On the problem of analyzing the dynamic properties of a layered half-space

An efficient method is developed for constructing the Green matrix functions of a layered inhomogeneous half-space. Matrix formulas convenient for programming are proposed, which make it possible to study the properties of a multilayered half-space with high accuracy. As an example of the problem of oscillations of a three-layer half-space, transformation of the dispersion characteristics of a three-layered medium is shown as a function of the relations of the mechanical and geometric parameters of its components. Study of the properties of the Green’s function of a medium with a low-velocity layered inclusion showed that each mode of a surface wave exists in a limited frequency range: in addition to the critical frequency of mode occurrence, the frequency of its disappearance exists—a frequency above which the mode is suppressed because of superposition of the zero of the Green’s function on its pole. A similar study conducted for a medium with a high-velocity layered inclusion has shown that in addition to the cutoff frequency (the frequency at which a surface wave propagating in the low-frequency range disappears), there is the frequency of its recurrent generation—the upper boundary of the “cutoff range” of the first mode. Beyond this range, the first mode propagates, and also the other propagating modes can appear. The critical relation of the geometric parameters of the medium determining the existence and boundaries of the cutoff range of a wave is established.     

The abstracted and integrated Green functions and OOP of BEM in soil dynamics

It has been generally recognized that Green function integrated with Boundary Element Method (BEM) has advantages in dimensional reduction, high accuracy and satisfaction of the radiation condition at infinity, etc. Recently, the computational technique has rapidly developed and the orient-object programming has been widely applied, whereas the attribute of abstraction and the integration of Green function employed in BEM have not been discovered yet. In this work the abstraction and integration of Green function are carried out for soil dynamics problems, and the procedure of the object-oriented calculation method is presented. Based on the Green function developed for the two-phase saturated medium, the response of the wave field to tunnel subjected to dynamic loading is calculated, and the transient solution as well as the time history of response is obtained. Supported by the National Natural Science Foundation of China (Grant No. 10572129)     

Error control in the perfectly matched layer based multilevel fast multipole algorithm

The development of efficient algorithms to analyze complex electromagnetic structures is of topical interest. Application of these algorithms in commercial solvers requires rigorous error controllability. In this paper we focus on the perfectly matched layer based multilevel fast multipole algorithm (PML-MLFMA), a dedicated technique constructed to efficiently analyze large planar structures. More specifically the crux of the algorithm, viz. the pertinent layered medium Green functions, is under investigation. Therefore, particular attention is paid to the plane wave decomposition for 2-D homogeneous space Green functions in very lossy media, as needed in the PML-MLFMA. The result of the investigations is twofold. First, upper bounds expressing the required number of samples in the plane wave decomposition as a function of a preset accuracy are rigorously derived. These formulas can be used in 2-D homogeneous (lossy) media MLFMAs. Second, a more heuristic approach to control the error of the PML-MLFMA’s Green functions is presented. The theory is verified by means of several numerical experiments.     

Calculation of a quasi-stationary field in the layered medium

A method of calculating the field of a single point charge in a layered medium is suggested. The method makes it possible to simplify the computational formulas for the potential. The solution of the problem employs the properties of Bessel and Struve functions. The results apply to a wide class of problems solvable in terms of Green’s function.     

Monte Carlo simulation in the theory of wave propagation in random media: I. Modified and Feynman path integrals

The purpose of this article (comprising parts I and II) is to develop and test the approach of combining a path-integral technique and a complex-valued Monte Carlo method to calculate the highest moments of the Green function of the stochastic wave equation for a random medium against the background of large-scale inhomogeneities. In part I, the new modified path-integral representations of the Green function moments of the stochastic wave equation have been developed. The limiting transition of these representations to the Feynman path integrals corresponding to the parabolic approximation is discussed. Path-integral representations for Green function moments are given for three models: a model of the stochastic wave equation and models of parabolic and Markov approximations. The Metropolis algorithm underlying the Monte Carlo method for calculating real and complex-valued path integrals is discussed in brief. Numerical results are presented in part II of the article.     

Modified physical optics algorithm for near field scattering

A novel modified physical optics algorithm is proposed to overcome the difficulties of near field scattering prediction for classical physical optics. The method is applied to calculating the near field radar cross section of electrically large objects by taking into account the influence of the distinct wave propagation vector, the near field Green function, and the antenna radiation pattern. By setting up local reference coordinates, each partitioned facet has its own distinct wave front curvature. The radiation gain for every surface element is taken into consideration based on the modulation of the antenna radiation pattern. The Green function is refined both in amplitude and phase terms and allows for near field calculation. The scattered characteristics of the near field targets are studied by numerical simulations. The results show that the approach can achieve a satisfactory accuracy.     

Calculation of the diffraction field in a layered medium illuminated through a phase mask

Optics and Spectroscopy - A local mode technique has been developed for calculating a diffraction field in a layered medium illuminated by the TE-polarized light through a phase mask with a...     

Calculation of the electroelastic Green’s function of the hexagonal infinite medium

The electroelastic 4 × 4 Green’s function of a piezoelectric hexagonal (transversely isotropic) infinitely extended medium is calculated explicitly in closed compact form ((73) ff. and (88) ff., respectively) by using residue calculation. The results can also be derived from Fredholm’s method [2]. In the case of vanishing piezoelectric coupling the derived Green’s function coincides with two well known results: Kröner’s expressions for the elastic Green’s function tensor [4] is reproduced and the electric part then coincides with the electric potential (solution of Poisson equation) which is caused by a unit point charge. The obtained electroelastic Green’s function is useful for the calculation of the electroelastic Eshelby tensor [16].     

