Investigation of regularized techniques for boundary knot method

Wang, Fuzhang; Chen, Wen; Jiang, Xinrong

doi:10.1002/cnm.1275

Cited by 25 publications

(14 citation statements)

References 19 publications

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“…In [27], the BKM is employed to find the numerical solution of inhomogeneous equation and it is applied to the Cauchy problem associated with the inhomogeneous Helmholtz equation. The authors of [63], investigated the regularization techniques for BKM and their experiments show that Thikhonov regularization with generalized cross-validation approach is better than other regularization techniques in the BKM solution of Helmholtz and modified Helmholtz problems. Also, in [12] the symmetric BKM is applied to the 2D and 3D Helmholtz and reaction-diffusion problems under complicated geometry.…”

Section: Boundary Knot Methodsmentioning

confidence: 99%

A boundary-only meshless method for numerical solution of the Eikonal equation

Dehghan

Salehi

2010

Comput Mech

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The radial basis function (RBF) collocation methods for the numerical solution of partial differential equation have been popular in recent years because of their advantage. For instance, they are inherently meshless, integration free and highly accurate. In this article we study the RBF solution of Eikonal equation using boundary knot method and analog equation method. The boundary knot method (BKM) is a meshless boundary-type radial basis function collocation technique. In contrast with the method of fundamental solution (MFS), the BKM uses the non-singular general solution instead of the singular fundamental solution to obtain the homogeneous solution. Similar to MFS, the RBF is employed to approximate the particular solution via the dual reciprocity principle. In the current paper, we applied the idea of analog equation method (AEM). According to AEM, the nonlinear governing operator is replaced by an equivalent nonhomogeneous linear one with known fundamental solution and under the same boundary conditions. Finally numerical results and discussions are presented to show the validity and efficiency of the proposed method.

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Section: Boundary Knot Methodsmentioning

confidence: 99%

A boundary-only meshless method for numerical solution of the Eikonal equation

Dehghan

Salehi

2010

Comput Mech

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“…For problems with noisy boundary data, the standard methods for solving matrix equations yield unstable results; thus, these authors used truncated singular value decomposition to solve the resulting matrix equation and obtained a stable and accurate numerical solution. Authors of investigated the regularization techniques for BKM, and their experiments show that Thikhonov regularization with generalized cross‐validation approach is more efficient than other regularization techniques for the BKM solution of Helmholtz and modified Helmholtz equations. Authors of used the geodesic distance instead of the standard isotropic Euclidean distance in the BKM to solve the anisotropic Helmholtz, diffusion and convection‐diffusion problems under 2D and 3D complicated geometries.…”

Section: Boundary Knot Methodsmentioning

confidence: 99%

A method based on meshless approach for the numerical solution of the two‐space dimensional hyperbolic telegraph equation

