基於微分方程式最大值原理，殘差關於解之單調性可被推論出，而本研究所探討之雙側逼近法正為一建構於此殘差與解單調關係上之數值方法。雙側逼近法首先採用傳統之加權殘值法，將一求解微分方程之問題，轉換為一具有限制條件之數學規劃問題，再結合最佳化法則從兩側逼近正確解，分別找出正確解之最小偏大近似解與最大偏小近似解。此方法不需先得知任何關於解之資訊即可求解且具有準確、程序簡便快速、計算量少之優點。最重要的是，不同於一般求解近似解之數值方法，雙側逼近法可給出近似解之誤差界，提供近似解之可靠度，為一極具貢獻之數值方法。於本文中，分別利用加權殘值法中兩種不同的子方法:配點法與子域法來轉換求解之微分方程，再進而採用基因演算法做為我們求解上下界之最佳化法則。本文將雙側逼近法分別應用至不同截面形狀管流之速度場分析，以及不同形狀薄板與文克勒基底薄板之靜態行為分析，將所求解之結果與解析解比對，均相當地令人滿意，驗證了此方法於邊界值問題上之可行性。而本文更特別針對了超級橢圓形做分析，於過去之文獻中，多為探討矩形、圓形、橢圓形等較一般形狀之問題，鑑於對於超級橢圓形相關資訊之缺乏，本文特別藉由此方法求解超級橢圓形管流以及超級橢圓形板分析，均得到合理之結果，提供了一求解此類形狀相關問題之有效數學工具。

Based on the maximum principle of differential equations, the monotonicity of residual versus solution can be concluded and the double side approach method is just a numerical method which is constructed on the basis of this monotonic relation. The double side approach method adopts the traditional method of weighted residuals to transform the problem of solving the differential equation into a mathematical programming problem with constraints and further to combine the optimization method to approach the solution from both sides, obtaining the minimal upper approximate solution and maximal lower approximate solution. By using this method, no information about the solution is needed in advance and it has the advantages of high accuracy, simple and fast procedure, less required calculation load and so on. Most important of all, unlike other numerical methods that are used to find the approximate solutions, the double side approach method can give the error bound of the approximate solution, providing the approximate solution with reliability, and this makes the method quite contributive. In this dissertation, two different submethods of method of weighted residuals, collocation method and subdomain method, are utilized to transform the differential equations, and then Genetic algorithms is adopted as our optimization method to find the upper and lower bounds. In this dissertation, the double side approach method is applied in analyzing the velocity field in ducts with various cross section shapes and the static behavior of thin plates with and without Winkler foundation under various shapes. Comparing our calculation results with analytical ones, satisfying results are obtained. The feasibility of this method can be verified. More particularly, this dissertation focuses on the analysis of the super ellipse problems. In the previous literatures, shapes such as rectangle, circle and elliptic received more attentions. In consideration of the lack of information about super ellipse problems, the double side approach method is utilized to solve the velocity field in super elliptical ducts, the static behavior of super elliptical plates with and without Winkler foundation, and reasonable results are obtained, providing an efficient tool to solve the problems of this kind of shapes.

[1]Crandall, S. H.: Engineering Analysis, McGraw-Hill, 1956.
[2]Finlayson, B. A.: The Method of Weighted Residuals and Variational Principles, Academic Press., New York and London, 1972.
[3]Finlayson, B. A. and Seroven, L. E., “The Method of Weighted Residuals,” A Review, Applied Mechanics Reviews, vol. 19, no. 9, 1968.
[4]徐次達, “固體力學加權殘值法,” 同濟大學出版社, 1987.
[5]徐次達, “加權殘值法計算力學在我國18年中的進展國際近期進展概要與展望,” 西安公路交通大學學報, 1997年增刊.
[6]傅作新, “結構與水體的偶合作用問題,” 工程數值方法學術會議論文, 1987.
[7]江理平, “彈性薄板彎曲、振動與穩定問題的殘值ODE方法,” 加權殘值法最新進展及其工程應用, pp. 174-179, 1992.
[8]趙志崗, “薄殼非線性穩定問題的加權殘數法初步研究,” 加權殘值法最新進展及其工程應用, pp. 237-240, 1992.
