Modeling three-dimensional elastic wave propagation in circular cylindrical structures using a finite-difference approach

Wave propagation along circular cylindrical structures is important for nondestructive-testing applications and shocks in tubes. To simulate elastic wave propagation phenomena in such structures the governing equations in cylindrical coordinates are solved numerically. To reduce the required amount of computer memory and the computational time, the stress components are eliminated in the equilibrium equations. In the resulting coupled partial differential equations, in which only the three displacement components are involved, the derivatives with respect to spatial coordinates and time are approximated using second order central differences. This leads to the present new approach, which is both accurate and efficient. In order to obtain a stable scheme the displacements must be allocated on a staggered grid. The von Neumann stability analysis is performed and the result is compared with an existing empirical criterion. Mechanical energies are observed in order to validate the finite-difference code. Since no material damping or energy dissipation is taken into account in the equations of motion, the total energy must remain constant over time. Only negligible variations are observed during long-term simulations. Dispersion relations are used to check the physical behavior of the waves calculated with the proposed finite-difference method: Theoretically calculated curves are compared with values obtained by a spectrum estimation method, applied to the results of a simulation.     

Features of normal wave dispersion in periodic structures

The dispersion dependences of normal waves in periodic structures are multivalued in nature. A physically justified choice of a single dispersion branch for each wave requires additional information. Questions are discussed on reducing the uncertainty in choosing dispersion, in particular, based on a limited transition from an analyzed periodic structure to another one, for which dispersion dependences are known with more certainty. Certain inaccuracies and errors in known publications connected with ambiguity of dispersion are noted and corrected.     

Modeling wave propagation in realistic heart geometries using the phase-field method

We present a novel algorithm for modeling electrical wave propagation in anatomical models of the heart. The algorithm uses a phase-field approach that represents the boundaries between the heart muscle and the surrounding medium as a spatially diffuse interface of finite thickness. The chief advantage of this method is to automatically handle the boundary conditions of the voltage in complex geometries without the need to track the location of these boundaries explicitly. The algorithm is shown to converge accurately in nontrivial test geometries with no-flux (zero normal current) boundary conditions as the width of the diffuse interface becomes small compared to the width of the cardiac action potential wavefront. Moreover, the method is illustrated for anatomically realistic models of isolated rabbit and canine ventricles as well as human atria.     

Measurement of the dispersion relation of guided non-axisymmetric waves in filament-wound cylindrical structures

To determine the dispersion relation, guided waves are excited in specimens over a broad frequency range. The surface displacements are measured over time and space. The recorded data are analysed using a quasi-three-dimensional spectrum estimation algorithm. In the time domain a fast Fourier transform is used to extract the frequencies. To obtain the wave numbers, in space a two-dimensional matrix-pencil approach is applied to the data set. Using a suitable constitutive model (transversely isotropic or orthotropic) dispersion curves are calculated. Good agreement was found between the experimental and the numerically calculated dispersion relations after adjusting the material parameters. Since the dispersion relation of a structure depends on the mechanical material properties frequency-dependent material parameters can be extracted from the above-mentioned relation between frequency and wave number.     

NDT response of spectral analysis of surface wave method to multi-layer thin high-strength concrete structures

This study presents the results of the non-destructive testing using spectral analysis of surface waves (SASW) based on high-strength concrete materials. This SASW method was used to evaluate the compressive strength of single-layer high-strength concrete slabs through a correlation with the surface wave velocities. This paper also presents the relationship between the theoretical and experimental compact dispersion curves when the SASW test is applied to multi-layer thin high-strength concrete slab systems with a finite thickness. The test results show that the surface wave velocity profile obtained from the theoretical dispersion curve has lower values than the profile obtained from the experimental compact dispersion curve under the condition of a finite thickness due to different boundary conditions and reflections from the boundaries. Based on the measured response, an experimental study was conducted to examine if the dispersive characteristics of Rayleigh wave exist in the multi-layer high-strength concrete slab systems. This study can be utilized in examining structural elements of high-strength concrete structures and can also be applied in the integrity analysis of high-strength concrete structures with a finite thickness.     

