Global Radial Basis Functions Research Articles

Stability estimates for radial basis function methods applied to linear scalar conservation laws

We derive stability estimates for three commonly used radial basis function (RBF) methods to solve hyperbolic time-dependent PDEs: the RBF generated finite difference (RBF-FD) method, the RBF partition of unity method (RBF-PUM) and Kansa's (global) RBF method. We give the estimates in the discrete ℓ2-norm intrinsic to each of the three methods. The results show that Kansa's method and RBF-PUM can be ℓ2-stable in time under a sufficiently large oversampling of the discretized system of equations. The RBF-FD method in addition requires stabilization of the spurious jump terms due to the discontinuous RBF-FD cardinal basis functions. Numerical experiments show an agreement with our theoretical observations.

A machine learning approach for entropy due to natural convection flow of a nanofluid under the uniform inclined magnetic field

In this study, entropy generation of natural convection flow of Cu-water nanofluid under the influence of a uniform inclined magnetic field (MF) is firstly modeled by a machine learning approach using the data obtained from a numerical process. The two dimensional, time dependent dimensionless equations are numerically solved by using global radial basis function (RBF) method in spatial derivatives and the second order backward differentiation formula (BDF2) in time derivatives. In a set of combined dimensionless problem parameters, numerical simulations are performed. In the obtained data, the inputs are a set of combined dimnesionless problem parameters and the outputs are entropy indicators. Then, global and local entropy indicators are modeled by Neural Network (NN). In view of smaller mean squared error metric results, small number of layers or layer sizes in NN seems enough in smaller data while trilayer NN (TNN) is a better modeler in case of larger data.

PARALLEL MESHLESS RADIAL BASIS FUNCTION COLLOCATION METHOD FOR NEUTRON DIFFUSION PROBLEMS

The meshless global radial basis function (RBF) collocation method is widely used to model physical phenomena in science and engineering. The method produces highly accurate solutions with an exponential convergence rate. However, due to the global approximation structure of the method, dense node distributions lead to long computation times and hinder the applicability of the technique. In order to overcome this issue, this study proposes a parallel meshless global RBF collocation algorithm. The algorithm is applied to 2-D neutron diffusion problems. The multiquadric is used as the RBF. The algorithm is developed with Mathematica and eight virtual processors are used in calculations on a multicore computer with four physical cores. The method provides accurate numerical results in a stable manner. Parallel speedup increases with the number of processors up to five and seven processors for external and fission source problems, respectively. The speedup values are limited by the constrained resource sharing of the multicore computer’s memory. On the other hand, significant time savings are achieved with parallel computation. For the four-group fission source problem, when 4316 interpolation nodes are employed, the utilization of seven processors instead of sequential computation decreases the computation time of the meshless approach by 716 s.

Modeling of Average Nusselt Number by Machine Learning and Interpolation Techniques

Abstract In this study, an important heat transfer, fluid flow parameter, and average Nusselt number Nu¯ are statistically modeled by using the data obtained from a numerical process. The two-dimensional, time-dependent dimensionless equations of natural convection (NC) flow either in the absence or in the presence of a uniform inclined magnetic field (MF) are numerically solved by using global radial basis function (RBF) method in spatial derivatives and the second-order backward differentiation formula (BDF2) in time derivatives. Numerical simulations are performed in a set of combined dimensionless problem parameters. A dataset with inputs Rayleigh number Ra, Prandtl number Pr, and output Nu¯ in the absence of MF and a dataset with inputs Ra, Pr, Hartmann number Ha, inclination angle γ, and output Nu¯ in the presence of inclined uniform MF are saved. The obtained data are separated into train and test sets. Then, Nu¯ is first modeled by Neural Networks (NN). Second, interpolation is also examined. In terms of mean squared error (MSE) metric, NN outputs give the best goodness of fit results compared to curve fitting on test data. On the other side, it is shown that interpolation is also an alternative for modeling. This modeling issue enables one to get the desired result without making heavy numerical calculations many times.

