摘要
Mathematica是美国Wolfram Research公司开发的优秀数学软件，它是当今世界用于科技计算的难得的一个完全集成环境下的符号运算系统。它拥有的不仅是数值计算能力，还有强大的符号推导能力，是科学计算、结构分析的有力工具。很多国内外科学家在各自的领域中应用它解决各类科学问题。
在结构分析中，矩形层合板的振动控制问题相对复杂。目前仅局限于有特殊边界条件的古典板理论，很少有文章涉到各种边界条件的高阶板理论。本文将利用Mathematica的计算环境，编制结构分析程序，在各种边界条件下，应用多种板理论求解各向异性矩形层合板的振动控制问题。
在数值计算领域，Fortran是最早使用和最通行的语言，程序能够高效执行。但它没有符号推导功能。Mathematica虽有强大的符号计算功能，但执行速度严重受到限制。本文将使Mathematica与Fortran语言相结合，在前者推导公式，到后者进行数值求解，集成双方优势，克服单一语言的不足。
随着计算技术的发展，有限元法在土木工程及其它许多领域中得到了越来越广泛的应用。但用C语言，或Fortran编写有限元程序少则上百行，多则数千行。不但编写修改麻烦而且冗长难懂。本文将使用Mathematica,Visual Fortran,Imsl的混合编程技术编制新型的有限元程序。充分发挥各种语言的优点，使程序具有代码少，精度高，结构化程度高，简单易懂的特点。
Wolfram Research is the creator of the excellent software-Mathematica, which is the world's only fully integrated environment for technical computing. It has the numerical-computation and powerful symbolic-manipulation capabilities and is the excellent tool for scientific computation. Many scientists apply Mathematica to solve all kinds of structural problems in engineering and other fields.
The problem of optimal design and control of composite laminated plates is very complexity. The research is presented based on the classical laminate theories for special boundary conditions and few papers have been formulated based on higher-order laminate theories with various cases of boundary conditions.The paper will create the structural-engineering program with Mathematica to solve the problem of composite laminated plates based on higher-order laminate theories with various cases of boundary conditions.
In the field of numerical-computation, Fortran is an old and practical computer language with great efficiency and speed, but it has not symbolic manipulation. However, although it has powerful symbolic-manipulation capabilities, Mathematica is awful limited in executable speed. The paper will combine Fortran with Mathematica to realize symbolic manipulation in the latter and numerical computation in the former .
With the rapid development of computer, the finite-element method is widely applied in civil engineering and other fields. However, it is very difficult to write the finite-element program with Fortran or C by reason of the excessive long sentences and complex structure. In the paper, one method was introduced to design the
finite-element program with three kinds of languages -- Mathematica , Visual
Fortran and Imsl. Because of the full advantage of these languages, the programs have the characteristics of brief sentences ,high accuracy and ordered texture.
引文
[1] Pavlovic MN, Sapountzakis E J, Computers and structures: non-numerical applications[J], Comput Struct,1986, 24:455-74
[2] Beltzer AI., Engineering analysis via symbolic computation-a breakthrough[J]. Appl Mech Rev, 1990, 43:119-27.
[3] Noor AK, Andersen CM, Computerized symbolic manipulation in structural mechanics-progress and potential[J], Comput Struct ,1979, 10:95 - 118.
[4] Pavelle R, Rothstein M, Fitch J. Computer algebra[J]. Sci Am, 1981, 245:102-13.
[5] Foster KR, Bau HH, Symbolic manipulation programs for the personal computer[J]. Science, 1989, 243:679- 84.
[6] Brown WS,ALTRAN user's manual[M],Murray Hill, NJ Bell Laboratories, 1977.
[7] Wolfram S., MATHEMATICA: a system for doing mathematics by computer[M], 2nd ed. New York, Addison Wesley, 1991.
[8] Mbakogu FC, Pavlovic cMN, Some applications of computer algebra to classical problems in plate theory [J], Int JMech Eng Educ, 1999, 27:348 - 65.
[9] Pavlovic MN, Poulton SM. On the computation of slab ective widths [J]. J Struct Eng (Proc ASCE) 1985;111:363 - 77.
