A local RBF method based on a finite collocation approach

Six strategies for defeating the Runge Phenomenon in Gaussian radial basis functions on a finite interval

Accurate prediction of photovoltaic performance hinges on resolving the electron density in the P-region and the hole density in the N-region. Motivated by this need, we present a comprehensive assessment of a meshless global radial basis function (RBF) collocation strategy for the steady current continuity equation, covering a one-dimensional two-region P–N junction and a two-dimensional single-region problem. The study employs Gaussian (GA) and generalized multiquadric (GMQ) bases, systematically varying shape parameter and node density, and presents a detailed performance analysis of the meshless method. Results map the accuracy–stability–computation-time landscape: GA achieves faster convergence but over a narrower stability window, whereas GMQ exhibits greater robustness to shape-parameter variation. We identify stability plateaus that preserve accuracy without severe ill-conditioning and quantify the runtime growth inherent to dense global collocation. A utopia-point multi-objective optimization balances error and computation time to yield practical node-count guidance; for the two-dimensional case with equal weighting, an optimum of 19 intervals per side emerges, largely insensitive to the RBF choice. Collectively, the results establish global RBF collocation as a meshless, accurate, and systematically optimizable alternative to conventional mesh-based solvers for high-fidelity carrier-density prediction in P-N junctions, thereby enabling more reliable performance analysis and design of photovoltaic devices.

The local radial basis function (RBF) method is a promising solver for variable‐order time fractional diffusion equation (TFDE), as it overcomes the computational burden of the traditional global method. Application of the local RBF method is limited to Fickian diffusion, while real‐world diffusion is usually non‐Fickian in multiple dimensions. This article is the first to extend the application of the local RBF method to two‐dimensional, variable‐order, time fractional diffusion equation in complex shaped domains. One of the main advantages of the local RBF method is that only the nodes located in the subdomain, surrounding the local point, need to be considered when calculating the numerical solution at this point. This approach can perform well with large scale problems and can also mitigate otherwise ill‐conditioned problems. The proposed numerical approach is checked against two examples with curved boundaries and known analytical solutions. Shape parameter and subdomain node number are investigated for their influence on the accuracy of the local RBF solution. Furthermore, quantitative analysis, based on root‐mean‐square error, maximum absolute error, and maximum error of the partial derivative indicates that the local RBF method is accurate and effective in approximating the variable‐order TFDE in two‐dimensional irregular domains.

Simulation of electromagnetic wave propagations in negative index materials by the localized RBF-collocation method

The data interpolation is an essential part of Bidimensional Empirical Mode Decomposition (BEMD) of an image. In the decomposition process, local maxima and minima of the image are extracted at each iteration and then interpolated to form the upper and lower envelopes, respectively. Because of the properties of radial basis function (RBF) interpolators, they are good candidates for use in BEMD. However, only one or two of the RBF interpolators have been utilized for BEMD so far. This paper employs many RBF interpolators for BEMD, compares their performances, and finds out the useful ones for BEMD especially in the image filtering application. We propose to apply the BEMD approach with the adequate interpolation function in the image denoising domain. After that, we combine the BEMD with the DWT to improve the BEMD denoising method. The analysis is done using real images. Simulations are made to focus mainly on the effect of interpolation methods by providing less or negligible control on the other parameters of the BEMD process. The study is believed to work as a guideline in the area of BEMD based real image in the filtering application.

In this article, a two-level meshless local Petrov Galerkin method (MLPG) is proposed to analyze nonlinear convection–diffusion equation based on the radial basis function (RBF) collocation method. Two-level method is employed to save the computation time, solving one small linearized convection–diffusion equation in a coarse nodal distribution by Newton iterative scheme and then processing one linear equation on a fine nodal distribution. The convergence analysis for the MLPG method and the two-level method is proven by applying the RBF interpolation method. Finally, the numerical examples provide a sufficient support and show that the proposed method is efficient for solving nonlinear convection–diffusion equation.

Simulation of linear and nonlinear advection-diffusion problems by the direct radial basis function collocation method

Preserving details while restoring images highly corrupted by impulsive salt and pepper noise remains a challenging problem. The authors proposed an algorithm based on radial basis functions (RBFs) interpolation which estimates the intensities of corrupted pixels by their neighbours. In this algorithm, intensity values of noisy pixels in the corrupted image are first estimated using RBFs. Next, the image is smoothed. The proposed algorithm can effectively remove the highly dense, impulsive salt and pepper noise. Experimental results show the superiority of the proposed algorithm both in noise suppression and details preservation in comparison to the recent similar methods. Extensive simulations show better results measured by peak signal-to-noise ratio and structural similarity index, especially when the image is corrupted by very highly dense impulse noise.