Calculating the forced response of cylinders and cylindrical shells using the wave and finite element method

The dynamic response of circular cylinders can be obtained analytically in very few (and simple) cases. For complicated (thick or anisotropic) circular cylinders, researchers often resort to the finite element (FE) method. This can lead to large models, especially at higher frequencies, which translates into high computational costs and memory requirements. In this paper, the response of axially homogenous circular cylinders (that can be arbitrarily complex through the thickness) is obtained using the wave and finite element (WFE) method. Here, the homogeneity of the cylinder around the circumference and along the axis are exploited to post-process the FE model of a small rectangular segment of the cylinder using periodic structure theory and obtain the wave characteristics of the cylinder. The full power of FE methods can be utilised to obtain the FE model of the small segment. Then, the forced response of the cylinder is posed as an inverse Fourier transform. However, since there are an integer number of wavelengths around the circumference of a closed circular cylinder, one of the integrals in the inverse Fourier transform becomes a simple summation, whereas the other can be resolved analytically using contour integration and the residue theorem. The result is a computationally efficient technique for obtaining the response to time harmonic, arbitrarily distributed loads of axially homogenous, circular cylinders with arbitrary complexity across the thickness.     

无限长粘弹性圆柱管中轴对称波的传播模式和衰减

在Kelvin-Voigt线性粘弹性模型框架内研究了无限长粘弹性圆柱管中轴对称波的传播和衰减。通过一个复根搜索程序对频散方程的求解得到无量纲化相速度频散曲线和衰减曲线。比较了引入损耗因子后粘弹性圆柱管与弹性圆柱管中轴对称波传播的特性。给出了圆柱管的外内径之比、损耗因子和材料参数对相速度和衰减的影响。分析结果表明与弹性圆柱管相比粘弹性圆柱管中轴对称波的各阶传播模式均存在衰减,高阶传播模式并无严格意义的截止频率。损耗因子对第一阶传播模式的相速度影响很小,而对衰减的影响则比较大,穿孔率对波传播的相速度和衰减有相当大的影响。     

Modeling elastic wave forward propagation and reflection using the complex screen method.

Formulation for calculating forward propagation and reflection in a 3D elastic structure based on the complex-screen method is given in this paper. The calculation of reflections is formulated based on the local Born approximation. When using a small angle approximation, the backscattering operator reduces to a screen operator which is similar to the forward screen propagator. Combining the forward propagator and backscattering operator together, the new method can properly handle the multiple forward scattering and single backscattering in a 3D heterogeneous model. Using a dual-domain technique, the new method is highly efficient in CPU time and memory savings. For models where reverberation and resonance scattering can be neglected, this method provides a fast and accurate algorithm. Synthetic seismograms for two-dimensional elastic models are calculated with this method and compared with those generated by the finite-difference method. The results show that the method works well for small to medium scattering angles and medium velocity contrasts.     

Multidomain spectral method for the helically reduced wave equation

We consider the 2+1 and 3+1 scalar wave equations reduced via a helical Killing field, respectively referred to as the 2-dimensional and 3-dimensional helically reduced wave equation (HRWE). The HRWE serves as the fundamental model for the mixed-type PDE arising in the periodic standing wave (PSW) approximation to binary inspiral. We present a method for solving the equation based on domain decomposition and spectral approximation. Beyond describing such a numerical method for solving strictly linear HRWE, we also present results for a nonlinear scalar model of binary inspiral. The PSW approximation has already been theoretically and numerically studied in the context of the post-Minkowskian gravitational field, with numerical simulations carried out via the “eigenspectral method.” Despite its name, the eigenspectral technique does feature a finite-difference component, and is lower-order accurate. We intend to apply the numerical method described here to the theoretically well-developed post-Minkowski PSW formalism with the twin goals of spectral accuracy and the coordinate flexibility afforded by global spectral interpolation.     

