矩形正交各向异性薄板弯曲受迫振动问题的分析解

通过恰当的辛内积定义,首先将矩形正交各向异性薄板弯曲受迫振动问题导入到辛对偶体系,并应用分离变量和辛本征展开的有效数学物理方法给出其受迫振动稳态解的一个解析求解方法.然后,具体讨论了对边简支和对边固支两种典型边界条件的正交各向异性薄板弯曲受迫振动问题的辛本征问题,并给出了对应的辛本征值超越方程和辛本征向量的解析表达式.最后,应用本文的方法分析求解了两个具体算例,并将受均布谐载作用的四边简支矩形板受迫振动稳态解的分析解与传统的Navier法进行了精度和收敛性的对比,结果表明本文的方法比传统的分析求解方法具有更好的精度和收敛性,尤其是对内力.     

Reissner模型中厚板边界元法的进一步应用

利用边界单元法求解Reissner模型中厚板弯曲问题以及对解的精度和收敛性的研究已在文献中作了详细的研究.在文献中主要求解了矩形板、菱形板和斜板的弯曲问题.本文主要研究悬臂板、中心穿孔、偏心穿孔的简支方板的弯曲问题并与文献的薄板边界元结果及其它已有文献的结果相对照.     

薄板的几何非线性求积元分析

针对薄板非线性迭代计算量很大的问题,依据von Kárman薄板非线性理论构造能量泛函,并用数值积分和数值微分进行离散,得到非线性方程组,从而利用求积元法(Quadrature Element Method,QEM)求解薄板的中等挠度的弯曲和非线性屈曲问题,得到可信的结果.算例表明:在处理薄板几何非线性问题上,QEM计算效率很高,应用潜力很大.     

四边固支弹性矩形薄板的自由振动

首先弹性矩形薄板的动力学方程表示成为Hamilton正则方程,然后采用辛几何方法对全状态相变量进行分离变量,并利用得到的共扼辛正交归一关系,求出四边固支弹性矩形薄板的固有频率和振型的解析解表达式．由于在求解过程中不需要事先人为的选取挠度函数,而是从弹性矩形薄板的动力学基本方程出发,直接利用数学的方法求出可以满足四边固支边界条件下薄板的固有频率和振型的解析解表达式,使得问题的求解更加理论化和合理化．此外,还给出了计算实例来验证本文所采用的方法以及所推导出公式的正确性．     

均布荷载下四边固支矩形薄板的挠度研究

为求解四边固支矩形薄板在均布荷载作用下的挠度表达式,本文利用叠加法以利维解为基础将复杂问题分解为多个简单问题,然后进行叠加,获得了四边固支矩形薄板在均布荷载作用下的挠度表达式。利用有限元模拟验证表达式的正确性,研究发现有限元结果与公式结果吻合较好,本文利用叠加法求得的挠度表达式具有较高的求解精度,且表达式具有良好的收敛性。基于现有资源,推导出均布荷载下固支矩形薄板的挠度表达式,可供相关人员参考。     

采用人工神经网络求解薄板的自由振动问题

本文提出利用神经网络的非线性表示能力求解薄板的自由振动问题,分析了标准的ＢＰ算法及多种改进措施,采用动量法和自适应调整学习率的ＢＰ算法求解,比较几种计算方法的计算结果,表明采用人工神经网络计算薄板自由振动问题方法可行,结果令人满意     

均布载荷下四边固支矩形薄板的挠度

为求解四边固支矩形薄板在均布载荷作用下的挠度表达式,以利维解为基础利用叠加法将复杂问题分解为多个简单问题,然后进行叠加,获得四边固支矩形薄板在均布载荷作用下的挠度表达式.利用有限元模拟验证表达式的正确性,发现有限元结果与公式结果吻合较好.     

采用人工神经网络求解薄板的自动振动问题

本文提出利用神经网络的非线性表示能力求解薄板的自由振动问题,分析了标准的ＢＰ算法及多种改进措施,采用动量法和自适应调整学习率的ＢＰ算法求解,比较几种计算方法的计算结果,表明采用人工网络计算薄板振动问题方法可行,结果令人满意。     

基于薄板样条插值算法的巨幅影像分块并行处理

薄板样条函数是空间插值中的一种重要方法。对于巨幅影像数据使用薄板样条函数进行空间插值时,可能会出现运行时间太长,以及计算机内存空间不足或程序运行无响应的问题。针对这些问题,根据薄板样条函数光滑、连续的特点,基于GDAL开源函数库,提出对巨幅影像数据的分块读取,在块内利用并行技术求解线性方程组,确定薄板样条函数,最后进行空间插值的方法。结果表明,该方法可以有效的解决这些问题。     

自由边界问题的非协调有限元逼近及其离散问题迭代解

1.连续问题稠密性定理 来源于力学、物理及工程中的薄板弯曲问题可归结为求解下面的变分不等式:其中,K为H~2(Ω)的非空闭凸集。 R.Glowinski et al.用混合法研究了固支障碍问题和带平均曲率约束问题,A.Fusciardi et al.用混合法研究了简支障碍问题,R.Glowinski et al.用非协调元法     