[10] Tahan N, Pavlovic MN, Kotsovos MD, Orthotropic rectangular plates under in-plane loading, Part 1: Closed-form solutions for stresses[J], Compos Struct, 1995, 33:35 -48.
[11] Tahan N, Pavlovic MN, Kotsovos MD, Orthotropie rectangular plates under in-plane loading. Part 2: Application of the series solution by reference to the problem of collinear compression[J], ComposStruct , 1995, 33:49-62.
[12] Tahan N, Pavlovic MN, Kotsovos MD, Shear-lag revisited:the use of single Fourier series for determining the ective breadth in plated structures[J]. Comput Struct , 1997, 63:759- 67.
[13] Pavlovic MN, Tahan N, Kotsovos MD, Non-collinearly loaded laminae. In: Drew HR, Pellegrino S, editors. New approaches to structural mechanics, shells and biological structures[M], Dordrecht, Kluwer Academic Publishers, 2002,189-202
[14] Kotsovos MD, Pavlovic MN, Structural concrete: .niteelement analysis for limit-state design[M], London, Thomas Telford, 1995.
[15] Luft RW, Roesset JM, Connor JJ, Automatic generation of .nite element matrices[J], J Struct Div (Proc ASCE) , 1971, 97:349 - 62.
[16] Gunderson RH, Cetiner A, Element stiffness matrix generator[J], J Struct Div (Proc ASCE), 1971, 97:363 - 75.
[17] Pedersen P, Megahed MM, Axisymmetric element analysis using analytical computing. [J] , Comput Struct , 1975;5:241 - 7.
[18] Pedersen P, On computer-aided analytic element analysis and the similarities of tetrahedronelements[J], Int J NumerMethEng , 1977, 11:611-22.
[19] Cecchi MM, Lami C, Automatic generation of stiness matrices for finite element analysis[J], Int J NumerMethEng, 1977, 11:396-400.
[20] Komco. AR, Fenves S J, Symbolic generation of finite element system matrices[J]. Comput Struct,1979,10:119 - 24.
[21] Noor AK, Andersen CM, Computerized symbolic manipulationin nonlinear finite elementanalysis[J], Comput Struct , 1981, 13:379 - 403.
[22] Bardell NS, The application of symbolic computing to the hierarchical finite element method[J], Int J Numer Meth Eng, 1989, 28:1181 - 204.
[23] KikuchiM, Application of the symbolic mathematics system to the .nite element problem - evaluation of the stiffness matrix of a 4-nodes isoparametric element[J], Comput Mech,1989,5:41 - 7.
[24] Beltzer AI, Variational and .nite element methods: a symbolic computation approach [M], Berlinx:Springer, 1990.
[25] Lee CK, Hobbs RE, Closed form sti.ness matrix solutions for some commonly used hybrid finite elements[J], Comput Struct, 1998, 67:463 - 82.
[26] Wilkins D J, Applications of a symbolic algebra manipulation language for composite structuresanalysis[J], ComputStruct , 1973, 3:801-7.
[27] Elishako. I. T.ang J, Buckling of polar orthotropic circular plates on elastic foundation by computerized symbolic algebra[J], Comput Meth Appl Mech Eng , 1988, 68:229 - 47.
[28] (美)D.尤金， 邓建松 彭冉冉译，Mathematica, 使用指南[M], 北京：科学出版社, 2002, 150-215
[29] Ioakimidis NI, Application of MATHEMATICA to the direct solution of torsion problems by the energy method[J], Comput Struct, 1992, 43:803 - 7.
[30] Pavlovic MN, Mbakogu FC, Rayleigh estimates of thefundamental frequencies of vibration of circular plates[J], J Sound Vib 1996, 198:389 - 94.
[31] Mbakogu FC, Pavlovic MN, Closed-form fundamentalfrequency estimates for polar orthotropic circular plates[J], Appl Acoust, 1998, 54:207 - 28.
[32] Mbakogu FC, Pavlovic MN, Bending of clamped orthotropic rectangular plates: a variational symbolic solution[J], Comput Struct, 2000, 77: 117 - 28.