The design optimization of acoustic metamaterials represents a critical challenge in modern noise control applications. Global sensitivity analysis (GSA) has emerged as a powerful tool for quantifying the correlation between metamaterials' design variables and corresponding sound quality metrics (loudness, sharpness, and speech intelligibility index). However, GSA's high computational intensity limits the practical application of design optimization, especially for sound quality metrics which vary with sound stimulus. This study introduces a computational framework using radial basis function (RBF) interpolation to accelerate the GSA of acoustic metamaterials, with a periodic arrangement of micro-perforated panels (MPPs) serving as an exemplary case study. The methodology employs RBF interpolation to create a surrogate model of metamaterial acoustic response, demonstrating the potential for computational cost reduction while maintaining accuracy. It is validated across three acoustic scenarios that represent common challenges in noise control: stationary signals (broad/narrowband noise), environmental sounds (mechanical noise, traffic), and dynamic signals (speech, music). The study systematically compares sensitivity indices derived from direct numerical simulation and interpolated predictions, assessing the RBF's impact on GSA accuracy. [The European Commission is gratefully acknowledged for supporting the Marie Sklodowska Curie program through the Horizon Europe DN METAVISION project (GA 101072415)]

Radial basis function based mesh deformation applied to simulation of flow around flapping wings

A 2‐D divergence‐free interpolator is presented which is demonstrated to give more realistic values between sparse velocity data than interpolating velocity components independently. The interpolator enforces physical dependence between the velocity components by ensuring mass is conserved. Tests using data for a synthetic eddy show that independent interpolation of components can give unrealistic velocities between widely spaced data. However, divergence‐free interpolation can reproduce the eddy almost as well with sparse data as with dense data. In the synthetic tests the divergence‐free radial basis function (RBF) interpolator performed as well as, or better than, optimal interpolation. A significant advantage of the “greedy fit” RBF interpolator implemented here is the small number of coefficients required to describe the interpolating surface. For large data sets this gives the interpolator a significant computational advantage over techniques, like Optimal Interpolation, which require one coefficient per data point. The RBF's economy of parameters and imposing physics with the divergence‐free constraint makes RBF interpolation of sparse data much more resistant to following the noise. The 2‐D divergence‐free interpolator can be used to interpolate geostrophic velocities or the horizontal transport of high‐frequency flows such as tidal currents. Tests with data from the local dynamics experiment moorings and from moving vessel acoustic Doppler current profiler measurements of tidal flow show the divergence‐free interpolator is better able to reproduce data left out of the interpolation than interpolating velocity components independently.

This paper explores the application of the mesh-free Local Radial Basis Function Collocation Method (LRBFCM) [1] in solution of coupled heat transfer and fluid flow problems associated with solidification of a binary alloy. The involved temperature, velocity, species and pressure fields are represented on overlapping sub-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBF’s. The energy and momentum equations are solved through explicit time stepping. The pressure-velocity coupling is calculated iteratively, with pressure correction, predicted from the local continuity equation violation [2,3]. The solution procedure is demonstrated in simulation of solidification of 8 wt % Pb-Sn alloy and 5 wt % Sn-Pb alloy in rectangular cavity at conditions, defined in [4]. The numerical examples include studies with different uniform discretization and its influence on final macrosegregation map. The solution is assessed by comparison with the reference results of the finite volume and finite element method. The advantages of the method are simplicity, accuracy and straightforward applicability in non-uniform node arrangements. The methodology will be use in complementing our heat transfer model [5] of the continuous casting of steel.

Kansa radial basis function method with fictitious centres for solving nonlinear boundary value problems

Eigenvalue stability of radial basis function discretizations for time-dependent problems

This paper proposes a partition of unity method (PUM) based on KDFCM (KDFcmPUM) that can be implemented to solve the dense matrix problem that occurs when the radial basis function (RBF) interpolation method deals with a large amount of scattered data. This method introduces a kernel fuzzy clustering algorithm to improve clustering accuracy and achieve the partition of unity. The local compact support RBF is used to construct the weight function, and local expression is obtained from the interpolation of the global RBF. Finally, the global expression is constructed by the weight function and local expression. In this paper, the method is applied to the orthogonally splitting imaging pose sensor to establish the mathematical model and the calibration and test experiments are carried out. The calibration and test accuracy both reached ±0.1 mm, and the number of operations was reduced by 4% at least. The experimental results show that KDFcmPUM is effective.