THE CONSTRUCTION OF WAVELET-BASED TRUNCATED CONICAL SHELL ELEMENT USING B-SPLINE WAVELET ON THE INTERVAL

Based on B-spline wavelet on the interval (BSWI), two classes of truncated conicalshell elements were constructed to solve axisymmetric problems, i.e. BSWI thin truncated conicalshell element and BSWI moderately thick truncated conical shell element with independent slope-deformation interpolation. In the construction of wavelet-based element, instead of traditionalpolynomial interpolation, the scaling functions of BSWI were employed to form the shape functionsthrough the constructed elemental transformation matrix,and then construct BSWI element viathe variational principle. Unlike the process of direct wavelets adding in the wavelet Galerkinmethod, the elemental displacement field represented by the coefficients of wavelets expansionwas transformed into edges and internal modes via the constructed transformation matrix. BSWIelement combines the accuracy of B-spline function approximation and various wavelet-basedelements for structural analysis. Some static and dynamic numerical examples of conical shellswere studied to demonstrate the present element with higher efficiency and precision than thetraditional element.     

区间B样条小波平面弹性及Mindlin板单元构造研究

基于二维张量积区间B样条小波及小波有限元理论,构造了一类用于分析弹性力学平面问题和中厚板问题的C0型区间B样条小波板单元。在二维小波单元的构造过程中,传统多项式插值被二维区间B样条小波尺度函数取代,进而构造形状函数和单元。与小波Galerkin方法不同,本文构造的区间B样条小波单元通过转换矩阵将无明确物理意义的小波插值系数转换到物理空间。区间B样条小波单元同时具有传统有限元和B样条函数数值逼近精度高及多种用于结构分析的基函数的优点。数值算例表明:与传统有限元和解析解相比,本文构造的二维小波单元具有求解精度高,单元数量和自由度少等优点。     

一种区间B样条小波平板壳单元及应用

基于二维张量积区间B样条小波,构造了一种件能良好的小波平板壳单元．在小波单元的构造过程中,用二维区间B样条小波尺度函数取代传统多项式插值,在所构造的区间B样条平面弹性单元和平面Mindlin板单元的基础上组合而成．区间B样条小波单元同时具有B样条函数数值逼近精度高和多种用于结构分析的基函数的特点．数值算例表明：与传统有限元和解析解相比,构造的小波平板壳单元具有求解精度高,单元数量和自由度少等优点．     

Study on spline wavelet finite-element method in multi-scale analysis for foundation

A new finite element method （FEM） of B-spline wavelet on the interval （BSWI） is proposed. Through analyzing the scaling functions of BSWI in one dimension, the basic formula for 2D FEM of BSWI is deduced. The 2D FEM of 7 nodes and 10 nodes are constructed based on the basic formula. Using these proposed elements, the multiscale numerical model for foundation subjected to harmonic periodic load, the foundation model excited by external and internal dynamic load are studied. The results show the pro- posed finite elements have higher precision than the tradi- tional elements with 4 nodes. The proposed finite elements can describe the propagation of stress waves well whenever the foundation model excited by extemal or intemal dynamic load. The proposed finite elements can be also used to con- nect the multi-scale elements. And the proposed finite elements also have high precision to make multi-scale analysis for structure.     

区间B样条小波薄壳截锥单元构造

利用区间B样条小波的尺度函数作为有限元插值函数,从轴对称壳的能量泛函出发,由变分原理导出了单元刚度矩阵和载荷列阵,构造了区间B样条小波薄壳截锥单元.区间B样条小波单元同时具有B样条函数数值逼近精度高和多种用于结构分析的变尺度基函数的特点.数值算例表明:与传统截锥单元相比,本文构造的小波单元具有求解精度高、单元数量和自由度少等优点.     

一维区间B样条小波单元的构造研究

基于区间B样条小波及小波有限元理论,提出了一种区间B样条小波有限元方法。传统有限元多项式插值被一维区间B样条小波尺度函数取代,进而构造形状函数和单元。与小波Galer-kin方法不同,本文构造的区间B样条小波单元通过转换矩阵将无明确物理意义的小波插值系数转换到物理空间。转换矩阵在小波单元构造过程中起到关键作用,为了保证求解的稳定性,转换矩阵必须非奇异。构造了以区间B样条尺度函数为插值函数的一系列一维区间B样条小波单元。数值算例表明,本文构造的区间B样条小波单元与传统有限元方法相比,在求解变截面,变载荷等问题时具有收敛快和精度高等优势;有效地丰富了小波有限元法单元库。     

THE ANALYSIS OF SHALLOW SHELLS BASED ON MULTIVARIABLE WAVELET FINITE ELEMENT METHOD

Based on the generalized variational principle and B-spline wavelet on the interval (BSWI), the multivariable BSWI elements with two kinds of variables (TBSWI) for hyperboloidal shell and open cylindrical shell are constructed in this paper. Different from the traditional method, the present one treats the generalized displacement and stress as independent variables. So differentiation and integration are avoided in calculating generalized stress and thus the precision is improved. Furthermore, compared with commonly used Daubechies wavelet, BSWI has explicit expression and excellent approximation property and thus further guarantee satisfactory results. Finally, the effciency of the constructed multivariable shell elements is validated through several numerical examples.     

