基于Hamilton体系的弹性力学问题的比例边界有限元方法

比例边界有限元方法是求解偏微分方程的一种半解析半数值解法。对于弹性力学问题,可采用基于力学相似性、基于比例坐标相似变换的加权余量法和虚功原理得到以位移为未知量的系统控制方程,属于Lagrange体系。但在求解时,又引入了表面力为未知量,控制方程属于Hamilton体系。因而,本文提出在比例边界有限元离散方法的基础上,利...     

Fully explicit Agmon's condition for general states of a special incompressible elastic material

Agmon's condition arises as a necessary condition at the boundary for minimizers in compressible and incompressible elasticity. It is commonly formulated as a statement concerning the solution set of a family of ODEs with constant coefficients. As such, it is algebraic “in principle”.In both the compressible and incompressible cases, Agmon's condition may be recast in a more overtly algebraic form, namely the requirement that a certain family of algebraic Riccati equations (parametrized over the tangent plane) should possess positive solutions. In order to reduce Agmon's condition to a fully explicit set of inequalities involving the components of the incremental elasticity tensor, one must be able to solve the algebraic Riccati equation explicitly. Known situations where this can be done tend to involve highly symmetric states of isotropic materials. It is therefore noteworthy that Agmon's condition may be rendered explicit for any boundary-point of an arbitrarily deformed incompressible neo-Hookean body.     

CORRESPONDENCE FUNCTION D(z)AND GENERALIZED IRREGULAR EQUATIONS

Extending Riemann’s idea of P function(using equation’s parameters to represent thefunction defined by the equation).we introduce correspondence functions (?)(z) to describeregular and irregular integrals in a unifying way.By explicit solution discuss monodromy group of non-Fuchsian equations.The explicitexpressions of exponent and expansion coefficients for Floquet solution are obtained.Method of correspondence functions permits us to obtain systematically the solutionsof generalized irregular equations. having regular,irregular poles,essential.algebraic,transcendental.logarithmic singularities as well as singular line.The representation of basic set of solutions by (?) function makes it possible to enlargethe scope of investgation of analytic theory.The significance of Poincaré’s conjecture is discussed,as(?)functions are automorphic.     

The small disturbance problems associated with the algebraic riccati equation of continuous linear time-invariant systems

The stability problem of the disturbed algebraic Riccati equation of continuous linear time-invariant systems is discussed in this paper. Through matrix norm analysis the estimation (expressed in terms of the disturbance range of the system parameters) of the disturbance range in the solution of the disturbed algebraic Riccati equation is established. Apparently this method is quite convenient for the practical computational purposes.     

The matrix sign function for solving surface wave problems in homogeneous and laterally periodic elastic half-spaces

The matrix sign function is shown to provide a simple and direct method to derive some fundamental results in the theory of surface waves in anisotropic materials. It is used to establish a shortcut to the basic formulas of the Barnett–Lothe integral formalism and to obtain an explicit solution of the algebraic matrix Riccati equation for the surface impedance. The matrix sign function allows the Barnett–Lothe formalism to be readily generalized for the problem of finding the surface wave speed in a periodically inhomogeneous half-space with material properties that are independent of depth. No partial wave solutions need to be found; the surface wave dispersion equation is formulated instead in terms of blocks of the matrix sign function of i$i$ times the Stroh matrix.     

Computing the Maslov Index from Singularities of a Matrix Riccati Equation

We study the Maslov index as a tool to analyze stability of steady state solutions to a reaction–diffusion equation in one spatial dimension. We show that the path of unstable subspaces associated to this equation is governed by a matrix Riccati equation whose solution S develops singularities when changes in the Maslov index occur. Our main result proves that at these singularities the change in Maslov index equals the number of eigenvalues of S that increase to \(+\infty \) minus the number of eigenvalues that decrease to \(-\infty \).     

Singular Solutions of Fully Nonlinear Elliptic Equations and Applications

We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of , and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragmén–Lindel?f result as well as a principle of positive singularities in certain Lipschitz domains.     

On the solution of the Chazy system of equations

We obtain a solution of the Chazy system that consists of nine nonlinear algebraic equations. This system gives a necessary condition for the class of nonlinear third-order differential equations with six singularities to be of P-type. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 92–98, January–March, 2009.     

