||Multi-objective optimization of functionally graded beams using a genetic algorithm with non-dominated sorting
||Department of Civil Engineering
functionally graded beams
layer-wise beam theories
本文應用混合分層(Layerwise, LW)高階剪切變形理論(Higher-Order Shear Deformation Theory, HSDT)進行受均勻溫度增量之功能性梯度(Functionally Graded, FG)簡支梁的熱挫屈分析。文中假設FG梁的材料性質取決於厚度和環境溫度變數，FG梁之有效材料性質可用二相材料混合法則(The Rule of Mixtures)或Mori-Tanaka微觀力學法則來進行計算。數值範例結果顯示本混合LW HSDT之臨界溫度增量參數解與文獻中提供的精確解相吻合。本文亦使用非支配排序遺傳演算法(Genetic Algorithm, GA)進行FG梁的多目標最佳化設計，以最大化其臨界溫度增量參數和最小化其總質量。在最佳化設計中，預先假設FG梁的成分體積分率為某種特定形式，例如單參數或三參數冪次律函數，前者用於FG梁的熱挫屈分析中，而後者用於FG梁的最佳化設計中。
A mixed layer-wise higher-order shear deformation theory is developed for the thermal buckling analysis of simply-supported, functionally graded beams subjected to a uniform temperature change. The material properties of the FG beam are assumed to be dependent on the thickness and temperature variables, and the effective material properties are estimated using either the rule of mixtures or the Mori-Tanaka scheme. The results shown in the numerical examples indicate the mixed LW HSDT solutions for critical temperature change parameters are in excellent agreement with the accurate solutions available in the literature. A multi-objective optimization of FG beams is presented to maximize the critical temperature change parameters and to minimize their total mass using a non-dominated sorting-based genetic algorithm (GA). Some specific forms for the volume fractions of the constituents of the FG beam are assumed in advance, such as the one- and three-parameter power-law functions. The former is used in the thermal buckling analysis of the FG beams for comparison purposes, and the latter is used in their optimal design.
Extended Abstract II
第一章 緒論 1
第二章 有效材料性質 5
2.1 二相材料混合法則 5
2.2 Mori-Tanaka 微觀力學法則 5
第三章 功能性材料梁的混合高階剪切變形理論 6
3.1 強形式數學方程式 6
3.2 應用 11
第四章 最佳化設計 14
4.1 最佳化設計問題說明 14
4.2 非支配排序GA 15
第五章 數值範例 19
5.1 複合材料梁與FG梁的熱挫屈分析 19
5.2 FG梁之材料組成最佳化設計 20
第六章 結論 22
Atashpaz-Gargari E, Lucas C (2007) Imperialist competitive algorithm: An algorithm for optimization inspired by imperialistic competition. IEEE Congress on Evolutionary Computation:4661-4667.
Aydogdu M (2007) Thermal buckling analysis of cross-ply laminated composite beams with general boundary conditions. Compos Sci Technol 67:1096-1104.
Bayat Y, Toussi HE (2017) Exact solution of thermal buckling and post buckling of composite and SMA hybrid composite beam by layerwise theory. Aero Sci Technol 67:484-494.
Bert CW, Malik M (1997) Differential quadrature: a powerful new technique for analysis of composite structures. Compos Struct 39:179-189.
Bouazza M, Benseddiq N, Zenkour AM (2019) Thermal buckling analysis of laminated composite beams using hyperbolic refined shear deformation theory. J Therm Stresses 42:332-340.
Brischetto S, Leetsch R, Carrera E, Wallmersperger T, Kroplin B (2008) Thermo-mechanical bending of functionally graded plates. J Therm Stresses 31:286-308.
Carrera E (2003) Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Arch Comput Methods Eng 10:215-296.
Carrera E, Giunta G, Petrolo M (2011) Beam Structures: Classical and Advanced Theories. John Wiley and Sons, New York.
Cho JR, Ha DY (2002) Volume fraction optimization for minimizing thermal stress in Ni-Al2O3 functionally graded materials. Mater Sci Eng A 334:147-155.
Coelhoa LS, Marianib VC (2012) Firefly algorithm approach based on chaotic Tinkerbell map applied to multivariable PID controller tuning. Comput Math Appl 64:2371-2384.
Deb K (2002) Multi-Objective Optimization Using Evolutionary Algorithms. John Wiley & Sons, Ltd, New York.
Deb K, Goyal M (1996) A combined genetic adaptive search (GeneAS) for engineering design. J Comput Sci Informatics 26:30-45.
Du H, Lim MK, Lin RM (1994) Application of generalized differential quadrature method to structural problems. Int J Numer Methods Eng 37:1881-1896.
Eshelby JD (1959a) The elastic field outside an ellipsoidal inclusion. Proc Roy Soc Lond Ser A 252:561-569.
Eshelby JD (1959b) The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc Roy Soc Lond Ser A 241:376-396.
Gen M, Cheng R (1997) Genetic Algorithms & Engineering Design. John Wiley & Sons, Inc, New York.
