Radial Basis Function Collocation Method Research Articles

Assessment of Local Radial Basis Function Collocation Method for Diffusion Problems Structured with Multiquadrics and Polyharmonic Splines

This paper aims to systematically assess the local radial basis function collocation method, structured with multiquadrics (MQs) and polyharmonic splines (PHSs), for solving steady and transient diffusion problems. The boundary value test involves a rectangle with Dirichlet, Neuman, and Robin boundary conditions, and the initial value test is associated with the Dirichlet jump problem on a square. The spectra of the free parameters of the method, i.e., node density, timestep, shape parameter, etc., are analyzed in terms of the average error. It is found that the use of MQs is less stable compared to PHSs for irregular node arrangements. For MQs, the most suitable shape parameter is determined for multiple cases. The relationship of the shape parameter with the total number of nodes, average error, node scattering factor, and the number of nodes in the local subdomain is also provided. For regular node arrangements, MQs produce slightly more accurate results, while for irregular node arrangements, PHSs provide higher accuracy than MQs. PHSs are recommended for use in diffusion problems that require irregular node spacing.

A novel global RBF direct collocation method for solving partial differential equations with variable coefficients

Conventional radial basis function (RBF) approximation techniques face two fundamental challenges: determining the optimal shape parameter and solving the ill-conditioned linear system, which result from the use of typical kernel functions such as the Multiquadric (MQ) RBF. The objective of this study is to address these challenges by employing a novel RBF kernel, termed the coupled MQ RBF (CMQ-RBF). The new kernel combines the MQ with an m-order conical spline, where m denotes the optimal value determined by the so-called back check method (BCM) introduced in this study. Leveraging the CMQ-RBF kernel, we present a novel global RBF collocation method that generates a well-conditioned linear system, enabling direct solver utilization and delivering highly accurate numerical results without the need for laborious shape parameter selection. Consequently, the previously challenging issues can be effectively circumvented by switching to the "better kernel" offered by the CMQ-RBF. The proposed methodology is characterized by its mathematical simplicity and ease of numerical implementation. Several benchmark examples are investigated to verify its performance.

An adaptive support domain for the in-compressible fluid flow based on the localized radial basis function collocation method

In this paper, an adaptive support domain scheme is proposed to the in-compressible fluid flow based on the localized radial basis function collocation method (LRBFCM). The idea of the adaptive support domain avoids complex mathematical derivations of the finite difference method (FDM). Unlike other upwind schemes in the LRBFCM, the support domain uses the sampling nodes similar to different finite difference schemes, and the improved LRBFCM is further applied to the in-compressible fluid flows. Considering the Kronecker tensor product and Cholesky decomposition techniques, the improved LRBFCM can be easily used to the lid-driven problems with Reynolds numbers up to 100,000 by a simple mathematical derivation. The proposed LRBFCM has an adaptive support domain similar to the finite difference schemes with much less CPU time costs.

Improving Numerical Accuracy of the Localized Oscillatory Radial Basis Functions Collocation Method for Solving Elliptic Partial Differential Equations in 2D

Recently, the localized oscillatory radial basis functions collocation method (L-ORBFs) has been introduced to solve elliptic partial differential equations in 2D with a large number of computational nodes. The research clearly shows that the L-ORBFs is very convenient and useful for solving large-scale problems, but this method is numerically less accurate. In this paper, we propose a numerical scheme to improve the accuracy of the L-ORBFs by adding low-degree polynomials in the localized collocation process. The numerical results validate that the proposed numerical scheme is highly accurate and clearly outperforms the results of the L-ORBFs.

A Novel ANN-Based Radial Basis Function Collocation Method for Solving Elliptic Boundary Value Problems

Elliptic boundary value problems (BVPs) are widely used in various scientific and engineering disciplines that involve finding solutions to elliptic partial differential equations subject to certain boundary conditions. This article introduces a novel approach for solving elliptic BVPs using an artificial neural network (ANN)-based radial basis function (RBF) collocation method. In this study, the backpropagation neural network is employed, enabling learning from training data and enhancing accuracy. The training data consist of given boundary data from exact solutions and the radial distances between exterior fictitious sources and boundary points, which are used to construct RBFs, such as multiquadric and inverse multiquadric RBFs. The distinctive feature of this approach is that it avoids the discretization of the governing equation of elliptic BVPs. Consequently, the proposed ANN-based RBF collocation method offers simplicity in solving elliptic BVPs with only given boundary data and RBFs. To validate the model, it is applied to solve two- and three-dimensional elliptic BVPs. The results of the study highlight the effectiveness and efficiency of the proposed method, demonstrating its capability to deliver accurate solutions with minimal data input for solving elliptic BVPs while relying solely on given boundary data and RBFs.

