Plate Spline Radial Basis Function Research Articles

Assessment of Three-Dimensional Interpolation Method in Hydrologic Analysis in the East China Sea

The water mass in the East China Sea (ECS) shelf has a complicated three-dimensional (3D) hydrologic structure. However, previous studies mostly concentrated on the sea surface based on the sparse in situ and incomplete satellite-derived observations. Therefore, the 3D interpolation technology was introduced for the reconstruction of hydrologic structure in the ECS shelf using in situ temperature and salinity observations in the summer and autumn of 2010 to 2011. Considering the high accuracy and good fitness of the radial basis function (RBF) methods, we applied the RBF methods to the in situ observations to completely reconstruct the 3D hydrologic fields. Other 3D interpolation methods and 2D methods were also tested for a comparison. The cubic and thin plate spline RBFs were recommended because their mean absolute error (MAE) in the 10-fold cross-validation experiments maintained the order of ~10−2. The 3D RBF reconstructions showed a reasonable 3D hydrologic structure and extra details of the water masses in the ECS shelf. It also helps evaluate regional satellite-derived sea surface temperature (SST). Comparisons between the interpolated and satellite-derived SST indicates that the large bias of satellite-derived SST in the daytime corresponds to weak mixing during low-speed wind and shows seasonal variation.

Wavelet collocation method based on Legendre polynomials and its application in solving the stochastic fractional integro-differential equations

This work is an extended version of the ICCS 2020 conference paper [1]. The paper aims to present an efficient numerical method to quantify the uncertainty in the solution of stochastic fractional integro-differential equations. The numerical method presented here is a wavelet collocation method based on Legendre polynomials, and their deterministic and stochastic operational matrix of integration. The operational matrices are used to convert the stochastic fractional integro-differential equation to a linear system of algebraic equations. The accuracy and efficiency of the proposed method are validated through numerical experiments. Moreover, the results are compared with the numerical methods based on the Gaussian radial basis function (GA RBF) and thin plate splines radial basis function (TBS RBF) to show the superiority of the proposed method. Finally, concerning the real-world application, a stock market model has been simulated and the results are demonstrated.

MHD natural convection in a cavity in the presence of cross partial magnetic fields and Al[formula omitted]O[formula omitted]-water nanofluid

In this study, a numerical investigation on nanofluid flow and heat transfer is presented in a cavity under the effects of cross partial magnetic fields and a partial heater. The governing stream function–vorticity equations are solved by thin plate spline radial basis functions (Rbfs) in space derivatives and backward Euler method in time derivatives. The vorticity transport equation involves either only buoyancy force or both buoyancy and cross partial magnetic field force. The results are depicted as streamlines, isotherms and vorticity contours as well as mean Nusselt number. The convective heat transfer increases with the rise in the length of the partial heater, in the concentration of nanoparticles and in Rayleigh number while it decreases with the augmentation of the Lorentz force.

A new variable-order fractional derivative with non-singular Mittag–Leffler kernel: application to variable-order fractional version of the 2D Richard equation

In this study, to overcome the limitations of non-singular fractional derivatives in the Caputo–Fabrizio and Atangana–Baleanu senses (especially in dealing with the variable-order (VO) fractional calculus), we introduce a new non-singular VO fractional derivative with Mittag–Leffler function as its kernel. Some useful results are derived from this fractional derivative. Moreover, this fractional derivative is used for introducing the VO fractional version of the 2D Richard equation. A meshless scheme based on the thin plate spline radial basis functions (RBFs) is developed for solving this equation. The validity of the formulated method is investigated through three numerical examples.