[33] (美)Richard J.Gaylord, Samuel N.Kanmin, 邵勇译, 数学软件Mathematica入门
[M]，北京：高等教育出版社，2001,407-431
[34] 荀飞,Mathematica4实例教程[M]，北京：中国电力出版社,2000,17-252
[35] 彭国伦，FORTRAN 95程序设计[M]，北京：中国电力出版社,2002，345-385
[36] O' Donoghue PE, Atluri SN, Control of dynamic response of acontinuum model of a large space structure[J], Comput & Structures, 1986, 23:199-209.
[37] Turvey GJ, Marshall IH, Buckling and Postbuckling of Composite Plates. London[M], Chapman & Hall, 1995.
[38] Rao SS, Optimum design of structures under shock and vibration environment[J], Shock Vibr Dig , 1989, 21:3 - 15.
[39] Yang JN, Soong TT, Recent advances in active control of civil engineering structures[J], Probab Engrg Mech , 1988, 3:179- 88.
[40] Miller RK, Masri SF, Dehganyer TJ, Caughey TK, Active vibration control of large civil structures[J], ASCE J Engrg Mech , 1988, 114:1542 - 70.
[41] Reinhom AM, Manolis GD, Recent advances in structural control[J], Shock Vibr Dig, 1989, 21:3- 8.
[42] Gabralyan MS, Krasoviskii NN, About stabilization of mechanical system problem[M], PMM, 1964, 28.
[43] Gabralyan MS, About stabilization of mechanical systems under continuous forces[J], YGUYervan , 1975, 2:47-56.
[44] M.N. Pavlovic, Symbolic computation in structural engineering [J], Computers and Structures 2003 (81) , 2121 - 2136
[45] 王元汉，李丽娟，李银平，有限元法基础与程序设计[M]，广州：华南理工大学出版社，2001，49—98
[46] (美)G．R．布查南，董文军，谢伟松译．有限元分析[M]，北京：科学出版社，2002，129-156
[47] Y.G. Youssif, M.E. Fares ,M.A. Hafiz, Optimal Control of the Dynamic Response of an Anisotropic Plate with Various Boundary Conditions[J], Mechanics Research Communications, 2001 (28):525-534
[48] Reddy JN, On the generalization of displacement based laminate theories[J], Appl MechRev, 1989, 42(11):213 - 22.
[49] 沈观林，复合材料力学，北京：清华大学出版社[M]，1995，223—253
[50] Reddy JN, Mechanics of Composite Materials and Structures Theory and Analysis[M],Boca Raton, FL,CRC Press, 1997
[51] M.E. Fares, Y.G. Youssif, A.E. Alamir , Optimal design and control of composite laminated plates with various boundary conditions using various plate theories[J], Composite Structures 2002, 56: 1- 12
[52] Gabralyan MS, About stabilization of mechanical systems under continuous
forces[J], YGU Yervan , 1975, 2:47 - 56.
[53] (美)R，M琼斯，朱颐龄译，复合材料力学[M]，上海：上海科学技术出版社出版，1981，18—53
[54] 龙驭球，包世华，结构力学教程[M]，北京：高等教育出版社，1988，1—70
[55] 曹国雄，弹性矩形薄板振动[M]，北京：中国建筑工业出版社，1983，50—93
[56] 罗祖道，王震鸣，复合材料力学进展[M]，北京：北京大学出版社，1992，110—154
[57] http ://library. wolfram.com/infocenter/TechNotes/174/ML_Tut.nb
[58] http://www.mathsource.com/Content/Enhancements/MathLink/0206-693
[59] Compaq Visaul Fortran 6.5A Online Help.
[60] IMSL with MPI Enhanced Subroutines and Functions for Distributed Scientific Applications, Fortran 90 MP Library User's Guide
[61] Compaq Array Visualizer 1.5 Online Help.
[62] 翟治年，王建国，基于MathLink的Fortran与Mathematica接口[J]，合肥工业大学学报，2003,26(3)：340-344
[63] David j.Kruglinski,Scot Wingo等，Programming Visual C++6.0技术内幕(第五版)[M]，北京：北京希望电子出版社，1999，49—509