Wave propagation in nonlinear and hysteretic media––a numerical study

This paper considers the problem of one dimensional wave propagation in nonlinear, hysteretic media. The constitutive law in the media is prescribed by an integral relationship based on the Duhem model of hysteresis. It is found that the well known nonlinear elastic stress–strain relationship is a special case of this integral relationship. It is also shown that the stress–strain relationship from the McCall and Guyer model of hyesteretic materials can also be derived from this integral stress–strain relationship. The first part of this paper focuses on a material with a quadratic stress–strain relationship, where the initial value problem is formulated into a system of conservation laws. Analytical solutions to the Riemann problem are obtained by solving the corresponding eigenvalue problem and serve as reference for the verification and illustration of the accuracy obtained using the applied numerical scheme proposed by Kurganov and Tadmor. The second part of this research is devoted to wave propagation in hysteretic media. Several types of initial excitations are presented in order to determine special characteristics of the wave propagation due to material nonlinearity and hysteresis. The results of this paper demonstrate the accuracy and the robustness of this numerical scheme to analyze wave propagation in nonlinear materials.     

The invariant manifold approach applied to nonlinear dynamics of a rotor-bearing system

The invariant manifold approach is used to explore the dynamics of a nonlinear rotor, by determining the nonlinear normal modes, constructing a reduced order model and evaluating its performance in the case of response to an initial condition. The procedure to determine the approximation of the invariant manifolds is discussed and a strategy to retain the speed dependent effects on the manifolds without solving the eigenvalue problem for each spin speed is presented. The performance of the reduced system is analysed in function of the spin speed.     

旋转薄壳稳定性的数值分析

用数值方法求解旋转薄壳稳定性问题的主要困难在于:对任意应力状态下的旋转壳进行稳定性分析,各谐波间彼此耦合,致使方程推导的复杂性和计算量都增大,本文采用[3]所提供的单元及经典的能量判据,在最一般的提法下构造了有限元列式,详细讨论了耦合情况,利用本文方案编制程序并进行了许多算例分析,结果令人满意。同时表明在某些情况下边界条件对临界载荷的影响很大,这可以部分地解释古典稳定性分析结果与实验偏差较大以及实验数据分散的原因。     

The influence of magnetic field upon the collapse of a cylindrical shock wave

In the present paper, the problem of propagation of collapsing cylindrical shock wave in an ideal gas permeated by a transverse magnetic field with infinite electrical conductivity is investigated. Here it is assumed that the medium ahead of the shock front is uniform and at rest. Also, its counter pressure concerning the motion of the wave front is neglected. This problem admits a self similar solution of second kind. The similarity exponent has been computed by solving a nonlinear eigenvalue problem and integrating numerically the self-similar equations for various values of adiabatic heat exponent and Cowling number. Numerical computations have been performed to determine the flow field behind the shock wave. The influence of magnetic field strength and adiabatic heat exponent on the flow parameters for various cases is presented.     

Analytical and numerical methods of symplectic system for Stokes flow in two-dimensional rectangular domain

In this paper, a new analytical method of symplectic system, Hamiltonian system, is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain. In the system, the fundamental problem is reduced to an eigenvalue and eigensolution problem. The solution and boundary conditions can be expanded by eigensolutions using adjoint relationships of the symplectic ortho-normalization between the eigensolutions. A closed method of the symplectic eigensolution is presented based on completeness of the symplectic eigensolution space. The results show that fundamental flows can be described by zero eigenvalue eigensolutions, and local effects by nonzero eigenvalue eigensolutions. Numerical examples give various flows in a rectangular domain and show effectiveness of the method for solving a variety of problems. Meanwhile, the method can be used in solving other problems.     

Analytical and numerical methods of symplectic system for Stokes flow in two-dimensional rectangular domain

In this paper,a new analytical method of symplectic system.Hamiltonian system,is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain.In the system,the fundamental problem is reduced to all eigenvalue and eigensolution problem.The solution and boundary conditions call be expanded by eigensolutions using ad.ioint relationships of the symplectic ortho-normalization between the eigensolutions.A closed method of the symplectic eigensolution is presented based on completeness of the symplectic eigensolution space.The results show that fundamental flows can be described by zero eigenvalue eigensolutions,and local effects by nonzero eigenvalue eigensolutions.Numerical examples give various flows in a rectangular domain and show effectivenees of the method for solving a variety of problems.Meanwhile.the method can be used in solving other problems.     

