界带分析的基本理论和计算方法

基于应用力学的辛数学理论,阐述了界带分析的基本理论与计算方法,主要包括:界带分析的基本概念、界带分析的变分原理、单个与不同原子周期链的波的色散关系、子结构界带分析计算方法以及数值算例等多方面内容。研究工作在表明辛数学理论优势和潜力的同时,也为相关问题的深入研究奠定了理论与算法基础。     

一维声子晶体带隙的非局部辛分析

周期性弹性复合结构(声子晶体)中传播的弹性波存在特殊的色散关系:弹性波只能在某段频率范围内无损耗的传播,该频率范围称为通带.一维声子晶体的色散问题可以看作分层介质中弹性波的传播问题,利用二维弹性理论予以分析.为了研究非局部效应对声子晶体带隙特性的影响,将Eringen的二维非局部弹性理论引入到Hamilton体系下,利用精细积分与扩展的Wittrick Williams算法可获取任意频率范围内的本征解.通过对不同算例的数值计算,分析和对比了非局部理论方法与传统局部理论方法的差别.并进一步指出了该套算法的适用性和优势所在.     

类石墨烯二维原子晶体的微态理论模型

基于微态(Micromorphic) 连续介质理论,提出了针对类石墨烯二维原子晶体的新力学模型. 该模型以有限大小的布拉维单胞为基元体,考虑基元粒子的宏观位移和微观变形,依据微态理论基本方程,推导了全局坐标系下模型的主导方程. 然后针对布拉维单胞中含有两个原子的类石墨烯晶体,通过分析单胞中声子振动模式与基元体自由度的关系,获得了微态形式下声子色散关系的久期方程,并根据二维晶体声子色散特性对久期方程进行了简化,进而确定了类石墨烯晶体模型的本构方程. 最后,以石墨烯和单层六方氮化硼为例,利用简化的表达式拟合了它们面内声子色散关系数据,计算了模型材料的常数,石墨烯模型的等效杨氏模量、泊松比分别为1.05 TPa 和0.197,氮化硼分别为0.766 TPa 和0.225,均与已有的实验值相符合.     

连续系统的界带分析方法

基于分析结构力学提出的界带分析方法，将子结构间的分界面延拓为有一定宽度的分界带/分界域，从而可以用于分析计算结构的非局部效应。界带分析方法首先在离散结构的分析计算中取得了成功，从而验证了该套理论算法的准确性。离散结构按界带宽度（影响域范围）划分子结构，因而限制了子结构区段积分计算的最小步长；而连续系统则要求可以实现任意步长的积分运算。通过引入步进的计算方法，界带分析方法可以实现任意步长的积分计算，进而可以解决连续系统的积分问题。通过数值算例验证了连续系统的界带分析方法的准确性和可行性，也为进一步研究该套计算方法在分析动力学中的应用打下基础。     

分级周期梁结构的振动带隙特性优化研究

针对分级周期梁结构,进行了振动带隙特性优化研究,以期提高结构的减振性能。采用谱元法计算分级周期梁的频响曲线,并结合传递矩阵法计算结构的色散关系,将两种方法相结合来研究结构的振动带隙特性。构建带隙占比函数作为优化目标函数,将单胞结构的尺寸作为优化参数进行带隙特性优化。经过优化,使得在研究频段内带隙特性大大提高。通过与有限元法和振动实验相对比,验证了谱元法计算和优化结果的正确性。研究内容对于提高周期结构的振动带隙特性和减振应用提供有益参考。     

碳纳米管的等价柔性节点空间框架模型

小变形情况下,碳纳米管C-C共价键间的相互作用可以用基于分子力学的宏观力学模型进行模拟.其中,基于分子结构力学的等价结构力学模型是最为有效的碳纳米管弹性参数的预测模型.现有的碳纳米管的等价结构力学模型是用具有刚性节点的空间框架结构模拟碳纳米管的原子晶格受力和变形的关系.根据碳纳米管的原子晶格的变形特点,本文首次提出了一个用柔性节点空间框架模拟碳纳米管原子晶格键角变化的分析模型,再通过应变能等价推导了柔性节点的等价抗弯刚度与分子力学中力常数的关系,从而给出了一个更精确的计算碳纳米管等价弹性参数的分子结构力学模型.文中用ANSYS计算了不同尺寸的锯齿型(zigzag)和扶手型(armchair)单壁碳纳米管的轴向杨氏模量、泊松比、剪切模量及径向杨氏模量,分析了碳纳米管的尺寸效应,并且与其它各种模型所得结果进行了比较.计算结果表明,本文所给碳纳米管的等价柔性节点空间框架模型不仅计算简单、高效,而且准确;并可以直接推广到多壁碳纳米管等价弹性模量的计算及碳纳米管的稳定和动力分析.     