Analysis of the band structure of transient in-plane elastic waves based on the localized radial basis function collocation method

In this paper, the localized radial basis function collocation method is applied to investigate the in-plane elastic wave propagation of 2D phononic crystals. The direct method with local support domain in one line is used to enhance the stability of the localized radial basis function collocation method when dealing with the derivative calculations in the boundary and interface conditions. The Houbolt method is employed to discretize the time, and then the Fast Fourier Transform is adopted to obtain the band structure according to the wave vectors in the first Brillouin Zone. Different acoustic impedance ratios, filling rates, and lattice forms are studied. The numerical results can be used to validate the band structures obtained directly from the frequency domain.

Modified space‐time radial basis function collocation method for long‐time simulation of transient heat conduction in3Danisotropic composite materials

AbstractIn this paper, the standard and the localized space‐time radial basis function (RBF) collocation methods are modified and combined with the time‐marching scheme and space‐time domain decomposition technique for simulating the long‐time transient heat conduction in 3D anisotropic composite materials. In the proposed approaches, we set the source points outside the whole space‐time domain in the standard space‐time RBF collocation method or outside the established subdomain in the localized approach by introducing the space and time magnification factors, instead of distributing them inside the original domain. In addition, the space‐time regulating factor is defined and added to the conventional schemes to improve the stability of numerical methods. The modified approaches are resulted in a simple and effective time‐marching process which can achieve long‐time simulation, resting on the property that the coefficient matrices generated by these two methods are only related to the space‐time distance between the collocation points and source points. Our ultimate aim is to develop a computing system for resolving dynamic problems in composite materials by designing a space‐time domain decomposition technique. Numerical experiments are conducted to demonstrate the accuracy, efficiency and stability of the presented methodologies.

A stabilized local RBF collocation method for incompressible Navier–Stokes equations

In this work, a stabilized local radial basis function (RBF) collocation method (LRBFCM) is proposed to solve the incompressible Navier–Stokes equations. An improved back ground fictitious grids or meshes technique is proposed to enhance the stability of the LRBFCM. The pressure is computed directly with the coupled velocity field in the Navier–Stokes equations. The explicit time-stepping scheme based on the forward Euler-method is employed for the time discretization, and the free surface of the flow is easy to capture as no numerical integration in the LRBFCM is needed. Numerical results show a high computational speed and a good stability of the proposed LRBFCM.

Meshless structure-preserving GRBF collocation methods for stochastic Maxwell equations with multiplicative noise

In this paper, motivated by the principle that numerical methods should preserve the intrinsic properties of the original system as much as possible, we propose two novel classes of structure-preserving methods for the stochastic Maxwell equations with multiplicative noise. More precisely, due to the advantages of high-order accuracy and the simplicity to deal with high-dimensional problems, the meshless global radial basis function (GRBF) collocation method is firstly utilized to discretize the stochastic Maxwell equations in space, the resulting semi-discretization preserves energy and symplectic structure. Then we apply Padé approximation, the splitting technique and the Runge–Kutta method to propose two kinds of efficient fully-discrete methods that are proved to be symplectic, multi-symplectic and energy-preserving simultaneously. Numerical experiments are given to indicate the validity of the proposed methods.

Local radial basis function collocation method preserving maximum and monotonicity principles for nonlinear differential equations

AbstractIn this paper, a hybrid numerical scheme based on combining exponential time differencing (ETD) and local radial basis function collocation method was constructed. Model problems with different boundary conditions were considered, and the resulting linear system was carefully analyzed. The relation between the number of points employed in the local radial basis function collocation method and the condition number of the coefficient matrix was given. For application, three typical differential equations were investigated, that is, the Allen–Cahn equation for checking the maximum‐preserving property, the combustion equation for checking the monotonicity‐preserving property, and the Gray–Scott system for checking the robustness of the proposed scheme. Numerical examples show the effectiveness of the proposed method.