Energy-Stable Global Radial Basis Function Methods on Summation-By-Parts Form

Radial basis function methods are powerful tools in numerical analysis and have demonstrated good properties in many different simulations. However, for time-dependent partial differential equations, only a few stability results are known. In particular, if boundary conditions are included, stability issues frequently occur. The question we address in this paper is how provable stability for RBF methods can be obtained. We develop a stability theory for global radial basis function methods using the general framework of summation-by-parts operators often used in the Finite Difference and Finite Element communities. Although we address their practical construction, we restrict the discussion to basic numerical simulations and focus on providing a proof of concept.

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.

The dynamic analysis of axisymmetric bodies with damping effects using the modified radial integration boundary elements method (MRIBEM)

This paper investigates the elastodynamic analysis of axisymmetric bodies under axisymmetric dynamic loads using the modified radial integration boundary elements method (MRIBEM). Purposely, the governing integral equations have been developed in cylindrical coordinates by employing the three-dimensional (3D) static fundamental solution. The axisymmetric bodies can be modeled as a two-dimensional (2D) problem, which is much simpler than 3D analysis and only requires a one-dimensional boundary mesh. In the proposed formulation, the effects of inertial and damping forces appear as 2D domain integrals. The domain variables are approximated using global radial basis functions (RBFs), and the modified radial integration method (MRIM) is employed to deal with the domain integral as the main idea of the present research. Various sample problems are provided in order to illustrate the capability of the presented method for analyzing axisymmetric bodies with concave geometry and damping effect under different types of transient loads. The Newmark and Houbolt methods were applied for time marching, and their efficiency was examined. The results of the present study were also validated by the axisymmetric FEM formulation and other available references.

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.

A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo’s Derivatives

This article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order 0<α<1, and 1<β<2. The numerical method is based on the Laplace transform technique combined with local radial basis functions. The method consists of three main steps: (i) first, the Laplace transform is used to transform the given time fractional model into an equivalent time-independent inhomogeneous problem in the frequency domain; (ii) in the second step, the local radial basis functions method is utilized to obtain an approximate solution for the reduced problem; (iii) finally, the Stehfest method is employed to convert the obtained solution from the frequency domain back to the time domain. The use of the Laplace transform eliminates the need for classical time-stepping techniques, which often require very small time steps to achieve accuracy. Additionally, the application of local radial basis functions helps overcome issues related to ill-conditioning and sensitivity to shape parameters typically encountered in global radial basis function methods. To validate the efficiency and accuracy of the proposed method, several test problems in regular and irregular domains with uniform and non-uniform nodes are considered.

A Surrogate-Assisted Differential Evolution Algorithm for High-Dimensional Expensive Optimization Problems.

The radial basis function (RBF) model and the Kriging model have been widely used in the surrogate-assisted evolutionary algorithms (SAEAs). Based on their characteristics, a global and local surrogate-assisted differential evolution algorithm (GL-SADE) for high-dimensional expensive problems is proposed in this article, in which a global RBF model is trained with all samples to estimate a global trend, and then its optima is used to significantly accelerate the convergence process. A local Kriging model prefers to select points with good predicted fitness and great uncertainty, which can effectively prevent the search from getting trapped into local optima. When the local Kriging model finds the best solution so far, a reward search strategy is executed to further exploit the local Kriging model. The experiments on a set of benchmark functions with dimensions varying from 30 to 200 are conducted to evaluate the performance of the proposed algorithm. The experimental results of the proposed algorithm are compared to four state-of-the-art algorithms to show its effectiveness and efficiency in solving high-dimensional expensive problems. Besides, GL-SADE is applied to an airfoil optimization problem to show its effectiveness.

Towards stability results for global radial basis function based quadrature formulas

Quadrature formulas (QFs) based on radial basis functions (RBFs) have become an essential tool for multivariate numerical integration of scattered data. Although numerous works have been published on RBF-QFs, their stability theory can still be considered as underdeveloped. Here, we strive to pave the way towards a more mature stability theory for global and function-independent RBF-QFs. In particular, we prove stability of these for compactly supported RBFs under certain conditions on the shape parameter and the data points. As an alternative to changing the shape parameter, we demonstrate how the least-squares approach can be used to construct stable RBF-QFs by allowing the number of data points used for numerical integration to be larger than the number of centers used to generate the RBF approximation space. Moreover, it is shown that asymptotic stability of many global RBF-QFs is independent of polynomial terms, which are often included in RBF approximations. While our findings provide some novel conditions for stability of global RBF-QFs, the present work also demonstrates that there are still many gaps to fill in future investigations.

