一类含分数阶阻尼的三维波导中的传播问题

本文研究一类三维波动方程,该方程含有分数阶小阻尼,边界含有小参数,并做正弦波动.我们利用多重尺度方法和Riemann-Liouville分数阶导数的定义及性质,对原边值问题应用泰勒公式,得到关于小参数的零阶和一阶方程边值问题.利用分离变量法,引入解谐参数,通过分析边值问题的可解性条件得到零阶近似解的振幅和相位的变化规律...     

外激励力作用下的轴向运动梁非线性振动的联合共振

研究轴向运动梁在外激励力作用下非线性振动的联合共振问题.利用哈密顿原理建立横向振动的轴向运动梁的振动微分方程,采用分离变量法分离时间变量和空间变量并利用Galerkin方法离散运动方程.采用IHB法进行非线性振动求解,分析在内共振条件且外激励作用下的联合共振问题,对周期解进行稳定性的判定.典型算例获得了不同外激励力振幅时系统非线性振动的复杂频幅响应曲线.     

时间分数阶Fokker-Planck方程的Jacobi谱配置方法

分数阶微分方程在工程、生物、金融等领域有广泛的应用．本文利用分数阶积分和微分公式的关系,针对一类带Dirichlet边值条件的时间分数阶Fokker-Planck方程,将其转化为与之等价的带有奇异核的积分微分方程,然后用高斯积分公式数值求解积分项,在时间和空间上都采用Jacobi谱配置法来离散求解积分微分方程．数值算例的结果表明,该方法是非常有效的,数值解具有谱精度,并且该方法容易推广到高维和非线性的情形．     

圆形薄膜自由振动的理论解

本文研究圆形薄膜的自由振动。首先根据哈密顿原理建立薄膜横向振动的动力学方程,然后采用分离变量法,导出时间t\、径向坐标r和环向坐标 变量分离的2个二阶常微分方程和1个贝塞尔方程并分别求解,求得周边固定圆形薄膜、扇形薄膜自由振动的理论解,从而得到固有频率及其振型的解析表达式。最后,应用ANSYS有限元计算软件计算上述几种类型自由振动的频率及其模态并与理论解比较。ANSYS有限元数值解与理论解二者十分接近,理论解是有限元数值解的下限。     

一类含时间分数阶导数的膜振动方程近似解析解的研究

研究一类含时间分数阶导数的膜振动方程,该方程边界正弦摄动变化。先对边界自变量应用泰勒级数展开,引入多重尺度到原方程及边界,利用Riemann-Liouville分数阶导数定义和性质得到关于小参数的零阶近似解。应用微分不等式理论证明了解的一致有效性。利用图形分析出各参数对解的影响。     

刚／粘塑性梁的强迫振动

本文依据粘塑性梁强迫振动的非齐次方程与非线性本构方程,提出采用分离变量的位移方法求解,获得该问题的应力和位移解.     

退化环面上的非线性jerk方程近似周期解的同伦分析方法

用同伦分析法求解退化环面上的非线性Jerk方程的近似周期和近似解析周期解。所得结果表明文中得到的一阶近似周期和一阶近似解析周期解与Gottlieb用低阶谐波平衡法求解得到的结果一样。当参数和初速度较大时,一阶近似周期与精确周期的百分比误差是4.831 8%,而二阶近似周期与精确周期的百分比误差小于0.219 9%。与数值方法给出的"精确"周期解比较,二阶近似解析周期解比一阶近似解析周期解要精确的多。因此,同伦分析法是求解非线性Jerk方程的一种非常有效的方法。     

缓增分数阶扩散方程的高阶时间离散LDG方法

研究了缓增分数阶扩散方程的高阶时间离散局部间断Galerkin (Local Discontinuous Galerkin, LDG)方法，不是直接求解缓增分数阶扩散方程，而是首先通过变换将其转化成Caputo型时间分数阶扩散方程。接着，采用L1-2差分逼近离散Caputo型分数阶导数，间断有限元离散空间变量，构造求解模型的全离散LDG格式。证明了所建立的全离散格式为无条件稳定的且具有最优误差阶，两个数值算了验证了所建立数值格式的精度和鲁棒性。数值实验结果表明所建立格式在时间和空间方向均具有高精度。     

高速光驱粘弹性阻尼减振框架结构的动力学方程及有限元数值解

对安装有粘弹性阻尼器的高速光驱框架结构进行了理论建模，粘弹性阻尼器采用分数Kelvin固体模型，建立了高速光驱粘弹性阻尼减振框架结构的分数阶动力学有限元方程，并利用Newmark数值积分法得到了数值解     

一类离散时间代数Riccati矩阵方程异类约束解的双迭代算法

本文研究在最优控制系统中遇到的离散时间代数Riccati矩阵方程(DTARME)异类约束解的数值计算问题.首先对多变量DTARME中的逆矩阵采用矩阵级数方法进行等价转化,然后采用牛顿算法求多变量DTARME的异类约束解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的异类约束解或者异类约束最小二乘解,建立求多变量DTARME的异类约束解的双迭代算法.双迭代算法仅要求多变量DTARME有异类约束解,不要求它的异类约束解唯一,也不对它的系数矩阵做附加限定.数值算例表明,双迭代算法是有效的.     