TM-modes in a planar optical waveguide with a graded index of the symmetric Epstein type

The eigenvalue problem for the TM-modes in a planar optical waveguide is rigorously solved by analytical and numerical methods in the case of a symmetric Epstein-type variation of the dielectric permittivity. It is shown that there exists a finite number of guided TM-modes. The TM-modal fields are expressed in terms of Heun functions and the corresponding eigenvalues (or propagation coefficients) are numerically determined from the minimal solution of a three-term recurrence relation. Upper and lower bounds are established for the difference between corresponding eigenvalues of the TE- and TM-modes.     

振动系统动力学设计迭代算法及解的存在性讨论

振动系统动力学设计被抽象为高维广义非线性特征值反问题。若系统构成以可变参数表示，则可构造一个非线性多元函数。基于多元函数极小原理，提出了一套求解这一特征值反问题的迭代算法。该算法不受系统给定阶数和给定方向的限制，也适用于具有重特征值的退化情况，系统或结构的构成材料可以是任意的。文中同时讨论了解的存在条件，且以显式表达，可方便地应用于工程实际。结合某直升机旋翼桨叶的动力学设计，给出了应用的数值算例。大量数值仿真结果及应用实践表明，本文算法具很好的收敛性，并有较高的计算精度。     

Diffraction of a mode close to its cut-off by a transversal screen in a planar waveguide

The problem of diffraction of a waveguide mode by a thin Neumann screen is considered. The incident mode is assumed to have frequency close to the cut-off. The problem is reduced to a propagation problem on a branched surface and then is considered in the parabolic approximation. Using the embedding formula approach, the reflection and transmission coefficients are expressed through the directivities of the edge Green’s function of the propagation problem. The asymptotics of the directivities of the edge Green’s functions are constructed for the case of small gaps between the screen and the walls of the waveguide. As the result, the reflection and transmission coefficients are found. The validity of known asymptotics of these coefficients is studied.     

Analysis of wave propagation in piezoelectric coupled cylinder affected by transverse shear and rotary inertia

In this paper, we examine the wave propagation in a piezoelectric coupled cylindrical shell affected by the shear effect and rotary inertia. A complete mathematical analysis of wave propagation solution in this piezoelectric coupled cylindrical shell is provided. The dispersion characteristics are derived through the solving an eigenvalue problem. Results are validated by the classical solution of a metallic cylinder. Besides providing and discussing the dispersion curves for different wave modes, we also examine the piezoelectric effects on the dispersion curves. Further to the above investigation, comparison of dispersion solutions from different shell theories is also conducted. This work may serve as a benchmark for wave propagation in piezoelectric coupled cylindrical shells.     

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.     

EXACT SOLUTION AND ITS BEHAVIOR CHARACTERISTIC OF NONLINEAR DUAL-POROSITY MODEL

IntroductionIn the computation of petroleum reservoir engineering design,the nonlinear quadraticgradient term is neglected by assuming small pressure gradient or small compressibility.Theassumption of small pressure gradient may cause significant errors i…     

经典梁热过屈曲问题的解析解

基于非线性经典梁理论,建立了控制轴向和横向变形的基本方程,将两个非线性方程化简为一个关于横向挠度的四阶非线性积分-微分方程。对于本文所考虑的三类边界条件,该方程与相应的边界条件构成了微分特征值问题;直接求解该问题,得到热过屈曲构形的解析解,该解是外加热载荷的函数。为考察热载荷以及边界条件的影响,根据得到的解析解给出了一些数值算例,讨论了梁过屈曲行为的性质。本文得到的解析解可用于验证或改进各类近似理论和数值方法。     

The general and basic boundary value problems of plane steady jet flows and the corresponding nonlinear equations

The general boundary value problem, including known plane steady jet flows of an ideal incompressible fluid, is formulated. The simplest problem retaining all the specific features of the general problem, known as the basic problem, is separated from the general problem. The solution of the basic problem is reduced to solving a non-linear integro-differential equation and also to solving nonlinear integral equations. Examples of flows whose determination is reduced to' solving the basic problem are cited.     