Giunta G, Belouettar S, Carrera E (2010) Analysis of beams by means of classical and advanced theories. Mech Adv Mater Struct 17:622-635.
Giunta G, Crisafulli D, Belouettar S, Carrera E (2011) Hierarchical theories for the free vibration analysis of functionally graded beams. Compos Struct 94:68-74.
Giunta G, Crisafulli D, Belouettar S, Carrera E (2013) A thermo-mechanical analysis of functionally graded beams via hierarchical modelling. Compos Struct 95:676-690.
Goldberg DE (1989) Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley Publishing Company, Inc, New York.
Goupee AJ, Vel S (2006) Optimization of natural frequencies of bidirectional functionally graded beams. Struct Multidisc Optim 32:473-484.
Hagan MT, Demuth HB, Beale M (1995) Neural Network Design. PWS Publishing Company, New York.
Jha DK, Kant T, Singh RK (2013) A critical review of recent research on functionally graded plates. Compos Struct 96:833-849.
Karama M, Afaq KS, Mistou S (2003) Mechanical behavior of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity. Int J Solids Struct 40:1525-1546.
Kamarian S, Yas MH, Pourasghar A, Daghagh M (2014) Application of firefly algorithm and ANFIS for optimization of functionally graded beams. J Exper Theor Artif Intell 26:197-209.
Khdeir AA (2001) Thermal buckling of cross-ply laminated composite beams. Acta Mech 149:201-213.
Kiani Y, Eslami MR (2010) Thermal buckling analysis of functionally graded material beams. Int J Mech Mater Des 6:229-238.
Koizumi, M (1993) Concept of FGM. Ceramic Trans 34:3_10.
Koizumi, M (1997) FGM activities in Japan. Compos Part B 28:1_4.
Konak A, Coit DW, Smith AE (2006) Multi-objective optimization using genetic algorithms: A tutorial. Reliability Eng Syst Safety 91:992-1007.
Lee HJ, Saravanos DA (1996) Coupled layerwise analysis of thermopiezoelectric composite beams. AIAA J 34:1231-1237.
Li ZM, Qiao P (2015) Thermal postbuckling analysis of anisotropic laminated beams with different boundary conditions resting on two-parameter elastic foundation. Eur J Mech A/Solids 54:30-43.
Liew KM, Ferreira AJM (2011) A review of meshless methods for laminated and functionally graded plates and shells. Compos Struct 93:2013-2041.
Liew KM, Pan ZZ, Zhang LW (2019) An overview of layerwise theories for composite laminates and structures: Development, numerical implementation and application. Compos Struct 216:240-259.
Mantari JL, Oktem AS, Soares CG (2012) A new higher order shear deformation theory for sandwich and composite laminated plates. Compos Part B 43:1489-1499.
Miyamoto Y, Kaysser WA, Rabin BH, Kawasaki A, Ford RG (1999) Functionally Graded Materials: Design, Processing, and Applications. Springer, New York.
Mori T, Tanaka K (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall 21:571-574.
Na KS, Kim JH (2009a) Optimization of volume fractions for functionally graded panels considering stress and critical temperature. Compos Struct 89:509-516.
Na KS, Kim JH (2009b) Volume fraction optimization of functionally graded composite panels for stress reduction and critical temperature. Fin Elem Anal Des 45:845-851.
Na KS, Kim JH (2010) Volume fraction optimization for step-formed functionally graded plates considering stress and critical temperature. Compos Struct 92:1283-1290.
Nguyen TT, Lee J (2017) Optimal design of thin-walled functionally graded beams for buckling problems. Compos Struct 179:459-467.
Nguyen TK, Nguyen TTP, Vo TP, Thai HT (2015) Vibration and buckling analysis of functionally graded sandwich beams by a new higher-order shear deformation theory. Compos Part B 76:273-285.
Pandey S, Pradyumna S (2018) Analysis of functionally graded sandwich plates using a higher-order layerwise theory. Compos Part B 153:325-336.
Punera D, Kant T (2019) A critical review of stress and vibration analyses of functionally graded shell structures. Compos Struct 210:787-809.
Ramirez F, Heyliger PR (2006) Discrete layer solution to free vibrations of functionally graded magneto-electro-elastic plates. Mech Adv Mater Struct 13:249-266.
Ramirez F, Heyliger PR, Pan E (2006) Static analysis of functionally graded elastic anisotropic plates using a discrete layer approach. Compos. Part B-Eng 37:10-20.
Reddy JN (1984a) A simple higher-order theory for laminated composite plates. J Appl Mech 51:745-752.
Reddy JN (1984b) Energy and Variational Methods in Applied Mechanics: with an Introduction to the Finite Element Method. John Wiley & Sons, Inc, New York.
Reddy JN (2000) Analysis of functionally graded plates. Int J Numer Methods Eng 47:663-684.
Reddy JN (2003) Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. CRC Press, New York.
Reddy JN (2006) Theory and Analysis of Elastic Plates and Shells. CRC Press, New York.
Reddy JN, Chin CD (1998) Thermomechanical analysis of functionally graded cylinders and plates. J Therm Stresses 21:593-626.
