Coupling of high order multiplication perturbation method and reduction method for variable coefcient singular perturbation problems

Based on the precise integration method(PIM), a coupling technique of the high order multiplication perturbation method(HOMPM) and the reduction method is proposed to solve variable coefcient singularly perturbed two-point boundary value problems(TPBVPs) with one boundary layer. First, the inhomogeneous ordinary diferential equations(ODEs) are transformed into the homogeneous ODEs by variable coefcient dimensional expansion. Then, the whole interval is divided evenly, and the transfer matrix in each sub-interval is worked out through the HOMPM. Finally, a group of algebraic equations are given based on the relationship between the neighboring sub-intervals, which are solved by the reduction method. Numerical results show that the present method is highly efcient.     

Coupling of high order multiplication perturbation method and reduction method for variable coefficient singular perturbation problems

Based on the precise integration method （PIM）, a coupling technique of the high order multiplication perturbation method （HOMPM） and the reduction method is proposed to solve variable coefficient singularly perturbed two-point boundary value prob lems （TPBVPs） with one boundary layer. First, the inhomogeneous ordinary differential equations （ODEs） are transformed into the homogeneous ODEs by variable coefficient dimensional expansion. Then, the whole interval is divided evenly, and the transfer ma trix in each sub-interval is worked out through the HOMPM. Finally, a group of algebraic equations are given based on the relationship between the neighboring sub-intervals, which are solved by the reduction method. Numerical results show that the present method is highly efficient.     

Precise integration method for solving singular perturbation problems

This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method.     

奇异Hamilton系统矩阵的精细积分法

常规位移有限元的结构振动方程是n个二阶常微分方程组.采用一般交分原理推导,将结构振动问题引入Hamiltoil体系,将得到2n个一阶常微分方程组.精细积分法宜于处理一阶方程,应用于线性定常结构动力问题求解,可以得到在数值上逼近精确解的结果.对于非齐次动力方程,当结构具有刚体位移时,系统矩阵将出现奇异.本文借鉴全元选大元高斯-约当法求解线性方程组的经验,提出全元选大元法求奇异矩阵零本征解的方法,该方法可以简便快速地寻求奇异矩阵零本征值对应的子空间.利用Hamiltoil体系已有研究成果及Hamilton系统的共轭辛正交归一关系,迅速将零本征值对应的子空间分离出来,通过投影排除奇异部分,然后用精细积分法求得问题的解.数值算例表明,该方法对Hamilton系统奇异问题,处理方便,计算量小,易于实现,同时保持了精细算法的优点.     

High order multiplication perturbation method for singular perturbation problems

This paper presents a high order multiplication perturbation method for sin- gularly perturbed two-point boundary value problems with the boundary layer at one end. By the theory of singular perturbations, the singularly perturbed two-point boundary value problems are first transformed into the singularly perturbed initial value problems. With the variable coefficient dimensional expanding, the non-homogeneous ordinary dif- ferential equations （ODEs） are transformed into the homogeneous ODEs, which are then solved by the high order multiplication perturbation method. Some linear and nonlinear numerical examples show that the proposed method has high precision.     

High order symplectic conservative perturbation method for time-varying Hamiltonian system

This paper presents a high order symplectic conservative perturbation method for linear time-varying Hamiltonian system.Firstly,the dynamic equation of Hamiltonian system is gradually changed into a high order perturbation equation,which is solved approximately by resolving the Hamiltonian coefficient matrix into a "major component" and a "high order small quantity" and using perturbation transformation technique,then the solution to the original equation of Hamiltonian system is determined through a series of inverse transform.Because the transfer matrix determined by the method in this paper is the product of a series of exponential matrixes,the transfer matrix is a symplectic matrix;furthermore,the exponential matrices can be calculated accurately by the precise time integration method,so the method presented in this paper has fine accuracy,efficiency and stability.The examples show that the proposed method can also give good results even though a large time step is selected,and with the increase of the perturbation order,the perturbation solutions tend to exact solutions rapidly.     

结构动力方程的增维精细积分法

对线性定常结构动力系统提出的精细积分方法,能够得到在数值上逼近于精确解的结果,但对于非齐次动力方程涉及到矩阵求逆的困难。提出采用增维的办法,将非齐次动力方程转化为齐次动力方程,在实施精细积分过程中不必进行矩阵求逆,这种方法对于程序实现和提高数值稳定性十分有利,而且在大型问题中计算效率较高,从而改进了精细积分方法的应用,数值例题显示了本文方法的有效性。     

