The global analysis of higher order nonlinear dynamical systems and the application of cell-to-cell mapping method

In this paper, the general characteristics and the topological consideration of the global behaviors of higher order nonlinear dynamical systems and the characteristics of the application of cell-to-cell mapping method in this analysis are expounded. Specifically, the global analysis of a system of two weakly coupled van der Pol oscillators using cell-to-cell mapping method is presented.The analysis shows that for this system, there exist two stable limit cycles in 4-dimensional state space, and the whole 4-dimensional state space is divided into two almost equal parts which are, respectively, the two asymototically stable domains of attraction of the two periodic motions of the two stable limit cycles. The validities of these conclusions about the global behaviors are also verified by direct long term numerical integration. Thus, it can be seen that the cell-to-cell mapping method for global analysis of fourth order nonlinear dynamical systems is quite effective.     

Closed-form analytic expressions for the pulsed-source radiated electromagnetic field in a class of media with time-varying wave speed

Closed-form analytic expressions are arrived at for the pulsed-source radiated electromagnetic field in a homogeneous, isotropic, lossless medium with constant wave impedance/admittance, but with time-varying wave speed. A particular choice of the excitation solves the corresponding initial-value problem. The results can be used to benchmark purely computational investigations.     

Multiresolution Symplectic Scheme for wave propagation in complex media

A fast adaptive symplectic algorithm named Multiresolution Symplectic Scheme (MSS) was first presented to solve the problem of the wave propagation (WP) in complex media, using the symplectic scheme and Daubechies‘ compactly supported orthogonal wavelet transform to respectively discretise the time and space dimension of wave equation. The problem was solved in multiresolution symplectic geometry space under the conservative Hamiltonian system rather than the traditional Lagrange system. Due to the fascinating properties of the wavelets and symplectic scheme, MSS is a promising method because of little computational burden, robustness and reality of long-time simulation.     

Finite element analysis of wave propagation in fluid-saturated porous media

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Propagation of a stress wave through a virtual functionally graded foam

Stress wave propagation through a Functionally Graded Foam Material (FGFM) is analysed in this paper using the finite element method. A finite element model of the Split Hopkinson Pressure Bar (SHPB) is developed to apply realistic boundary conditions to a uniform density foam and is validated against laboratory SHPB tests. Wave propagation through virtual FGFMs with various gradient functions is then considered. The amplitude of the stress wave is found to be shaped by the gradient functions, i.e., the stress can be amplified or diminished following propagation through the FGFMs. The plastic dissipation energy in the specimens is also shaped by the gradient functions. This property of FGFMs provides significant potential for such materials to be used for cushioning structures.     

Application of a wave transform to pressure transient testing in porous media

Solutions to the diffusion equation for nonuniform media are difficult to obtain in a form that can be easily evaluated. Often the solutions are written as the inverse Laplace transform of an inverse Fourier transform. In this paper, I show that the wave transform of Bragg and Dettman (1968) coupled with the Cagniard-de Hoop method for solving the wave propagation problem results in simplified solutions to the problem of pressure transient testing in linear composite reservoirs. The potential usefulness of an inverse wave transform, which would transform measured pressure data (smooth) into a wave signal propagating at the velocity of the square root of the diffusivity, is demonstrated by a synthetic example. In the example, diffusivity of the source region is estimated from the time of the direct wave arrival, while diffusivity of a second, higher diffusivity region is estimated from the velocity of the head wave. In the wave domain the time-like variable has units of (time)^1/2 which makes the units of velocity equal to L T-^(1/2). I also demonstrate, using synthetic data, that it is difficult, but perhaps possible, to invert the wave transform numerically.     

Propagation of nonlinear waves in an inhomogeneous gas-liquid medium. Derivation of wave equations in the Korteweg-de Vries approximation

Equations describing the propagation of waves of small but finite amplitude in a liquid with gas bubbles are derived. The bubble distribution density is a continuous function of bubble size and spatial coordinates. It is found that, for a uniform bubble distribution, the obtained equations become the Korteweg-de Vries, Kadomtsev-Petviashvili and Khokhlov-Zabolotskaya equations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 188–197, March–April, 2009.     

Fast Fourier homogenization for elastic wave propagation in complex media

In the context of acoustic or elastic wave propagation, the non-periodic asymptotic homogenization method allows one to determine a smooth effective medium and equations associated with the wave propagation in a given complex elastic or acoustic medium down to a given minimum wavelength. By smoothing all discontinuities and fine scales of the original medium, the homogenization technique considerably reduces meshing difficulties as well as the numerical cost associated with the wave equation solver, while producing the same waveform as for the original medium (up to the desired accuracy). Nevertheless, finding the effective medium requires one to solve the so-called “cell problem”, which corresponds to an elasto-static equation with a finite set of distinct loadings. For general elastic or acoustic media, the cell problem is a large problem that has to be solved on the whole domain and its resolution implies the use of a finite element solver and a mesh of the fine scale medium. Even if solving the cell problem is simpler than solving the wave equation in the original medium (because it is time and source independent, based on simple tetrahedral meshes and embarrassingly parallel) it is still a challenge. In this work, we present an alternative method to the finite element approach for solving the cell problem. It is based on a well-known method designed by H. Moulinec and P. Suquet in 1998 in structural mechanics. This iterative technique relies on Green functions of a simple reference medium and extensively uses Fast Fourier Transforms. It is easy to implement, very efficient and relies on a simple regular gridding of the medium. Through examples we show that the method gives excellent results, even, under some conditions, for discontinuous media.     

