Radial Basis Function Collocation Method Research Articles

Application of the local RBF collocation method to natural convection in a 3D cavity influenced by a magnetic field

This paper explores, for the first time, the application of the novel mesh-free local radial basis function collocation method (LRBFCM) to the solution of a multi-physics problem in three dimensions. A related benchmark problem is solved by considering the natural convection of an incompressible Newtonian fluid in a differentially heated cubic cavity with and without the application of a magnetic field. The research is limited to typical magnetic fields used in the magnetohydrodynamic processing of liquid metals. For this purpose the assumption of small magnetic Reynolds numbers Rem ≪ 1 is made. Spatial discretization is performed by local non-uniform collocation with scaled multiquadrics radial basis functions (RBFs) with the shape parameter set to a constant value and the explicit Euler formula used to perform the time stepping. The involved temperature, velocity and pressure fields are represented on overlapping seven-nodded sub-domains. The pressure-velocity coupling is resolved by the fractional step method. The originality of the contribution represents LRBFCM solution of the classic three-dimensional steady natural convection benchmark for Rayleigh numbers from 105 to 107 and Prandtl number 0.71, and its extension to Prandtl number 0.1, and Hartman numbers 0, 10, 50 and 100. The accuracy of the LRBFCM is found to be comparable with the published benchmark results obtained using established numerical methods.

On Multiscale RBF Collocation Methods for Solving the Monge–Ampère Equation

This paper considers some multiscale radial basis function collocation methods for solving the two-dimensional Monge–Ampère equation with Dirichlet boundary. We discuss and study the performance of the three kinds of multiscale methods. The first method is the cascadic meshfree method, which was proposed by Liu and He (2013). The second method is the stationary multilevel method, which was proposed by Floater and Iske (1996), and is used to solve the fully nonlinear partial differential equation in the paper for the first time. The third is the hierarchical radial basis function method, which is constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. Compared with the first two methods, the hierarchical radial basis function method can not only solve the present problem on a single level with higher accuracy and lower computational cost but also produce highly sparse nonlinear discrete system. These observations are obtained by taking the direct approach of numerical experimentation.

A Novel Meshfree Approach with a Radial Polynomial for Solving Nonhomogeneous Partial Differential Equations

In this article, a novel radial–based meshfree approach for solving nonhomogeneous partial differential equations is proposed. Stemming from the radial basis function collocation method, the novel meshfree approach is formulated by incorporating the radial polynomial as the basis function. The solution of the nonhomogeneous partial differential equation is therefore approximated by the discretization of the governing equation using the radial polynomial basis function. To avoid the singularity, the minimum order of the radial polynomial basis function must be greater than two for the second order partial differential equations. Since the radial polynomial basis function is a non–singular series function, accurate numerical solutions may be obtained by increasing the terms of the radial polynomial. In addition, the shape parameter in the radial basis function collocation method is no longer required in the proposed method. Several numerical implementations, including homogeneous and nonhomogeneous Laplace and modified Helmholtz equations, are conducted. The results illustrate that the proposed approach may obtain highly accurate solutions with the use of higher order radial polynomial terms. Finally, compared with the radial basis function collocation method, the proposed approach may produce more accurate solutions than the other.

A two-stage adaptive scheme based on RBF collocation for solving elliptic PDEs

In this paper we present a new adaptive two-stage algorithm for solving elliptic partial differential equations via a radial basis function collocation method. Our adaptive meshless scheme is based at first on the use of a leave-one-out cross validation technique, and then on a residual subsampling method. Each of phases is characterized by different error indicators and refinement strategies. The combination of these computational approaches allows us to detect the areas that need to be refined, also including the chance to further add or remove adaptively any points. The resulting algorithm turns out to be flexible and effective through a good interaction between error indicators and refinement procedures. Several numerical experiments support our study by illustrating the performance of our two-stage scheme. Finally, the latter is also compared with an efficient adaptive finite element method.