Multidomain spectral method for the helically reduced wave equation

We consider the 2+1 and 3+1 scalar wave equations reduced via a helical Killing field, respectively referred to as the 2-dimensional and 3-dimensional helically reduced wave equation (HRWE). The HRWE serves as the fundamental model for the mixed-type PDE arising in the periodic standing wave (PSW) approximation to binary inspiral. We present a method for solving the equation based on domain decomposition and spectral approximation. Beyond describing such a numerical method for solving strictly linear HRWE, we also present results for a nonlinear scalar model of binary inspiral. The PSW approximation has already been theoretically and numerically studied in the context of the post-Minkowskian gravitational field, with numerical simulations carried out via the “eigenspectral method.” Despite its name, the eigenspectral technique does feature a finite-difference component, and is lower-order accurate. We intend to apply the numerical method described here to the theoretically well-developed post-Minkowski PSW formalism with the twin goals of spectral accuracy and the coordinate flexibility afforded by global spectral interpolation.     

Acoustic wave dispersion in a cylindrical elastic tube filled with a viscous liquid

This paper deals with the study of the velocity and the attenuation of an acoustic wave propagating inside a cylindrical elastic tube filled with a viscous liquid. A theory describing the propagation of the axisymmetrical modes in such waveguides is presented, with special attention given to the absorption produced by the viscous mechanisms in the liquid. One of these mechanisms is related to the momentum transfer between the compression and rarefaction regions of a propagating wave. The other viscous mechanism is due to the momentum transport inside the viscous boundary layer, close to the tube wall. Numerical calculations were carried out to investigate the influence of different parameters (frequency, tube radii, viscosity coefficient) on the propagation of acoustic waves.     

Analysis on band gap properties of periodic structures of bar system using the spectral element method

In this paper, the band gap properties of the periodic structures of bar system, which include the rod-joint, truss, and frame structures, are studied using spectral element method (SEM). The spectral equations of the rod, beam, and joint elements are established and the spectral equations of the whole structures are further assembled. The frequency responses of the whole structures are calculated and the results are compared with those calculated by the finite element method (FEM). It can be observed that the SEM is more accurate in high frequency ranges. The band gap properties of the three types of periodic structures are studied, respectively. Furthermore, the effects of structural length, unit cell number, structural configurations, load conditions, and structural damping on the band gap properties are investigated.     

Detection of defects in cylindrical structures using a time reverse method and a finite-difference approach

The detection and characterization of defects in structures is an important issue in non-destructive testing. To avoid the scanning of large samples, guided elastic waves, which propagate along the structure, are excited. These waves interact with a defect, which results in a scattered wave field. In an experiment, the displacements of these scattered waves are recorded over time for a fixed axial coordinate at a number of circumferential positions of a circular cylindrical tube. Since in complex structures it is difficult to determine the axial and particularly circumferential position of the defect directly from the time signals, a time reversed numerical simulation is performed. There the measured displacement histories are reversed in time and used as displacement excitations in a simulation of the tested structure. A three-dimensional code in cylindrical coordinates, based on a velocity-stress finite-difference method, is used to simulate the wave propagation. As long as the geometric and material parameters are chosen equivalent to the performed experiment, the scattered waves travel back through the simulated structure and interfere, even if no defect is present in the numerical model. The result is an increase of the amplitudes of the stress and displacement components at the location where the defect was in the tested sample.     

Broadband near-infrared fibers dispersion measurement using white-light spectral interferometry

A new technique for an experimental determination of the effective refractive index, group refractive index and dispersion of fibers in a broad near-infrared spectral range is presented. The method is based on a white-light spectral interference which utilizes an unbalanced Michelson interferometer. The effective refractive index is obtained by a direct fitting the cosine function to the spectral interference pattern recorded by a low resolution spectrometer. The method has been tested in the spectral range of 1000-1700 nm both with standard telecommunication fibers and a sample of a photonic fiber. The accuracy of dispersion measurement () exceeds those from the previously reported near-infrared white-light spectral interference methods.     