Van der Pol type self-excited micro-cantilever probe of atomic force microscopy

A technique for dimensional reduction of nonlinear delay differential equations (DDEs) with time-periodic coefficients is presented. The DDEs considered here have a canonical form with at most cubic nonlinearities and periodic coefficients. The nonlinear terms are multiplied by a perturbation parameter. Perturbation expansion converts the nonlinear response problem into solutions of a series of nonhomogeneous linear ordinary differential equations (ODEs) with time-periodic coefficients. One set of linear nonhomogeneous ODEs is solved for each power of the perturbation parameter. Each ODE is solved by a Chebyshev spectral collocation method. Thus we compute a finite approximation to the nonlinear infinite-dimensional map for the DDE. The linear part of the map is the monodromy operator whose eigenvalues characterize stability. Dimensional reduction on the map is then carried out. In the case of critical eigenvalues, this corresponds to center manifold reduction, while for the noncritical case resonance conditions are derived. The accuracy of the nonlinear Chebyshev collocation map is demonstrated by finding the solution of a nonlinear delayed Mathieu equation and then a milling model via the method of steps. Center manifold reduction is illustrated via a single inverted pendulum including both a periodic retarded follower force and a nonlinear restoring force. In this example, the amplitude of the limit cycle associated with a flip bifurcation is found analytically and compared to that obtained from direct numerical simulation. The method of this paper is shown by example to be applicable to systems with strong parametric excitations.     

分层介质中Love波的扩展W-W算法

应用精细积分法(PIM)和扩展Wittrick-Williams(W-W)算法求解横观各向同性分层半空间中的Love波问题.Love波对应于波数-频率域线性常微分方程的本征值问题.精细积分法是求解线性常微分方程两端边值问题和初值问题的高精度算法.利用本征值计数技术,扩展W-W算法可以不遗漏地找到所有本征值.因此,文中使用的方法可以得到计算机精度意义下的精确解.     

ADAPTIVE INTERVAL WAVELET PRECISE INTEGRATION METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS

IntroductionThepreciseintegrationmethod(PIM) [1],whichwasproposedforsolvingstructuraldynamicequations.Thismethodissimplerandpossesseshigherprecision .Forlinearsteadystructuraldynamicsystems,itsnumericalresultsattheintegrationpointsarealmostequaltothatoftheexactsolutioninmachineaccuracy .InthepreciseintegrationmethodforsolvingPDEs,theequationsshouldbediscretizedinthephysicalspaceforobtainingthesystemofODEsintime ,whichisoftenexecutedbythefinitedifferencemethodorthefiniteelementmethod .Inrec…     

基于精细积分技术的非线性动力学方程的同伦摄动法

将精细积分技术（PIM）和同伦摄动方法（HPM）相结合,给出了一种求解非线性动力学方程的新的渐近数值方法。采用精细积分法求解非线性问题时,需要将非线性项对时间参数按Taylor级数展开,在展开项少时,计算精度对时间步长敏感;随着展开项的增加,计算格式会变得越来越复杂。采用同伦摄动法,则具有相对筒单的计算格式,但计算精度较差,应用范围也限于低维非线性微分方程。将这两种方法相结合得到的新的渐近数值方法则同时具备了两者的优点,既使同伦摄动方法的应用范围推广到高维非线性动力学方程的求解,又使精细积分方法在求解非线性问题时具有较简单的计算格式。数值算例表明,该方法具有较高的数值精度和计算效率。     

TRANSFER MATRIX METHOD FOR ANALYZING VIBRATION AND DAMPING CHARACTERISTICS OF ROTATIONAL SHELL WITH PASSIVE CONSTRAINED LAYER DAMPING TREATMENT

The first order differential matrix equations of the host shell and constrained layer for a sandwich rotational shell are derived based on the thin shell theory.Employing the layer wise principle and first order shear deformation theory, only considering the shearing deformation of the viscoelastic layer, the integrated first order differential matrix equation of a passive constrained layer damping rotational shell is established by combining with the normal equilibrium equation of the viscoelastic layer.A highly precise transfer matrix method is developed by extended homogeneous capacity precision integration technology.The numerical results show that present method is accurate and effective.     

Stationary random waves propagation in 3D viscoelastic stratified solid

Propagation of stationary random waves in viscoelastic stratified transverse isotropic materials is investigated. The solid was considered multi-layered and located above the bedrock, which was assumed to be much stiffer than the soil, and the power spectrum density of the stationary random excitation was given at the bedrock. The governing differential equations are derived in frequency and wave-number domains and only a set of ordinary differential equations ( ODEs) must be solved. The precise integration algorithm of two-point boundary value problem was applied to solve the ODEs. Thereafter, the recently developed pseudo-excitation method for structural random vibration is extended to the solution of the stratified solid responses.     

比例边界等几何分析方法Ⅰ:波导本征问题

提出比例边界等几何方法 (scaled boundary isogeometric analysis, SBIGA), 并用以求解波导本征值问题. 在比例边界等几何坐标变换的基础上, 利用加权余量法将控制偏微分方程进行离散处理, 半弱化为关于边界控制点变量的二阶常微分方程, 即 TE 波或 TM 波波导的比例边界等几何分析的频域方程以及波导动刚度方程, 同时利用连分式求解波导动刚度矩阵. 通过引入辅助变量进一步得出波导本征方程. 该方法只需在求解域的边界上进行等几何离散, 使问题降低一维, 计算工作量大为节约, 并且由于边界的等几何离散, 使得解的精度更高, 进一步节省求解自由度. 以矩形和 L 形波导的本征问题分析为例, 通过与解析解和其他数值方法比较, 结果表明该方法具有精度高、计算工作量小的优点.     

