Comparing RBF-FD approximations based on stabilized Gaussians and on polyharmonic splines with polynomials

2018 ◽  
Vol 115 (4) ◽  
pp. 462-500 ◽  
Author(s):  
L. G. C. Santos ◽  
N. Manzanares-Filho ◽  
G. J. Menon ◽  
E. Abreu
Keyword(s):  
Polyharmonic Splines

A Spline Framework for Estimating the EEG Surface Laplacian Using the Euclidean Metric

2011 ◽  
Vol 23 (11) ◽  
pp. 2974-3000 ◽  
Author(s):  
Claudio G. Carvalhaes ◽  
Patrick Suppes
Keyword(s):  
Basic Problem ◽  
Algebraic Surfaces ◽  
Eeg Analysis ◽  
Sphere Model ◽  
Surface Laplacian ◽  
Three Sphere ◽  
Euclidean Metric ◽  
Low Degree ◽  
Euclidean Setting ◽  
Polyharmonic Splines

This letter develops a framework for EEG analysis and similar applications based on polyharmonic splines. This development overcomes a basic problem with the method of splines in the Euclidean setting: that it does not work on low-degree algebraic surfaces such as spherical and ellipsoidal scalp models. The method’s capability is illustrated through simulations on the three-sphere model and using empirical data.

scholarly journals A stabilized well-balanced RBF-FD meshless method for shallow-water equations

2019 ◽  
Author(s):  
Joel Antonio Godoy de Moraes ◽  
Eduardo Cardoso de Abreu ◽  
Luis Guilherme Cunha Santos
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Bottom Topography ◽  
Point Clouds ◽  
General Criterion ◽  
Modified Method ◽  
Modified Method Of Characteristics ◽  
Polyharmonic Splines ◽  
Convection Dominated ◽  
Careful Design

In this work, we are concerned with the study and computing of stabilized radial basis function-generated finite difference (RBF-FD) approximations for shallow-water equations. In order to obtain both stable and highly accurate numerical approximations of convection-dominated shallow-water equations, we use stabilized flat Gaussians (RBFSGA-FD) and polyharmonic splines with supplementary polynomials (RBFPHS-FD) as basis functions, combined with modified method of characteristics. These techniques are combined with careful design for the spatial derivative operators in the momentum flux equation, according to a general criterion for the exact preservation of the “lake at rest” solution in general mesh-based and meshless numerical schemes for the strong form of the shallow-water equations with bottom topography. Both structured and unsructured point clouds are employed for evaluating the influence of cloud refinement, size of local supports and maximal permissible degree of the polynomials in RBFPHS-FD.

Fast evaluation of polyharmonic splines in three dimensions

2006 ◽  
Vol 27 (3) ◽  
pp. 427-450 ◽  
Author(s):  
R. K. Beatson ◽  
M. J. D. Powell ◽  
A. M. Tan
Keyword(s):  
Three Dimensions ◽  
Fast Evaluation ◽  
Polyharmonic Splines

scholarly journals The Method of Fundamental Solutions with Dual Reciprocity for thin Plates on Winkler Foundations with Arbitrary Loadings

2008 ◽  
Vol 24 (2) ◽  
pp. 163-171 ◽  
Author(s):  
C. C. Tsai
Keyword(s):  
Lateral Displacement ◽  
Fundamental Solutions ◽  
Solution Procedure ◽  
Real Coefficient ◽  
Method Of Fundamental Solutions ◽  
Thin Plates ◽  
Complex Coefficient ◽  
Particular Solutions ◽  
Analytical Formulas ◽  
Polyharmonic Splines

ABSTRACTThis paper describes the combination of the method of fundamental solutions (MFS) and the dual reciprocity method (DRM) as a meshless numerical method to solve problems of thin plates resting on Winkler foundations under arbitrary loadings, where the DRM is based on the augmented polyharmonic splines constructed by splines and monomials. In the solution procedure, the arbitrary distributed loading is first approximated by the augmented polyharmonic splines (APS) and thus the desired particular solution can be represented by the corresponding analytical particular solutions of the APS. Thereafter, the complementary solution is solved formally by the MFS. In the mathematical derivations, the real coefficient operator in the governing equation is decomposed into two complex coefficient operators. In other words, the solutions obtained by the MFS-DRM are first treated in terms of these complex coefficient operators and then converted to real numbers in suitable ways. Furthermore, the boundary conditions of lateral displacement, slope, normal moment, and effective shear force are all given explicitly for the particular solutions of APS as well as the kernels of MFS. Finally, numerical experiments are carried out to validate these analytical formulas.

Sign in / Sign up

Export Citation Format

Share Document