Radial Basis Function Collocation Method Research Articles

Local radial basis function collocation method for stokes equations with interface conditions

In the present work, local radial basis function (RBF) collocation method is used to solve the Stokes problem numerically. Benefits of the local meshless method is its flexibility of handling complex computational domain and interface. Local meshless method produces sparse coefficient matrix, though its structure is depending on the placement of nodes, and type of boundary conditions. The Stokes equations are collocated in a rectangular region with different interface conditions and direct solvers are used instead of iterative algorithm. Due to the unknown boundary conditions for the pressure term in the Stokes equations, value of the pressure is obtained by integrating the Stokes equations and eliminating the constant of integration. In the numerical implementation, circular, elliptic and Deltoid like interfaces are taken into consideration to test performance of the proposed approach for complex problems. Meshless solution of the Stokes equations for the first two problems are compared with the published results in the literature. The last problem is different form the first two problems in the sense that the continuity equation is also discontinuous and the velocity has nonzero value in both the inner and outer sub-domains.

Solving the Nonlinear Heat Equilibrium Problems Using the Local Multiquadric Radial Basis Function Collocation Method

In this article, the nonlinear heat equilibrium problems are solved by the local multiquadric (MQ) radial basis function (RBF) collocation method. The system of nonlinear algebraic equations is solved by iteration based on the residual norm-based algorithm, in which the direction of evolution is determined by a linear equation. In addition, the role of the collocation point and source point is clearly defined such that in our proposed method the field value of any interested point can be expressed. Six numerical examples are shown to check the performance of the proposed method. As the number of supporting points (mp) increases, the accuracy of numerical solution increases. Among all examples, mp = 50 can perform well. In addition, the selection of shape parameter, c, affects the accuracy. However, as c < 2 the maximum relative absolute error percentage is less than 1%.

Symmetric method of approximate particular solutions for solving certain partial differential equations

In this study, a symmetric method of approximate particular solutions (MAPS) is proposed for solving certain partial differential equations (PDEs). Inspired by the unsymmetric MAPS and symmetric radial basis function collocation method (RBFCM), the symmetric MAPS is developed by using the bi-particular solutions of the multiquadrics (MQ). Similar to the unsymmetric MAPS, the right-hand-side of the governing equation is mainly approximated by the MQ in the proposed method. In addition, the system matrix of the prescribed method is symmetric. Numerical examples are solved by the unsymmetric & symmetric RBFCM and MAPS for different problems with different types of governing equations and boundary conditions. Numerical results with different shape parameters are analyzed to show that the symmetric methods are more stable. In addition, the accuracy improvement of the symmetric MAPS is studied. Finally, the stability performance of the symmetric MAPS is further studied for convection-diffusion problems at high Péclet numbers.

Noniterative Localized and Space-Time Localized RBF Meshless Method to Solve the Ill-Posed and Inverse Problem

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.

Divergence-free radial kernel for surface Stokes equations based on the surface Helmholtz decomposition

In this paper, we develop and analyze a new divergence-free kernel approximation method for the time-dependent incompressible Stokes equations on surfaces. The novelty of our proposed method comes from the surface Helmholtz decomposition, which can convert the surface Stokes equations into a coupled equations in which the velocity is deadly divergence-free so that no inf–sup conditions have to be satisfied. Spatial discretization of velocity is implemented by the radial kernel divergence-free approximation spaces, which only need scatter nodes on surfaces. Using the radial basis function collocation method in space, we derive the rigorous stability and convergence result. Numerical examples are presented, demonstrating the efficiency of some model problems on more general surfaces.

Solving Backward Heat Conduction Problems Using a Novel Space–Time Radial Polynomial Basis Function Collocation Method

In this article, a novel meshless method using space–time radial polynomial basis function (SRPBF) for solving backward heat conduction problems is proposed. The SRPBF is constructed by incorporating time-dependent exponential function into the radial polynomial basis function. Different from the conventional radial basis function (RBF) collocation method that applies the RBF at each center point coinciding with the inner point, an innovative source collocation scheme using the sources instead of the centers is first developed for the proposed method. The randomly unstructured source, boundary, and inner points are collocated in the space–time domain, where both boundary as well as initial data may be regarded as space–time boundary conditions. The backward heat conduction problem is converted into an inverse boundary value problem such that the conventional time–marching scheme is not required. Because the SRPBF is infinitely differentiable and the corresponding derivative is a nonsingular and smooth function, solutions can be approximated by applying the SRPBF without the shape parameter. Numerical examples including the direct and backward heat conduction problems are conducted. Results show that more accurate numerical solutions than those of the conventional methods are obtained. Additionally, it is found that the error does not propagate with time such that absent temperature on the inaccessible boundaries can be recovered with high accuracy.

A Radial Basis Function Collocation Method for Space-dependent Inverse Heat Problems

In this study, a radial basis function collocation method (RBFCM) is proposed for the numerical treatment of inverse space-wise dependent heat source problems. Multiquadric radial basis function is applied for spatial discretization whereas for temporal discretization Runge-Kutta method of order four is employed. Numerical experiments for one, two and three-dimensional cases are included to test the efficiency and accuracy of the suggested method. Both non-rectangular and rectangular geometries with uniform and non-uniform points are taken into consideration and the obtained results are compared with the exact as well as with the techniques presented in recent literature.

A local meshless method for the numerical solution of space‐dependent inverse heat problems

In this paper, a local radial basis function collocation method is proposed for the numerical solution of inverse space‐wise dependent heat source problems. Multiquadric radial basis function is used for spatial discretization. The method accuracy is tested in terms of absolute root mean square and relative root mean square error norms. Numerical tests on a noisy data are performed on both regular domain and irregular domain. To test the efficiency and accuracy of the proposed method, numerical experiments for one‐, two‐, and three‐dimensional cases are performed. Both regular and irregular geometries with uniform and nonuniform points are taken into consideration, and the numerical results are also compared with the existing methods reported in literature.

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.