共查询到20条相似文献,搜索用时 193 毫秒
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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. 相似文献
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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. 相似文献
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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。 相似文献
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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。 相似文献
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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. 相似文献
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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 相似文献
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《Engineering Analysis with Boundary Elements》2012,36(12):1894-1907
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. 相似文献
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Yongbin Ma 《先进材料力学与结构力学》2019,26(6):552-558
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. 相似文献
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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. 相似文献
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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. 相似文献
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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. 相似文献
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P. Zhuang Y. T. Gu F. Liu I. Turner P. K. D. V. Yarlagadda 《International journal for numerical methods in engineering》2011,88(13):1346-1362
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. 相似文献
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S. P. Singh P. Nithiarasu P. F. Eng R. W. Lewis A. K. Arnold 《International journal for numerical methods in engineering》2008,73(8):1137-1152
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. 相似文献
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New Finite Difference Methods Based on IIM for Inextensible Interfaces in Incompressible Flows
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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. 相似文献