An Element-Free Galerkin Method for Time-Fractional Diffusion-Wave Equations北大核心CSCD

利用无单元Galerkin法,对Caputo意义下的时间分数阶扩散波方程进行了数值求解和相应误差理论分析。首先用L1逼近公式离散该方程中的时间变量,将时间分数阶扩散波方程转化成与时间无关的整数阶微分方程;然后采用罚函数方法处理Dirichlet边界条件,并利用无单元Galerkin法离散整数阶微分方程;最后推导该方程无单元Galerkin法的误差估计公式。数值算例证明了该方法的精度和效果。     

Spectral Collocation Method for Caputo Fractional Multi-point Value Problems北大核心CSCD

Spectral collocation method is investigated for the nonlinear Caputo fractional multi-point value problems. The main idea of the presented method is to solve the corresponding nonlinear weakly singular Volterra–Fredholm integral equations obtained from the nonlinear Caputo fractional multi-point value problems. In order to carry out convergence analysis for the presented method, we investigate the Gronwall type inequality with Volterra–Fredholm integral terms. The provided convergence analysis shows that the presented method has spectral convergence, which is confirmed by the provided numerical experiments. At present, numerical methods for fractional multi-point value problems are rarely studied. The method and convergence analysis in this paper are useful references for the researches of related subjects. © 2022 Chinese Academy of Sciences. All rights reserved.     

Topology Optimization Design of Heat Convection Problems With Variable-Density Cells北大核心CSCD

为获得优异的散热结构设计,发展了一种基于腐蚀-扩散算子的变密度胞元层级结构设计方法.通过腐蚀-扩散算子得到了一系列拓扑相似但体积分数不同的变密度微结构,计算并拟合得到变密度微结构等效热传导系数曲线.在此基础上,采用移动渐近线法更新宏观设计变量,将变密度微结构植入相应体积分数的宏观单元中完成装配.通过数值算例对不同优化方法下温度场的热柔顺度、平均温度、方差等参数进行了比较分析,结果表明,变密度胞元层级结构比传统单尺度胞元结构和周期胞元结构具有更好的散热性能.     

Thermal Buckling Analysis of FGM Sandwich Circular Plates Under Transverse Nonuniform Temperature Field Actions北大核心CSCD

Based on the von Kármán geometric nonlinear plate theory, the displacement⁃type geometric nonlinear governing equations for FGM sandwich circular plates under transverse nonlinear temperature field actions were derived. With the immovable clamped boundary condition, the analytical formula for dimensional critical buckling temperature differences of the system was obtained from the solution of the linear eigenvalue problem. Moreover, the 2⁃point boundary value problem of ordinary differential equations was solved with the shooting method. The effects of geometric parameters, constituent material properties, gradient indexes, temperature field parameters and layer⁃thickness ratios on the critical buckling temperature differences, the thermal postbuckling equilibrium paths, and the buckling equilibrium configurations of FGM sandwich circular plates, were investigated. The results show that, with the increases of the thickness⁃radius ratio, the relative thickness of the FGM layer and the gradient index, the FGM sandwich circular plate's critical buckling temperature difference will increase monotonically. Given a fixed radius and a fixed total thickness, the postbuckling deformation of the FGM sandwich circular plate will decrease significantly with the relative thickness of the FGM layer. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Multi-Scale Prediction of Thermal and Mechanical Properties of C/SiC Braided Composites北大核心CSCD

C/ SiC composites have been widely used in aerospace, national defense, and chemical industries due to their excellent mechanical and thermal properties. Accurate knowledge about the mechanical/ thermal properties of C/ SiC composites is very important for their efficient application in related fields. Based on the representative volume element (RVE) and periodic boundary conditions, a micro/ meso single-cell model for C/ SiC composites was established in view of the non-uniform and multi-scale characteristics of fiber bundles, such as the volume fraction, the interweaving mode, and the weaving dimension. The finite element software ABAQUS was used to predict the micro-scale thermal and mechanical properties of the fiber bundle, and the fiber bundle properties were introduced into the mesoscopic model to analyze and obtain the macroscopic thermal and mechanical properties of the composite. Based on this multi-scale correlation analysis method, the thermal conductivity and thermal expansion coefficient of fiber bundles and C/ SiC composites were further studied at the operating temperatures ranging from 27~1 227 ℃. The study has certain guiding significance for the application of C/ SiC composites in engineering. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

A Complex Mode Method for Wind-Induced Responses of 6-Parameter Practical Viscoelastic Damping Energy Dissipation Structures Based on the Davenport Wind Speed Spectrum北大核心CSCD

针对六参数实用黏弹性阻尼耗能结构,基于Davenport风速谱系列响应问题进行了系统的研究.首先,利用六参数黏弹性阻尼器的微分型本构关系,建立了耗能结构基于Davenport风速谱激励下的运动方程;然后,运用复模态法将耗能结构的运动方程由二阶微分方程转化为一阶方程,获得了耗能结构系统对风振激励响应的频域解和功率谱密度函数表达式;最后,利用数学恒等式,基于随机振动理论获得了耗能结构系统在Davenport风速谱激励下的响应和阻尼器受力的解析解.该文方法不仅考虑了结构系统在风振激励作用下全振型展开的结果,表达式较现有结果更为简便,效率及精度更高,且适用于非经典阻尼结构.     

交错排列结构的多尺度渐近分析

针对一类交错排列结构上的具有快速振荡系数的椭圆问题进行了多尺度渐近分析.证明了多尺度渐近展开方法的相关基础定理和多尺度解的误差估计.数值算例验证了所提出的多尺度有限元算法的有效性.进一步地,讨论了不同交错排列方式对材料等效性能的影响.     

