A meshless method for Kirchhoff plate bending problems

In this work a meshless method for the analysis of bending of thin homogeneous plates is presented. This meshless method is based on the use of radial basis functions to build an approximation of the general solution of the partial differential equations governing the Kirchhoff plate bending problem. In order to obtain a symmetric and non‐singular linear equation system the Hermite collocation method is used. To assess the formulation a series of plates with different boundary conditions are analysed. Comparisons are made with other results available in the literature. Copyright © 2001 John Wiley & Sons, Ltd.     

Automatic Vehicle License Plate Recognition Using Optimal Deep Learning Model

The latest advancements in highway research domain and increase inthe number of vehicles everyday led to wider exposure and attention towards the development of efficient Intelligent Transportation System (ITS). One of the popular research areas i.e., Vehicle License Plate Recognition (VLPR) aims at determining the characters that exist in the license plate of the vehicles. The VLPR process is a difficult one due to the differences in viewpoint, shapes, colors, patterns, and non-uniform illumination at the time of capturing images. The current study develops a robust Deep Learning (DL)-based VLPR model using Squirrel Search Algorithm (SSA)-based Convolutional Neural Network (CNN), called the SSA-CNN model. The presented technique has a total of four major processes namely preprocessing, License Plate (LP) localization and detection, character segmentation, and recognition. Hough Transform (HT) is applied as a feature extractor and SSA-CNN algorithm is applied for character recognition in LP. The SSA-CNN method effectively recognizes the characters that exist in the segmented image by optimal tuning of CNN parameters. The HT-SSA-CNN model was experimentally validated using the Stanford Car, FZU Car, and HumAIn 2019 Challenge datasets. The experimentation outcome verified that the presented method was better under several aspects. The projected HT-SSA-CNN model implied the best performance with optimal overall accuracy of 0.983%.     

A meshfree Hermite‐type radial point interpolation method for Kirchhoff plate problems

A meshfree computational method is proposed in this paper to solve Kirchhoff plate problems of various geometries. The deflection of the thin plate is approximated by using a Hermite‐type radial basis function approximation technique. The standard Galerkin method is adopted to discretize the governing partial differential equations which were derived from using the Kirchhoff's plate theory. The degrees of freedom for the slopes are included in the approximation to make the proposed method effective in enforcing essential boundary conditions. Numerical examples with different geometric shapes and various boundary conditions are given to verify the efficiency, accuracy, and robustness of the method. Copyright © 2005 John Wiley & Sons, Ltd.     

非线性拟双曲方程的有限元配置法数值分析

基于有限元配置法,采用分片双三次Hermite插值多项式空间作为逼近函数空间,本文对粘性振动及神经传播过程中涉及的一类非线性拟双曲方程的初边值问题建立了二维半离散和全离散格式.并对两种格式证明了数值解的存在唯一性,应用微分方程先验估计的理论和技巧得到了L2模最佳阶误差估计.数值实验结果表明:所提方法在保证整体误差估计要求且不增加计算量的前提下,比传统有限元方法有更高的逼近精度,并扩展了配置法的应用范围.     

时间分数阶Fokker-Planck方程的Jacobi谱配置方法

分数阶微分方程在工程、生物、金融等领域有广泛的应用．本文利用分数阶积分和微分公式的关系,针对一类带Dirichlet边值条件的时间分数阶Fokker-Planck方程,将其转化为与之等价的带有奇异核的积分微分方程,然后用高斯积分公式数值求解积分项,在时间和空间上都采用Jacobi谱配置法来离散求解积分微分方程．数值算例的结果表明,该方法是非常有效的,数值解具有谱精度,并且该方法容易推广到高维和非线性的情形．     

A new discrete Kirchhoff quadrilateral element based on the third-order theory for composite plates

