The role of the multiquadric shape parameters in solving elliptic partial differential equations

Wertz, John; Kansa, Edward J.; Ling, Leevan

doi:10.1016/j.camwa.2006.04.009

Cited by 86 publications

(55 citation statements)

References 22 publications

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“…Since many application papers (e.g. [9,21]) stress the importance of proper scaling, we want to track the influence of the dilation k carefully.…”

Section: Kernels and Convolutionsmentioning

confidence: 99%

“…If r and s are fixed, the user should first choose a fine and quasi-uniform set Yr of trial centers, because its fill distance hr in Ω r will in the end drive the convergence rate. The final choice of test centers Ys must then be made to satisfy (21) for a suitable fill distance hs of Ys in Ω s . But in view of the term h −µ−2σ s in the final estimate, the choice of hs should be not too small unless we pick µ = −2σ and ρ ≥ σ.…”

Section: Error Analysismentioning

confidence: 99%

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Recovery of functions from weak data using unsymmetric meshless kernel-based methods

Schaback¹

2008

Applied Numerical Mathematics

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“…Since many application papers (e.g. [9,21]) stress the importance of proper scaling, we want to track the influence of the dilation k carefully.…”

Section: Kernels and Convolutionsmentioning

confidence: 99%

Section: Error Analysismentioning

confidence: 99%

Recovery of functions from weak data using unsymmetric meshless kernel-based methods

Schaback¹

2008

Applied Numerical Mathematics

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“…In the paper of Fedoseyev et al [8], the computational domain, Ω was extended slightly beyond the boundaries, ∂Ω k given by the line segments of the unit square: (0,1) (0,0), (0,1) (1,1) (0,1), The domain extended 0.020 beyond the unit square in the x 1 and x 2 directions. The parameters on the boundaries, see Wertz et al [9], were increased by a factor of 30. Consequently, numerical integration such as Gauss-Legendre (GL) needs to be performed, by first interpolating this integrand onto the appropriate GL zeros.…”

Section: Numerical Resultsmentioning

confidence: 99%

Collocation and optimization initialization

Kansa¹,

Ling²

2014

WIT Transactions on Modelling and Simulation

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Integrated volumetric methods such as finite elements and their "meshless" variations are typically smoother than the strong form finite difference and radial basis function collocation methods. Numerical methods decrease their convergence rates with successively higher orders of differentiation along with improved conditioning. In contrast, increasing order of integration increases the convergence rate at the expense of poorer conditioning. In the study presented, a two-dimensional Poisson equation with exponential dependency is solved. The solution of the point collocation problem becomes the initial estimate for an integrated volumetric minimization process. Global, rather than local integration, is used since there is no need to construct any meshes for integration as done in the "meshless" finite element analogs. The root mean square (RMS) errors are compared. By pushing the shape parameter to very large values, using extended precision, the RMS errors show that spatial refinement benefits are relatively small compared to pushing shape parameters to increasing larger values. The improved Greedy Algorithm was used to optimize the set of data and evaluation centers for various shape parameters. Finally, extended arithmetic precision is used to push the range of the shape parameters.

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“…To increase the accuracy and the convergence rate of MQ, several approaches have been proposed such as a) increasing a or decreasing h or both [22], b) integrated methods of Mai-Duy and Tran-Cong [1,[40][41][42] and c) using higher order MQ, e.g. ϕ i = (r 2 i + a 2 i ) β , where β > 1 2 [43]. To avoid the reduction of convergence rates due to differentiation and enhance the stability of the collocation-based numerical schemes in the case of Neumann type boundary value problems, in the present work we use Cartesian grid technique to discretise governing equations obtained by first-order formulation as follows.…”

Section: Numerical Formulationsmentioning

confidence: 99%

A Cartesian‐grid collocation technique with integrated radial basis functions for mixed boundary value problems

Mai‐Duy

Tran-Cong

et al. 2009

Numerical Meth Engineering

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In this paper, high order systems are reformulated as first order systems which are then numerically solved by a collocation method. The collocation method is based on Cartesian discretisation with 1D-integrated radial basis function networks (1D-IRBFN) [1]. The present method is enhanced by a new boundary interpolation technique based on 1D-IRBFN which is introduced to obtain variable approximation at irregular points in irregular domains. The proposed method is well suited to problems with mixed boundary conditions on both regular and irregular domains. The main results obtained are (a) the boundary conditions for the reformulated problem are of Dirichlet type only; (b) the integrated RBFN approximation avoids the well known reduction of convergence rate associated with differential formulations; (c) the primary variable (e.g. displacement, temperature) and the dual variable (e.g. stress, temperature gradient) have similar convergence order; (d) the volumetric locking effects associated with incompressible materials in solid mechanics are alleviated. Numerical experiments show that the proposed method achieves very good accuracy and high convergence rates.

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The role of the multiquadric shape parameters in solving elliptic partial differential equations

Cited by 86 publications

References 22 publications

Recovery of functions from weak data using unsymmetric meshless kernel-based methods

Recovery of functions from weak data using unsymmetric meshless kernel-based methods

Collocation and optimization initialization

A Cartesian‐grid collocation technique with integrated radial basis functions for mixed boundary value problems

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