A k-Space Method for Moderately Nonlinear Wave Propagation

Parabolic approximation in ray coordinates for high-frequency nonlinear waves in a inhomogeneous and high speed moving fluid

A proper dielectric coating can reduce the electromagnetic scattering of the conducting object significantly so that it cannot be detected by the radar. However, when the object contains a thin coating, fine grid size is needed to discretize the thin coating in the conventional finite-difference time-domain (FDTD) algorithm, which increases the amount of memory and computational time significantly. To overcome this dilemma, we present a transformation optics-based FDTD (TO-FDTD) algorithm to accelerate the solution of electromagnetic scattering from objects with thin dielectric coatings. Two kinds of novel TO-FDTD models are proposed in this paper for a coated cylinder and a coated arbitrary polygonal cylinder, respectively. Through coordinate transformation, the size of the object remains unchanged while its thin coating is enlarged to a thicker one, meaning that it can be simulated by the FDTD algorithm with uniform coarse grids instead of fine grids. The transformed material parameters become inhomogeneous and anisotropic in the transformed region, which can be obtained by solving a Jacobian transformation matrix. We then develop a stable FDTD algorithm for solving anisotropic Maxwell’s equations. Bistatic scatterings of coated cylinders and a coated polygonal cylinder are solved by the TO-FDTD algorithm proposed in this paper, respectively. The result of the TO-FDTD algorithm matches well with the exact value and the result of the commercial software Comsol. The computational efficiency and accuracy of the proposed TO-FDTD algorithm are validated by numerical experiments. Numerical results show that the TO-FDTD algorithm has higher computational accuracy than the conventional FDTD algorithm that fails to simulate the absorbing property of the coating, when the same coarse grid size is used in the simulation. Under the same level of accuracy, the proposed TO-FDTD method can improve the computational efficiency by 62-63 times than the conventional FDTD method with fine grids in the simulations in the paper.

In this paper, a numerical approach based on weighted Laguerre polynomials finite-difference time-domain (WLP-FDTD) method is proposed to analyze noise coupling in power/ground planes between package and printed circuit board (PCB). Compared to the traditional finite-difference time-domain (FDTD) method, which is limited by Courant-Friedrich-Lecy stability condition, the proposed method has the unconditionally stable characteristic. Examples of a ball grid array package mounted on a PCB were simulated, and the numerical results show that the computation time of the proposed WLP-FDTD method can be 4.4% of that of the FDTD method at the same accuracy condition. Therefore, the proposed method has higher efficiency than the conventional FDTD method without sacrificing precision.

In this study, a combination of two fundamental frequencies f1=485 KHz and f2 = 515 KHz. The source transducers are placed on the surface of the sand 25 cm above the bottom and 20 cm below the water surface in the tank of dimension 74 × 47 × 43.5 cm simulating a seabed with sand of maximum particle size of 5 mm. Burger's equation is the simplest model for describing the second order nonlinear effects in the propagation of high amplitude plane waves and, in addition, the dissipative effects in media. Typically, the interactions of two acoustic waves at a discontinuity in a dispersive but homogeneous medium will generate harmonics and intermodulation terms. This is numerically modeled in Matlab using the finite difference time domain method with Neumann boundary conditions and practically verified with measurements from the tank. The measured and simulated results have shown significant agreement with the theory. For instance, second order intermod term show 2 dB decreases in amplitude for 7 mm increase in thickness at constant density of plywood. Also if the thickness is kept same, e.g., 5 mm, there is 10 dB drop in same intermod term for change in density from plywood 550 kg/m3 to copper 8930 kg/m3.

A multiscale model for the simulation of two‐dimensional nonlinear wave propagation in microcracked materials exhibiting hysteretic nonlinearity is presented. We use trigger‐like elements with a two state nonlinear stress‐strain relation to simulate microcracks at the microlevel. A generalized Preisach space approach, based on the eigenstress‐eigenstrain formulation, upscales the microscopic state relation to the mesoscopic level. The macroscopic response of the sample to an arbitrary excitation signal is then predicted using a staggered grid Elastodynamic Finite Integration Technique (EFIT) formalism. We apply the model to investigate spectral changes of a pulsed signal traversing a localized microdamaged region with hysteretic nonlinearity in a plate, and to study the influence of a superficial region with hysteretic nonlinearity on the nonlinear Rayleigh wave propagation.