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

In this paper, the band structures of nanoscale phononic crystals based on the nonlocal elasticity theory are calculated by using a meshfree local radial basis function collocation method (LRBFCM). The direct method is applied to enhance the stability of the derivative calculations in the LRBFCM. A simple summation in the LRBFCM is proposed to deal with the integration related to the nonlocal stresses or tractions. The LRBFCM for the band structure calculations is validated by the results obtained with the first-principle and the transfer matrix (TM) method for one-dimensional (1D) phononic crystals, as well as the comparison of the frequency responses of the two-dimensional (2D) periodic structures.

An investigation of global radial basis function collocation methods applied to Helmholtz problems

Global radial basis function (RBF) collocation methods with inifinitely smooth basis functions for partial differential equations (PDEs) work in general geometries, and can have exponential converg ...

A semi-Lagrangian meshless framework for numerical solutions of two-dimensional sloshing phenomenon

This paper presents a semi-Lagrangian meshless framework based on the localized radial basis function collocation method and the second-order explicit Runge–Kutta method (RK2-LRBFCM) for analyzing the sloshing phenomenon in a 2D tank. In the present method, the semi-Lagrangian approach is adopted to constrain the lateral movements of inner collocation nodes. And the LRBFCM is used to obtain the velocity potentials at each time instant governed by Laplace equation, then second-order explicit Runge–Kutta method is introduced to calculate the free-surface elevation and velocity potential at the next time instant. In comparison with the reference results, the efficiency and accuracy of the proposed RK2-LRBFCM are demonstrated in the solutions of several benchmark examples including horizontally excited tanks, vertically excited tanks, horizontally and vertically excited tanks. Moreover, the effect of the protrusion (shape and size) of the bottom wall on the sloshing phenomenon in horizontally excited tank is investigated.

Preconditioning RBF collocation method by adapted finite difference matrix

In this paper, we shall consider an adapted finite difference preconditioning technique for radial basis function (RBF) method. This technique, for the system of equations that arises from solving Helmholtz equation with Dirichlet boundary conditions by RBF collocation method, is considered. We prove that the eigenvalues of preconditioned matrices are bounded for one and two-dimensional cases. We also show that RBF interpolating matrix can be well-conditioned by using appropriate number of polynomial bases in augmented term of RBF approximation. Numerical experiments show the efficiency and robustness of the preconditioning procedure, particularly when there are non-zero boundary conditions.

Solving fractional Laplacian visco-acoustic wave equations on complex-geometry domains using Grünwald-formula based radial basis collocation method

We propose a radial basis function collocation method (RBF method) to solve fractional Laplacian visco-acoustic wave equation for the Earth media having heterogeneous velocity model and complex geometry. Unlike the fractional Laplacian wave equation proposed in Zhu and Harris (2014), the wave equation we consider has a different definition for the fractional Laplacian. Specifically, spectral and Riesz fractional Laplacians are considered in Zhu and Harris (2014) and the present paper, respectively. Accordingly, the Fourier pseudospectral method (FPS method) and the RBF method are employed to solve the spectral and the Riesz fractional Laplacian wave equations. The two wave equations are observed to produce obviously different wavefields. We demonstrate the validity and flexibility of the proposed RBF method by considering five benchmarks of seismic forward modeling: (1) two-dimensional Earth media with four types of velocity models (homogeneous, two-layer, homogeneous but complex-geometry, and heterogeneous models) and (2) a spherical medium with homogeneous velocity model. We make a three-way comparison among numerical solutions to the Riesz fractional Laplacian, the spectral fractional Laplacian, and the integer-order visco-acoustic wave equations, and observe that when wave attenuation is weak the Riesz wave equation yields more similar wavefield to that of the integer-order wave equation than the spectral wave equation does. Furthermore, uniform and quasi-uniform layouts for collocation points of the RBF method are considered, and the latter layout turns out to be economical since it can preserve the solution accuracy with the minimum number of collocation points. The RBF method is truly mesh-free and dimension-free and can easily handle high-dimensional, irregular domains. Additionally, the method is easier to implement than element-based methods, such as finite element and spectral element methods, for discretizing the Riesz fractional Laplacian.