Numerical Inverse Laplace Transform Methods for Advection-Diffusion Problems

Partial differential equations arising in engineering and other sciences describe nature adequately in terms of symmetry properties. This article develops a numerical method based on the Laplace transform and the numerical inverse Laplace transform for numerical modeling of diffusion problems. This method transforms the time-dependent problem to a corresponding time-independent inhomogeneous problem by employing the Laplace transform. Then a local radial basis functions method is employed to solve the transformed problem in the Laplace domain. The main feature of the local radial basis functions method is the collocation on overlapping sub-domains of influence instead of on the whole domain, which reduces the size of the collocation matrix; hence, the problem of ill-conditioning in global radial basis functions is resolved. The Laplace transform is used in comparison with a finite difference technique to deal with the time derivative and avoid the effect of the time step on numerical stability and accuracy. However, using the Laplace transform sometimes leads to a solution in the Laplace domain that cannot be converted back into the real domain by analytic methods. Therefore, in such a case, the Laplace transform is inverted numerically. In this investigation, two inversion techniques are utilized; (i) the contour integration method, and (ii) the Stehfest method. Three test problems are used to evaluate the proposed numerical method. The numerical results demonstrate that the proposed method is computationally efficient and highly accurate.

The solution of elastodynamic problems using modified radial integration boundary element method (MRIBEM)

In this paper, the modified radial integration boundary elements method (MRIBEM) is applied to analyze two-dimensional (2D) elastodynamic problems. The boundary and domain variables are approximated by Lagrangian shape functions and global radial basis functions (RBFs), respectively. As a key idea of this study, the domain integrals that result from implying the inertia force effect are computed by means of a modified radial integration method (MRIM). With the MRIBEM, we are able to deal with problems with concave shapes when the domain variables are approximated by global RBFs by defining an auxiliary point or, in case of complex domain shapes, two or more auxiliary points as the origin for radial integration. Two-time marching schemes, i.e. Newmark method and the Houbolt method, are implemented for the solution of governing differential equations along time. The performance of the proposed method is examined in the solution of four numerical examples.

High-Order Semi-Lagrangian Schemes for the Transport Equation on Icosahedron Spherical Grids

The transport process is an important part of the research of fluid dynamics, especially when it comes to tracer advection in the atmosphere or ocean dynamics. In this paper, a series of high-order semi-Lagrangian methods for the transport process on the sphere are considered. The methods are formulated entirely in three-dimensional Cartesian coordinates, thus avoiding any apparent artificial singularities associated with surface-based coordinate systems. The underlying idea of the semi-Lagrangian method is to find the value of the field/tracer at the departure point through interpolating the values of its surrounding grid points to the departure point. The implementation of the semi-Lagrangian method is divided into the following two main procedures: finding the departure point by integrating the characteristic equation backward and then interpolate on the departure point. In the first procedure, three methods are utilized to solve the characteristic equation for the locations of departure points, including the commonly used midpoint-rule method and two explicit high-order Runge–Kutta (RK) methods. In the second one, for interpolation, four new methods are presented, including (1) linear interpolation; (2) polynomial fitting based on the least square method; (3) global radial basis function stencils (RBFs), and (4) local RBFs. For the latter two interpolation methods, we find that it is crucial to select an optimal value for the shape parameter of the basis function. A Gauss hill advection case is used to compare and contrast the methods in terms of their accuracy, and conservation properties. In addition, the proposed method is applied to standard test cases, which include solid body rotation, shear deformation of twin slotted cylinders, and the evolution of a moving vortex. It demonstrates that the proposed method could simulate all test cases with reasonable accuracy and efficiency.

Determination of an Extremal in Two-Dimensional Variational Problems Based on the RBF Collocation Method

This paper introduces a direct method derived from the global radial basis function (RBF) interpolation over arbitrary collocation nodes occurring in variational problems involving functionals that depend on functions of a number of independent variables. This technique parameterizes solutions with an arbitrary RBF and transforms the two-dimensional variational problem (2DVP) into a constrained optimization problem via arbitrary collocation nodes. The advantage of this method lies in its flexibility in selecting between different RBFs for the interpolation and parameterizing a wide range of arbitrary nodal points. Arbitrary collocation points for the center of the RBFs are applied in order to reduce the constrained variation problem into one of a constrained optimization. The Lagrange multiplier technique is used to transform the optimization problem into an algebraic equation system. Three numerical examples indicate the high efficiency and accuracy of the proposed technique.

