Spline Weight Function Research Articles

Meshless Error Recovery Parametric Investigation in Incompressible Elastic Finite Element Analysis

The meshless displacement error-recovery parametric investigation in finite element method-based incompressible elastic analysis is presented in this study. It investigates key parameters such as interpolation schemes, patch configurations, dilation indexes, weight functions, and meshing patterns. The study evaluates error recovery effectiveness (local and global), convergence rates, and adaptive mesh improvement for triangular/quadrilateral discretization schemes. It uses meshless moving least squares (MLS) interpolation with rectangular and circular support regions and solves benchmark plate and cylinder problems. It is observed that a circular influence region, a cubic spline weight function, and regular mesh patterns yield a better performance of than an MLS-based error recovery method. The study also concludes that lower dilation index values with rectangular influence regions are preferable for regular meshes, while higher dilation index values with radial influence regions are suitable for preferable meshes to enhance MLS error recovery.

Elastoplastic Analysis of Metallic Parts Employing a Meshless Method.

The use of finite element method-based approaches has been popular in studying the elastoplastic behavior of metal parts. However, there has been a growing demand for meshless methods. In response, researchers have developed a meshless solution for 2D elastoplastic evaluation of metal components. This approach obtains the locally symmetric weak form of the governing elastoplastic integral equations at each node throughout the problem area and boundary. The elastoplastic constitutive relationships consider a small deformation rate independent associative flow theory applicable to isotropic hardening materials. The proposed solution algorithm can handle loading, unloading, and reverse loading. Numerical results were computed using Gaussian and spline weight functions, and the presented meshless solution proved to be robust and accurate for conducting the elastoplastic investigation of metallic parts. Furthermore, the Gaussian weight function was found to be more robust than the spline weight function. In conclusion, this paper presents a reliable meshless solution for elastoplastic analysis and highlights the advantages of using Gaussian weight functions.

Comparison of meshfree displacement and stress error recovery of finite element solutions using moving least squares interpolation

The moving least square (MLS) interpolation based the recovery procedures have been successfully applied to recover the finite element solution errors in the analysis of elastic plates and pipes problems and can be advantageously applied for large deformation and fracture problems. The study presents the displacement and stress error recovery characteristics in the error estimation analysis employing Moving Least Squares interpolation approach. The study considers quartic spline, cubic spline, and exponential weight function with three different order of basis function in Moving Least Squares interpolation based error recovery analysis. The displacement/stress errors in finite element solution are quantified in energy norm. The cylinder and plate benchmark examples using triangular and quadrilateral elements are analyzed to compare the convergence, effectivity and adaptively improved meshes obtained using the various displacement/stress recovery procedures. The study shows that cubic spline weight function and quadratic basis function found to perform better in MLS based meshfree recovery technique for stress as well as displacement errors recovery of finite element solution. It is observed from the study that increasing the order of basis function will enhance the error estimation quality that is, rate of convergence become faster with improved effectivity of the results. The increase in convergence rate with the increase of the order of basis function is more in displacement recovery technique as compared to stress recovery technique. It is observed in the analysis of benchmark example with linear triangular meshing that the error reduction using meshfree MLS interpolation based displacement and stress recovery is about 10% and 150% respectively for the displacement and stress recovery over the mesh dependent least square based displacement and stress recovery. The study concludes that the effectiveness and efficiency of meshfree displacement/stress error recovery technique strongly depends on the weight and basis functions of MLS method to recover the errors.

Extended Peridynamics and Parameter Optimization Study Based on Moving Least Squares Method

Based on the theory of peridynamics, the least squares and the moving least squares method are proposed to fit the physical information at nondiscrete points. It makes up for the shortcomings of the peridynamic method that only solves the discrete nodes and cannot obtain the physical information of other blank areas. The extended method is used to fit the one-way vibration problem of the rod, and the curve of the displacement of a nondiscrete node in the rod is extracted with time. The fitted displacement results are compared with the theoretical results to verify the feasibility of the fitting method. At the same time, the parameters in the fitting of the moving least squares method are optimized, and the effects of different tight weight functions and influence ranges on the results are analyzed. The results show that when the weight function is a power exponential function, the fitting effect increases with the decrease in the coefficient. When the weight function is a cubic spline weight function, a better fitting effect is obtained. And in the case of ensuring the fitting result, the affected area should be reduced as much as possible, and the calculation efficiency and precision can be improved.

