https://msvlab.hre.ntou.edu.tw/mcs2008-dehghan.pdf

A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions

Rotation free isogeometric thin shell analysis using PHT-splines

An extended isogeometric thin shell analysis based on Kirchhoff-Love theory

Radial basis functions (RBFs) are constructed in terms of one-dimensional distance variable and appear to have certain advantages over the traditional coordinates-based functions. In contrast to the traditional meshed-based methods, the RBF collocation methods are mathematically simple and truly meshless, which avoid troublesome mesh generation for high-dimensional problems involving irregular or moving boundary. This opening chapter begins with the introduction to RBF history and its applications in numerical solution of partial differential equations and then gives a general overview of the book.

Recent Advances in Radial Basis Function Collocation Methods

https://msvlab.hre.ntou.edu.tw/cite/JCP-Young-2005.pdf

Novel meshless method for solving the potential problems with arbitrary domain

In this paper, it is proved that the two approaches, known in the literature as the method of fundamental solutions (MFS) and the Trefftz method, are mathematically equivalent in spite of their essentially minor and apparent differences in formulation. In deriving the equivalence of the Trefftz method and the MFS for the Laplace and biharmonic problems, it is interesting to find that the complete set in the Trefftz method for the Laplace and biharmonic problems are embedded in the degenerate kernels of the MFS. The degenerate scale appears using the MFS when the geometrical matrix is singular. The occurring mechanism of the degenerate scale in the MFS is also studied by using circulant. The comparison of accuracy and efficiency of the two methods was addressed.

https://msvlab.hre.ntou.edu.tw/cma2007-ok.pdf

On the equivalence of the Trefftz method and method of fundamental solutions for Laplace and biharmonic equations

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.

https://msvlab.hre.ntou.edu.tw/paper/eabe/eabe-13.pdf

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

In this paper, the method of fundamental solutions (MFS) of single and double-layer potential approaches for solving the eigenfrequencies of multiply connected membranes is proposed. By employing the fundamental solution, the coefficients of influence matrices are easily determined. The spurious eigensolution accompanied by the true eigensolution appears. It is found that the spurious eigensolution using the MFS depends on the location of the inner boundary where the sources are distributed. To verify this finding, the true and spurious eigenvalues in an annular domain are analytically studied using the degenerate kernels and circulants for an annular membrane. In order to obtain the true eigensolution, the singular value decomposition (SVD) updating techniques and the Burton and Miller method are utilized to filter out the spurious eigensolutions. Two examples are demonstrated analytically and numerically to see the validity of the present method.

https://msvlab.hre.ntou.edu.tw/EABE2005.pdf

Eigensolutions of multiply connected membranes using the method of fundamental solutions

In this paper, the Green's function for the annular Laplace problem is first derived by using the image method which can be seen as a special case of method of fundamental solutions. Three cases, fixed–fixed, fixed–free and free–fixed boundary conditions are considered. Also, the Trefftz method is employed to derive the analytical solution by using T-complete sets. By employing the addition theorem, both solutions are found to be mathematically equivalent when the number of Trefftz base and the number of image points are both infinite. On the basis of the same number of degrees of freedom, the convergence rate of both methods is compared with each other. In the successive image process, the final two images freeze at the origin and infinity, where their singularity strengths can be analytically and numerically determined in a consistent manner.

https://msvlab.hre.ntou.edu.tw/paper/new/EABE_2134%20(1).pdf

Equivalence between the Trefftz method and the method of fundamental solution for the annular Green's function using the addition theorem and image concept.

http://msvlab.hre.ntou.edu.tw/paper/new/vibaco14-ian.pdf

Ying-Te Lee

Papers

On the equivalence of the Trefftz method and method of fundamental solutions for Laplace and biharmonic equations

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

Eigensolutions of multiply connected membranes using the method of fundamental solutions

Equivalence between the Trefftz method and the method of fundamental solution for the annular Green's function using the addition theorem and image concept.

Free vibration analysis of circular plates with multiple circular holes using indirect BIEMs