The nonlinear wave propagation in a high Mach number, low-density region of a spherically symmetric expanding gas flow is considered. It is well known that classical inviscid gas dynamics is inadequate in describing such flow regimes and consequently the investigation is initiated with Boltzmann's equation. The asymptotic theory describing such a flow has the effect of truncating any order moment equation. This and other simplifications inherent in the approach enable the time dependent moment equations to be integrated and the solutions used in the study of boundary and initial value problems. The theory is applied to the study of a pulse propagating along the centerline of a steady axially symmetric free jet exhausting into vacuum.

We discuss linear and nonlinear optical wave propagation in a left-handed medium (LHM) or medium of negative refraction (NRM). We use the approach of characterizing the medium response totally by a generalized electric polarization [with a dielectric permittivity $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\varepsilon}}(\ensuremath{\omega},\stackrel{\ensuremath{\rightarrow}}{k})]$ that can be decomposed into curl and noncurl parts. The description has a one-to-one correspondence with the usual approach characterizing the LHM response with a dielectric permittivity $\ensuremath{\varepsilon}<0$ and a magnetic permeability $\ensuremath{\mu}<0.$ The latter approach is less physically transparent in the optical frequency region because the usual definition of magnetization loses its physical meaning. Linear wave propagation in a LHM or NRM is characterized by negative refraction and negative group velocity that could be clearly manifested by ultrashort pulse propagation in such a medium. Nonlinear optical effects in a LHM can be predicted from the same calculations adopted for ordinary media using our general approach.

A wave-vector-frequency-domain method is presented to describe one-directional forward or backward acoustic wave propagation in a nonlinear homogeneous medium. Starting from a frequency-domain representation of the second-order nonlinear acoustic wave equation, an implicit solution for the nonlinear term is proposed by employing the Green's function. Its approximation, which is more suitable for numerical implementation, is used. An error study is carried out to test the efficiency of the model by comparing the results with the Fubini solution. It is shown that the error grows as the propagation distance and step-size increase. However, for the specific case tested, even at a step size as large as one wavelength, sufficient accuracy for plane-wave propagation is observed. A two-dimensional steered transducer problem is explored to verify the nonlinear acoustic field directional independence of the model. A three-dimensional single-element transducer problem is solved to verify the forward model by comparing it with an existing nonlinear wave propagation code. Finally, backward-projection behavior is examined. The sound field over a plane in an absorptive medium is backward projected to the source and compared with the initial field, where good agreement is observed.

A new explicit and nondissipative finite-difference time-domain (FDTD) method in two and three dimensions is proposed for the relaxation of the Courant condition. The third-degree spatial difference terms with second- and fourth-order accuracies are added with coefficients to the time-development equations of FDTD(2,4). Optimal coefficients are obtained by a brute-force search of the dispersion relations, which reduces phase velocity errors but satisfies the numerical stabilities as well. The new method is stable with large Courant numbers, whereas the conventional FDTD methods are unstable. The new method also has smaller numerical errors in the phase velocity than conventional FDTD methods with small Courant numbers.

In this chapter we shall discuss a variety of intracellular Ca2+ wave phenomena, but always from the perspective that the distance scales of interest are large enough that Ca2+ transport is well-modeled by conservation equations based on a continuum description of matter (recall Chapter 7). Although recent experimental and theoretical work suggests that the macroscopic behavior of propagating Ca2+ waves (e.g., wave speed) may depend in subtle ways on the density and distribution of intracellular Ca2+ release channels, we postpone consideration of intracellular heterogeneities such as clusters of Ca2+ release channels until later in the chapter. This makes sense because both the mathematics and simulation methods used to study nonlinear wave propagation in homogeneous media are simpler than the heterogeneous case. This simplicity should facilitate the development of intuition regarding nonlinear wave propagation. Throughout the chapter a recurring theme will be the manner in which Ca2+ buffers, through their important association with free Ca2+, can influence wave phenomena dependent on diffusion. The chapter concludes with calculations of localized Ca2+ elevations due to intracellular Ca2+ release, i.e., Ca2+ “puffs” or “sparks,” elementary events that sum to produce Ca2+ waves.