Crack detection in a shaft by combination of wavelet-based elements and genetic algorithm

A new crack detection method is proposed for detecting crack location and depth in a shaft. Rotating Rayleigh-Euler and Rayleigh-Timoshenko beam elements of B-spline wavelet on the interval (BSWI) are constructed to discretize slender shaft and stiffness disc, respectively. According to linear fracture mechanics theory, the localized additional flexibility in crack vicinity can be represented by a lumped parameter element. The cracked shaft is modeled by wavelet-based elements to gain precise frequencies. The first three measured frequencies are used in crack detection process and the normalized crack location and depth are detected by means of genetic algorithm. To investigate the robustness and accuracy of the proposed method, some numerical examples and experimental cases of cracked shaft are conducted. It is found that the method is capable of detecting crack in a shaft.     

薄板的多变量小波有限元分析

基于区间B样条小波和广义变分原理,提出了多变量小波有限元法,构造了一种新的薄板多变量小波有限单元.由广义变分原理推导结构的多变量有限元列式,区间B样条小波尺度函数作为插值函数构造的多变量小波有限元法中,广义应力和应变被作为独立变量进行插值,避免了传统方法中应力应变求解的微分运算,减小了计算误差.区间B样条小波良好的数值...     

Isogeometric cable elements based on B-spline curves

Meccanica - In this paper, the isogeometric cable elements based on B-spline curves are developed for the static analysis of cable structures under conservative static loads. For this, the...     

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运用von Neumann稳定性分析方法对采用3次B-样条核函数的一维标准SPH进行了稳 定性分析,在此基础上对扰动波长的影响进行了讨论,并用一维弹性杆的SPH 计算对分析结论进行了数值验证. 理论分析和数值计算表明, von Neumann分析所得的稳定性随扰动波长变化的规律与实际计算结果完全相符, SPH稳定性与扰 动波长密切相关. 对于所分析的SPH方法,在扰动波长最小处稳定性最差,而在其它扰 动波长下,SPH的计算则可以在拉伸状态下稳定.     

三角形夹芯板夹心层的等效弹性常数

推导了三角形夹芯板夹心层的等效弹性常数,为验证所推导的弹性常数的合理性,用NASTRAN有限元计算程序对两种模型进行了静力计算比较,一种是用所推导的弹性常数将夹心等效后,以层合板的形式进行计算的模型,另一种是对整个结构直接采用板单元进行计算的模型.计算结果表明:两种模型结果吻合良好.     

On the instability of an axially moving elastic plate

Problems of stability of an axially moving elastic band travelling at constant velocity between two supports and experiencing small transverse vibrations are considered in a 2D formulation. The model of a thin elastic plate subjected to bending and tension is used to describe the bending moment and the distribution of membrane forces. The stability of the plate is investigated with the help of an analytical approach. In the frame of a general dynamic analysis, it is shown that the onset of instability takes place in the form of divergence (buckling). Then the static forms of instability are investigated, and critical regimes are studied as functions of geometric and mechanical problem parameters. It is shown that in the limit of a narrow strip, the 2D formulation reduces to the classical 1D model. In the limit of a wide band, there is a small but finite discrepancy between the results given by the 1D model and the full 2D formulation, where the discrepancy depends on the Poisson ratio of the material. Finally, the results are illustrated via numerical examples, and it is observed that the transverse displacement becomes localised in the vicinity of free boundaries.     

薄板小波有限元理论及其应用

利用样条小波尺度函数构造了常用的三角形和矩形薄板单元的位移函数,得到了利用小波函数表示的形函数。采用合理的局部坐标,对单元进行压缩,使单元在局部坐标区间上有其值,成功地推导出了分域的三角形和矩形薄板小波有限元列式。在此基础上,提出了弹性地基薄板的小波有限元求解方法。通过两个算例对薄板的挠度和弯矩进行了计算,数值结果表明,求解结果具有收敛快、精度高的特点。     

Non-linear dynamic behaviour of multi-folding microstructure systems based on origami skill

This paper presents the theoretical basis for both static and dynamic numerical approaches to the elastic stability and elasto-plastic stability of a folding multi-layered truss. Both analyses are based on bifurcation theory and include geometrical non-linearity. The dynamic analysis includes an allowance for contact between nodes. Comparisons are made between elastic behaviour and elasto-plastic behaviour obtained from both numerical dynamic methods in which bifurcations are demonstrated as a theoretically elastic analysis. Our analysis reveals that several folding behaviours in the pantographic truss are identified during the elastic and elasto-plastic instabilities, where the folding behaviour of the truss is shown to be a function of the initial geometry and velocity of the dynamic loading. The authors suggest that understanding this behaviour will be very useful for developing light-weight structures subject to dynamic loading based on static bifurcation analysis and dynamic analysis.     