A time-domain method for load identification using moving weighted least square technique

Based on the thought of Green’s kernel function method (GKFM), an improved time-domain load identification method using moving weighted least square technique (MWLST) which can accurately fit dynamic load is proposed. Better than the traditional shape function method using moving least square fitting (SFM_MLSF), the proposed method considers continuity and correlation of dynamic load between two adjacent sampling points, and involves the weighted contribution of sampling points to the fitting point. In numerical examples, Gauss, Cubic and Quartic spline weight functions are utilized in the proposed method to realize the reconstruction of kernel matrix. It is found that the accuracies of load identification are almost same when their optimum supported domain radii are adopted. Furthermore, the numerical results illustrate that the proposed method can identify dynamic load more accurately and smoothly than GKFM and SFM_MLSF significantly by the same regularization method for ill-posedness, and the proposed method has excellent stability and robustness. Additionally, a special technique combining both the whole identification and the truncated-processing identification is proposed to identify external dynamic loads during hoisting process, which solves the oscillation problem caused by using inversing methods directly.

Meshless Method for Nonuniform Heat-Transfer/Solidification Behavior of Continuous Casting Round Billet

Numerical simulation is the primary approach to evaluate the complex solidification behavior for the continuous casting (CC) process, in which the methods based on the mesh and topology technique have been widely used to solve field variables. For the traditional techniques, such as the finite difference, finite element, and boundary element methods, it is arduous or even impossible to deal with complicated problems that involve multiphase coupling, interface tracking/reconstruction, and self-adaptation due to the inherent weaknesses in the grid structure and mesh dependence. Hence, the present work explores a meshless calculation method for the two-dimensional unsteady heat-transfer problem and proposes an element-free Galerkin (EFG) model for solving heat transfer inside the CC mold based on the moving least-squares approximation. The temperature functions are approximated and constructed by a linear basis and cubic spline weight function over a set of rectangular supporting domain; then, the discrete heat-transfer governing equation based on the EFG method is deduced. The heat flux measured in the real casting process is set as the boundary condition to calculate the nonuniform solidification of the round billet. The calculated results demonstrate that the shell thickness is consistent with that obtained by the square root law of solidification. In addition, the high heat flux region near the meniscus directly determines the growth characteristics of the initial billet shell, which ultimately results in the overall nonuniformity of the shell. The EFG method has the characteristics of fast convergence, high computational accuracy, and great discrete flexibility; it also provides a novel and effective approach for subsequent thermomechanical coupling and crack propagation analysis in the CC process.

Isogeometric analysis on implicit domains using weighted extended PHT-splines

The urging quest for diminishing the gap between finite element methods and computer-aided design is growing every day. Isogeometric analysis is driving that search a step forward in contrast with traditional finite element methods. One of the areas that need more investigation is using isogeometric analysis on domains that contain singularities. In this case, the parameterization process is occasionally non-trivial, and some of the inherited basis functions are not well defined producing rough solutions. Moreover, NURBS basis functions are not the best choice to perform local refinement since adaptivity is indispensable. In this study, a new approach that serves as a generalization for some of the ideas in both the isogeometric analysis and the WEB method is introduced. Isogeometric analysis on implicit domains replaces mesh generation with implicit spline weight function generation and construct weighted extended PHT-splines for analysis. The geometry and the analysis both are built starting from the same function space avoiding parameterization. The PHT-splines are adaptive and allow local refinement naturally leading to a higher accuracy while dealing with singularities. An illustration with numerical examples demonstrates the efficiency and performance of the proposed method.