结构工程科学中若干计算结构力学问题的研究展望

本文概述了计算结构力学的形成过程，评述了结构工程科学中以下７个需要重点研究的计算结构力学问题：数值方法基本理论问题（误差估计理论、网格自适应加密技术、多变量有限元理论和半解析数值方法）、工程结构优化设计、结构施工力学、计算机数值模拟和仿真技术、本构模型、计算机技术新发展的影响、计算机辅助设计。并论述了数值计算、理论和试验这三者之间相互依赖、相互促进、相互交叉又相互制约的辩证关系。     

基于双边Lanczos算法的声子晶体复模态重分析

针对声子晶体拓扑结构修改时模态计算效率低的问题,提出基于双边Lanczos算法的复模态重分析。与全分析不同,本方法利用声子晶体原始结构的模态分析结果,通过双边Lanczos算法构建投影向量矩阵,将广义特征值方程映射进子空间来压缩矩阵规模,求解方程后再利用近似模态关系得到最终解。通过对尺寸和形状发生修改的声子晶体进行分析,验证了方法所求结果具有高精度,与全分析相比缩减了约35%的计算时间,在处理拓扑修改变化量大和计算规模大的声子晶体模态分析问题上有很大潜力。     

非结构网格热化学非平衡辐射流场数值计算

针对三维热化学非平衡辐射流场设计了基于非结构网格的数值计算方法.根据原子分子光谱理论逐条计算了100～1 500nm间N,O,N+,O+的谱线以及N2,O2,NO,N+2等分子的10个谱带,特别分析了NO的β'带,γ'带,δ带和ε带的辐射特性.采用耦合辐射的双温模型计算热化学非平衡流场,辐射源项通过直接求解辐射输运方程RTE(radiative transport equation)获得.在空间和方向上分别离散后,利用有限体积法求解不同方向上的辐射输运方程.计算得出了再入飞行器前驻点的辐射强度分布.采用该数值方法计算了MUSES-C模型在速度为11.6km/s时的绕流流场及前驻点处的辐射热流密度.并通过对比分析了热辐射对流场的影响.     

一维固液介质准周期结构声子晶体的传输特性

针对一维固液介质的Fibonacci序列准周期声子晶体，利用传输矩阵法研究了弹性纵波在准周期结构中的透射特性。计算结果表明：准周期结构具有比周期结构更宽的禁带，并伴随带边谐振；序列越高，带边谐振越强，并在较低序列的禁带中心出现谐振模；弹性波以0.1rad斜入射时，透射谱向高频方向移动；当固液介质波阻抗接近时，出现等间距的谐振模，同时禁带消失。     

Phonon dispersion analysis of carbon nanotubes based on inter-belt model and symplectic solution method

The objective of the study is to develop a totally new theory and structural mechanics model for phonon dispersion analysis of carbon nanotubes. The fundamental theory and computational algorithm for phonon dispersion analysis of carbon nanotubes are developed based on the symplectic theory and algorithm established in applied mechanics in recent years. Carbon nanotubes are simulated by two kinds of structural mechanics models, i.e. the conventional sub-structure model and the inter-belt model. The variational principle for wave dissipation analysis of periodic structure is given on the basis of the symplectic-mathematical theory. Numerical examples are carried out to demonstrate the validity of the theory and algorithm developed. By the comparison of the results obtained by the two kinds of structural mechanics models, it can be found that the inter-belt model has more advantages than the conventional sub-structure model in the calculation of phonon spectra of nanotubes. As a basic research work, the present study illustrates well the potential of the symplectic-mathematical theory as well as the inter-belt model and is valuable for the further research in computational nanomechanics.     

