Thin-plate Spline Radial Basis Function Research Articles

Experimental and numerical analysis of hyperelastic plates using Mooney-Rivlin strain energy function and meshless collocation method

In this research, the flexural behavior of a hyperelastic plate made of silicone under uniformly distributed loading was investigated. The displacements and strains were formulated using the first-order shear deformation plate theory (FSDPT). The right Cauchy-Green tensor and Mooney-Rivlin strain energy function were employed to formulate hyperelastic silicone plate governing equations. The strong-form of the governing equations of the silicone plate using Mooney-Rivlin strain energy function and Euler-Lagrange relations were derived for the first time. In the governing equations, the effects of geometrical and material nonlinearity effects were considered simultaneously. The meshless collocation method (MCM) and thin-plate spline radial basis function were utilized to discretize the governing equations. The arc-length continuation algorithm was also used for analyzing the system of non-linear equations. MCM results were compared to those of the experimental test to validate the results of the meshless method. For this purpose, a square silicone plate with simply supported boundary conditions under uniformly distributed loading was investigated via the digital image correlation (DIC) technique. According to the obtained results, the MCM based on radial basis function is an adequate method for non-linear analysis of hyperelastic plates with Mooney-Rivlin strain energy function under uniformly distributed loading.

Meshless method and experimental analysis of hyperelastic plates using Mooney-Rivlin strain energy function subjected to concentrated loading

The purpose of this research is to investigate a strong-form meshless method for experimentally and numerically analyzing hyperelastic plates subjected to concentrated loading. The silicone hyperelastic plate was formulated utilizing Mooney-Rivlin strain energy function and Lagrangian strains. The displacements and Lagrangian strains of the plate are also calculated using the first-order shear deformation plate theory (FSDPT). The hyperelastic plate’s governing equations are deduced in strong form and analyzed using the logarithmic thin-plate spline radial basis function (LTPS-RBF) and the meshless method for the first-time. The nonlinear system of governing equations is solved using the arc-length continuation algorithm. In the meshless method, the concentrated loading is modeled using the Kronecker delta function in the expression of external work. A silicone plate is employed experimentally to evaluate the accuracy of the meshless method’s results. Furthermore, the finite element method is utilized to compare the contours obtained from the meshless method. Concentrated loading is considered at various points to validate the meshless methods and results from the experimental analysis. The most significant difference between the meshless method and the experimental results is less than 15%. The results indicate that the meshless method is highly accurate when analyzing hyperelastic plates subjected to concentrated loading.

Assessment of torsional behavior of bone section using meshfree method based on TPS radial basis function

The torsional study of bone cross-sections consisting of porous and functionally graded material (FGM) is presented in this paper. The modeling of the bone can be thought of as an FGM. For the modeling of FGM material, a classical power law is used. The strength of the bone is reduced when the microstructure of the bone changes with age. Saint-Venant torsion theory is used to formulate the governing equation. Thin plate spline (TPS) radial basis function-based collocation meshfree method is used for discretization and solution of the governing differential equations. The discretized partial differential equations are solved using MATLAB programming. To assess the effectiveness and accuracy of the current TPS-based meshfree approach, numerical examples for circular and hollow circular sections were used to conduct a convergence and validation test. Finally, the torsional rigidity and shear stress of the FGM and porous bone sections are estimated using an actual cross-section of the bone. After conducting a convergence study, a method is developed to produce fresh findings for the actual bone shape section.

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.

On the convergence of cardinal interpolation by parameterized radial basis functions

We consider cardinal interpolation on gridded data by using various radial basis functions associated with one or two parameters, one of which leads asymptotically to so-called flat-limits. Previously it had been shown that the classical Paley–Wiener functions can be recovered by such cardinal interpolations as the parameter tends to infinity. In this article, we extend the results by relaxing the requirements on the approximand functions from several points of view. The radial basis functions that we are concerned with and which are of special interest contain the celebrated multiquadrics, inverse multiquadrics and shifted thin-plate spline radial basis functions for instance. We also generalise the classes of admitted approximands as well as the radial basis functions to generalised multiquadrics in place of the well-known ordinary or for example inverse multiquadrics. An interesting analytical aspect of this work is that – unlike the classical Whittaker–Shannon theorem – functions (approximands) may be reproduced for the parameter c→∞ in the generalised multiquadrics cardinal approximands, where the usual Shannon series does not converge with theses approximands due to the slow decay of the sinc-function which does not allow e.g. polynomials as approximands. In contrast to the latter, the generalised multiquadrics cardinal functions employed here decay sufficiently fast for each fixed parameter c that even polynomials may be admitted as approximands and are reproduced when then the parameter tends to infinity.