[9]孫博華, “擾動權餘法及在薄板大撓度問題上的應用,” 固體力學學報, 1986.
[10]Abstract of WCCM Vol. I & Vol. II, “Proceedings of the First Congress on Computational Mechanics,” Sept., 20-45, 1986, Austin U.S.A.
[11]邱吉寶, “加權殘值法的理論與應用,” 宇航出版社, 1991.
[12]朱寶安, “力學問題—現代數學規劃加權殘值法,” 天津科學技術出版社.
[13]朱寶安, “數學規劃加權殘值法動態,” 天津大學分校, 1995.
[14]Appl, F. C. and Hing, H. M., “A principle for convergent upper and lower bounds,” Int. J. Mech. Sci., vol. 6, pp 381-389, 1964.
[15]Baluch, M. H., Mohsen, M. F. and Ali, A. I., "Method of weighted residuals as applied to nonlinear differential equation," Appl. Math. Modelling, vol. 7, pp. 362-365, 1983.
[16]Gupta, U. S., Jain, S. K. and Jain, D., "Method of collocation by derivatives in the study of axisymmetric vibrations of circular plates," Computer & Structures, vol. 57, pp. 841-845, 1995.
[17]Segerlind, L. J., "Weighted residual solutions in the time domain," International Journal for Numerical Methods in Engineering, vol. 28, pp. 679-685, 1989.
[18]Neuman, C. P. and Schonbach, D. I., "Discrete weighted residual methods : Multi-interval methods," Int. J. Systems Sci., vol. 8, no. 11, pp. 1281-1298, 1977.
[19]Spall, R., "Spectral Collocation methods for one-dimensional phase-change problems," Int. J. Heat Mass Transfer, vol. 38, no. 15, pp. 2743-2748, 1995.
[20]Lepik, U., "Axisymmetric vibrations of elastic-plastic cylindrical shells by Galerkin's method," Int. J. Impact Eng., vol. 18, no. 5, pp. 489-504, 1996.
[21]Adomaitis, R. A. and Lin, Y. H., "A technique for accurate collocation residual calculations," Chemical Engineering Journal, vol. 71, pp. 127-134, 1998.
[22]Hoon, K. H. and Khong, P. W., "The semi-energy and the lower bound methods to the post-buckling of plate," Computers & Structures, vol. 58, no. 1, pp. 107-113, 1996.
[23]Mahajerin, E. and Burgess, G., "Fundamental Collocation method applied to plane thermoelasticity problems," Computers & Structures, vol. 57, no. 5, pp. 795-797, 1995.
[24]Vitanov, N. K., "Upper bounds on the heat transport in a porous layer," Physica D, vol. 136, pp. 322-339, 2000.
[25]林金木, “數學規劃加權殘值法在工程上的應用,” 博士論文, 成功大學, 2000.
[26]李宗乙, “遺傳算則之双側逼近加權殘值法在工程上之應用”, 博士論文, 成功大學, 2001.
[27]Chen, C. K., Chen, C. L. and Lin, J. M., “Error bound estimate of weighted residuals method using genetic algorithms,” Applied Mathematics and Computation, vol. 81, pp. 207-219, 1997.
[28]Young, J. D., "Linear program approach to linear differential problems," Int. J. Eng. Sci., vol. 2, pp. 413-416, 1964.
[29]Kobayashi, S. and Thomsen, E. G., "Upper and lower solutions to axisymmetric compression and extension problems," Int. J. Mech. Sci., vol. 7, pp. 127-143, 1965.
[30]Gorla, R. S., Canter, M. S. nd Pallone, P. J., "Variational approach to unsteady flow and heat transfer in a channel," Heat and Mass Transfer, vol. 33, pp. 439-442, 1998.
[31]Chen, S., Qiu, Z. and Song, D., "A new method for computing the upper and lower bounds on frequencies of structures with interval parameters," Mechanics Research Communications, vol. 22, no. 5, pp. 431-439, 1995.
[32]Collatz, L.: The numerical treatment of differential equations, Springer-Verlag, Berlin, 1960.
