Numerical investigation on convergence of boundary knot method in the analysis of homogeneous Helmholtz, modified Helmholtz, and convection–diffusion problems

Abstract This paper concerns a numerical study of convergence properties of the boundary knot method (BKM) applied to the solution of 2D and 3D homogeneous Helmholtz, modified Helmholtz, and convection–diffusion problems. The BKM is a new boundary-type, meshfree radial function basis collocation technique. The method differentiates from the method of fundamental solutions (MFS) in that it does not need the controversial artificial boundary outside physical domain due to the use of non-singular general solutions instead of the singular fundamental solutions. The BKM is also generally applicable to a variety of inhomogeneous problems in conjunction with the dual reciprocity method (DRM). Therefore, when applied to inhomogeneous problems, the error of the DRM confounds the BKM accuracy in approximation of homogeneous solution, while the latter essentially distinguishes the BKM, MFS, and boundary element method. In order to avoid the interference of the DRM, this study focuses on the investigation of the convergence property of the BKM for homogeneous problems. The given numerical experiments reveal rapid convergence, high accuracy and efficiency, mathematical simplicity of the BKM.

[1]  Božidar Šarler.  Towards a mesh-free computation of transport phenomena , 2002 .

[2]  C. S. Chen,et al.  Some comments on the use of radial basis functions in the dual reciprocity method , 1998 .

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

[4]  J. H. Kane.  Boundary Element Analysis in Engineering Continuum Mechanics , 1994 .

[5]  L. Wrobel,et al.  On the convergence of the dual reciprocity boundary element method , 1994 .

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

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

[8]  T. Kitagawa,et al.  Asymptotic stability of the fundamental solution method , 1991 .

[9]  W. Chen,et al.  A note on radial basis function computing , 2001, ArXiv.

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

[11]  Palghat A. Ramachandran,et al.  A Particular Solution Trefftz Method for Non-linear Poisson Problems in Heat and Mass Transfer , 1999 .

[12]  Chang Shu,et al.  SIMULATION OF NATURAL CONVECTION IN ECCENTRIC ANNULI BETWEEN A SQUARE OUTER CYLINDER AND A CIRCULAR INNER CYLINDER USING LOCAL MQ-DQ METHOD , 2005 .

[13]  M. Golberg.  Boundary integral methods : numerical and mathematical aspects , 1999 .

[14]  Ming-Wan Lu,et al.  Least‐squares collocation meshless method , 2001 .

[15]  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.

[16]  R. L. Hardy.  Multiquadric equations of topography and other irregular surfaces , 1971 .

[17]  A. Bogomolny.  Fundamental Solutions Method for Elliptic Boundary Value Problems , 1985 .

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

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

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

[21]  P. W. Partridge,et al.  The method of fundamental solutions with dual reciprocity for diffusion and diffusion–convection using subdomains , 2000 .

[22]  A. Duarte,et al.  Investigations on the hp-Cloud Method by solving Timoshenko beam problems , 2000 .

[23]  Yinglong Zhang,et al.  On the dual reciprocity boundary element method , 1993 .

[24]  P. W. Partridge,et al.  The dual reciprocity boundary element method , 1991 .

[25]  Xiong Zhang,et al.  A 2-D meshless model for jointed rock structures , 2000 .

[26]  C. Brebbia,et al.  A new approach to free vibration analysis using boundary elements , 1983 .

[27]  Ching-Shyang Chen,et al.  A numerical method for heat transfer problems using collocation and radial basis functions , 1998 .

[28]  R. Schaback,et al.  Numerical Techniques Based on Radial Basis Functions , 2000 .

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

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

[31]  Larry L. Schumaker,et al.  Curve and Surface Fitting: Saint-Malo 1999 , 2000 .

[32]  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.

[33]  E. Kansa.  Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates , 1990 .

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

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

[36]  E. Kansa.  MULTIQUADRICS--A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS-- II SOLUTIONS TO PARABOLIC, HYPERBOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 1990 .

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

[38]  K. Balakrishnan,et al.  The method of fundamental solutions for linear diffusion-reaction equations , 2000 .

[39]  Jichun Li.  Mathematical justification for RBF-MFS , 2001 .