Comparison of nonparametric methods for static visual field interpolation

Visual field testing with standard automated perimetry produces a sparse representation of a sensitivity map, sometimes called the hill of vision (HOV), for the retina. Interpolation or resampling of these data is important for visual display, clinical interpretation, and quantitative analysis. Our objective was to compare several popular interpolation methods in terms of their utility to visual field testing. We evaluated nine nonparametric scattered data interpolation algorithms and compared their performances in normal subjects and patients with retinal degeneration. Interpolator performance was assessed by leave-one-out cross-validation accuracy and high-density interpolated HOV surface smoothness. Radial basis function (RBF) interpolation with a linear kernel yielded the best accuracy, with an overall mean absolute error (MAE) of 2.01 dB and root-mean-square error (RMSE) of 3.20 dB that were significantly better than all other methods (p ≤ 0.003). Thin-plate spline RBF interpolation yielded the best smoothness results (p < 0.001) and scored well for accuracy with overall MAE and RMSE values of 2.08 and 3.28 dB, respectively. Natural neighbor interpolation, which may be a more readily accessible method to some practitioners, also performed well. While no interpolator will be universally optimal, these interpolators are good choices among nonparametric methods.

Estimation of space deformation model for non-stationary random functions

Stationary Random Functions have been successfully applied in geostatistical applications for decades. In some instances, the assumption of a homogeneous spatial dependence structure across the entire domain of interest is unrealistic. A practical approach for modeling and estimating non-stationary spatial dependence structure is considered. This consists in transforming a non-stationary Random Function into a stationary and isotropic one via a bijective continuous deformation of the index space. So far, this approach has been successfully applied in the context of data from several independent realizations of a Random Function. In this work, we propose an approach for non-stationary geostatistical modeling using space deformation in the context of a single realization with possibly irregularly spaced data. The estimation method is based on a non-stationary variogram kernel estimator which serves as a dissimilarity measure between two locations in the geographical space. The proposed procedure combines aspects of kernel smoothing, weighted non-metric multi-dimensional scaling and thin-plate spline radial basis functions. On a simulated data, the method is able to retrieve the true deformation. Performances are assessed on both synthetic and real datasets. It is shown in particular that our approach outperforms the stationary approach. Beyond the prediction, the proposed method can also serve as a tool for exploratory analysis of the non-stationarity.

Free Vibration Analysis of Carbon Nanotubes by Timoshenko Beam Model and Thin-Plate Spline Radial Basis Function

Timoshenko beam model and the meshless method based on thin-plate spline radial basis function are used to analyze the free vibration of carbon nanotubes. The natural frequencies of the carbon nanotubes with different length-to-diameter ratios and boudary conditions are compared with the results of published literatures which demonstrate the high accuracy of present method.

Three-dimensional vibration analysis of isotropic plates by multiquadric and thin-plate spline radial basis functions

In this paper, the meshless collocation method based on global multiquadric and thin-plate spline radial basis functions is presented to discretize the governing equations and boundary conditions of the three-dimensional isotropic plate. Several numerical examples are used to show convergence of the present method. It is found that the natural frequencies computed by multiquadric radial basis function with shape parameter c=0.3/Nx (Nx is the number of node at the x side) converge most rapidly, the natural frequencies computed by thin-plate spline radial basis function with shape parameter m=2 converge most rapidly. The results of the present paper are in good agreement with the available published results.

A 2D contaminant transport model for unsaturated soils

This paper presents a numerical model of two-dimensional (2D) contaminant transport through unsaturated porous media using a mesh-free method, called the radial point interpolation method (RPIM), with polynomial reproduction. A 2D form of the advection– dispersion equation, involving a reactive contaminant with linear first-order degradation, is considered in the analysis. In the RPIM, the Galerkin weak form is used to establish the system equation using 2D mesh-free shape functions constructed using thin plate spline radial-basis functions. A Matlab code is developed to obtain the numerical solution. Experimental results are used to validate the proposed RPIM. The applicability and performance of the RPIM are studied using three numerical examples. The results of the RPIM are compared with those obtained from the finite-element method. The RPIM has generated results with no oscillations, and which are insensitive to Péclet constraints.

Error Estimates for Thin Plate Spline Approximation in the Disk

This paper is concerned with approximation properties of linear combinations of scattered translates of the thin-plate spline radial basis function |·|2log|·| where the translates are taken in the unit disk D in R2. We show that the Lp approximation order for this kind of approximation is 2 + 1/p (for sufficiently smooth functions), which matches Johnson's upper bound and, thus, gives the saturation order. We also show that when one increases the density of the centers at the boundary, approximation order 4 - the best possible order in the absence of a boundary - can be obtained.

Cooperative-competitive genetic evolution of radial basis function centers and widths for time series prediction

In a radial basis function (RBF) network, the RBF centers and widths can be evolved by a cooperative-competitive genetic algorithm. The set of genetic strings in one generation of the algorithm represents one REP network, not a population of competing networks. This leads to moderate computation times for the algorithm as a whole. Selection operates on individual RBFs rather than on whole networks. Selection therefore requires a genetic fitness function that promotes competition among RBFs which are doing nearly the same job while at the same time promoting cooperation among RBFs which cover different parts of the domain of the function to be approximated. Niche creation resulting from a fitness function of the form |w(i)|(beta)/E(|w(i')|(beta)), 1<beta<2 can facilitate the desired cooperative-competitive behavior. The feasibility of the resulting algorithm to evolve networks of Gaussian, inverse multiquadric, and thin-plate spline RBFs is demonstrated by predicting the Mackey-Glass time series. For each type of RBF, and for networks of 25, 50, 75, 100, 125, and 150 RBF units, prediction errors for the evolved Gaussian RBF networks are 50-70% lower than RBF networks obtained by k-means clustering.