A new discrete Kirchhoff quadrilateral element based on the refined third-order theory is developed for the analysis of composite plates. The element has seven degrees of freedom per node, namely, the three displacements, two rotations and two transverse shear strain components at the mid-surface. The inplane displacements and the shear strains are interpolated using bilinear interpolation functions and the mid-surface rotations are interpolated using bi-quadratic functions based on the discrete Kirchhoff technique. The element stiffness matrix and the consistent load vector are developed using the principle of virtual work. The finite element formulation is validated by comparing the results for simply-supported plate with the analytical Navier solution. Comparison of the present results with those using other available elements based on the TOT establishes the superiority of the present element in respect of simplicity, accuracy and computational efficiency. The element is free from shear locking     

A Multiple-Precision Study on the Modified Collocation Trefftz Method

Recently, Liu (CMES 21(2007), 53) developed the modified collocation Trefftz method (MCTM) by setting a characteristic length slightly larger than the maximum radius of the computational domain. In this study, we find that the range of admissible characteristic length can be significantly enlarged if the LU decomposition is applied for solving the resulted dense unsymmetric matrix. Furthermore, we discover a range formula for admissible characteristic length, in which the number of the T-complete functions, the shape of the computation domain, and the exponent bits of the involved floating-point arithmetic have been taken into consideration. In order to validate the prescribed formula for different exponent bits, the multiple precision floating-point reliable (MPFR) library is used. In addition, we find that the MCTM is a numerical method of exponential convergence. In other words, increasing the numbers of the T-complete functions can reduce the logarithmic error proportionally till the precision limit, which can be set up for the MPFR library. Numerical experiments are carried out to demonstrate that the proposed MCTM with the LU decomposition can solve the Laplace equation stably and accurately, even for a Cauchy problem. A multiple-precision comparison between the MCTM and the method of fundamental solution is also preformed.     

周边固支缝合复合材料层板的小波配置精细积分法

建立了缝合复合材料面内纤维弯曲几何描述模型,将拟Shannon小波配置法和缝合单层板的刚度矩阵应用到缝合层合板的Hamilton正则方程中,构造了缝合板平面方向离散、而厚度方向解析的Hamilton正则方程,然后用精细积分法求解。用拟Shannon尺度函数表示的近似解很适于求解固支边界问题。数值算例结果表明,小波配置精细积分法在缝合复合材料层板位移、应力分析方面,较低网格密度下即可获得较精确的结果。从而为缝合层合板静力学问题分析提供了一种方法。     

New finite element model for analysis of Kirchhoff plate

A new kind of approach to formulate an isotropic thin plate bending element is presented. The strain energy of the element is decomposed into two parts: an integral concerning the first strain invariant and a line integral around the elemental boundary. The former can be discretized by quasi-conforming technique¹ and the latter can be directly calculated using the interpolation of the deflection and its normal slope along the element boundary. By this method, an assumed first strain invariant quadrilateral element (AFSIQ) is derived. The procedure of formulating the element and the numerical examples show that the new kind of element not only simplifies the formulation considerably but also has excellent accuracy.     

Local Collocation Method for Prediction of Natural Frequency of Functionally Graded Cylindrical Shells

In this article, the local thin plate spline collocation method and first-order shear deformation shell theory are used to predict the natural frequency of functionally graded cylindrical shells. The local collocation method does not require special treatment for essential boundary condition nor time-consuming integration of weak form. Natural frequency of functionally graded cylindrical shells with clamped and simply supported boundary conditions are solved and compared with available published results to assess the performance of the present method.     

A Galerkin symmetric and direct BIE method for Kirchhoff elastic plates: formulation and implementation

A variational Boundary Element formulation is proposed for the solution of the elastic Kirchhoff plate bending problem. The stationarity conditions of an augmented potential energy functional are first discussed. After addressing the topic of the choice of the test functions, a regularization process based on integrations by parts is developed, which allows to express the formulation in terms of double integrals, the inner being at most weakly singular and the outer regular. Standard integration procedures may then be applied for their numerical evaluation in the presence of both straight and curved boundaries. The normal slope and the vertical displacement must be C⁰ and C¹ continuous, respectively. Numerical examples show, through comparisons with analytical solutions, that a high accuracy is achieved. © 1998 John Wiley & Sons, Ltd.     