一种四边形广义协调薄板弯曲单元

为了降低位移型任意四边形薄板弯曲有限元构造的难度,基于修正最小势能变分原理,提出了一种新的广义协调位移的设计方法,并且构造出一种任意四边形网格下的位移型薄板弯曲单元Q8P。这种单元能够通过常应力分片检查,当弯矩,扭矩和横向剪力沿着单元边界的分布为常值时,由于位移在单元边界非逐点协调而产生的附加能量为零。这种新型的广义协调单元保证了有限元解的收敛性,计算效率高,具有较高的数值精度,并且对网格畸变不敏感。     

Evolutionary optimization of maximizing the difference between two natural frequencies of a vibrating structure

This paper deals with the evolutionary optimization of maximizing the difference between two natural frequencies of a vibrating structure. Two new criteria, namely the material efficiency criterion and the smooth change criterion, are derived for solving this kind of evolutionary optimization problem. Using these two new criteria, the evolutionary optimization method has been further extended and applied to maximize the difference between the fundamental and the second natural frequencies of a structure under both plane stress and thin plate flexural bending conditions. The related results demonstrated that the extended evolutionary structural optimization method is useful in and applicable to dealing with the evolutionary optimization of maximizing the difference between two natural frequencies of a vibrating structure. Moreover, the results also indicated that owing to the different mechanism between plane stress and thin plate flexural bending conditions, the optimal topologies, the normalized difference between two natural frequencies and the normalized material efficiency are different for a vibrating structure under these two different situations.     

A family of rectangular bending elements

By using separate independent transverse and rotational displacement variables in terms of a polynomial it is possible to produce high order conforming elements for plate bending and, at the same time, to include the effect of shear deformation in the analysis. The procedure for constructing a family of conforming rectangular plate bending elements with any number of nodes and the derivation of the stiffness matrix are illustrated. A computer programme is developed to generate the stiffness coefficients of the elements in this family; whereupon the characteristics of elements with as many as 17, 21 or 25 nodes and so on can be investigated. It is demonstrated that accurate results can be obtained for thin and moderately thick plates with various boundary conditions under bending by using just one or a few high order elements in this family. Hence the procedure for solving a problem in plate bending can be much simplified and the total number of nodes in a problem can be much reduced. Highlight in this family is the 17-node element which yields good results without involving too many nodes for many plate bending problems.     

哈密顿体系下矩形薄板自由振动的一般解

根据弹性薄板自由振动问题的基本方程，把问题引入到哈密顿对偶体系中．x方向模拟为时间，选取弯矩，等效剪力，转角和挠度为对偶向量，得到了在不同边界条件时关于x轴对称和反对称时的解析解．算例研究了四边固支薄板的自由振动情形，从而推广了哈密顿体系的应用范围，验证了哈密顿体系求解方法在自由振动问题中的有效性．     

厚板动力响应分析的一种辛算法

将厚板动力分析从Lagrange体系改换为Hamilton体系.根据古典阴阳互补和现代对偶互补的基本思想,首次建立了线性阻尼情形下厚板动力学的相空间非传统Hamilton变分原理.这种变分原理不仅能反映这种动力学初值—边值问题的全部特征,而且它的欧拉方程具有辛结构的特征.基于该变分原理,提出一种称之为辛空间有限元—时间子域法的辛算法.这种新方法是由空间域采用有限元法与时间子域采用La-grange插值多项式插值的时间子域法相结合而成.文中用这种辛算法分析了四种支承条件下厚板的动力响应问题.算例的计算结果表明,这种新方法的稳定性、收敛性、计算精度和效率都明显高于国际上常用的W ilson-θ法和Newm ark-β法.     

A New Family of Mixed Methods for the Reissner-Mindlin Plate Model Based on a System of First-Order Equations

The mixed method for the biharmonic problem introduced in (Behrens and Guzmán, SIAM J. Numer. Anal., 2010) is extended to the Reissner-Mindlin plate model. The Reissner-Mindlin problem is written as a system of first order equations and all the resulting variables are approximated. However, the hybrid form of the method allows one to eliminate all the variables and have a final system only involving the Lagrange multipliers that approximate the transverse displacement and rotation at the edges of the triangulation. Mixed finite element spaces for elasticity with weakly imposed symmetry are used to approximate the bending moment matrix. Optimal estimates independent of the plate thickness are proved for the transverse displacement, rotations and bending moments. A post-processing technique is provided for the displacement and rotations variables and we show numerically that they converge faster than the original approximations.     

A spline wavelet finite element formulation of thin plate bending

The wavelet scaling functions of spline wavelets are used to construct the displacement interpolation functions of triangular and rectangular thin plate elements. The displacement shape functions are then expressed by spline wavelet functions. A spline wavelet finite element formulation of thin plate bending is developed by using the virtual work principle. Two numerical examples have shown that the bending deflections and moments of thin plates agree well with those obtained by the differential equations and conventional elements. It is demonstrated that the current spline wavelet finite element method (FEM) can achieve a high numerical accuracy and converges fast. The proposed spline wavelet finite element formulation has a wide range of applicability since it is developed in the same way like conventional displacement-based FEM.     

A boundary integral method applied to plates of arbitrary plan form and arbitrary boundary conditions

An integral equation method which has been applied to thin elastic clamped plates of arbitrary plan form[1] has been extended to include arbitrary boundary conditions. Numerical difficulties which arise in the case of free boundaries have been avoided by defining an integration contour which differs from the actual plate boundary. Thus second order singularities which arise in the integrand of the equations are avoided.The method is applied to two rectangular plates with mixed boundary conditions (i.e. clamped, simply-supported and free edges). Average errors are 1.5% for displacements and 3% for bending moments.