Optimization of natural frequencies of bidirectional functionally graded beams

We propose a methodology to optimize the natural frequencies of functionally graded structures by tailoring their material distribution. The element-free Galerkin method is used to analyze the two-dimensional steady-state free and forced vibration of functionally graded beams. To optimize the material composition, the spatial distribution of volume fractions of the material constituents is defined using piecewise bicubic interpolation of volume fraction values that are specified at a finite number of grid points. Subsequently, we use a real-coded genetic algorithm to optimize the volume fraction distribution for three model problems. In the first problem, we seek material distributions that maximize each of the first three natural frequencies of a functionally graded beam. The goal of the second model problem is to minimize the mass of a functionally graded beam while constraining its natural frequencies to lie outside certain prescribed frequency bands. The last problem aims to minimize the mass of a functionally graded beam by simultaneously optimizing its thickness and material distribution such that the fundamental frequency is greater than a prescribed value.

[1]  R. Hill.  A self-consistent mechanics of composite materials , 1965 .

[2]  K. Tanaka,et al.  Average stress in matrix and average elastic energy of materials with misfitting inclusions , 1973 .

[3]  R. Cook,et al.  Concepts and Applications of Finite Element Analysis , 1974 .

[4]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[5]  P. Lancaster,et al.  Surfaces generated by moving least squares methods , 1981 .

[6]  Peter Lancaster,et al.  Curve and surface fitting - an introduction , 1986 .

[7]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[8]  Kalyanmoy Deb,et al.  A Comparative Analysis of Selection Schemes Used in Genetic Algorithms , 1990, FOGA.

[9]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[10]  Ken Brodlie,et al.  Visualization of surface data to preserve positivity and other simple constraints , 1995, Comput. Graph..

[11]  Kalyanmoy Deb,et al.  Simulated Binary Crossover for Continuous Search Space , 1995, Complex Syst..

[12]  Subra Suresh,et al.  Curvature changes during thermal cycling of a compositionally graded NiAl2O3 multi-layered material , 1996 .

[13]  Viggo Tvergaard,et al.  MICROMECHANICAL MODELS FOR GRADED COMPOSITE MATERIALS , 1997 .

[14]  G. Dvorak,et al.  Micromechanical models for graded composite materials : ii. thermomechanical loading , 1998 .

[15]  Kalyanmoy Deb,et al.  A Niched-Penalty Approach for Constraint Handling in Genetic Algorithms , 1999, ICANNGA.

[16]  A. Kawasaki,et al.  Functionally graded materials : design, processing and applications , 1999 .

[17]  J. N. Reddy,et al.  Vibration of functionally graded cylindrical shells , 1999 .

[18]  Romesh C. Batra,et al.  EXACT CORRESPONDENCE BETWEEN EIGENVALUES OF MEMBRANES AND FUNCTIONALLY GRADED SIMPLY SUPPORTED POLYGONAL PLATES , 2000 .

[19]  K. Deb.  An Efficient Constraint Handling Method for Genetic Algorithms , 2000 .

[20]  S. Gasik.  THERMAL-ELASTO-PLASTIC ANALYSIS OF W-CU FUNCTIONALLY GRADED MATERIALS SUBJECTED TO A UNIFORM HEAT FLOW BY MICROMECHANICAL MODEL , 2000 .

[21]  Gregory J. Wagner,et al.  Application of essential boundary conditions in mesh-free methods: a corrected collocation method , 2000 .

[22]  S. Vel,et al.  Exact Solution for the Cylindrical Bending Vibration of Functionally Graded Plates , 2002 .

[23]  S. Vel,et al.  Three-dimensional exact solution for the vibration of functionally graded rectangular plates , 2004 .

[24]  Isaac Elishakoff,et al.  Analytical Polynomial Solutions for Vibrating Axially Graded Beams , 2004 .

[25]  L. Qian,et al.  Static and dynamic analysis of 2‐D functionally graded elasticity by using meshless local petrov‐galerkin method , 2004 .

[26]  Romesh C. Batra,et al.  Design of bidirectional functionally graded plate for optimal natural frequencies , 2005 .