Natural convection in a cavity under partial magnetic field applied from different corners

Abstract In this study, natural convection in a differentially heated square cavity is investigated under the influence of a partial uniform magnetic field applied from either the left bottom corner of the heated wall or the right bottom corner of the cold wall. The governing dimensionless equations are numerically solved by thin plate spline radial basis functions (Rbf) based pseudo spectral method. Relevant parameter variations, Hartmann and Rayleigh numbers, the lengths of the left or right bottom corners of the partial magnetic field and the angle of the applied magnetic field are analyzed in terms of streamlines, isotherms and vorticity contours as well as average Nusselt number through the hot wall. The inhibition of Lorentz force is realized if the corner partial magnetic field of length less than a half is applied from the left bottom corner at Ra = 105 and Ha = 50. In addition, in case of the applied magnetic field from the left bottom corner, the average Nusselt number and the maximum absolute value of the stream function also decrease 8.49% and 5.85%, respectively, more than the applied magnetic field from the right bottom corner at Hartmann number of 100 and Rayleigh number of 106.

Implicit RBF Meshless Method for the Solution of Two-dimensional Variable Order Fractional Cable Equation

In the present work, the numerical solution of two-dimensional variable-order fractional cable (VOFC) equation using meshless collocation methods with thin plate spline radial basis functions is considered. In the proposed methods, we first use two schemes of order O(τ2) for the time derivatives and then meshless approach is applied to the space component. Numerical results obtained from solving considered model on regular and irregular domains, demonstrate the accuracy and efficiency of the proposed schemes.

A hybrid FE-MLPG method to simulate stationary dynamic and propagating quasi-static cracks

This article describes a combined way based on FE and MLPG1 methods to numerically solve stationary dynamic fracture and quasi-static linear elastic crack propagation problems. The main goal of this research is to take the advantages of two numerical techniques in fracture mechanics. So, the Local Petrov–Galerkin meshfree method with thin plate spline radial basis function is employed for the discretization of non-classic linear elastic thermoelasticity governing equations. The method is equipped with triangular background cells organized throughout a solid to assist in the construction of the polygonal integration boundary-domain and defining the total geometry. The grid is also used to propose a neighbor point determination rule at Gauss points without using average nodal spacing computations, geometry parameters and the well-known visibility method around the crack tip. Several examples including stationary and propagating cracks are presented to show the effectiveness of the hybrid numerical method. Computational accuracy is investigated by error estimation analyses and qualitatively comparing the results of this method with different analytical and numerical solutions available in the literature. Wave propagation is also considered by providing the temporal distribution of stress and thermal waves in different cracked structures.

Computational Analysis of Linear Elastic Crack Growth in Functionally Graded Bodies Using Non-Uniform Steps Integrated in the MLPG

This paper describes the estimation of mixed-mode crack growth path in functionally graded materials under mechanical and thermal loads by the meshless method. Simple and complex geometries with single and double cracks are provided to verify the merits of a recently proposed modified meshless local Petrov–Galerkin method and to show the applicability of this method in crack growth simulation. Polygon test function domains are constructed locally at every crack growth step in order to numerically calculate contour and domain integrals. The classical form of thin plate spline radial basis function without enrichment with a regular distribution of points in the vicinity of crack tips are used. In this efficient method, the total number of domain points are kept fixed during crack growth steps forever. For this end, a simple method of new crack tip determination at every growth step is proposed. The MLPG results are compared with reference solutions including experimental, finite element, boundary element and other meshless methods and the accuracy of this method is investigated in different numerical test problems.

Meshless Solution of the Problem on the Static Behavior of Thin and Thick Laminated Composite Beams

For the first time, the static behavior of laminated composite beams is analyzed using the meshless collocation method based on a thin-plate-spline radial basis function. In the approximation of a partial differential equation by using a radial basis function, the shape parameter has an important role in ensuring the numerical accuracy. The choice of a shape parameter in the thin plate spline radial basis function is easier than in other radial basis functions. The governing differential equations are derived based on Reddy’s third-order shear deformation theory. Numerical results are obtained for symmetric cross-ply laminated composite beams with simple-simple and cantilever boundary conditions under a uniform load. The results found are compared with available published ones and demonstrate the accuracy of the present method.