饱和土中爆炸波传播问题的数值模拟

应用流体弹塑性力学的框架描述了爆炸荷载作用下饱和土介质的本构模型,包括体积压缩关系和强度关系。应用LS-dyna软件分析了三相饱和土介质中的爆炸波传播及其与结构的相互作用问题,并与先前完成的试验结果进行了对比,两者吻合较好。饱和土中爆炸波传播的一些特殊现象被重现,表明本构模型的描述是可行的。     

INSTABILITY AND DISPERSIVITY OF WAVE PROPAGATION ININELASTIC SATURATED/UNSATURATED POROUS MEDIA

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Stationary discontinuity and flutter instability of wave propagation in elasto-plastic saturated porous media

In the present work a model based on the Biot theory for simulating coupled hydrodynamic behavior in saturated porous media is utilized with integration of the inertial coupling effect between the solid-fluid phases of the media into the model. The non-associated Drucker-Prager criterion to describe nonlinear constitutive behavior of pressure dependent elasto-plasticity for the media is particularly considered. With no consideration of compressibility of solid grains and the pore fluid, the discontinuity and instability of the wave propagation in saturated porous media are analyzed for the plane strain problems in detail. The critical conditions of stationary discontinuity and flutter instability in the wave propagation are given. The results and conclusions obtained by the present work will provide some bases or clues for overcoming the difficulties in numerical modeling of wave propagation in the media subjected to dynamic loading. The project supported by the National Natural Science Foundation of China (19832010)     

Body wave propagation in rotating elastic media

We investigate the propagation of elastic waves through an elastic medium submitted to an angular rotation Ω. Wave propagation is shown to be directly related to the Kibel number Ki=ω/Ω, where ω is the wave frequency. Two dispersive waves W₁ and W₂ are obtained which tend to the classical dilatational and shear waves, respectively, when Ki tends to infinity. Wave W₁ shows a cutoff frequency ω_c=Ω below which it does not propagate. The case of small angular rotation Ω is also studied. The corrections to be introduced to dilatational and shear waves are then shown to be of order O(Ki⁻¹).     

Propagation,reflection, and transmission of SH-waves in slightly compressible,finitely deformed elastic media

The propagation, reflection, and transmission of SH waves in slightly compressible, finitely deformed elastic media are considered in this paper. The dispersion relation for SH-wave propagation in slightly compressible, finitely deformed layer overlying a slightly compressible, finitely deformed half-space is derived. The present paper also deals with the reflection and refraction (transmission) phenomena due to the SH wave incident at the plane interface between two distinct slightly compressible, finitely deformed elastic media. The closed form expressions for the amplitude ratios of reflection and refraction coefficients of the reflected and refracted SH waves are obtained from suitable boundary conditions. For the numerical discussions, we consider the Neo-Hookean form of a strain energy function. The phase speed curves, the variations of reflection, and transmission coefficients with the angle of incidence, and the plots of the slowness sections are presented by means of graphs.     

均质土中半埋入变截面桩的瞬态波动响应

基于回传射线矩阵法,求得单位脉冲作用下均质土中半埋入变截面桩的桩顶速度导纳,然后利用Fourier逆变换和卷积定理得到瞬态半正弦激振力作用下的桩顶时域速度响应。比较半埋入缩颈桩、半埋入扩颈桩与半埋入完整桩的速度导纳和反射波,分析缺陷类型和长度、缺陷埋深和程度对速度导纳和反射波的影响,并进行缩颈-扩颈桩与扩颈-缩颈桩的桩顶速度导纳和反射波曲线的对比。结果表明,桩侧介质变化对桩顶速度导纳和反射波的影响小于桩身缺陷。     

Propagation of Elastic Waves in Periodically Inhomogeneous Media

The structural theory of waves and vibrations in periodically inhomogeneous media is set out. Relevant research results are presented. Emphasis is on the principles of the theory, surface waves, and other surface effects     

Thermoelastic wave propagation in inhomogeneous media

Summary A novel approach to the modeling of thermoelastic wave propagation is presented, based on the thermodynamics of discrete systems. The first novelty includes the representation of integral balance laws for thermoelasticity in terms of contact quantities that describe the nonequilibrium state of elements. The next new aspect is a modification of the recently proposed wave-propagation algorithm, which is used as a tool for determining the contact quantities in a finite-volume scheme for the numerical simulation of two-dimensional thermoelastic wave propagation in inhomogeneous media. Such a modification is needed to provide the satisfaction of the thermodynamic consistency conditions between adjacent discrete elements. Results of computations for certain test problems show the efficiency and physical consistency of the algorithm. Received 11 November 1999; accepted for publication 2 May 2000     

地震波倾斜入射下两相饱和土层响应数值模拟

基于Biot两相饱和多孔介质理论建立了频域内饱和土层的刚度矩阵,对从基岩倾斜入射地震波下饱和土层的波动响应进行了数值研究,入射波采取修正的Kanai-Tajimi谱。通过土层地表面处的水平以及竖直放大系数研究了在不同工况条件下土层的波动响应,研究结果表明,土层的波动响应在入射角接近临界入射角时达到峰值;土层的波动响应随着基岩埋深基以及土层阻尼比的增大显著减小;而土层与基岩的波速比对土层的波动响应的影响存在临界波速比,使土层响应达到极值。对于含有夹层的地质环境,研究了波动响应沿深度的分布,通过与均质土层下响应对比,显示非均质夹层能显著减小土层地表处的波动响应,尤其软夹层能较好地阻止入射波向上的传播。     

简单周期结构波传播主动控制研究

研究了在梁结构插入压电材料的周期结构波传播的一般规律。以梁的弯曲波传播方程的解析解为基础，进一步分析了简单周期结构的波传播的一些固有特性。并在此基础上，以压电材料作为作动器，采用主动控制方法研究了调整结构传播波的带通、带阻的频率范围可行性和有效性。