An FMM for waveguide problems of 2-D Helmholtz’ equation and its application to eigenvalue problems

This paper discusses an FMM for solving waveguide problems and associated eigenvalue problems for Helmholtz’ equation in a two dimensional infinite strip with homogeneous Neumann boundary condition on the sides. Layer potentials with Green’s function for this problem are evaluated efficiently with the help of the method of images and FMM. We apply FMM to solve some boundary value problems in waveguides and associated resonance frequency problems using the Sakurai–Sugiura projection method after discussing the required analytic continuation of the solutions to complex frequencies. Some numerical examples show the accuracy and the efficiency of the proposed method.     

Influences of large deformation and rotary inertia on wave propagation in piezoelectric cylindrically laminated shells in thermal environment

Influences of large deformation (geometrical non-linear) and rotary inertia on wave propagation in a long, piezoelectric cylindrically laminated shell in thermal environment is presented in this paper. Nonlinear dynamic governing equations of piezoelectric cylindrically laminated shells are derived by means of Hamilton’s principle. The wave propagation modes are obtained by solving an eigenvalue problem. Numerical examples show that the characteristics of wave propagation in piezoelectric cylindrically laminated shells are relates to the large deformation, rotary inertia and thermal environment of the piezoelectric cylindrically laminated shells. The effect of large deformation, rotary inertia and thermal load on wave propagation in the piezoelectric cylindrically laminated shells is discussed by comparing with the result from the small deformation (geometrical linear shell theory). This method may be used to investigate wave propagation in various laminated material, layers numbers and thickness of piezoelectric cylindrically laminated shells under large deformation. The results carried out can be used in the ultrasonic inspection techniques and structural health monitoring.     

Natural frequencies and mode shapes of statically deformed inclined risers

We investigate numerically the linear vibrations of inclined risers using the Galerkin approach. The riser is modeled as an Euler–Bernoulli beam accounting for the nonlinear mid-plane stretching and self-weight. After solving for the initial deflection of the riser due to self-weight, we use a Galerkin expansion employing 15 axially loaded beam mode shapes to solve the eigenvalue problem of the riser around the static equilibrium configuration. This yields the riser natural frequencies and corresponding exact mode shapes for various values of inclination angles and tension. The obtained results are validated against a boundary-layer analytical solution and are found to be in good agreement. This constitutes a basis to study the nonlinear forced vibrations of inclined risers.     

FFT-based spectral analysis methodology for one-dimensional wave propagation in poroelastic media

The general one-dimensional equilibrium equations describing the dynamic behaviour of a porous medium form a system of coupled hyperbolic partial differential equations. A transition from the time to the frequency domain is made by spectral decomposition of the displacements. The equations simplify to a set of coupled ordinary differential equations. A solution can be obtained by solving a frequency-dependent eigenvalue problem. The characteristic equation clarifies the double wave-pattern and the attenuation of each wave. A spectrally formulated element uses the frequency-dependent eivenvectors as shape functions. The mass distribution is treated exactly without the need of subdividing a member into smaller elements and therefore wave propagation within an element is also treated exactly.     

Axisymmetric waves in layered hollow cylinders with axially polarized piezoceramic layers

The propagation of axisymmetric waves in layered hollow cylinders with axially polarized layers is studied. Use is made of an efficient numerical analytic method. The original problem of electroelasticity for partial differential equations is reduced to a boundary-value eigenvalue problem for ordinary differential equations by representing the components of the stress tensor, displacement vectors, electric-flux density, and electric potential as waves traveling in the axial direction. The problem is solved with the stable discrete-orthogonalization method in combination with incremental search. Results of a numerical analysis of the dispersion equations over a wide range of geometrical characteristics of the cylinders are presented     

Nonlinear dispersion of long internal waves

The propagation of long weakly nonlinear waves in an atmospheric waveguide is considered. A model system of Kadomtsev-Petviashvili equations [1], which describes the propagation of such waves, is derived. In the case of one excited wave mode the system of model equations goes over into the Kadomtsev-Petviashvili equation, in which, however, the variables x and t are interchanged. The reasons for this are clarified. In the two-dimensional case an approximate solution of the model equations is constructed, and steady nonlinear waves and their interaction in a collision are considered. The results of a numerical verification of the stability of the approximate steady solutions and of the solution to the problem of decay of the wave into quasisolitons are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 151–157, May–June, 1988.