Abstract
The new double side approach method combining the mathematical programming and the subdomain method in the method of weighted residual is presented in this article. Under the validation of maximum principle, and up on the subdomain method, the differential equation can be transferred into a bilateral inequality problem. Applying the genetic algorithms helps to find optimal solutions of upper and lower bounds which satisfy the inequalities. Here, the method is first verified by analyzing the deflection of elliptical clamped plate problem under various aspect ratios and further apply it to analyze the clamped super-elliptical plates problem. By using this method, the good approximate solution can be obtained accurately.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Finlayson B.A., Scriven L.E.: The method of weighted residuals—a review. Appl. Mech. Rev. 19(9), 735–748 (1966)
Zhang Y.C., He X.: Analysis of free vibration and buckling problems of beams and plates by discrete least-squares method using B5-spline as trial functions. Comput. Struct. 31(2), 115–119 (1989)
Chen C.K., Lin C.L., Chen C.L.: Error bound estimate of weighted residuals method using genetic algorithms. Appl. Math. Comput. 81, 207–219 (1997)
**ng S.Y., Li Z.Q., Zhu B.A.: Two methods of mathematical programming and weighted residuals. J. **’an Highway Univ. 17, 61–65 (1997)
Appl F.C., Hung H.M.: A principle for convergent upper and lower bounds. Int. J. Mech. Sci. 6, 289–381 (1964)
Goldberg D.E.: Genetic Algorithms in Search. Optimization and Machine Learning. Wesley, Reading (1989)
Davis L.: Handbook of Genetic Algorithms. Van Nostrand Reinhold, New York (1991)
Holland J.H.: Adaptation in Natural and Artificial System. MIT Press, Cambridge (1992)
Srinivas M., Patmaik L.M.: Genetic algorithms—a survey. Computer 27(6), 17–26 (1994)
Jenkis W.M.: Towards structural optimization via the genetic algorithm. Comput. Struct. 40, 1321–1327 (1991)
Christopher L.H., Donald E.B.: A parallel heuristic for quadratic assignment problems. Comput. Oper. Res. 18, 275–289 (1991)
Reeves C.R.: Modern Heuristic Technique for Combinational Problems. Halsted Press, New York (1993)
Lee Z.Y., Chen C.K., Hung C.I.: Upper and lower bounds of the solution for an elliptic plate problem using a genetic algorithm. Acta Mech. 157, 201–212 (2002)
Wang C.M., Wang L., Liew K.M.: Vibration and buckling of super elliptical plates. J. Sound Vib. 171, 301–314 (1994)
Çeribaşı S., Altay G., Dökmeci M.C.: Static analysis of superelliptical clamped plates by Galerkin’s method. Thin-Walled Struct. 46, 122–127 (2008)
Protter M.H.: Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs (1967)
Zhu B.A.: Upper and lower bounds of the solutions for plate and shell problems. J. Solid Mech. 15, 91–94 (1994) (in Chinese)
Leipholz H.: Theory of Elasticity. Noordhoff, Leyden (1974)
Timoshenko S., Woinowsky-Krieger S.: Theory of Plates and Shells. McGraw-Hill, New York (1959)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tang, HW., Yang, YT. & Chen, CK. Application of new double side approach method to the solution of super-elliptical plate problems. Acta Mech 223, 745–753 (2012). https://doi.org/10.1007/s00707-011-0592-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-011-0592-x