Simulation of the interaction of light with 3-D metallic nanostructures using a proper orthogonal decomposition-Galerkin reduced-order discontinuous Galerkin time-domain method

In this artice, we report on a reduced-order model (ROM) based on the proper orthogonal decomposition (POD) technique for the system of 3-D time-domain Maxwell's equations coupled to a Drude dispersion model, which is employed to describe the interaction of light with nanometer scale metallic structures. By using the singular value decomposition (SVD) method, the POD basis vectors are extracted offline from the snapshots produced by a high order discontinuous Galerkin time-domain (DGTD) solver. With a Galerkin projection and a second order leap-frog (LF₂) time discretization, a discrete ROM is constructed. The stability condition of the ROM is then analyzed. In particular, when the boundary is a perfect electric conductor condition, the global energy of the ROM is bounded, which is consistent with the characteristics of global energy in the DGTD method. It is shown that the ROM based on Galerkin projection can maintain the stability characteristics of the original high dimensional model. Numerical experiments are presented to verify the accuracy, demonstrate the capabilities of the POD-based ROM and assess its efficiency for 3-D nanophotonic problems.     

Weighted isoperimetric inequalities in warped product manifolds

We prove some sharp isoperimetric type inequalities in warped product manifolds, or more generally, multiply warped product manifolds. We also relate them to inequalities involving the higher order mean-curvature integrals. Applications include some sharp eigenvalue estimates, Pólya-Szegö inequality, Faber-Krahn inequality, Sobolev inequality and some sharp geometric inequalities in some warped product spaces.     

Transition mechanisms of translational motions of bubbles in an ultrasonic field

The translation behaviors of oscillating bubbles are closely related to the polymerizations and dispersions between them, which are crucial for the ultrasonic cavitation effect. In this study, six types of translational motion of bubbles with a wide range of sizes (2–100 μm) in the R₀₁-R₀₂ plane are investigated. Our results demonstrate that in addition (to the 2nd order harmonic), the 1/2 order subharmonic can change the bubble pairs from the three states of the attraction, stable after attraction, and repulsion to that of the repulsion, coalescence, and attraction, respectively. Furthermore, within the range of the main resonance radius and the 1/2 order subharmonic resonance radius, the chaotic bubble pairs with alternating attractive and repulsive forces appear in the region between the coalescence pairs and stable pairs after attraction. Finally, the corresponding physical mechanisms of the chaotic translational motions are also revealed.     

八节点等参元滑动接触算法的改进

分析认为采用八节点等参元计算滑动接触问题时,往往存在较大的接触应力计算误差的主要原因在于这种单元具有两种不同的节点。为此,本文提出了双模拟接触边界法,即分别采用两种不同的节点,形成两个模拟接触边界,并限制“从接触边界”上的节点只能和“主接触边界”上其同类节点形成的模拟接触边界形成接触关系,从而有效的减小了采用八节点等参元计算滑动接触问题时的误差。     

高阶奇异项及J积分表达式

从４种典型裂纹情况的特征展开式出发，运用其微分性质和伪正交性质得到了含有高阶奇异项的Ｊ积分表达式．它们是通用的显函数表征的．本研究支持了Ｈｕｉ和Ｒｕｉｎａ的杰出研究和重要观点，即高阶奇异项和非奇异项一样在小范围屈服描述中扮演着重要角色．研究表明，常规柔度方法确定的Ｊ积分已隐含了高阶奇异项的贡献．断裂准则有可能由含高阶奇异项和非奇异项的Ｊ积分的临界值来描述     

饱和地基上含刚核弹性圆板的竖向振动分析

基于作者提出的饱和土弹性波动方程，研究了饱和地基上含刚核弹性圆板在竖向集中谐和力作用下的振动特性。首先应用Hankel积分变换求解该饱和土波动方程，然后按混合边值条件建立起一组描述含刚核弹性圆板振动的对偶积分方程，并将其化为第二类Fredholm积分方程进行数值求解。文末给出了含刚核弹性圆板在饱和地基上振动的阻抗函数随无量纲频率a0的变化曲线，并考察了土的渗透系数，弹性板含刚域的大小以及板的柔度等参数对阻抗函数的影响，得出了一些有意义的结论。     

一种变厚度的精化壳体理论及其应用

在Reddy型高阶壳体理论的基础上，采用沿壳体厚度方向的剪切应变呈抛物线分布并且能够满足在壳体的上下表面为零的假设，发展出了一种适合于对变厚度壳体进行非线性分析的方法。该方法利用Ritz原理得到问题的控制方程。通过对一种典型的变厚度壳体结构（子午线轮胎）的结构分析，该方法的计算结果与商用有限元软件的三维分析结果能够很好的吻合。表明了该方法的适用性和有效性。     

有限厚度砂床对波浪载荷的响应

本文采用水-土两相介质混合物的连续介质力学理论分析了有限厚度砂床对波浪载荷的响应。土床看作饱和孔隙弹性介质,用Biot理论来描述流动和变形的耦合效应,波浪用线性水波描述。在上述假设下得到了描述土床对波浪载荷响应的基本方程。用解沂-数值方法给出了土床中位移、应力和孔隙水压力的分布并将理沦计算结果与波浪-砂床模型试验的实验资料进行了比较。文章还对影响土床对波浪响应的参量特性进行了分析和讨论。根据参量研究、理论计算与实验资料比较的结果,可以得出结论:本文所提供的理论模型和计算方法能用来估算海床对波浪载荷的影响。     

非线性动力系统的规范形和余维3退化分叉

本文里我们利用矩阵表示法计算了具有Z_(2~-)对称性时非线性动力系统的高阶规范形,求出了余维2退化和余维3退化情况下相应规范形的普适开折。最后利用所得到的规范形和普适开折讨论了非线性动力系统的余维3退化分叉。     

一阶非线性项、四阶色散项的Boussinesq类方程

推导了由一阶色散项Ｏ（β２）表示的Ｂｏｕｓｉｎｅｓｑ类方程，方程中保留了一阶非线性项Ｏ（α）及四阶色散项Ｏ（β８），其中α＝Ａ／ｈ０，β＝ｈ０／Ｌ，Ａ为特征波高，Ｌ为特征波长，ｈ０为特征水深从理论上证明了Ｂｏｕｓｉｎｅｓｑ改善型方程对色散性精度的提高，阐明了此类方程对色散项所保留的精度为Ｏ（β８），而并非是此类方程推导之初的假设为Ｏ（β２）这一点，将改变人们传统的认识