A Meshless Method Using Radial Basis Functions for the Numerical Solution of Two-Dimensional ZK–BBM Equation

The main idea of this article is obtain the numerical solution of the two-dimensional ZK–BBM equation. We construct the solution by using different approach, that is based on using collocation points. The solution is based on using the thin plate splines radial basis function, which builds an approximated solution with discretizing the time and the space to small steps. we use a predictor–corrector scheme to avoid solving the nonlinear system. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme.

A meshless method using radial basis functions for numerical solution of the two-dimensional KdV-Burgers equation

The aim of this article is to obtain the numerical solution of the two-dimensional KdV-Burgers equation. We construct the solution by using a different approach, that is based on using collocation points. The solution is based on using the thin plate splines radial basis function, which builds an approximated solution with discretizing the time and the space to small steps. We use a predictor-corrector scheme to avoid solving the nonlinear system. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme.

A machine-learning approach for structural damage detection using least square support vector machine based on a new combinational kernel function

Health assessment and monitoring of engineered systems have become one of the fastest growing multi-disciplinary research areas over the last two decades. One of the largest concerns in structural health monitoring is how to infer structural conditions from the measurements and the data collected by sensors. The ultimate aim is to detect the structural damages with a high level of certainty and hence to extend the life of structures. In this study, a new strategy for structural damage detection is proposed using least square support vector machines based on a new combinational kernel. Thin plate spline Littlewood–Paley wavelet kernel function introduced in this article is a novel combinational kernel function, which combines thin plate spline radial basis function kernel with local characteristics and a modified Littlewood–Paley wavelet kernel function with global characteristics. During the process of structural damage detection, a social harmony search algorithm optimizes the parameters of least square support vector machine and the thin plate spline Littlewood–Paley wavelet kernel. The results obtained by this method are compared with least square support vector machine based on the other combinational and conventional kernels. These results show that the accuracy of damage detection based on least square support vector machine with thin plate spline Littlewood–Paley wavelet kernel is higher than other methods that utilize conventional kernels under similar conditions. In comparison with other combinational kernels, least square support vector machine with thin plate spline Littlewood–Paley wavelet kernel possesses a better dissemination and learning ability by incorporating the advantages of radial basis function kernel and wavelet kernel functions.

A Numerical Algorithm Based on RBFs for Solving an Inverse Source Problem

In this paper, a meshless numerical scheme for solving an inverse source problem is considered. The proposed scheme is based on approximating the solution employing the thin plate spline (TPS) radial basis function (RBF). Applying this radial basis function results in a badly ill-condition system of equations. The Tikhonov regularization method is employed for solving this system of equations. Determination of regularization parameter is based on generalized cross-validation (GCV) technique. Some numerical examples are presented to demonstrate the accuracy and ability of this method.

Extended Radial Point Interpolation Method for crack analysis in orthotropic media

Orthotropic materials are particular type of anisotropic materials; In contrast with isotropic materials, their properties depend on the direction in which they are measured. Orthotropic composite materials and their structures have been extensively used in a wide range of engineering applications. Studies on their physical behaviors under in-work loading conditions are essential. In this present, we apply an extended meshfree radial point interpolation method (XRPIM) for analyzing crack behaviour in 2D orthotropic materials models. The thin plate spline (TPS) radial basis function (RBF) is used for constructing the RPIM shape functions. Typical advantages of using RBF are the satisfaction of the Kronecker’s delta property and the high-order continuity. To calculate the stress intensity factors (SIFs), Interaction integral method with orthotropic auxiliary fields are used. Numerical examples are performed to show the accuracy of the approach; the results are compared with available refered results. Our numerical experiments have shown a very good performance of the present method.