Response of a Porous Seabed around an Immersed Tunnel under Wave Loading: Meshfree Model

Seabed instability surrounding an immersed tunnel is a vital engineering issue regarding the design and maintenance for submarine tunnel projects. In this study, a numerical model based on the local radial basis function collocation method (LRBFCM) is developed to evaluate the seabed behaviour in a marine environment, in which the seabed is treated as the porous medium and governed by Biot’s “ u − p ” approximation. As for the flow field above the seabed, the VARANS equations are used to simulate the fluid motion and properties. The present model is validated with analytical solutions and experimental data which show a good capacity of the integrated model. Both wave and current loading are considered in this study. Parametric studies are carried out to investigate the effects of wave characteristics and soil properties. Based on the numerical results, the maximum liquefaction depth around the immersed tunnel could be deeper under the wave loading with long wave period (T) and large wave height (H). Moreover, a seabed with lower permeability ( K s ) and degree of saturation ( S r ) is more likely to be liquefied.

Fully meshless solution of the one-dimensional multigroup neutron transport equation with the radial basis function collocation method

In this paper a fully meshless method is proposed for the numerical solution of the one-dimensional multigroup neutron transport equation with anisotropic scattering. Both first-order and even-parity forms of the transport equation are studied. The radial basis function collocation method is chosen for the spatial treatment, and Legendre polynomials are used to approximate the angular variable. The selection of the Legendre polynomials instead of discrete ordinates approach resulted with a fully meshless algorithm in both independent variables. Multiquadric is utilized as the radial function. Seven problems are considered to evaluate the performance of the method. The results show that the method converges exponentially, and it is possible to obtain high levels of accuracies for the multiplication factor and neutron flux with a good stability in both spatial and angular domains. For the one-group isotropic benchmark problem, discrete ordinates solutions employing discontinuous linear finite elements for the spatial variable are also provided, and a comparison of the methods revealed that the fully meshless method produced more accurate results than the discrete ordinates-finite element scheme when the shape parameter is properly chosen.

A local radial basis function collocation method for band structure computation of 3D phononic crystals

In this paper, we further extend the local radial basis function collocation method (LRBFCM) for efficient computation of band structures of phononic crystals from 2D to 3D. The proposed LRBFCM uses one fictitious node to tackle instability problems caused by calculation of derivatives of the wave pressure. A few examples of sound pressure wave propagation are modelled to validate the developed method. Comparisons with finite element modeling demonstrate the high stability and efficiency of the new method in computation of band structures of 3D phononic crystals.

Gaussian Radial Basis Function for Unsteady Groundwater Flow

The radial basis function (RBF) collocation methods are meshless with spectrally accurate; its approximation is an extremely powerful tool for solving various types of partial differential equations (PDEs). In this paper, the Gaussian RBF (G-RBF) collocation method is applied to solving unsteady groundwater flow governed by nonlinear elliptic PDEs. During the whole solution process, no iterations are required for a kind of nonlinear PDEs. Numerical results are compared with the well-known Kansa’s method, also namely as Multiquadrics radial basis function (MQ-RBF). The results obtained show good performance of the G-RBF as solution to unsteady groundwater flow problems.

A meshless procedure for analysis of fluid flow and heat transfer in an internally finned square duct

Application of the method of fundamental solutions in combination with the global radial basis function collocation method for analysis of fluid flow and heat transfer in an internally finned square duct is presented in the paper. Fluid flow problem is solved using the modified method of fundamental solutions. After that, the average fluid velocity and product of friction factor and Reynolds number can be determined. Heat transfer problem in the fluid is governed by a nonlinear equation with linear boundary conditions. The Picard iteration method is employed in the paper in order to transform the nonlinear problem into a sequence of inhomogeneous problems. At each iteration step, the general solution is obtained using the modified method of fundamental solutions and the particular solution is obtained using the global radial basis function collocation method. When the iteration process is stopped, the Nusselt number can be determined.