Post-buckling analysis of viscoelastic plates with fractional derivative models

The post-buckling response of thin plates made of linear viscoelastic materials is investigated. The employed viscoelastic material is described with fractional order time derivatives. The governing equations, which are derived by considering the equilibrium of the plate element, are three coupled nonlinear fractional partial evolution type differential equations in terms of three displacements. The nonlinearity is due to nonlinear kinematic relations based on the von Kármán assumption. The solution is achieved using the analog equation method (AEM), which transforms the original equations into three uncoupled linear equations, namely a linear plate (biharmonic) equation for the transverse deflection and two linear membrane (Poisson’s) equations for the inplane deformation under fictitious loads. The resulting initial value problem for the fictitious sources is a system of nonlinear fractional ordinary differential equations, which is solved using the numerical method developed recently by Katsikadelis for multi-term nonlinear fractional differential equations. The numerical examples not only demonstrate the efficiency and validate the accuracy of the solution procedure, but also give a better insight into this complicated but very interesting engineering plate problem     

Vibrations of inhomogeneous anisotropic viscoelastic bodies described with fractional derivative models

The dynamic response of plane inhomogeneous anisotropic bodies made of linear viscoelastic materials is investigated. The mechanical behavior of the viscoelastic material is described by differential constitutive equations with fractional order derivatives. The governing equations, which are derived by considering the dynamic equilibrium of the plane body element, are two coupled linear fractional evolution partial differential equations in terms of the displacements, whose order is in general greater than two with respect to time derivatives. A method is presented to establish the additional required initial conditions beside the described initial displacements and velocities. Using the Analog Equation Method (AEM) in conjunction with the Domain Boundary Element Method (D/BEM) the governing equations are transformed into a system of multi-term ordinary fractional differential equations (FDEs), which are solved using the numerical method for multi-term FDEs developed recently by Katsikadelis. Numerical examples are presented, which not only demonstrate the efficiency of the solution procedure and validate its accuracy, but also permit a better understanding of the dynamic response of plane bodies described by different viscoelastic models.     

Investigation on a thermo-piezoelectric problem with temperature-dependent properties under fractional order theory of thermoelasticity

The dynamic response of a one-dimensional thermo-piezoelectric problem with variable material properties is investigated in the context of fractional order theory of thermoelasticity. The system of governing equations for the thermo-piezoelectric problem is formulated and then solved by means of Laplace transform together with its numerical inversion. The distributions of the considered non-dimensional displacement, temperature, and stress are obtained and illustrated graphically. The effects of fractional order parameter and temperature-dependent properties on the considered quantities are investigated and the results show that they have significant influence on the variations of the considered quantities.     

考虑瞬态响应的薄板结构阻尼材料层拓扑优化

讨论了敷设阻尼材料的薄板结构考虑瞬态响应时阻尼材料层的最优布局问题。基于SIMP方法构造人工阻尼材料惩罚模型和结构拓扑优化模型,以阻尼材料的相对密度作为设计变量,在给定阻尼材料用量的条件下,最小化结构瞬态位移响应的时间积分。由于结构整体呈现非比例阻尼特性,采用逐步积分法对结构的振动方程进行求解。通过伴随变量法得到目标函数对设计变量的灵敏度表达式,在此基础上采用基于梯度的移动渐近线方法求解。数值算例验证了优化模型与算法的合理性和有效性。     

Discrete Fourier-series method in problems of bending of variable-thickness rectangular plates

The approach to the solution of the boundary-value problems of bending of elastic rectangular plates of variable thickness is presented. It is proposed to introduce into the resolving system of partial differential equations additional functions which enables the variables to be formally separated and the problem to be reduced to a unidimensional one by representing all the functions as a Fourier series in a single coordinate. In this case the problem can be solved by the stable numerical method of discrete orthogonalization. To calculate the additional functions, Fourier series of discretely assigned functions with allowance for variations in the plate thickness are used. The boundary-value problems for rectangular plates of variable thickness were solved assuming that their weight is unchanged.     

Nonlinear free vibration of a composite rectangular specially-orthotropic plate with variable fiber spacing

The geometrically nonlinear free vibration of a composite rectangular plate with variable fiber spacing is investigated. The investigation is limited to a single ply composite having straight and parallel fibers. The fibers are distributed more densely in the central region where high stiffness is needed than in other regions. The assumptions of von Karman’s nonlinear thin plate theory are made. The problem is solved numerically using the hierarchical finite element method. The nonlinear equations of free motion are mapped from the time domain to the frequency domain using the harmonic balance method. The resultant nonlinear equations are solved iteratively using the linearized updated mode method. Results for the fundamental linear and nonlinear frequencies are obtained for simply supported and clamped composite square plates with three variable distributions of E-Glass, Graphite, and Boron fibers in Epoxy matrices. The efficiency of the hierarchical finite element procedure is demonstrated through convergence and comparison with published data. The variable fiber spacing, fiber volume fraction, type of fiber material, and boundary conditions are shown to influence the hardening behavior.     