多周期双啁啾镜结构的空间解波分复用器

以穿透深度依赖于入射波长的基本双啁啾结构为基础，通过对其中的布拉格膜对进行“周期 重复”的方法来提高空间线性侧向位移，从而实现不同波长的入射光的空间分离，用于高集成度的薄膜空间解波分复用器件的设计.比较了总层数相同的直接双啁啾结构和重复周期为n倍的改进结构，发现后者在理论设计更具有应用优势，同时在实验制作上有更高的容差性 .对空间色散特性曲线在周期重复数较大、入射波长较大的情况下所出现的强烈振荡做出了定性分析. 关键词： 双啁啾结构 空间色散 波分复用 薄膜器件     

Local spectral time-domain method for electromagnetic wave propagation

We explore the feasibility of using a local spectral time-domain (LSTD) method to solve Maxwell's equations that arise in optical and electromagnetic applications. The discrete singular convolution (DSC) algorithm is implemented in the LSTD method for spatial derivatives. Fourier analysis of the dispersive error of the DSC algorithm indicates that its grid density requirement for accurate simulations can be as low as approximately two grid points per wavelength. The analysis is further confirmed by numerical experiments. Our study reveals that the LSTD method has the potential to yield high resolution for solving large-scale electromagnetic problems.     

Adapting the spectral vanishing viscosity method for large-eddy simulations in cylindrical configurations

During the last decade the spectral vanishing viscosity (SVV) method has been adopted successfully for large-eddy-type simulations (LES) with high-order discretizations in both Cartesian and cylindrical coordinate systems. For the latter case, however, previous studies were confined to annular domains. In the present work, we examine the applicability of SVV in cylindrical coordinates to flows in which the axis region is included, within the setting of an exponentially convergent spectral element–Fourier discretization. In addition to the ‘standard’ SVV viscosity kernel, two modified kernels with enhanced stabilization in the axis region are considered. Three fluid flow examples are considered, including turbulent pipe flow. The results, on the one hand, show a surprisingly small influence of the SVV kernel, while on the other, they reveal the importance of spatial resolution in the axis region.     

New exact solution structures and nonlinear dispersion in the coupled nonlinear wave systems

In this Letter, to further understand the role of nonlinear dispersion in coupled nonlinear wave systems in both real and complex fields, we study the coupled Klein–Gordon equations with nonlinear dispersion in real field (called CKG(m,n,k)$\mathrm{CKG}(m,n,k)$ equation) and (2+1)$(2+1)$-dimensional generalization of coupled nonlinear Schrödinger equation with nonlinear dispersion in complex field (called GCNLS(m,n,k)$\mathrm{GCNLS}(m,n,k)$ equation) via some transformations. As a consequence, some types of solutions are obtained, which contain compactons, solitary pattern solutions, envelope compacton solutions, envelope solitary pattern solutions, solitary wave solutions and rational solutions.     

Simulation of elastic wave propagation in cylindrical structures including excitation by piezoelectric transducers

The development and optimization of non-destructive testing procedures usually needs experimental data. As experiments are time-consuming and expensive to conduct, we would like to use numerical data instead. This is admissible, if the simulation describes the physical experiments accurately. A three-dimensional displacement-stress finite-difference model is presented for a piezoelectric transducer coupled to an anisotropic tube. The allocation of the displacement and stress components on a staggered grid leads to a stable scheme. A full piezoelectric model of the transducer is used, including transverse isotropy in the elastic, dielectric, and piezoelectric constants. Similar to an experiment, elastic waves are excited in the corresponding simulation by applying a voltage signal to the electrodes of the piezoelectric transducer. Predictions of the simulation model for a piezoelectric ring transducer coupled to a carbon-fibre-reinforced shell are compared to experimental results to test the validity of the numerical data.