The Boundary Knot Method for Simulating Unsteady Incompressible Fluid Flow

This article presents a numerical algorithm using the Boundary knot (BK) method for the incompressible Navier-Stokes equations. To deal with time derivatives, the forward time differences are employed yielding the Poisson's equation. The BK method with the general solution of the Helmholtz equation and dual reciprocity (DR) principle is chosen to solve the Poisson's equation. In numerical examples the reverse principle with thin plate spline (TPS) radial basis functions (RBF) is used to construct the particular solution. It is found that BKM with a classical DR principle gives reasonable results with the advantage of BKM spectral convergence; however the accuracy of the solution is affected by the accuracy of particular solution created using DR.

A nth-Order Shear Deformation Theory for the Free Vibration Analysis of the Isotropic Plates on Elastic Foundations

A nth-order shear deformation theory for free vibration of the isotropic plates resting on a two-parameter Pasternak foundations is developed. The present theory does not require shear correction factor, and satisfies the zero transverse shear stress boundary conditions on the top and bottom surface of the plate. Governing equations are derived from the principle of virtual displacements. Meshless global collocation method based on the thin plate spline radial basis function is used to solve the governing differential equations. The accuracy of the present theory is demonstrated by comparing the present results with available published results.

Meshless local collocation method for natural frequencies and mode shapes of laminated composite shells

Meshless local collocation method produces much better conditioned matrices than meshless global collocation methods. In this paper, the meshless local collocation method based on thin plate spline radial basis function and first-order shear deformation theory are used to calculate the natural frequencies and mode shapes of laminated composite shells. Through numerical experiments, the accuracy and efficiency of present method are demonstrated.

Neural Network Based Hybrid Adaptive Controller for an Autonomously Driving Car Using Thin Plate Spline Radial Basis Activation Function

– This paper presents a hybrid lateral and longitudinal controller for a self-driving passenger car. The controller comprises a Proportional Derivative (PD) controller as a closed loop controller and Neural Network (NN) based adaptive compensator as a feed forward controller. The activation function of the NN adaptive stage is based on a poly-harmonic Thin Plate Spline (TPS) Radial Basis Function (RBF), which promises better accuracy, smoother interpolation and closed form solutions. The controller development and testing has been performed using a non-linear vehicle dynamics model, which has been developed using the Matlab / Simulink tool. The Controller performance in terms of vehicle lane following (lateral deviation control) and safe cruising control (longitudinal spacing error control) have been verified through simulations. Reductions of lateral deviation error by 15% and longitudinal spacing error by 7% have been achieved.

Inpainting for satellite imagery using thin plate spline radial basis function neural networks in shearlet domain

Remotely sensed images are occasionally subject to the problem of dead pixels. Removing or filling dead pixels is known as image inpainting. In this paper, thin plate spline radial basis function based inpainting in the Shearlet domain is presented to solve this problem. The proposed algorithm takes the advantage of a Discrete Shearlet Transform (DST) which has a good performance with textured and non textured images. Additionally, the algorithm works well for multichannel remote sensing data. The algorithm is tested with the QuickBird Satellite images for inpainting. The test results and the computed Peak Signal to Noise Ratio (PSNR) values demonstrate the feasibility of the use of Radial Basis Function (RBF) method for inpainting of remotely sensed images.

An improved local radial point interpolation method for transient heat conduction analysis

The smoothing thin plate spline (STPS) interpolation using the penalty function method according to the optimization theory is presented to deal with transient heat conduction problems. The smooth conditions of the shape functions and derivatives can be satisfied so that the distortions hardly occur. Local weak forms are developed using the weighted residual method locally from the partial differential equations of the transient heat conduction. Here the Heaviside step function is used as the test function in each sub-domain to avoid the need for a domain integral. Essential boundary conditions can be implemented like the finite element method (FEM) as the shape functions possess the Kronecker delta property. The traditional two-point difference method is selected for the time discretization scheme. Three selected numerical examples are presented in this paper to demonstrate the availability and accuracy of the present approach comparing with the traditional thin plate spline (TPS) radial basis functions.