The local radial basis function collocation method for elastic wave propagation analysis in 2D composite plate

In this paper, the local radial basis function collocation method (LRBFCM) is applied to simulate the in-plane elastic wave propagation in composite plate. In order to reduce the influence caused by large aspect ratio, an improved LRBFCM based on direct method is developed so that the governing equations can be numerically decoupled by using bilinear nodes in different directions. Several numerical examples are given to demonstrate the accuracy and efficiency of the proposed technique in simulating the elastic wave propagation with large aspect ratio.

A Modified RBF Collocation Method for Solving the Convection-Diffusion Problems

The main purposes of this study are to propose the modified radial basis function (RBF) collocation method using a hybrid radial basis function to solve the convection-diffusion problems numerically and to choose the optimal shape parameter of radial basis functions. The modified numerical scheme is tested on a benchmark problem with varying shape parameters. The root mean square error and maximum error are used to validate the accuracy and efficiency of the method. The proposed method can be a good alternative to the radial basis function collocation method to improve accuracy and results.

Localized oscillatory radial basis functions collocation method for solving elliptic partial differential equations in 2D

In recent years, localized methods are proven to be very effective for solving various types of problems in scientific computing. Many researchers have successfully implemented localized approaches to solve large-scale problems. Oscillatory radial basis functions collocation method, a global method, is a meshless numerical method in the literature. The novelty of this article is to address the computational efficiency issues of the oscillatory radial basis functions collocation method using a localized approach for solving elliptic partial differential equations in 2D. We carry out a number of experiments to validate our proposed numerical scheme. Numerical results clearly demonstrate that our scheme is highly accurate and computationally efficient.

Fractional Dual-Phase-Lag Model for Nonlinear Viscoelastic Soft Tissues

The primary goal of this paper is to create a new fractional boundary element method (BEM) model for bio-thermomechanical problems in functionally graded anisotropic (FGA) nonlinear viscoelastic soft tissues. The governing equations of bio-thermomechanical problems are briefly presented, including the fractional dual-phase-lag (DPL) bioheat model and Biot’s model. The more complex shapes of nonlinear viscoelastic soft tissues can be handled by the boundary element method, which also avoids the need for the interior domain to be discretized. The fractional dual-phase-lag bioheat equation was solved using the general boundary element method (GBEM) based on the local radial basis function collocation method (LRBFCM). The poroelastic fields are then calculated using the convolution quadrature boundary element method (CQBEM) The numerical findings show that our proposed numerical model is valid, efficient, and accurate.

Simulation of generalized time fractional Gardner equation utilizing in plasma physics for non-linear propagation of ion-acoustic waves

In this work, radial basis function collocation method (RBFCM) is implemented for generalized time fractional Gardner equation (GTFGE). The RBFCM is meshless and easy-to-implement in complex geometries and higher dimensions, therefore, it is highly demanding. In this work, the Caputo derivative of fractional order ? ? (0, 1] is used to approximate the first order time derivative whereas, Crank-Nicolson scheme is hired to approximate space derivatives. The numerical solutions are presented and discussed, which demonstrate that the method is effective and accurate.

Lagrange radial basis function collocation method for boundary value problems in $ 1 $D

<abstract><p>This paper introduces the Lagrange collocation method with radial basis functions (LRBF) as a novel approach to solving 1D partial differential equations. Our method addresses the trade-off principle, which is a key challenge in standard RBF collocation methods, by maintaining the accuracy and convergence of the numerical solution, while improving the stability and efficiency. We prove the existence and uniqueness of the numerical solution for specific differential operators, such as the Laplacian operator, and for positive definite RBFs. Additionally, we introduce a perturbation into the main matrix, thereby developing the perturbed LRBF method (PLRBF); this allows for the application of Cholesky decomposition, which significantly reduces the condition number of the matrix to its square root, resulting in the CPLRBF method. In return, this enables us to choose a large value for the shape parameter without compromising stability and accuracy, provided that the perturbation is carefully selected. By doing so, highly accurate solutions can be achieved at an early level, significantly reducing central processing unit (CPU) time. Furthermore, to overcome stagnation issues in the RBF collocation method, we combine LRBF and CPLRBF with multilevel techniques and obtain the Multilevel PLRBF (MuCPLRBF) technique. We illustrate the stability, accuracy, convergence, and efficiency of the presented methods in numerical experiments with a 1D Poisson equation. Although our approach is presented for 1D, we expect to be able to extend it to higher dimensions in future work.</p></abstract>