Wiener-Hopf analysis of an acoustic plane wave in a trifurcated waveguide

In this paper, we have made Wiener-Hopf analysis of an acoustic plane wave by a semi-infinite hard duct that is placed symmetrically inside an infinite soft/hard duct. The method of solution is integral transform and Wiener-Hopf technique. The imposition of boundary conditions result in a 2 × 2 matrix Wiener-Hopf equation associated with a new canonical scattering problem which is solved by using the pole removal technique. In the solution, two infinite sets of unknown coefficients are involved that satisfy two infinite systems of linear algebraic equations. These systems of linear algebraic equations are solved numerically. The graphs are plotted for sundry parameters of interest. Kernel functions are also factorized.     

矩阵黎卡提(Riccati)微分方程的分析解

相应哈密顿矩阵本征解的基础上,本文给出了黎卡提微分方程的分析解,对于最优控制以及卡尔曼－布西滤波的黎卡提微分方程分别给出了分析解的公式。     

Axisymmetric nuclei of strain and their applications for multilayered solids

In this paper, the effect of several axisymmetric elastic singularities (i.e., point forces, double forces, sum of two double forces and centers of dilatation) on the elastic response of a multilayered solid is investigated. The boundary conditions in an infinite solid at the plane passing through the singularity are derived first using Papkovich–Neuber harmonic functions. Then, a Green’s function solution for multilayered solids is obtained by solving a set of simultaneous linear algebraic equations using both the boundary conditions for the singularity and the layer interfaces. Finally, the elastic solutions in a single layer on an infinite substrate due to point defects and infinitesimal prismatic dislocation loops are presented to illustrate the application of these Green’s function solutions.     

Special Cases in Optimization Problems for Stationary Linear Closed-Loop Systems

Consideration is given to problems of solving the algebraic Riccati equation (ARE)—J-factorization of matrix polynomials and J-factorization of rational matrices—to which traditional solution algorithms are not applicable. In this connection, solution algorithms for these problems are discussed where the eigenvalues of the Hamiltonian matrix corresponding to the ARE and the zeros of matrix polynomials are located on the imaginary axis. Moreover, a procedure is set forth for asymptotic expansion of a stabilizing solution of the ARE in the neighborhood of a point at which the ARE has no stabilizing solution. It is shown how this expansion can be used for constructing canonical J-factorization of matrix polynomials that is nearly a noncanonical J-factorization. It is pointed out that the algorithms described can be implemented with the help of MATLAB routines     

Stress singularities in an anisotropic body of revolution

To fill the gap in the literature on the application of three-dimensional elasticity theory to geometrically induced stress singularities, this work develops asymptotic solutions for Williams-type stress singularities in bodies of revolution that are made of rectilinearly anisotropic materials. The Cartesian coordinate system used to describe the material properties differs from the coordinate system used to describe the geometry of a body of revolution, so the problems under consideration are very complicated. The eigenfunction expansion approach is combined with a power series solution technique to find the asymptotic solutions by directly solving the three-dimensional equilibrium equations in terms of the displacement components. The correctness of the proposed solution is verified by convergence studies and by comparisons with results obtained using closed-form characteristic equations for an isotropic body of revolution and using the commercial finite element program ABAQUS for orthotropic bodies of revolution. Thereafter, the solution is employed to comprehensively examine the singularities of bodies of revolution with different geometries, made of a single material or bi-materials, under different boundary conditions.     

代数黎卡提方程的求解与辛子空间迭代法

综合共轭辛子空间迭代法以及２＾Ｎ消元迭代算法，利用接触变换下黎卡提方程解的变换公式，给出了代数黎卡提方程的有效解法。即使有│μ│＝１的本征根，解法依然有效。数例表明了方法的有效性。     

Error Estimates for Approximate Solutions of the Riccati Equation with Real or Complex Potentials

A method is presented for obtaining rigorous error estimates for approximate solutions of the Riccati equation, with real or complex potentials. Our main tool is to derive invariant region estimates for complex solutions of the Riccati equation. We explain the general strategy for applying these estimates and illustrate the method in typical examples, where the approximate solutions are obtained by gluing together WKB and Airy solutions of corresponding one-dimensional Schrödinger equations. Our method is motivated by, and has applications to, the analysis of linear wave equations in the geometry of a rotating black hole.     

Analytical approximation to large-amplitude oscillation of a non-linear conservative system

This paper deals with non-linear oscillation of a conservative system having inertia and static non-linearities. By combining the linearization of the governing equation with the method of harmonic balance, we establish analytical approximate solutions for the non-linear oscillations of the system. Unlike the classical harmonic balance method, linearization is performed prior to proceeding with harmonic balancing, thus resulting in a set of linear algebraic equations instead of one of non-linear algebraic equations. Hence, we are able to establish analytical approximate formulas for the exact frequency and periodic solution. These analytical approximate formulas show excellent agreement with the exact solutions, and are valid for small as well as large amplitudes of oscillation.     

