Numerical Modeling of Photovoltaic Cells with the Meshless Global Radial Basis Function Collocation Method

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

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

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

Phononic band structures and stability analysis using radial basis function method with consideration of different interface models

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

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

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.

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.

Radial basis functions method for solving the fractional diffusion equations

A study is presented using a mesh-free approach and a radial basis function generated finite difference (RBF-FD) method for numerically modelling 3-D gravity data. The gravity responses, that is, vertical gravity and gravity gradients, are obtained by solving the partial differential equation (PDE), that is, the Poisson’s equation for gravitational potential. The mesh-free approach discretizes PDEs using exclusively a cloud of unconnected nodes, instead of traditional tessellated meshes as used by mesh-based numerical methods such as finite difference, finite element and finite volume. Thus, the potentially computationally expensive and unstable creation and manipulation of 3-D meshes can be entirely avoided. A new type of finite-smoothness radial basis functions (RBFs), namely, the quintic-order polyharmonic spline (PHS) RBF, is proposed here in the RBF-FD frame for solving the gravity problem. Previous geophysical data modelling studies using RBF-FD have employed the infinitely smooth RBFs, such as the popular Gaussian (GA) RBFs. Here, both GA and PHS RBFs were tested with different numbers of nodes per mesh-free subdomain and with various shape parameter values (only GA RBFs have a shape parameter). The test results show that the PHS RBF-FD method is more computationally efficient than the GA RBF-FD counterpart. To achieve more efficiency, unstructured node distributions are proposed in discretizing the density models. For both quasi-uniform and unstructured node distributions, numerical results from the proposed PHS RBF-FD demonstrate that the computed vertical gravity and gravitational potential values agree well with analytical solutions with a reasonable number of degrees of freedom. A comparison study of modelling a complex density model with the PHS RBF-FD scheme and nodal finite-element method shows that the RBF-FD scheme generates sparse, asymmetric, linear systems of equations, supports unstructured nodal discretization and local refinement, and can have nonlinear h-convergence under refinement. Finally, the proposed RBF-FD method was applied to obtain the vertical gravity and gravity gradients over a real-world density model, where the benefits of mesh-free discretization are clearly illustrated.

This paper describes the solution of a steady-state natural convection problem in porous media by the radial basis function collocation method (RBFCM). This mesh-free (polygon-free) numerical method is for a coupled set of mass, momentum, and energy equations in two dimensions structured by the Hardy's multiquadrics with different shape parameter and different order of polynomial augmentation. The solution is formulated in primitive variables and involves iterative treatment of coupled pressure, velocity, pressure correction, velocity correction, and energy equations. Numerical examples include convergence studies with different collocation point density and arrangements for a two-dimensional differentially heated rectangular cavity problem at filtration Rayleigh numbers Ra*=25, 50 and 100, and aspect ratios A=1/2, 1, and 2. The solution is assessed by comparison with reference results of the fine-mesh finite volume method in terms of mid-plane velocity components, mid-plane and insulated surface temperatures, streamfunction minimum, and Nusselt number.

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

In this article, we study a two point boundary value problem of non linear differential equation on a semi infinitedomain that describes the unsteady flow of gas through a porous medium. Under special transform, we convert thisproblem to boundary value problem in compactly supported domain [0,1]. An algorithm provided for obtainingsolution by Legendre wavelet collocation method. This method is effectively used to determine y (t) and its initialslope at the origin. The convergence and stability analysis is provided. The results thus obtained are compared withthe those obtained from modified decomposition method [5], Variational iterational method [6], rational Chebyshevfunctions method (RCM) [7] and radial basis function (RBF) collocation method [10]. It has been observed thatthe proposed method provide better results with lesser computational complexity. Keywords: Convergence and stability analysis, Legendre Wavelets, Legendre wavelet collocation method, Kidder's equation.

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 ...