Solving biharmonic equation as an optimal control problem using localized radial basis functions collocation method

Solving fourth or higher order differential equations using localized numerical methods as the finite element, the finite difference and the localized radial basis functions (LRBFs), show difficulties to get accurate results. So many authors adopted iterative methods to deal with biharmonic equation by decoupling it into two Poisson’s problems. In this paper we investigate the formulation of a mixed fourth order boundary value problem as an optimal control one. Then, we establish a new iterative method by coupling an optimization iterative scheme and localized radial basis functions meshless collocation method to deal with the numerical solution of such problem. To transform the problem into an optimal control one, we firstly construct the constraints functions by splitting the biharmonic equation into two coupled Laplace equations. The Neumann boundary condition is used as energy-like error functional to be minimized. Theoretical analysis of the existence and uniqueness of the solution of such formulated optimization problem and its equivalence to the initial biharmonic problem are also demonstrated. Finally we show the effectiveness of the proposed method by solving problems in both convex and non-convex regular and irregular domain.

Hot Rolling Simulation System for Steel Based on Advanced Meshless Solution

In this work, a rolling simulation system for the hot rolling of steel is elaborated. The system is capable of simulating rolling of slabs and blooms, as well as round or square billets, in different symmetric or asymmetric forms in continuous, reversing, or combined rolling. Groove geometries are user-defined and an arbitrary number of rolling stands and distances between them may be used. A slice model assumption is considered, which allows the problem to be efficiently coped with. The related large-deformation thermomechanical problem is solved by the novel meshless Local Radial Basis Function Collocation Method. A compression test is used to compare the simulation results with the Finite Element Method. A user-friendly rolling simulation application has been created for the industrial use based on C# and .NET framework. Results of the simulation, directly taken from the system, are shown for each type of the rolling mill configurations.

Meshless methods for solving Dirichlet boundary optimal control problems governed by elliptic PDEs

In this paper, two meshless schemes are proposed for solving Dirichlet boundary optimal control problems governed by elliptic equations. The first scheme uses radial basis function collocation method (RBF-CM) for both state equation and adjoint state equation, while the second scheme employs the method of fundamental solution (MFS) for the state equation when it has a zero source term, and RBF-CM for the adjoint state equation. Numerical examples are provided to validate the efficiency of the proposed schemes.

Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters

The meshless radial basis function collocation method is an efficient numerical technique for solving partial differential equations. The multiquadric is the most widely utilized radial function for this purpose; but it contains a shape parameter, which has a significant effect on the performance of the method. In this study, the meshless collocation method employing multiquadric as the radial function with optimum shape parameters is applied to the numerical solution of the multigroup neutron diffusion equation. The optimization of the shape parameter is performed by minimizing the Madych-Nelson function. One external and two fission source problems are solved to investigate the performance of the method. The results show that the meshless collocation method with optimized shape parameters yield a high level of accuracy with an exponential convergence rate.

Numerical approximations for the Riesz space fractional advection-dispersion equations via radial basis functions

The radial basis functions collocation method is developed to solve Riesz space fractional advection-dispersion equation (RSFADE). To do this, we first provide the Riemann-Liouville fractional derivatives of the five kinds of RBFs, including the Powers, Gaussian, Multiquadric, Matérn and Thin-plate splines, in one dimension. Then a method of lines, implemented as a meshless method based on spatial trial spaces spanned by the RBFs is developed for the numerical solution of the RSFADE. Numerical experiments are presented to validate the newly developed method and to investigate accuracy and efficiency. The numerical rate of convergence in space is computed. The stability of the linear systems arising from discretizing the Riesz fractional derivative with RBFs is also analysed.

A new LRBFCM-GBEM modeling algorithm for general solution of time fractional-order dual phase lag bioheat transfer problems in functionally graded tissues

The main objective of this article is to propose a new hybrid modeling algorithm based on combining local radial basis function collocation method (LRBFCM) and general boundary element method (GBEM) for solving time fractional-order dual phase lag bioheat transfer problems in functionally graded tissues. The LRBFCM was developed and implemented using an implicit time-stepping technique and Caputo time fractional derivative for solving the fractional-order governing equation without dual phase lags. Due to suitability of the GBEM for modeling of bioheat transfer in functionally graded tissues. Therefore, GBEM is applied for solving the dual phase lags governing equation without fractional-order derivative. The numerical results are depicted graphical forms to show the effects of functionally graded parameter, fractional-order parameter and anisotropy on the nonlinear temperature distribution. Also, these numerical results demonstrate the validity and accuracy of the proposed algorithm and technique.