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

Simulation of laminar backward facing step flow under magnetic field with explicit local radial basis function collocation method

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.

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.

Using a local radial basis function collocation method to approximate radiation boundary conditions

A meshless collocation method for band structure simulation of nanoscale phononic crystals based on nonlocal elasticity theory

In this work we present an overview of the RBF finite collocation approach. The RBF-FC method is an alternative decomposition for the localisation of the radial basis function (RBF) collocation method for the solution of PDEs. In contrast to the popular finite difference approach, in which the PDE governing operator is reconstructed from simple RBF interpolants, the finite collocation formulation instead enforces all governing and boundary PDEs within the local RBF collocation systems, and assembles the field variable directly into a sparse global assembly. In this way the solution of the PDE is enforced by collocation, rather than by differencing. By including the PDE governing operator within the local collocation systems any data extracted from the RBF interpolation field naturally respects the local physics of the PDE, including information about the local convective field. This gives rise to an “implicit upwinding” effect, which allows stable solutions to be obtained in convective-dominant scenarios when using centred stencils. The approach also allows high rates of spatial convergence to be obtained. We present here an overview of the RBF-FC formulation, demonstrating the performance of the method with benchmark examples for linear elasticity and convection-diffusion PDEs. The adaptation of the method for unsteady problems is described, including a Richardson extrapolation method to improve rates of temporal convergence. The application of the method to the capture and transport

Meshless approach to the large-eddy simulation of the continuous casting process

This article presents a local knot method (LKM) to solve inverse Cauchy problems of Helmholtz equations in arbitrary 2D and 3D domains. The Moore–Penrose pseudoinverse using the truncated singular value decomposition is employed in the local approximation of supporting domain instead of the moving least squares method. The developed approach is a semi‐analytical and local radial basis function collocation method using the non‐singular general solution as the basis function. Like the traditional boundary knot method, the LKM is simple, accurate and easy‐to‐program in solving inverse Cauchy problems associated with Helmholtz equations. Unlike the boundary knot method, the new scheme can directly reconstruct the unknowns both inside the physical domain and along its boundary by solving a sparse linear system, and can achieve a satisfactory solution. Numerical experiments, involving the complicated geometry and the high noise level, confirm the accuracy and reliability of the proposed method for solving inverse Cauchy problems of Helmholtz equations.

In many references, both the ill-posed and inverse boundary value problems are solved iteratively. The iterative procedures are based on firstly converting the problem into a well-posed one by assuming the missing boundary values. Then, the problem is solved by using either a developed numerical algorithm or a conventional optimization scheme. The convergence of the technique is achieved when the approximated solution is well compared to the unused data. In the present paper, we present a different way to solve an ill-posed problem by applying the localized and space-time localized radial basis function collocation method depending on the problem considered and avoiding the iterative procedure. We demonstrate that the solution of certain ill-posed and inverse problems can be accomplished without iterations. Three different problems have been investigated: problems with missing boundary condition and internal data, problems with overspecified boundary condition, and backward heat conduction problem (BHCP). It has been demonstrated that the presented method is efficient and accurate and overcomes the stability analysis that is required in iterative techniques.

In this paper, adaptive meshfree methods using local nodes and radial basis functions (RBFs) which based on strong-form formulation is presented. In this present formulation, radial basis functions are used in the function approximation for the discretization of the governing system equations. Regularization techniques are suggested and examined to stabilize the solutions in order to obtain stable and accurate results. Different schemes for constructing regularization matrix are compared and discussed. As stability is restored, meshfree strong-form method using local nodes and RBFs can facilitate an easier implementation for adaptive analysis to achieve desired accuracy. Residual based error indicator is devised in the adaptive scheme. A simple and practical node refinement procedure is presented for node insertion at each adaptive step.

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

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.

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

Localized radial basis function scheme for multidimensional transient generalized newtonian fluid dynamics and heat transfer