Boundary element analysis of Kirchhoff plates with direct evaluation of hypersingular integrals

The typical Boundary Element Method (BEM) for fourth‐order problems, like bending of thin elastic plates, is based on two coupled boundary integral equations, one strongly singular and the other hypersingular. In this paper all singular integrals are evaluated directly, extending a general method formerly proposed for second‐order problems. Actually, the direct method for the evaluation of singular integrals is completely revised and presented in an alternative way. All aspects are dealt with in detail and full generality, including the evaluation of free‐term coefficients. Numerical tests and comparisons with other regularization techniques show that the direct evaluation of singular integrals is easy to implement and leads to very accurate results. Copyright © 1999 John Wiley & Sons, Ltd.     

A moving Kriging interpolation‐based meshless method for numerical simulation of Kirchhoff plate problems

This paper mainly proposes an alternative way for numerical implementation of thin plates bending based on a new improvement of meshless method, which is combined between the standard element‐free Galerkin method and one different shape functions building technique. The moving Kriging (MK) interpolation is applied instead of the traditional moving least‐square approximation in order to overcome Kronecker's delta property where the standard method does not satisfy. Obviously, the deflection of the thin plates is approximated via the MK interpolation. To illustrate this approach, numerical analysis is examined in both regular and irregular systems. Three examples with different geometric shapes of thin plates undergoing a simply supported boundary are performed. In addition, two important parameters of the present method are also analyzed. A good agreement can be found among the proposed, analytical and finite element methods. Copyright © 2008 John Wiley & Sons, Ltd.     

A general degree semihybrid triangular compatible finite element formulation for Kirchhoff plates

We revisit compatible finite element formulations for Kirchhoff plates and propose a new general degree hybridized approach that strictly imposes C¹ continuity. These new elements are triangular and based on nodal polynomial approximation functions that only use displacement and rotation degrees of freedom for assembly, and thereby “nearly” impose C¹ continuity. This condition is then strictly enforced by adding appropriately chosen hybrid constraints and the corresponding Lagrange multipliers. Unlike all other existing approaches, this formulation allows for the definition of elements of arbitrary degree considering a single polynomial basis for each element, without using degrees of freedom associated with second-order derivatives. The convergence is compared with that of alternative approaches in terms of numbers of elements and degrees of freedom.     

用Fourier边界展开的方法研究具有曲线边界的厚板的自由振动

应用Fourier边界展开的方法研究具有曲线边界的厚板的自由振动问题。主要是应用一次剪切变形理论研究实心厚板的自由振动特性。首先，在极坐标系里表示出任意具有曲线边界厚板的边界条件，导出所需公式。其次，用Fourier边界展开的方法分析问题，并作相应的阶次截断，得到固有频率方程。这里，就可数值求得无因次的固有频率。由于使用的解精确满足板振动的微分方程，此方法能给出高精度的结果，而计算量小、分析过程较简单且编程也较容易。     

Jacobi Spectral Collocation Method for the Time Variable-Order Fractional Mobile-Immobile Advection-Dispersion Solute Transport Model

An efficient high order numerical method is presented to solve the mobile-immobile advection-dispersion model with the Coimbra time variable-order fractional derivative, which is used to simulate solute transport in watershed catchments and rivers. On establishing an efficient recursive algorithm based on the properties of Jacobi polynomials to approximate the Coimbra variable-order fractional derivative operator, we use spectral collocation method with both temporal and spatial discretisation to solve the time variable-order fractional mobile-immobile advection-dispersion model. Numerical examples then illustrate the effectiveness and high order convergence of our approach.     

