A meshless, integration-free, and boundary-only RBF technique

By W. Chen, Masataka Tanaka. arXiv.org, 2002

298 citations recorded by the source corpus.

Based on the radial basis function (RBF), nonsingular general solution, and dual reciprocity method (DRM), this paper presents an inherently meshless, integration-free, boundary-only RBF collocation technique for numerical solution of various partial differential equation systems. The basic ideas behind this methodology are very mathematically simple. In this study, the RBFs are employed to approximate the inhomogeneous terms via the DRM, while nonsingular general solution leads to a boundary-only RBF formulation for homogenous solution. The present scheme is named as the boundary knot method (BKM) to differentiate it from the other numerical techniques. In particular, due to the use of nonsingular general solutions rather than singular fundamental solutions, the BKM is different from the method of fundamental solution in that the former does not require the artificial boundary and results in the symmetric system equations under certain conditions. The efficiency and utility of this new technique are validated through a number of typical numerical examples. Completeness concern of the BKM due to the sole use of the nonsingular part of complete fundamental solution is also discussed.

Subfields: Civil and Structural Engineering, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/d4cabb05f5179575fb91dd7e0bfb2efebabfd5c5

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Citations in the corpus

Numerical investigation on convergence of boundary knot method in the analysis of homogeneous Helmholtz, modified Helmholtz, and convection–diffusion problems (2003, methodology): “successfully in some inhomogeneous problems [11, 12 ].”
A highly accurate collocation Trefftz method for solving the Laplace equation in the doubly connected domains (2008, methodology): “An improved method than the MFS is the so-called boundary knot method [17, 18] or the boundary collocation method [19,20].”
Computation of Incompressible Navier-Stokes Equations by Local RBF-based Differential Quadrature Method (2005, background): “Subsequently, Fornberg [5], Hon and Wu [6], Chen et al [7], Fasshauer [8], Chen and Tanaka [9] also made a great contribution in this development.”
Radial basis function collocation method solution of natural convection in porous media (2004, unknown): “Their proposal has been further generalised by Chen and Tanaka (2000, 2002) to cope with the Dirichlet and Neumann boundary conditions as well as include the symmetric collocation.”
A Modified Method of Fundamental Solutions with Source on the Boundary for Solving Laplace Equations with Circular and Arbitrary Domains (2007, methodology): “…an improved technique which is the so called as boundary knot method or boundary collocation method was introduced by Chen and his collaborators (Chen & Tanaka 2002; Chen & Hon 2003), Kang and his coworkers (Kang et al. 1999; Kang & Lee 2000) as well as Chen and his coworkers (Chen et al. 2000,…”
A survey of applications of the MFS to inverse problems (2011, methodology): “The boundary knot method (BKM), which is a boundary method very closely related to the MFS introduced by Chen and Tanaka 64 , 65 , was also applied to the Cauchy problem ( 3.”
A method based on meshless approach for the numerical solution of the two‐space dimensional hyperbolic telegraph equation (2012, methodology): “(22) But Chen and Tanaka [34] introduced a different approach to overcome this difficulty.”
Potential Problems by Singular Boundary Method Satisfying Moment Condition (2009, methodology): “This study focuses on the boundary-type meshless numerical techniques, for instance, method of fundamental solutions (MFS) [Chen, Golberg and Hon (1998); Cheng, Young and Tsai (2000); Fairweather and Karageorghis (1998); Karageorghis (1992); Liu (2008); Marin (2008,2010a,2010b); Poullikkas, Karageor”
Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations (2003, methodology): “Other great contributions in using RBFs to solve PDEs include the work of Fornberg [13], Hon and Wu [14], Chen and Tanaka [ 15 ], Chen et al. [16].”