The nonlinear acoustic problem of a finite amplitude plane wave propagating along the axial direction in a traveling wave tube is studied. Based on the one-dimensional Westervelt equation, a one-dimensional nonlinear wave equation is derived in which the cross section of the traveling wave tube is considered. The two-order finite difference scheme is used to solve the nonlinear wave equation. The nonlinear propagation characteristics of a finite amplitude wave in the traveling wave tube is analyzed. In the expanding transition section, the acoustic pressure amplitude of the acoustic wave decreases with the increase of the cross-sectional area of the pipeline. The nonlinear characteristics of the acoustic wave show waveform distortion and harmonic growth. The waveform distortion becomes more serious in the rear of traveling wave tube than in the front of the tube. Considering the acoustic reflection condition at the mouth, the influence of differently shaped diffusion sections on the acoustic pressure distribution in the test section is investigated. The larger the change rate of the diffusion section in an area, the less amplitude of the sound pressure, and the nonlinear effect of the sound wave propagation is weakened. These nonlinear wave propagation characteristics in a travelling wave tube provide important guidance for both designing a uniform sound pressure distribution in the test section and determining the optimal measuring points for different sizes of structures in spacecraft.

The nonlinear propagation of different types of DANW acoustic dust nonlinear waves has been investigated in a magnetized dusty plasma consisting of negatively charged dust particles, Maxwellian electrons, and ions. Application of the standard reductive perturbation theory is used to derive the corresponding three-dimensional nonlinear a complex Ginzburg–Landau (3D-CGLE) equation which governs the dynamics of the dust-acoustic wave packets. The stationary analytical solutions of the CGLE are numerically analysed where the effect of the physical parameters of the dusty plasma model on the wave’s propagation is taken into account. It has been found that there can be a relationship between the appearance of soliton waves and electromagnetic waves, as well as between shock-like waves and periodic travelling waves. Expression of the importance of these findings is the cornerstone of explaining the true relationship between the propagation of nonlinear waves in the physics of space, for example, the Earth’s magnetic field.

The finite-difference time-domain (FDTD) method is performed on discrete rectangular cells in the computational domain, and non-uniform cells are often used to save computational costs. Non-uniform cells increase the truncation error at the boundary of two domains whose cell sizes differ. We apply a high-order implicit scheme at the boundary to reduce this error. Numerical formulations and error criteria of the scheme are described and evaluated in practical applications. Compared with the conventional FDTD method, the new technique is more accurate.

Finite-difference time domain (FDTD) methods, which have been widely employed in three-dimensional transient electromagnetic (TEM) modeling, require very small time steps to simulate the electromagnetic fields and this will be time consuming. We present an efficient numerical method for three-dimensional TEM forward modeling. Its key features are based on a correspondence principle between the diffusive and fictitious wave fields. The diffusive Maxwell’s equations are transformed and solved in a so-called fictitious wave domain. This scheme allows larger time steps than conventional FDTD methods, allows including air layers, and allows simulating topography. The need for initial field calculations is avoided by including an electric current source in the governing equations. This also avoids a traditional assumption of a flat earth surface in TEM modeling. We test the accuracy of the electromagnetic fields’ responses using our method with the spectral differential difference (SLDM) solutions. The results show good agreement even under the existence of air layers and topography in the model.

A method for fast simulation of nonlinear wave propagation has been developed. It is based on quasi-linear approximation. By using the angular spectrum method, the contribution from an arbitrarily plane was found by back propagation of the wave field from the focal plane, squaring the field, and forward propagation of the result to the observation plane. In the spatial/time Fourier-domain, propagation is performed by a complex exponential factor, whereas the square operation corresponds to a 3D convolution. It turned out that the integration along the ultrasound beam could be solved analytically. The computation in the Fourier-domain was then reduced to a triple-integral. The resulting second harmonic wave in the observation plane was found by (3D) inverse FFT. A simulation program has been implemented in Matlab, using a laptop PC with 1.2 GHz Pentium III processor. Applied to a phased array transducer the simulation time was less than 5 minutes. The implementation of the algorithm was verified by measurements. We observed satisfactory accordance between simulations and measurements.