Q345 中厚板 V 形折弯应力 应变分析

目的研究Q345中厚板在V形折弯时的应力应变中性层内移规律。方法通过对中厚板135°折弯过程进行数值模拟,分析了板料在成形过程中的等效应力应变及其中性层内移现象,测得了应变中性层与几何中间层之间的距离变化。在此基础上,采用厚度为16 mm的Q345板料进行物理实验,并与模拟结果进行了对比分析。结果在成形过程中,等效应力应变中性层均产生了内移现象,且应力中性层的内移量明显大于应变中性层的内移量;成形结束之后,应变中性层与几何中间层的距离为3.3 mm;实验结果显示,应变中性层的内移量为3.1 mm左右,与模拟值较为一致。结论数值模拟方法可以为应力应变中性层计算提供指导。对于中厚板折弯,采用几何中间层展开求得的坯料尺寸并不准确。     

基于纹理及小波分析的车牌定位方法

提出了一种基于纹理和小波分析的车牌定位方法。针对图像背景复杂,且车牌所占比例较小的特点,提出了一种确定基元分类阈值的二值化方法;根据车牌字符分布规律,提出了二值纹理基元分析方法,提取车牌候选区域;基于小波分析提取车牌区域竖笔画特征,采用隶属度定量表征车牌竖笔画特征、位置特征及形状特征,给出综合这些特征、从候选区域提取车牌区域的方法。测试结果表明,该方法正确定位率超过96%     

Free Vibration Analysis of Folded Plate Structures by the FSDT Mesh-free Method

Based on the first-order shear deformation theory, a mesh-free Galerkin method for free vibration analysis of stiffened and un-stiffened folded plates and plate structures is presented in this paper. The folded plate or plate structure is modelled as a composite structure that consists of flat plates. The stiffness and mass matrices of the flat plates are derived based on the mesh-free formulation. To avoid the failure of direct superposition, a treatment is introduced to modify the stiffness and mass matrices. The global stiffness and mass matrices of the entire folded plate or plate structure are then obtained by superposing the modified stiffness and mass matrices of the flat plates. The analysis of the stiffened folded plates or plate structures proceeds in a similar fashion, as they are regarded as composite structures of stiffened and un-stiffened flat plates. The stiffness and mass matrices of the stiffened flat plates are also given by the mesh-free method. As no meshes are used in deriving the stiffness and mass matrices, the proposed method is more flexible in studying problems for which remeshing is inevitable with the finite element methods. Several numerical examples are computed with the proposed method to demonstrate its accuracy and convergence. The results show good agreement with the solutions that have been given by other researchers and ANSYS.     

A Meshless Numerical Method for Kirchhoff Plates under Arbitrary Loadings

This paper describes the combination of the method of fundamental solutions (MFS) and the dual reciprocity method (DRM) as a meshless numerical method to solve problems of Kirchhoff plates under arbitrary loadings. In the solution procedure, a arbitrary distributed loading is first approximated by either the multiquadrics (MQ) or the augmented polyharmonic splines (APS), which are constructed by splines and monomials. The particular solutions of multiquadrics, splines and monomials are all derived analytically and explicitly. Then, the complementary solutions are solved formally by the MFS. Furthermore, the boundary conditions of lateral displacement, slope, normal moment, and effective shear force are all given explicitly for the particular solutions of multiquadrics, splines and polynomials as well as the kernels of MFS. Finally, numerical experiments are carried out to validate these analytical formulas. In these numerical experiments, homogeneous problems are first considered to find the best location of the MFS sources by the way proposed by Tsai, Lin, Young and Atluri (2006). Then the corresponding nonhomogeneous problems are solved by the DRM based on both the MQ and APS. The numerical results demonstrate that the MQ is in general more accurate than the thin plate spline, or the first order APS, but less accurate than the high order APSs. Overall, this paper derives a meshless numerical method for solving problems of Kirchhoff plates under arbitrary loadings with all kinds of boundary conditions by both the MQ and APS.