 | |
0.15957798 | 0.11588562 | 0.09558908 | 0.08375115 | ||
9.53036311 | 11.97168174 | 13.53757952 | 5.46066961 | ||
IPA | 11.12198460 | 15.31530709 | 18.56722348 | 21.32430073 | |
0.15957798 | 0.11588562 | 0.09558908 | 0.08323011 | ||
9.53036311 | 11.97168204 | 13.69778710 | 15.02075228 | ||
ALGA | 11.12198680 | 15.31531111 | 18.56722310 | 21.32430035 | |
0.15957795 | 0.11588559 | 0.09558908 | 0.08323011 | ||
9.53036500 | 11.97168518 | 13.69778682 | 15.02075202 | ||
ALPSA | 11.12199160 | 15.31530987 | 18.56722509 | 21.32430176 | |
0.15957788 | 0.11588560 | 0.09558907 | 0.08323010 | ||
9.53036911 | 11.97168421 | 13.69778829 | 15.02075301 |
[1] |
Jiayue Zheng, Shangbin Cui. Bifurcation analysis of a tumor-model free boundary problem with a nonlinear boundary condition. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4397-4410. doi: 10.3934/dcdsb.2020103 |
[2] |
Francesca Bucci, Igor Chueshov, Irena Lasiecka. Global attractor for a composite system of nonlinear wave and plate equations. Communications on Pure and Applied Analysis, 2007, 6 (1) : 113-140. doi: 10.3934/cpaa.2007.6.113 |
[3] |
Junjiang Lai, Jianguo Huang. A finite element method for vibration analysis of elastic plate-plate structures. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 387-419. doi: 10.3934/dcdsb.2009.11.387 |
[4] |
Mei Li, Hongjun Gao, Bingjun Wang. Analysis of a non-autonomous mutualism model driven by Levy jumps. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1189-1202. doi: 10.3934/dcdsb.2016.21.1189 |
[5] |
Shun Li, Peng-Fei Yao. Modeling of a nonlinear plate. Evolution Equations and Control Theory, 2012, 1 (1) : 155-169. doi: 10.3934/eect.2012.1.155 |
[6] |
Jesús Ildefonso Díaz, L. Tello. On a climate model with a dynamic nonlinear diffusive boundary condition. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 253-262. doi: 10.3934/dcdss.2008.1.253 |
[7] |
Liping Zhang. A nonlinear complementarity model for supply chain network equilibrium. Journal of Industrial and Management Optimization, 2007, 3 (4) : 727-737. doi: 10.3934/jimo.2007.3.727 |
[8] |
Benoît Jubin, Norbert Poncin, Kyosuke Uchino. Free Courant and derived Leibniz pseudoalgebras. Journal of Geometric Mechanics, 2016, 8 (1) : 71-97. doi: 10.3934/jgm.2016.8.71 |
[9] |
Boguslaw Twarog, Robert Pekala, Jacek Bartman, Zbigniew Gomolka. The changes of air gap in inductive engines as vibration indicator aided by mathematical model and artificial neural network. Conference Publications, 2007, 2007 (Special) : 1005-1012. doi: 10.3934/proc.2007.2007.1005 |
[10] |
Shijin Deng, Linglong Du, Shih-Hsien Yu. Nonlinear stability of Broadwell model with Maxwell diffuse boundary condition. Kinetic and Related Models, 2013, 6 (4) : 865-882. doi: 10.3934/krm.2013.6.865 |
[11] |
Andrzej Nowakowski. Variational analysis of semilinear plate equation with free boundary conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3133-3154. doi: 10.3934/dcds.2015.35.3133 |
[12] |
Yongqin Liu. Asymptotic behavior of solutions to a nonlinear plate equation with memory. Communications on Pure and Applied Analysis, 2017, 16 (2) : 533-556. doi: 10.3934/cpaa.2017027 |
[13] |
Wenzhen Gan, Peng Zhou. A revisit to the diffusive logistic model with free boundary condition. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 837-847. doi: 10.3934/dcdsb.2016.21.837 |
[14] |
Yang Li, Yonghong Ren, Yun Wang, Jian Gu. Convergence analysis of a nonlinear Lagrangian method for nonconvex semidefinite programming with subproblem inexactly solved. Journal of Industrial and Management Optimization, 2015, 11 (1) : 65-81. doi: 10.3934/jimo.2015.11.65 |
[15] |
Aliki D. Muradova, Georgios K. Tairidis, Georgios E. Stavroulakis. Adaptive Neuro-Fuzzy vibration control of a smart plate. Numerical Algebra, Control and Optimization, 2017, 7 (3) : 251-271. doi: 10.3934/naco.2017017 |
[16] |
Shaojun Zhang, Zhong Wan. Polymorphic uncertain nonlinear programming model and algorithm for maximizing the fatigue life of V-belt drive. Journal of Industrial and Management Optimization, 2012, 8 (2) : 493-505. doi: 10.3934/jimo.2012.8.493 |
[17] |
Yi Xu, Wenyu Sun. A filter successive linear programming method for nonlinear semidefinite programming problems. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 193-206. doi: 10.3934/naco.2012.2.193 |
[18] |
Chengxia Lei, Fujun Li, Jiang Liu. Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4499-4517. doi: 10.3934/dcdsb.2018173 |
[19] |
Yongjian Yang, Zhiyou Wu, Fusheng Bai. A filled function method for constrained nonlinear integer programming. Journal of Industrial and Management Optimization, 2008, 4 (2) : 353-362. doi: 10.3934/jimo.2008.4.353 |
[20] |
Chuanhao Guo, Erfang Shan, Wenli Yan. A superlinearly convergent hybrid algorithm for solving nonlinear programming. Journal of Industrial and Management Optimization, 2017, 13 (2) : 1009-1024. doi: 10.3934/jimo.2016059 |
2020 Impact Factor: 1.801
Tools
Article outline
Figures and Tables
[Back to Top]