Analysis of Two-Level Mesh Method for Partial Integro-Differential Equation

In this paper, we present two-level mesh scheme to solve partial integro-differential equation. The proposed method is based on a finite difference method. For the first step, we use finite difference method in time and global radial basis function (RBF) finite difference (FD) in space. For the second step, we use the finite difference method to solve the proposed problem. This two-level mesh scheme is obtained by combining the radial basis function with finite difference. We prove the stability and convergence of scheme and show that the convergence order is O τ 2 + h 2 , where τ and h are the time step size and space step size, respectively. The results of numerical examples are compared with analytical solutions to show the efficiency of proposed scheme. The numerical results are in good agreement with theoretical ones.

Radial Basis Function Regression Of Lagrangian Three-Dimensional Particle Tracking Data

Flow characterization by means of Particle Tracking Velocimetry (PTV) has gained significant importance in recent years. This is especially true in microfluidics, where the limited optical access only allows for using one camera. A commonly used technique is the Astigmatism Particle Tracking Velocimetry (APTV), which provides reliable 3D3C measurements. However, the resulting data is available on scattered points and usually requires interpolation onto regular grids for further processing. In this work, we test Radial Basis Function (RBF) with the Partition of Unity Method (PUM) for the regression and mesh-free derivative evaluation in large velocity fields. The RBF-PUM approach is first benchmarked on a synthetic test case against the classic Gaussian Window interpolation (AGW) and global RBF. Then, we test the RBF-PUM approach on a three-dimensional experimental dataset consisting of 500.000 data points in a vortex flow. The results prove that the RBF-PUM allows for accurate regression at an accessible computational cost.

Meshless symplectic and multi-symplectic algorithm for Klein–Gordon–Schrödinger system with local RBF collocation

This work proposes a novel meshless symplectic and multi-symplectic approximation for Klein–Gordon–Schrödinger system. The main features of this method are: (I) the application of local radial basis function (RBF) collocation method (LRBFCM) in spatial discretization; (II) the application of symplectic integrator in time discretization. LRBFCM method can deal with the ill-conditioned problem and shape-parameter sensitivity of global RBF method, and is simple and effective. The conservatism of this method is discussed and its accuracy is evaluated. Numerical simulations are given for one dimension (1D)/2D with regular and irregular domains to verify the effectiveness of our new method.

A meshless multi-symplectic local radial basis function collocation scheme for the “good” Boussinesq equation

A novel meshless multi-symplectic scheme is proposed for the “good” Boussinesq equation. The scheme consists of a local radial basis function (RBF) collocation method (LRBFCS) in space and a symplectic integrator in time. The LRBFCS is simple and efficient, since only a sparse banded linear system has to be solved. Moreover, it can avoid the ill-conditioned problem and shape-parameter-sensitivity of the global RBF method. The multi-symplectic LRBFCS (MLRBFCS) is more accurate than traditional methods. Numerical experiments with uniform knots and random knots are designed to verify the effectiveness of the method.

Robust optimal design of low frequency metamaterials considering the fabrication imperfections and parameter fluctuations

PurposeThe purpose of this paper is to develop a robust optimization methodology for metamaterial (MM) unit designs to minimize the effect of manufacturing and operational uncertainties.Design/methodology/approachA new robustness quantification function, applicable to both convex and nonconvex relationships between the mean and the standard deviation, is introduced. A distance-based local radial basis function network surrogate model is proposed to substitute the global radial basis function network to reduce the heavy computational cost without any scarification on the solution accuracy.FindingsThe optimized results of a prototype MM unit demonstrate the feasibility and merit of the proposed methodology. The proposed methodology outperforms the existing ones in both performance and robust parameters in the design of a prototype MM unit.Originality/valueIt provides a robust optimization methodology for MM units when considering the imperfections in fabrications and fluctuations in operation and environment conditions in engineering applications.