车桥耦合系统随机振动的时域显式解法

在桥面和轨道随机不平顺作用下,车桥耦合系统振动是一个典型的非平稳随机振动问题.笔者分别建立表征物理演变机制的车辆系统和桥梁系统的动力响应显式表达式,然后利用车桥之间的运动相容条件,建立车桥之间接触力关于桥面不平顺的显式表达式.在此基础上,即可直接利用统计矩运算法则,获得车桥接触力的统计矩演化规律,并进一步计算车辆系统和桥梁系统关键响应的演变统计矩.此外,也可以基于车桥接触力关于桥面不平顺的显式表达式,高效地进行随机模拟（即Monte Carlo模拟, MCS）,以获得车桥耦合系统关键响应的演变统计矩及其他统计信息.在上述过程中,由于实现了车桥耦合系统物理演变机制和概率演化规律的相对分离,在响应统计矩计算中,无需反复求解车桥耦合系统的运动微分方程,且可以仅针对车桥接触力及其他所关注的关键响应开展降维计算,大幅提高了车桥耦合系统随机振动的计算效率.数值算例表明,所提出的方法具有理想的计算精度和计算效率.     

陀螺系统随机振动分析的辛本征展开方法

探讨了受随机载荷作用下陀螺阻尼系统随机动力响应问题.虚拟激励法作为随机振动分析的一种高效、精确方法已经广泛应用于结构抗震、抗风等工程领域.在以单类物理变量描述的Lagrange(拉格朗日)体系框架下,振型分解方法已被有效应用于上述随机振动问题的模型自由度缩减.然而,对于陀螺系统的随机振动问题,由于陀螺效应的存在,基于Rayleigh商本征值的振型分解方法受到很大限制.对此,首先给出了陀螺系统辛本征值问题的一般形式.然后对于受平稳随机载荷激励的陀螺系统(无阻尼或有阻尼)引入虚拟激励法,基于辛本征空间展开推导了系统随机振动响应功率谱的求解列式;对于仅考虑陀螺效应的保守系统(无阻尼),该求解列式可以表述为一个显式表达式.在数值算例中,应用该文提出的方法分析了平稳随机载荷作用下一类阻尼陀螺系统的随机振动响应问题,通过与其它方法进行对比,验证了该方法的精确性和有效性.     

Explicit Bounds for the Return Probability of Simple Random Walks

We give an exact computation of the second order term in the asymptotic expansion of the return probability, P_2n^d(0,0), of a simple random walk on the d-dimensional cubic lattice. We also give an explicit bound on the remainder. In particular, we show that P_2n^d(0,0) < 2 (d/4n)^d/2 where n M=M(d) is explicitly given.     

A Multiscale Multilevel Monte Carlo Method for Multiscale Elliptic PDEs with Random Coefficients

We propose a multiscale multilevel Monte Carlo(MsMLMC) method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting. Our method consists of offline and online stages. In the offline stage,we construct a small number of reduced basis functions within each coarse grid block, which can then be used to approximate the multiscale finite element basis functions. In the online stage, we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis.The MsMLMC method can be applied to multiscale RPDE starting with a relatively coarse grid, without requiring the coarsest grid to resolve the smallestscale of the solution. We have performed complexity analysis and shown that the MsMLMC offers considerable savings in solving multiscale elliptic PDEs with random coefficients. Moreover, we provide convergence analysis of the proposed method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.     

随机结构动力反应和可靠性分析

本研究随机动力荷载作用下随机结构反应有限元分析方法，提出了随机结构动力反应和基于随机抗力可靠性计算公式。     

分析非线性系统随机响应的一种等效非线性化方法*

本文提出了分析非线性系统随机响应的一种新的等效非线性化方法。文中阐述了该方法的基本思想和处理方法,并对几种常见类型的非线性系统进行了分析和计算,结果表明,利用本文提出方法所得的均方位移响应与精确解或者Monte Carlc模拟解之间具有较好的一致性,并比等效线性化方法有更高的精度。     

模糊随机桁架结构的动力特性双因子分析法

针对模糊随机桁架结构的动力特性分析,提出了一种新的模糊随机有限元方法．当结构的物理参数和几何尺寸同时具有模糊随机性时,利用模糊因子法和随机因子法建立了结构刚度矩阵和质量矩阵;从结构振动的Rayleigh商表达式出发,利用区间运算推导出结构动力特性模糊随机变量的计算表达式;之后利用随机变量的矩法和代数综合法,推导出结构特征值的数字特征的计算式．通过算例分析了模糊随机桁架结构参数的模糊随机性对其动力特性的影响．该方法的优点是能准确反映结构某一参数的模糊随机性对结构特征值及其数字特征的影响．     

Analysis of the heterogeneous multiscale method for elliptic homogenization problems

A comprehensive analysis is presented for the heterogeneous multiscale method (HMM for short) applied to various elliptic homogenization problems. These problems can be either linear or nonlinear, with deterministic or random coefficients. In most cases considered, optimal estimates are proved for the error between the HMM solutions and the homogenized solutions. Strategies for retrieving the microstructural information from the HMM solutions are discussed and analyzed.

     

重复结构振动控制的降维方法

提出了对称结构、旋转周期结构和链式结构的振动控制的降维方法．以某种对称的方式设置广义坐标的凝聚、传感器、驱动器的位置以及输入与控制力的关系,即可使控制系统具有和结构同样的重复性．对凝聚了的广义坐标和系统输入采用适当的变换,即可通过执行一些子结构的控制问题实现整体系统的振动控制,从而使控制问题的维度显著降低．