A Novel Localized Meshless Method for Solving Transient Heat Conduction Problems in Complicated Domains

This paper first attempts to solve the transient heat conduction problem by combining the recently proposed local knot method (LKM) with the dual reciprocity method (DRM). Firstly, the temporal derivative is discretized by a finite difference scheme, and thus the governing equation of transient heat transfer is transformed into a non-homogeneous modified Helmholtz equation. Secondly, the solution of the non-homogeneous modified Helmholtz equation is decomposed into a particular solution and a homogeneous solution. And then, the DRM and LKM are used to solve the particular solution of the non-homogeneous equation and the homogeneous solution of the modified Helmholtz equation, respectively. The LKM is a recently proposed local radial basis function collocation method with the merits of being simple, accurate, and free of mesh and integration. Compared with the traditional domain-type and boundary-type schemes, the present coupling algorithm could be treated as a really good alternative for the analysis of transient heat conduction on high-dimensional and complicated domains. Numerical experiments, including two- and three-dimensional heat transfer models, demonstrated the effectiveness and accuracy of the new methodology.

Adaptive selection strategy of shape parameters for LRBF for solving partial differential equations

Radial basis function (RBF) is a basis function suitable for scattered data interpolation and high dimensional function interpolation, wherein the independent shape parameters have a direct impact on the accuracy of calculation results. Research in this field is mostly concerned with the shape parameter selection strategy based on the premise of global distribution or block regional distribution. In this paper, a shape parameter selection strategy is proposed, which is used for the local RBF collocation method (LRBF) for solving partial differential equations. It overcomes many limitations of the traditional methods applied to LRBF. In this strategy, a set of twin matrices similar to the interpolation matrix are constructed to evaluate the error of the model. In addition, the penalty term contained in the twin matrix is used to relax the influence of the far end region on the target node. Since the objective problem is nonlinear, a particle swarm optimization algorithm (PSO) is employed to minimize the training objective and adjust the shape parameters of the basis function at each iteration. Extensive numerical results showed the effectiveness of the error estimation strategy, by providing a good shape parameter and better solution accuracy. At the end of the paper, the generality of the shape parameter optimization framework based on this strategy is discussed through three examples.

A test of backward stochastic differential equations solver for solving semilinear parabolic differential equations in 1D and 2D

Backward stochastic differential equation solver was first introduced by Han et al in 2017. A semilinear parabolic partial differential equation is converted into a stochastic differential equation, and then solved by the backward stochastic differential equation (BSDE) solver. The BSDE solver uses the backward Euler scheme for time discretization and stochastic process and neural network to learn the derivative functions at each time step. The algorithm is ideal for solving high-dimensional PDEs, especially in dimensions higher than 3-dimension problems, whereas the traditional numerical techniques fail to produce any simulations. We modified the BSDE solver so that is works for one-dimensional problems as well. The focus of this paper is to understand how the BSDE solver works in comparison with the traditional numerical techniques in low dimensional spaces (1D and 2D). We examined the BSDE solver in terms of accuracy, efficiency and convergence. Through five classical differential equations, we discovered that the solver works for low dimensional spaces, as accurate as it can be in high dimensional spaces. It is more accurate than the radial basis function collocation method reported in literature and the results by the Picard method. However, the BSDE solver is time consuming. This however, can be solved by parallel computing if needed.

A Novel RBF Collocation Method Using Fictitious Centre Nodes for Elasticity Problems

The traditional radial basis function collocation method (RBFCM) has poor stability when solving two-dimensional elastic problems, and the numerical results are very sensitive to shape parameters, especially in solving elastic problems. In this paper, a novel radial basis function collocation method (RBFCM) using fictitious centre nodes is applied to the elastic problem. The proposed RBFCM employs fictitious centre nodes to interpolate the unknown coefficients, and is much less sensitive to the shape parameter compared with the traditional RBFCM. The details of the shape parameters are discussed for the novel RBFCM in elastic problems. Elastic problems with and without analytical solutions are given to show the effectiveness of the improved RBFCM.