Dehghan

Salehi

2012

Math Methods in App Sciences

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Communicated by S. GeorgievIn the current article, we investigate the RBF solution of second-order two-space dimensional linear hyperbolic telegraph equation. For this purpose, we use a combination of boundary knot method (BKM) and analog equation method (AEM). The BKM is a meshfree, boundary-only and integration-free technique. The BKM is an alternative to the method of fundamental solution to avoid the fictitious boundary and to deal with low accuracy, singular integration and mesh generation. Also, on the basis of the AEM, the governing operator is substituted by an equivalent nonhomogeneous linear one with known fundamental solution under the same boundary conditions. Finally, several numerical results and discussions are demonstrated to show the accuracy and efficiency of the proposed method.where @ is the boundary of , which is a closed subset of R 2 and T is a positive number.When˛andˇare constants and˛>ˇ> 0, A D B D 1, Eqn (1) represents the two-dimensional telegraph equation which will be studied in the current investigation.Equation (1), referred as the second-order telegraph equation with constant coefficients, models the mixture between diffusion and wave propagation by introducing a term that accounts for the effects of finite velocity to standard heat or mass transport equation [1]. Problem (1) is commonly used in signal analysis for transmission and propagation of the electrical signals [2].Authors of [33,34] introduced the BKM, which eliminates the disadvantage of applying fictitious boundary and preserves all benefits of the MFS. The BKM employs the non-singular general solutions in place of the singular fundamental solutions. This method is truly meshfree, integration-free, mathematically simple, boundary-type and easy to implement. The BKM has been employed for a wide range of physical and engineering problems. In [35], the BKM is extended to solve the 2D Helmholtz and convection-diffusion problems under rather complex-shaped interior and exterior contours, which shows that unlike the MFS, the BKM is insensitive to geometric irregularity. This method is used in [36] for solving the Poisson equation . In that article, authors used the high-order general solutions of the Helmholtz and modified Helmholtz equations to evaluate the particular solution. In [37], the BKM is employed for solving the Cauchy problem associated with the inhomogeneous Helmholtz equation. For problems with noisy boundary data, the standard methods for solving matrix equations yield unstable results; thus, these authors [37] used truncated singular value decomposition to solve the resulting matrix equation and obtained a stable and accurate numerical solution. Authors of [38] investigated the regularization techniques for BKM, and their experiments show that Thikhonov regularization with generalized cross-validation approach is more efficient than other regularization techniques for the BKM solution of Helmholtz and modified Helmholtz equations. Authors of [39] used the geodesic distance instead of the standard isotropic Euc...

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“…The BKM has been applied to various problems such as Helmholtz [44], convection-diffusion [45,46], membrane vibration [47], plate vibration [48]. The key issue in the BKM is to construct the nonsingular RBF general solutions satisfying the governing equation, which has been discussed in Sect.…”

Section: ð4:9þmentioning

confidence: 99%

Boundary-Type RBF Collocation Methods

Chen

2013

Recent Advances in Radial Basis Function Collocation Methods

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The mesh generation in the standard BEM is still not trivial as one may imagine, especially for high-dimensional moving boundary problems. To overcome this difficulty, the boundary-type RBF collocation methods have been proposed and endured a fast development in the recent decade thanks to being integration-free, spectral convergence, easy-to-use, and inherently truly meshless. First, this chapter introduces the basic concepts of the method of fundamental solutions (MFS). Then a few recent boundary-type RBF collocation schemes are presented to tackle the issue of the fictitious boundary in the MFS, such as boundary knot method (BKM), regularized meshless method, and singular boundary method. Following this, an improved multiple reciprocity method (MRM), the recursive composite MRM (RC-MRM), is introduced to establish a boundary-only discretization of nonhomogeneous problems. Finally, numerical demonstrations show the convergence rate and stability of these boundary-type RBF collocation methods for several benchmark examples.Keywords Meshless Á Integration-free Á Collocation Á Fundamental solutions Á Singularity Á Method of fundamental solutions Á Boundary knot method Á Regularized meshless method Á Singular boundary method Á Boundary particle method During the past two decades we have witnessed a research boom on the boundarytype meshless techniques since the construction of a mesh in the standard BEM is not trivial. Among the typical techniques are the boundary node method, the local boundary integral equation method, the boundary cloud method, the boundary point method, the boundary point interpolation method (BPIM), and the method of fundamental solutions (MFS). The essence of all these techniques, excluding the MFS and BPIM, is basically a combination of the moving least square (MLS) technique with various boundary element schemes, whereas the MFS is a boundary-type RBF collocation scheme.Such MLS-based methods involve singular integration and are mathematically complicated, and their low-order approximations also depress computational efficiency. The numerical integration requires a background mesh, and thus the

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Investigation of regularized techniques for boundary knot method

Cited by 25 publications

References 19 publications

A boundary-only meshless method for numerical solution of the Eikonal equation

A boundary-only meshless method for numerical solution of the Eikonal equation

A method based on meshless approach for the numerical solution of the two‐space dimensional hyperbolic telegraph equation

Boundary-Type RBF Collocation Methods

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