The improved complex variable element-free Galerkin method for the analysis of Kirchhoff plates

Based on the improved complex variable moving least-squares (ICVMLS) approximation, the improved complex variable element-free Galerkin (ICVEFG) method for the bending problem of Kirchhoff plate is presented. Compared with the moving least-squares (MLS) approximation, in the ICVMLS approximation, the approximation function of two-dimensional problems can be obtained with one-dimensional basis function, then the computational efficiency of the shape functions is higher. Compared with the meshless methods based on the MLS approximation, under the same node distributions, the ones using the ICVMLS approximation can obtain the solutions with higher computational accuracy; and under the similar computational accuracy, the ones using the ICVMLS approximation have higher computational efficiency. The ICVMLS approximation is used to form the approximation function of the deflection of a Kirchhoff plate, the Galerkin weak form of the bending problem of Kirchhoff plates is adopted to obtain the discretized system equations, and the penalty method is employed to enforce the essential boundary conditions, then the corresponding formulae of the ICVEFG method for the bending problem of Kirchhoff plates are presented. By computing and analyzing four typical examples, it is shown that the ICVEFG method of Kirchhoff plates in this paper is efficient. And the computational precision of the numerical solutions is analyzed to select the basis function, weight function, scaling factor, node distribution and penalty factor in the ICVEFG method. Numerical examples show that the method in this paper has better convergence and higher accuracy. When the quadratic polynomial basis function and the cubic or quartic spline weight function are used, and d m a x = 4.2−4.4 , the ICVEFG method of Kirchhoff plates in this paper can obtain the solutions with high computational accuracy. And more nodes are distributed in the problem domain, higher computational accuracy of the solution will obtained, which shows that the method in this paper has better convergence.

Postbuckling of carbon nanotube reinforced functionally graded plates with edges elastically restrained against translation and rotation under axial compression

Abstract This paper presents, to the authors’ knowledge, a first known postbuckling analysis of carbon nanotube (CNT) reinforced functionally graded plates with edges elastically restrained against translation and rotation. The plate considered is of moderate thickness and, hence, the first-order shear deformation theory (FSDT) and von Karman assumption are adopted to incorporate the effects of transverse shear strains, rotary inertia and moderate rotations. The element-free IMLS-Ritz method is employed. The cubic spline weight function and linear basis are utilized in the approximation. In this study, the bending stiffness is evaluated through the nodal integration scheme. The postbuckling path is traced using the arc-length method combined with the modified Newton–Raphson technique. Parametric studies on the postbuckling behavior of CNT reinforced functionally graded plates are conducted to examine the effects of CNT content by volume, plate width-to-thickness ratio and plate aspect ratio by varying the elastically restrained parameters of translation and rotation on the boundaries. The results of the present study are obtained for simplified cases so that comparison studies can be undertaken using the values reported in the literature.

Computation of vibration solution for functionally graded carbon nanotube-reinforced composite thick plates resting on elastic foundations using the element-free IMLS-Ritz method

This paper explores the element-free IMLS-Ritz method for computation of vibration solution of thick functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates resting on elastic foundations. The shear deformation effect is incorporated through the first-order shear deformation theory (FSDT). The cubic spline weight function and linear basis are utilized in the approximation. Regular node arrangements and cell background meshes are employed in the numerical integration. The penalty method is adopted to impose the essential boundary conditions. Numerical stability and applicability of the IMLS-Ritz method are examined through solving a few numerical example problems. The influence of Winkler modulus parameters on the vibration behavior of FG-CNTRC plates is studied. Besides, the effects of CNT volume fraction, CNT distribution, plate thickness-to-width ratio, plate aspect ratio on FG-CNTRC plates are investigated under different boundary conditions. The vibration frequencies and mode shapes of the FG-CNTRC plates on different Winkler foundations are presented.

Environmental Quality Assessment Based on a New Kind of Neural Network Using Rational Spline Weight Functions

As we all known, artificial neural network can be used in the process of environmental quality assessment. To improve the accuracy and science of assessment, a method of environmental quality assessment is presented in this paper, which is based on spline weight function (SWF) neural networks. The weigh functions of the neural network are composed of rational spline functions with cubic numerator and linear denominator (3/1 rational SWF). The simulation results show that, compared with the conventional BP neural networks, this method can get very high precision and accuracy. This case demonstrates that SWF neural networks can offer a very prospective tool for environmental quality assessment.