[33]Potter, M. H. and Weinberger, H. F.: Maximum principles in differential equations, Prentice-Hall, 1967.
[34]Sperb, R. P.: Maximum principles and their applications, Academic Press, New York, 1981.
[35]Bagley, J. D.: The behavior of adaptive system which employ genetic and correlation algorithms, PhD. Dissertation, University of Michigan, 1967.
[36]Holland, J.: Adaptation in natural and artificial system, The University of Michigan Press, 1975.
[37]Lawrence, D.: Handbook of genetic algorithms, Van Nostrand Reinhold, New York, 1991.
[38]Goldberg, D. E.: Genetic Algorithms in Search. Optimization and Machine Learning, Addison-Wesley, Reading, Ma, 1989.
[39]Davis, L.: Handbook of Genetic Algorithms, Van Nostrand Reinhold, New York, 1991.
[40]Jenkis, W. M., “Towards structural optimization via the genetic algorithm”, Computers & Structures, vol. 40, pp. 1321-1327, 1991.
[41]Yokota, T., Gen, M. and Li, Y. X., "Genetic algorithm for non-linear mixed integer programming problems and its applications," Computers and Eng., vol. 30, no. 4, pp. 905-917, 1996.
[42]Riche, R. L. and Haftka, R. T., "Improved genetic algorithm for minimum thickness composite laminate design," Composites Engineering, vol. 5, no. 2
[43]Wang, B. P. and Chen, J. L., "Application of genetic algorithm for the support location optimization of beams," Computers & Structures, vol. 58, no. 4, pp. 797-800, 1996.
[44]Zhang, Y. E., Sun, J. P. and Ji, A. B., "A solution of fuzzy possibilistic linear program problems with equality constraint," Journal of Hebei University, vol. 19, no. 2, pp. 116-118, 1999. (in Chinese)
[45]Gao, F., Hu, Q., Wang, Z. and Wang, D., "Improved evolutionary direction way for genetic algorithms and structural design," Journal of Northeastern University, vol. 19, no. 1, pp. 79-82, 1998. (in Chinese)
[46]Liu, S., Zheng, B. and Wang, M., "Algorithm design based on genetic algorithm for integer programming problem," Journal of Northeastern University, vol. 19, no. 2, pp. 198-200, 1998. (in Chinese)
[47]Chen, Y. H. and Fang, S. C., "Solving convex programming problems with equality constraints by Neural Networks," Computers Math. Appl., vol. 36, no. 7, pp. 41-68, 1998.
[48]Srinivas, M. and Patmaik, L. M., “Genetic algorithms-A survey”, Computer, Vol. 27 (6), pp. 17-26, 1994.
[49]Gen, M. and Cheng, R., “Genetic algorithms and engineering design,” John Wiley & Sons, New York, 1997.
[50]雲慶夏, 黃光球, 王戰權, “遺傳算法和遺傳規劃,” 北京, 冶金工業出版社.
[51]祁載康, 萬耀青, 梁嘉玉, “工程優化原理及應用,” 北京理工大學出版社.
[52]Dryden, H. L., Murnaghan, F. D. and Bateman, H.: Hydrodynamics, Dover, New York, 1956.
[53]Shah, R. K. and London, A. L.: Laminar Flow Forced Convection in Ducts, Academic Press, New York, 1978.
[54]Spiga, M. and Morini, G. L., “A symmetric solution for velocity profile in laminar flow through rectangular ducts,” Int. Comm. in Heat and Mass Transfer, vol. 21 (4), pp.469-475, 1994.
[55]Wang, C. Y., “Heat transfer and flow through a super-elliptic duct-Effect of corner rounding”, Mechanics Research Communications, vol. 36, pp. 509-514, 2009.
[56]Leipholz, H.: Theory of elasticity. Noordhoff international publishing leyden, 1974.
[57]Timoshenko, S. and Woinowsky-Krieger, S.: Theory of plates and shells, McGraw-Hill, New York, 1959.
[58]Çeribaşı, S., Altay, G. and Dökmeci, M. C., “Static analysis of superelliptical clamped plates by Galerkin’s method,” Thin-Walled Structures, vol. 46, pp. 122-127, 2008.