GUIDED CIRCUMFERENTIAL WAVES IN DOUBLE-WALLED CARBON NANOTUBES

A model of guided circumferential waves propagating in double-walled carbon nan- otubes is built by the theory of wave propagation in continuum mechanics,while the van der Waals force between the inner and outer nanotube has been taken into account in the model.The dispersion curves of the guided circumferential wave propagation are studied,and some dispersion characteristics are illustrated by comparing with those of single-walled carbon nanotubes.It is found that in double-walled carbon nanotubes,the guided circumferential waves will propagate in more dispersive ways.More interactions between neighboring wave modes may take place.In particular,it has been found that a couple of wave modes may disappear at a certain frequency and that,while a couple of wave modes disappear,another new couple of wave modes are excited at the same wave number.     

Parametric variational principle and quadratic programming method for van der Waals force simulation of parallel and cross nanotubes

A parametric variational principle for van der Waals force simulation between any two adjacent nonbonded atoms and the corresponding improved quadratic programming method for numerical simulation of mechanical behaviors of carbon nanotubes are developed. Carbon nanotubes are modeled and computed based on molecular structural mechanics model. van der Waals force is simulated by the network of bars (called bar network) with a special nonlinear mechanical constitutive law (called generalized parametric constitutive law) in the finite element analysis. Compared with conventional numerical methods, the proposed method does not depend on displacement and stress iteration, but on the base exchanges in the solution of a standard quadratic programming problem. Thus, the model and method developed present very good convergence behavior in computation and provide accurate predictions of the mechanical behaviors and displacement distributions in the nanotubes. Numerical results demonstrate the validity and the efficiency of the proposed method.     

Molecular structural mechanics approach to carbon nanotubes on graphics processing units

A molecular structural mechanics approach to carbon nanotubes on graphics processing units (GPUs) is reported. As a powerful parallel and relatively low cost processor, the GPU is used to accelerate the computations of the molecular structural mechanics approach. The data structures, matrix-vector multiplication algorithm, texture reduction algorithm, and ICCG method on the GPU are presented. The computations for Young's moduli of carbon nanotubes by the molecular structural mechanics approach on the GPU show its accuracy. The running times of large degree of freedom (DOF) carbon nanotubes, whose DOF is larger than 100,000, on the GPU are compared against those on the CPU, proving the GPU can accelerate the computations of the molecular structural mechanics approach to carbon nanotubes.     

PARAMETRIC VARIATIONAL PRINCIPLE BASED ELASTIC-PLASTIC ANALYSIS OF COSSERAT CONTINUUM

A new algorithm is developed based on the parametric variational principle for elastic-plastic analysis of Cosserat continuum. The governing equations of the classic elastic-plastic problem are regularized by adding rotational degrees of freedom to the conventional translational degrees of freedom in conventional continuum mechanics. The parametric potential energy principle of the Cosserat theory is developed, from which the finite element formulation of the Cosserat theory and the corresponding parametric quadratic programming model are constructed. Strain localization problems are computed and the mesh independent results are obtained.     

基于参数变分原理的Cosserat连续体弹塑性分析

基于参数变分原理,提出了Cosserat模型弹塑性计算的算法,给出了基于Cosserat理论的参数最小势能原理,基于所提出的变分方程,建立了Cosserat理论弹塑性分析的参数二次规划模型,进一步将算法应用于平面应变软化问题计算中,获得的结果具有良好的非网格依赖性.     

Prediction of buckling characteristics of carbon nanotubes

In this paper, to investigate the buckling characteristics of carbon nanotubes, an equivalent beam model is first constructed. The molecular mechanics potentials in a C–C covalent bond are transformed into the form of equivalent strain energy stored in a three dimensional (3D) virtual beam element connecting two carbon atoms. Then, the equivalent stiffness parameters of the beam element can be estimated from the force field constants of the molecular mechanics theory. To evaluate the buckling loads of multi-walled carbon nanotubes, the effects of van-der Waals forces are further modeled using a newly proposed rod element. Then, the buckling characteristics of nanotubes can be easily obtained using a 3D beam and rod model of the traditional finite element method (FEM). The results of this numerical model are in good agreement with some previous results, such as those obtained from molecular dynamics computations. This method, designated as molecular structural mechanics approach, is thus proved to be an efficient means to predict the buckling characteristics of carbon nanotubes. Moreover, in the case of nanotubes with large length/diameter, the validity of Euler’s beam buckling theory and a shell model with the proper material properties defined from the results of present 3D FEM beam model is investigated to reduce the computational cost. The results of these simple theoretical models are found to agree well with the existing experimental results.