Complex wavenumber Fourier analysis of the B-spline based finite element method

We present the results of one-dimensional complex wavenumber Fourier analysis of the B-spline variant of Finite Element Method (FEM). Generally, numerical results of elastic wave propagation in solids obtained by FEM are polluted by dispersion and attenuation. It was shown for the higher-order B-spline based FEM, that the optical modes did not occur in the case of infinite domains, unlike the higher-order Lagrangian and Hermitian finite elements, and also the dispersion errors are smaller. The paper’s main focus is on the wave propagation through B-spline multi-patch/segment discretization with the C⁰$C^0$ connection of B-spline segments and, chiefly, to the determining of dispersion and attenuation dependences. The numerical approach employed leads to substantial minimization of dispersion errors. Furthermore, the errors decrease in line with the increasing order of the B-spline elements/segments, with the local refinement, and also by the particular choice of the positions of control points through the optimizing procedure.     

爆炸荷载作用下地下拱形结构动力响应样条小波有限元研究

采用小波有限元方法研究爆炸荷载作用下地下结构的动力响应,克服在模拟过程中由于材料的奇异性、地质条件的复杂性和加载的快速性出现的应力集中和计算效率低下,根据样条有限点法,构造了单向和双向区间B样条圆环扇形小波单元,用尺度函数作为插值函数;结合工程实例,通过Matlab软件编程对爆炸荷载作用地下拱形结构的动力响应进行了数值模拟,并与应用传统有限元程序模拟结果进行对比。结果表明,小波有限元用很少单元取得了较高的精度,计算效率比传统有限元提高一倍。     

Wavelet basis expansion-based spatio-temporal Volterra kernels identification for nonlinear distributed parameter systems

In practical industries, there are many systems belong to nonlinear distributed parameter systems (DPS); unfortunately, modeling of nonlinear DPS is a challenging task because of the infinite-dimensional and nonlinear properties. To model the nonlinear DPS, a spatio-temporal Volterra model is presented with a series of spatio-temporal kernels. It can be considered as a spatial extension of the traditional Volterra model. One question involved in modeling a spatio-temporal functional relationship between the input and output of nonlinear distributed parameter systems using spatio-temporal Volterra series is to identify the spatio-temporal Volterra kernel functions. In addition, in order to derive a low-order model, the Karhunen–Loève (KL) decomposition is used for the time/space separation. The basic routine of the approach is that, first, from the system outputs, KL decomposition is used for the time/space separation, where the spatio-temporal output is decomposed into a few dominant spatial basis functions with temporal coefficients. Second, according to temporal coefficients of outputs under multilevel excitations, the Volterra series outputs of different orders are estimated with the wavelet balance method. Third, the Volterra kernel functions of different orders are separately estimated through their corresponding Volterra series outputs by expanding them with four-order B-spline wavelet on the interval (BSWI). Finally, the spatio-temporal Volterra model can be reconstructed using the time/space synthesis. The simulation studies verify the effectiveness of the presented identification method.     

Use of B-spline surface to model large-deformation continuum plates: procedure and applications

The absolute nodal coordinate formulation (ANCF) has been used in the analysis of large deformation of flexible multibody systems that encompass belt drive, rotor blade, and cable applications. As demonstrated in the literature, the ANCF finite elements are ideal for isogeometric analysis. The purpose of this investigation is to establish a relationship between the B-splines, which are widely used in the geometric modeling, and the ANCF finite elements in order to construct continuum models of large-deformation geometries. This paper proposes a simplified approach to map the B-spline surfaces into ANCF thin plate elements. Matrix representation of the mapping process is established and examined through numerical examples successfully. The matrix representation of the mapping process is used because of its suitability of computer coding and to minimize the calculation time. The error estimation is carried out by analyzing the gap between the points of each ANCF element and the corresponding points of the portion of the B-spline surface. The Hausdorff distance is used to study the effect of the number of control points, the degree of interpolation, and the knot multiplicity on the mapped geometry. It is found that cubic interpolation is recommended for optimizing the accuracy of mapping the B-spline surface to ANCF thin plate elements. It is found that thin plate element in ANCF missing a number of basis functions which considered a source of error between the two surfaces, as well as it does not allow to converting the ANCF thin plate elements model to B-spline surface. In this investigation, an application example of modeling large-size wind turbine blade with uniform structure is illustrated. The use of the continuum plate elements in modeling flexible blades is more efficient because of the relative scale between the plate thickness and its length and width and the high flexibility of its structure. The numerical results are compared with the results of ANSYS code with a good agreement. The dynamic simulation for mapped surface model shows a numerical convergence, which ensures the ability of using the proposed approach for applications of dynamics for design and computer-aided design.     

变截面弹性直杆纵振动分析的小波——DQ法

在经典微分求积(DQ)法基础上, 根据多分辨分析理论, 以尺度函数为基础构造插 值基函数, 形成了新的微分方程边值问题的求解方法------小波--DQ法, 并应用该方法分析了变截面弹性直杆的纵振动问题, 给出了其频率方程, 计算出了不同参数下固 支--固支, 自由--自由楔形直杆和锥形直杆的固有频率, 数值结果表明该方法是一个简单易行高精度的方法, 该方法可以推广应用于其他力学问题的数值分析.