FREE VIBRATION OF ANISOTROPIC RECTANGULAR PLATES BY GENERAL ANALYTICAL METHOD

According to the differential equation for transverse displacement function of anisotropic rectangular thin plates in free vibration, a general analytical solution is established. This general solution, composed of the composite solutions of trigonometric function and hyperbolic function, can satisfy the problem of arbitrary boundary conditions along four edges. The algebraic polynomial with double sine series solutions can also satisfy the problem of boundary conditions at four corners. Consequently, this general solution can be used to solve the vibration problem of anisotropic rectangular plates with arbitrary boundaries accurately. The integral constants can be determined by boundary conditions of four edges and four corners. Each natural frequency and vibration mode can be solved by the determinate of coefficient matrix from the homogeneous linear algebraic equations equal to zero. For example, a composite symmetric angle ply laminated plate with four edges clamped has been calculated and discussed.     

Recursive geometric integrators for wave propagation in a functionally graded multilayered elastic medium

The differential equations governing transfer and stiffness matrices and acoustic impedance for a functionally graded generally anisotropic magneto-electro-elastic medium have been obtained. It is shown that the transfer matrix satisfies a linear 1st order matrix differential equation, while the stiffness matrix satisfies a nonlinear Riccati equation. For a thin nonhomogeneous layer, approximate solutions with different levels of accuracy have been formulated in the form of a transfer matrix using a geometrical integration in the form of a Magnus expansion. This integration method preserves qualitative features of the exact solution of the differential equation, in particular energy conservation. The wave propagation solution for a thick layer or a multilayered structure of inhomogeneous layers is obtained recursively from the thin layer solutions. Since the transfer matrix solution becomes computationally unstable with increase of frequency or layer thickness, we reformulate the solution in the form of a stable stiffness-matrix solution which is obtained from the relation of the stiffness matrices to the transfer matrices. Using an efficient recursive algorithm, the stiffness matrices of the thin nonhomogeneous layer are combined to obtain the total stiffness matrix for an arbitrary functionally graded multilayered system. It is shown that the round-off error for the stiffness-matrix recursive algorithm is higher than that for the transfer matrices. To optimize the recursive procedure, a computationally stable hybrid method is proposed which first starts the recursive computation with the transfer matrices and then, as the thickness increases, transits to the stiffness matrix recursive algorithm. Numerical results show this solution to be stable and efficient. As an application example, we calculate the surface wave velocity dispersion for a functionally graded coating on a semispace.     

The creeping motion of multiple spherical liquid drops

This paper deals with the drag factor of the multiple spherical liquid drops in the creeping motion by means of the Sampson singularities and collocation technique. The drag factors of the drops are calculated under distinct conditions: different number of liquid drops in the chain and different sphere spacing. From the results the influence of the viscosity ratio on the shielding effect and end effect are revealed. The convergence of the method is also studied in this paper.In this paper the collocation technique developed by Gluckman et al. in treating the rigid sphere case is applied to deal with the creeping motion of multiple spherical liquid drops which has improtant applications in bioengineering and chemical engineering. Writing the general solutions in inner and outer regions of the spheres and satisfying the kinematic and dynamic matching conditions at the collocation points on the interfaces, a set of linear algebraic equations is obtained to determine the unknown coefficients in the solutions. By means of any matrix inversion technique the approximate solutions are presented. In the first section of this paper the mathematic formulation of the problem is given and then in the second section the numerical results are introduced and analysed.     

An analytical approximate technique for a class of strongly non-linear oscillators

An analytical approximate technique for large amplitude oscillations of a class of conservative single degree-of-freedom systems with odd non-linearity is proposed. The method incorporates salient features of both Newton's method and the harmonic balance method. Unlike the classical harmonic balance method, accurate analytical approximate solutions are possible because linearization of the governing differential equation by Newton's method is conducted prior to harmonic balancing. The approach yields simple linear algebraic equations instead of non-linear algebraic equations without analytical solution. With carefully constructed iterations, only a few iterations can provide very accurate analytical approximate solutions for the whole range of oscillation amplitude beyond the domain of possible solution by the conventional perturbation methods or harmonic balance method. Three examples including cubic-quintic Duffing oscillators are presented to illustrate the usefulness and effectiveness of the proposed technique.