Integration of Ordinary Differential Equations via Nonlocal Symmetries

We present a nonlocal symmetry method to reduce scalar first- and second-orderordinary differential equations (ODEs) to quadratures. It is shown that a second-orderODE admitting a non-Abelian two-dimensional Lie algebra of point symmetriesis reducible to quadratures via a nonideal of the algebra. We also providea direct method of integration for a first-order ODE admitting an exponential nonlocal symmetry which satisfies an additional property.Moreover, we give examples, two on double reduction and several on Abel equations of the second kind, that illustrate ourapproaches.     

Parametric resonance in non-linear elastodynamics

We show that finite amplitude shearing motions superimposed on an unsteady simple extension are admissible in any incompressible isotropic elastic material. We show that the determining equations for these shearing motions admit a general reduction to a system of ordinary differential equations (ODEs) in the remarkable case of generalized circularly polarized transverse waves. When these waves are standing and the underlying unsteady simple extension is composed of a harmonic perturbation of a static stretch it is possible to reduce the determining ODEs to linear or non-linear Mathieu equations. We use this property for a detailed study of the phenomenon of parametric resonance in non-linear elastodynamics.     

病态代数方程的精细积分解法

基于精细积分思想,提出了一种有效的病态代数方程组求解方法。类似于稳态热传导方程可视为瞬态热传导方程的极限形式,将具有正定对称实系数矩阵的病态代数方程组归结为一个常微分方程组初值问题的极限形式,并在此基础上建立了病态代数方程组的精细积分解法。该方法不仅精度高,而且能以指数速度收敛,具有较高的效率。本文还讨论了病态代数方程...     

Transient heat conduction analysis using the NURBS-enhanced scaled boundary finite element method and modified precise integration method

The Non-uniform rational B-spline(NURBS)enhanced scaled boundary finite element method in combination with the modified precise integration method is proposed for the transient heat conduction problems in this paper.The scaled boundary finite element method is a semi-analytical technique,which weakens the governing differential equations along the circumferential direction and solves those analytically in the radial direction.In this method,only the boundary is discretized in the finite element sense leading to a reduction of the spatial dimension by one with no fundamental solution required.Nevertheless,in case of the complex geometry,a huge number of elements are generally required to properly approximate the exact shape of the domain and distorted meshes are often unavoidable in the conventional finite element approach,which leads to huge computational efforts and loss of accuracy.NURBS are the most popular mathematical tool in CAD industry due to its flexibility to fit any free-form shape.In the proposed methodology,the arbitrary curved boundary of problem domain is exactly represented with NURBS basis functions,while the straight part of the boundary is discretized by the conventional Lagrange shape functions.Both the concepts of isogeometric analysis and scaled boundary finite element method are combined to form the governing equations of transient heat conduction analysis and the solution is obtained using the modified precise integration method.The stiffness matrix is obtained from a standard quadratic eigenvalue problem and the mass matrix is determined from the low-frequency expansion.Finally the governing equations become a system of first-order ordinary differential equations and the time domain response is solved numerically by the modified precise integration method.The accuracy and stability of the proposed method to deal with the transient heat conduction problems are demonstrated by numerical examples.     

插值矩阵法分析与复合材料界面相交的平面裂纹奇异应力场

提出了用插值矩阵法分析与各向异性材料界面相交的平面裂纹应力奇异性。基于V形切口尖端附近区域位移场渐近展开,将位移场的渐近展开式的典型项代入线弹性力学基本方程,得到关于平面内与复合材料界面相交的裂纹应力奇异性指数的一组非线性常微分方程的特征值问题,运用插值矩阵法求解,获得了平面内各向异性结合材料中与界面以任意角相交的裂纹尖端的应力奇异性指数随裂纹角的变化规律,数值计算结果与已有结果比较表明,本文方法具有很高的精度和效率。     

Analytical solution for bending problem of moderately thick composite annular sector plates with general boundary conditions and loadings using multi-term extended Kantorovich method

An analytical solution for bending of composite sector plates is presented using multi-term extended Kantorovich method (MTEKM). The governing equations are derived using the displacement field of the first-order shear deformation theory and converted into two sets of coupled ordinary differential equations (ODEs) utilizing MTEKM. Next, an analytical iterative procedure is presented for solving the derived sets of ODEs based on state-space method. Numerous examples are studied by the present method, and as special cases, solid sector and rectangular plates are also investigated. Next, the results obtained by the present method are compared to those of finite element method and other results available in the literature. It is found that the present method has a high convergence rate as well as good accuracy in all cases.