Sleep Sensor Temperature Prediction Based on Neural Network Training Algorithm with Rational Spline Weight Functions

In wireless sensor network, it is necessary to make effective prediction of sensor node’s data during its sleep period. In this paper a model of rational cubic spline weight function (SWF) neural network with linear denominator was established for sensor node’s temperature prediction. This kind of rational spline function is denoted by 3/1 rational splines. Then we trained and tested the network, the simulation results showed that, compared to the traditional BP neural network, the training speed is higher and the error is smaller. Therefore the prediction model can effectively predict the sensor’s temperature.

Error Analysis of Neural Network with Multiply Neurons Using Rational Spline Weight Functions

In order to describe the generalization ability, this paper discusses the error analysis of neural network with multiply neurons using rational spline weight functions. We use the cubic numerator polynomial and linear denominator polynomial as the rational splines for weight functions. We derive the error formula for approximation, the results can be used to algorithms for training neural networks.

Passenger Flow Analysis in Subway Using a Kind of Neural Network

This paper aims to analyze passenger flow in subway based on a kind of rational spline weight function neural network, in which the numerator of the spline is a cubic polynomial and the denominator of the spline is a quadratic polynomial, and this kind spline is denoted by 3/2 rational splines. There are many factors affecting the passenger flow. Combined the main influential factors with the self-learning method of neural network, we establish the neural network model of passenger flow in subway. This paper introduces the spline weight function neural network and the passenger flow model based on this neural network. Finally MATLAB simulation verifies that the 3/2 rational spline weight function neural network can be applied to analyze the passenger flow in subway with high accuracy.

Population Analysis Based on Neural Network of Rational Spline Weight Functions with Cubic Numerator and Quadratic Denominator

With the steady growth of the population in our country, the analysis of population is essential to the distribution of social resources and the improvement of the population quality. In this paper, we use the neural network of rational spline weight functions with cubic numerator and quadratic denominator to establish the model of Chinese population by four indicators, gender ratio, birth rate, elderly dependency ratio and natural increase rate. Finally we come to the conclusion through MATLAB simulation that the training speed is fast and the error is small. This kind of neural network can be applied to population analysis and it could be used in various fields.

Error Analysis of Neural Network with Rational Spline Weight Function

Training neural network by spline weight function (SWF) has overcomed many defects of traditional neural networks (such as local minima, slow convergence and so on). It becomes more important because of its simply topological structure, fast learning speed and high accuracy. To generalize the SWF algorithm, this paper introduces a kind of rational spline weight function neural network and analyzes the performance of approximation for this neural network.

Approximation Analysis for a New Kind of Neural Network Using Rational Spline Weight Functions

To describe the performance for a new kind of neural network, This paper discusses the approximation of the neural network using a kind of rational spline weight function. The rational spline consists of piecewise rational functions with cubic numerators and linear denominators. The theoretic formula of approximation is proposed and an example is also given. It can be concluded that this new neural network can get very high training accuracy.

Complexity Analysis of Neural Network Based on Rational Spline Weight Functions

This paper aims to obtain the time complexity for a new kind of neural network using rational spline weight functions. In this paper, we introduce the architecture of the neural network, and analyze the time complexity in detail. Finally, some examples are also given to verify the theoretical analysis. The results show that the time complexity depends on the number of patterns, the input and out dimensions of the neural networks.

Implementation of Mail Classification Using Neural Networks of the Second Type Spline Weight Functions

How to effectively filter out spam is a topic worthy of further study for the growing proliferation of spam. The main purpose of this paper is to apply a new neural network algorithm to the classification of spam. In this paper, we introduce a second type of spline weight function neural network algorithm, as well as e-mail feature extraction and vectorization, and then introduced the mail sorting process. Experiments show that it can get a relatively high accuracy and recall rate on the spam classification. Therefore, with this new algorithm, we can achieve better classification results.

Chinese Word Segmentation Based on the First Kind of Spline Weight Function Neural Networks

With the continuous development of information technology, word segmentation technology becomes an important link in dealing with the increasing amount of information conveniently. Different from English word segmented by spaces, Chinese writing is continuous, and there is no space between words, this brings a lot of trouble to word segmentation. In this article, through the analysis of different property of words to get a code that can be used for training and combined it with the first kind of spline weight function neural network, then by training a large number of existing rules encoding to generate a study method that can divide the statement correctly.