Analyzing three-dimensional potential problems with the improved element-free Galerkin method

The potential problem is one of the most important partial differential equations in engineering mathematics. A potential problem is a function that satisfies a given partial differential equation and particular boundary conditions. It is independent of time and involves only space coordinates, as in Poisson’s equation or the Laplace equation with Dirichlet, Neumann, or mixed conditions. When potential problems are very complex, both in their field variable variation and boundary conditions, they usually cannot be solved by analytical solutions. The element-free Galerkin (EFG) method is a promising method for solving partial differential equations on which the trial and test functions employed in the discretization process result from moving least-squares (MLS) interpolants. In this paper, by employing improved moving least-squares (IMLS) approximation, we derive the formulas for an improved element-free Galerkin (IEFG) method for three-dimensional potential problems. Because there are fewer coefficients in the IMLS approximation than in the MLS approximation, and in the IEFG method, fewer nodes are selected in the entire domain than in the conventional EFG method, the IEFG method should result in a higher computing speed.     

Time‐dependent fractional advection–diffusion equations by an implicit MLS meshless method

Recently, many new applications in engineering and science are governed by a series of fractional partial differential equations (FPDEs). Unlike the normal partial differential equations (PDEs), the differential order in an FPDE is with a fractional order, which will lead to new challenges for numerical simulation, because most existing numerical simulation techniques are developed for the PDE with an integer differential order. The current dominant numerical method for FPDEs is finite difference method (FDM), which is usually difficult to handle a complex problem domain, and also difficult to use irregular nodal distribution. This paper aims to develop an implicit meshless approach based on the moving least squares (MLS) approximation for numerical simulation of fractional advection–diffusion equations (FADE), which is a typical FPDE The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless strong‐forms. The stability and convergence related to the time discretization of this approach are then discussed and theoretically proven. Several numerical examples with different problem domains and different nodal distributions are used to validate and investigate the accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of the FADE. Copyright © 2011 John Wiley & Sons, Ltd.     

An implicit–explicit solution method for electro‐osmotic flow through three‐dimensional micro‐channels

This paper presents a comprehensive finite‐element modelling approach to electro‐osmotic flows on unstructured meshes. The non‐linear equation governing the electric potential is solved using an iterative algorithm. The employed algorithm is based on a preconditioned GMRES scheme. The linear Laplace equation governing the external electric potential is solved using a standard pre‐conditioned conjugate gradient solver. The coupled fluid dynamics equations are solved using a fractional step‐based, fully explicit, artificial compressibility scheme. This combination of an implicit approach to the electric potential equations and an explicit discretization to the Navier–Stokes equations is one of the best ways of solving the coupled equations in a memory‐efficient manner. The local time‐stepping approach used in the solution of the fluid flow equations accelerates the solution to a steady state faster than by using a global time‐stepping approach. The fully explicit form and the fractional stages of the fluid dynamics equations make the system memory efficient and free of pressure instability. In addition to these advantages, the proposed method is suitable for use on both structured and unstructured meshes with a highly non‐uniform distribution of element sizes. The accuracy of the proposed procedure is demonstrated by solving a basic micro‐channel flow problem and comparing the results against an analytical solution. The comparisons show excellent agreement between the numerical and analytical data. In addition to the benchmark solution, we have also presented results for flow through a fully three‐dimensional rectangular channel to further demonstrate the application of the presented method. Copyright © 2007 John Wiley & Sons, Ltd.     

New Finite Difference Methods Based on IIM for Inextensible Interfaces in Incompressible Flows

In this paper, new finite difference methods based on the augmented immersed interface method (IIM) are proposed for simulating an inextensible moving interface in an incompressible two-dimensional flow. The mathematical models arise from studying the deformation of red blood cells in mathematical biology. The governing equations are incompressible Stokes or Navier-Stokes equations with an unknown surface tension, which should be determined in such a way that the surface divergence of the velocity is zero along the interface. Thus, the area enclosed by the interface and the total length of the interface should be conserved during the evolution process. Because of the nonlinear and coupling nature of the problem, direct discretization by applying the immersed boundary or immersed interface method yields complex nonlinear systems to be solved. In our new methods, we treat the unknown surface tension as an augmented variable so that the augmented IIM can be applied. Since finding the unknown surface tension is essentially an inverse problem that is sensitive to perturbations, our regularization strategy is to introduce a controlled tangential force along the interface, which leads to a least squares problem. For Stokes equations, the forward solver at one time level involves solving three Poisson equations with an interface. For Navier-Stokes equations, we propose a modified projection method that can enforce the pressure jump condition corresponding directly to the unknown surface tension. Several numerical experiments show good agreement with other results in the literature and reveal some interesting phenomena.