Saravanos DA, Heyliger PR (1995) Coupled electromechanical response of composite beams with embedded piezoelectric sensors and actuators. J Intell Mater Syst Struct 6:350-363.
Sayyad AS, Ghugal YM (2018) Modeling and analysis of functionally graded sandwich beams: A review. Mech Adv Mater Struct, https://doi.org/10.1080/15376494.2018.1447178.
She GL, Yuan FG, Ren YR (2017) Thermal buckling and post-buckling analysis of functionally graded beams based on a general higher-order shear deformation theory. Appl Math Modell 47:340-357.
Shen HS (2009) Functionally Graded Materials: Nonlinear Analysis of Plates and Shells. CRC Press, Boca Raton.
Shimpi RP, Ainapure AV (2001) A beam finite element based on layerwise trigonometric shear deformation theory. Compos Struct 53:153-162.
Shimpi RP, Ghugal YM (2001) A new layerwise trigonometric shear deformation theory for two-layered cross-ply beams. Compos Sci Technol 61:1271-1283.
Simsek M (2010) Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nuclear Eng Des 240:697-705.
Soldatos KP, Elishakoff, I (1992) A transverse shear and normal deformable orthotropic beam theory. J Sound Vib 155:528-533.
Swaminathan K, Naveenkumar DT, Zenkour AM, Carrera E (2015) Stress, vibration and buckling analyses of FGM plates-A state-of-the-art review. Compos Struct 120:10-31.
Tahani M (2007) Analysis of laminated composite beams using layerwise displacement theories. Compos Struct 79:535-547.
Takagi H, Hayashi I (1999) NN-Driven Fuzzy Reasoning. Int J Appl Res 5:191-212.
Tauchert TR (1987) Thermal buckling of antisymmetric angle-ply laminates. J Therm Stresses 10:113-124.
Thai HT, Kim SE (2015) A review of theories for the modeling and analysis of functionally graded plates and shells. Compos Struct 128:70-86.
Thai HT, Vo TP (2012) Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories. Int J Mech Sci 62:57-66.
Timarci T, Soldatos KP (1995) Comparative dynamic studies for symmetric cross-ply circular cylindrical shells on the basis of a unified shear deformable shell theory. J Sound Vib 187:609-624.
Tornabene F, Ceruti A (2013) Mixed static and dynamic optimization of four-parameter functionally graded completely doubly curved and degenerate shells and panels using GDQ method. Math Problems Eng 2013:867079 (33 pages).
Tornabene F, Viola E (2007) Vibration analysis of spherical structural elements using the GDQ method. Comput Math Appl 53:1538-1560.
Tornabene F, Viola E (2008) 2-D solution for free vibrations of parabolic shells using generalized differential quadrature method. Eur J Mech A/Solids 27:1001-1025.
Tornabene F, Viola E (2009) Free vibrations of three parameter functionally graded parabolic panels of revolution. Mech Res Commun 2009;36(5):587-94
Touratier M (1991) An efficient standard plate theory. Int J Eng Sci 29:901-916.
Tran TT, Nguyen NH, Do TV, Minh PV, Duc ND (2019) Bending and thermal buckling of unsymmetric functionally graded sandwich beams in high-temperature environment based on a new third-order shear deformation theory. J Sandw Struct Mater, https://doi.org/10.1177/1099636219849268.
Vlasov VZ (1961) Thin-Walled Elastic Beams. Program for Scientific Translation: Jerusalem, Israel.
Walker M, Smith RE (2003) A technique for the multiobjective optimization of laminated composite structures using genetic algorithms and finite element analysis. Compos Struct 62:123-128.
Washizu K (1982) Variational Methods in Elasticity and Plasticity. Pergamon Press, New York.
Wattanasakulpong N, Prusty BG, Kelly DW (2011) Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams. Int J Mech Sci 53:734-743.
Wu CP, Chen WY (1994) Vibration and stability of laminated plates based on a local high order plate theory. J Sound Vib 177:503-520.
Wu CP, Chiu KH, Wang YM (2008) A review on the three-dimensional analytical approaches of multilayered and functionally graded piezoelectric plates and shells. CMC-Comput Mater Continua 8:93-132.
Wu CP, Kuo HC (1992) Interlaminar stresses analysis for laminated composite plates based on a local high order lamination theory. Compos Struct 20:237-247.
Wu CP, Lee CY (2001) Differential quadrature solution for the free vibration analysis of laminated conical shells with variable stiffness. Int J Mech Sci 43:1853-1869.
Wu CP, Liu YC (2016) A review of semi-analytical numerical methods for laminated composite and multilayered functionally graded elastic/piezoelectric plates and shells. Compos Struct 147:1-15.
Yas MH, Kamarian S, Jam JE, Pourasghar A (2011) Optimization of functionally graded beams resting on elastic foundations. J Solid Mech 3:365-378.
Yas MH, Kamarian S, Pourasghar A (2014) Application of imperialist competitive algorithm and neural networks to optimise the volume fraction of three-parameter functionally graded beams. J Exper Theor Artif Intell 26:1-12.