A meshless method for free vibration analysis of circular and rectangular clamped plates using radial basis function

In this paper, a meshless method for solving the eigenfrequencies of clamped plates using the radial basis function (RBF) is proposed. The coefficients of influence matrices are easily determined by the two-point function. By employing the RBF in the imaginary-part fundamental solution, true eigensolutions instead of spurious one are obtained for plate vibration. In order to obtain the eigenvalues and boundary modes at the same time, singular value decomposition technique is utilized. Two examples, circular and rectangular clamped plates, are demonstrated to see the validity of the present method.

[1]  P. K. Banerjee,et al.  A new BEM formulation for the acoustic eigenfrequency analysis , 1988 .

[2]  J. L. Goldberg.  Matrix theory with applications , 1991 .

[3]  Norio Kamiya,et al.  A new complex-valued formulation and eigenvalue analysis of the Helmholtz equation by boundary element method , 1996 .

[4]  Shin-Hyoung Kang,et al.  FREE VIBRATION ANALYSIS OF ARBITRARILY SHAPED PLATES WITH CLAMPED EDGES USING WAVE-TYPE FUNCTIONS , 2001 .

[5]  G. de Mey,et al.  A simplified integral equation method for the calculation of the eigenvalues of Helmholtz equation , 1977 .

[6]  F. C. Wong,et al.  Analytical derivations for one-dimensional eigenproblems using dual boundary element method and multiple reciprocity method , 1997 .

[7]  J. Monaghan,et al.  Smoothed particle hydrodynamics: Theory and application to non-spherical stars , 1977 .

[8]  Y. P. Chiu,et al.  On the pseudo-differential operators in the dual boundary integral equations using degenerate kernels and circulants , 2002 .

[9]  Youssef F. Rashed,et al.  A MESH-FREE METHOD FOR LINEAR DIFFUSION EQUATIONS , 1998 .

[10]  S. N. Atluri,et al.  Numerical integration of singularities in meshless implementation of local boundary integral equations , 2000 .

[11]  Dimitri E. Beskos,et al.  Boundary Element Methods in Dynamic Analysis , 1987 .

[12]  S. Mukherjee,et al.  THE BOUNDARY NODE METHOD FOR POTENTIAL PROBLEMS , 1997 .

[13]  Young-Seok Kang,et al.  VIBRATION ANALYSIS OF ARBITRARILY SHAPED MEMBRANES USING NON-DIMENSIONAL DYNAMIC INFLUENCE FUNCTION , 1999 .

[14]  Wing Kam Liu,et al.  Reproducing kernel particle methods , 1995 .

[15]  K. H. Chen,et al.  The Boundary Collocation Method with Meshless Concept for Acoustic Eigenanalysis of Two-Dimensional Cavities Using Radial Basis Function , 2002 .

[16]  K. H. Chen,et al.  Boundary collocation method for acoustic eigenanalysis of three-dimensional cavities using radial basis function , 2002 .

[17]  Y. C. Hon,et al.  Boundary knot method for 2D and 3D Helmholtz and convection–diffusion problems under complicated geometry , 2003 .

[18]  Y. C. Cheng,et al.  Comment on "Eigenmode analysis of arbitrarily shaped two-dimensional cavities by the method of point matching". , 2002, The Journal of the Acoustical Society of America.

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

[20]  Wen Chen.  Meshfree boundary particle method applied to Helmholtz problems , 2002 .

[21]  N. Kamiya,et al.  A note on multiple reciprocity integral formulation for the Helmholtz equation , 1993 .

[22]  Richard Paul Shaw,et al.  Helmholtz‐equation eigenvalues and eigenmodes for arbitrary domains , 1974 .

[23]  W. H. Wittrick,et al.  An automatic computational procedure for calculating natural frequencies of skeletal structures , 1970 .

[24]  A. Nowak,et al.  The Multiple reciprocity boundary element method , 1994 .

[25]  Jeng-Tzong Chen,et al.  A study on the multiple reciprocity method and complex-valued formulation for the Helmholtz equation , 1998 .

[26]  S. M. Dickinson.  THE BUCKLING AND FREQUENCY OF FLEXURAL VIBRATION OF RECTANGULAR ISOTROPIC AND ORTHOTROPIC PLATES USING RAYLEIGH'S METHOD , 1978 .

[27]  Gene H. Golub,et al.  Matrix computations , 1983 .

[28]  A nonsingular integral formulation for the Helmholtz eigenproblems of a circular domain , 1999 .

[29]  F. C. Wong,et al.  DUAL FORMULATION OF MULTIPLE RECIPROCITY METHOD FOR THE ACOUSTIC MODE OF A CAVITY WITH A THIN PARTITION , 1998 .

[30]  K. H. Chen,et al.  Comments on “Free vibration analysis of arbitrarily shaped plates with clamped edges using wave-type function” , 2003 .

[31]  Shin-Hyoung Kang,et al.  APPLICATION OF FREE VIBRATION ANALYSIS OF MEMBRANES USING THE NON-DIMENSIONAL DYNAMIC INFLUENCE FUNCTION , 2000 .

[32]  Shyh-Rong Kuo,et al.  Analytical study and numerical experiments for true and spurious eigensolutions of a circular cavity using the real‐part dual BEM , 2000 .

[33]  W. Chen,et al.  A meshless, integration-free, and boundary-only RBF technique , 2002, ArXiv.

[34]  D. E. Beskos,et al.  Some studies on dual reciprocity BEM for elastodynamic analysis , 1996 .

[35]  K. H. Chen,et al.  Determination of spurious eigenvalues and multiplicities of true eigenvalues using the real-part dual BEM , 1999 .

[36]  Lee,et al.  Eigenmode analysis of arbitrarily shaped two-dimensional cavities by the method of point-matching , 2000, The Journal of the Acoustical Society of America.

[37]  Wen Chen,et al.  New Insights in Boundary-only and Domain-type RBF Methods , 2002, ArXiv.

[38]  Wen Chen,et al.  Relationship between boundary integral equation and radial basis function , 2002, ArXiv.

[39]  K. H. Chen,et al.  COMMENTS ON &&VIBRATION ANALYSIS OF ARBITRARY SHAPED MEMBRANES USING NON-DIMENSIONAL DYNAMIC INFLUENCE FUNCTION'' , 2000 .

[40]  Xiong Zhang,et al.  Meshless methods based on collocation with radial basis functions , 2000 .

[41]  Mahadevan Ganesh,et al.  Particular solutions of 3D Helmholtz-type equations using compactly supported radial basis functions , 2000 .

[42]  S. Atluri,et al.  A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics , 1998 .

[43]  Wen Chen,et al.  Symmetric boundary knot method , 2002, ArXiv.