The Casimir pressure regularization in two-dimensional field models

A method for Casimir pressure calculation with the help of the regular part of the Green surface function is considered in the two-dimensional case. Also, a method for the approximate calculation of the regular part of the Green surface function using a Born-type series is suggested. It is tested for a problem for which the exact solution is known.     

Investigation on electromagnetic scattering from rough soil surface of layered medium using the small perturbation method

Electromagnetic scattering from a rough surface of layered medium is investigated, and the formulae of the scattering coefficients for different polarizations are derived using the small perturbation method. A rough surface with exponential correlation function is presented for describing a rough soil surface of layered medium, the formula of its scattering coefficient is derived by considering the spectrum of the rough surface with exponential correlation function; the curves of the bistatic scattering coefficient of HH polarization with variation of the scattering angle are obtained by numerical calculation. The influence of the permittivity of layered medium, the mean layer thickness of intermediate medium, the roughness surface parameters and the frequency of the incident wave on the blstatic scattering coefficient is discussed. Numerical results show that the influence of the permittivity of layered medium, the mean layer thickness of intermediate medium, the rms and the correlation length of the rough surface, and the frequency of the incident wave on the bistatic scattering coefficient is very complex.     

The green function technique and the Pauli principle in intermediate states

Arguments are presented to demonstrate that in calculating higher-order transition amplitudes by the Green function technique, terms due to occupied intermediate levels need not be subtracted. The inclusion of spurious contributions exactly compensates for ignoring core excitations.     

Calculation of the bistatic target strength of a complex multiresonant shell by the finite element method

A method for calculating the bistatic target strength of a complex elastic structure with characteristic linear dimensions of one to ten wavelengths is developed and tested. The proposed method is based on the finite-element approach to modeling the mechanoacoustic properties of the object under consideration, which is complemented with the algorithms of incident plane wave generation and far-field calculation using the Green function of free space. The results of testing the accuracy of the proposed computational method in application to bodies with a simple shape and a known analytical solution are presented. Specifically, a deviation of less than 3 dB from the theoretical value is obtained for the main lobe of the directional pattern of the target strength. The results of calculations are compared with experimental data obtained for a thinwalled reinforced shell used as an example.     

Elastic interaction of point defects in cubic and hexagonal crystals

The elastic interaction of two point defects in cubic and hexagonal structures has been considered. On the basis of the exact expression for the tensor Green’s function of the elastic field obtained by the Lifschitz–Rozentsveig for a hexagonal medium, an exact formula for the interaction energy of two point defects has been obtained. The solution is represented as a function of the angle of their relative position on the example of semiconductors such as III-nitrides and α-SiC. For the cubic medium, the solution is found on the basis of the Lifschitz–Rozentsveig Green’s tensors corrected by Ostapchuk, in the weak-anisotropy approximation. It is proven that the calculation of the interaction energy by the original Lifschitz–Rozentsveig Green’s tensor leads to the opposite sign of the energy. On the example of the silicon crystal, the approximate solution is compared with the numerical solution, which is represented as an approximation by a series of spherical harmonics. The range of applicability of the continual approach is estimated by the quantum mechanical calculation of the lattice Green’s function.     

Generalized numerical renormalization group for dynamical quantities

In this paper we introduce a new approach for calculating dynamical properties within the numerical renormalization group. It is demonstrated that the method previously used fails for the Anderson impurity in a magnetic field due to the absence of energy scale separation. The problem is solved by evaluating the Green function with respect to the reduced density matrix of the full system, leading to accurate spectra in agreement with the static magnetization. The new procedure provides a unifying framework for calculating dynamics at any temperature and represents the correct extension of Wilson's original thermodynamic calculation.     

Reflection characterization of multilayer surface films

The reflection of linearly polarized light from a multilayer system of ultrathin dielectric surface films is investigated both analytically in the long-wavelength limit and numerically by the standard way of calculating the reflection characteristics for the layered medium. The second-order approximate formulas for reflection coefficients and characteristic reflection angles are derived and their accuracy is estimated. It is shown that approximate expressions obtained for reflection parameters of multilayer system in the long-wavelength limit are of immediate interest to the solution of the inverse problem for ultrathin layered surface structures. Innovative possibilities for optical diagnostics are generated by means of polarizing and principal angles. For determining the parameters of multiple surface layers an appropriate method is found by combining differential reflectance with ellipsometry.     

Semiclassical and quantum Liouville theory on the sphere

We solve the Riemann–Hilbert problem on the sphere topology for three singularities of finite strength and a fourth one infinitesimal, by determining perturbatively the Poincaré accessory parameters. In this way we compute the semiclassical four point vertex function with three finite charges and a fourth infinitesimal. Some of the results are extended to the case of n finite charges and m infinitesimal. With the same technique we compute the exact Green function on the sphere with three finite singularities. Turning to the full quantum problem we address the calculation of the quantum determinant on the background of three finite charges and the further perturbative corrections. The zeta function technique provides a theory which is not invariant under local conformal transformations. Instead by employing a regularization suggested in the case of the pseudosphere by Zamolodchikov and Zamolodchikov we obtain the correct quantum conformal dimensions from the one loop calculation and we show explicitly that the two loop corrections do not change such dimensions. We expect such a result to hold to all order perturbation theory.     

Acoustic cloaking by the wave flow method

The methods for calculating acoustic cloaking that is implemented by the wave flow method are reviewed, and the efficiency of this technique for cloaking regions with symmetries of three types is analyzed. It is shown that the main problem in implementing acoustic cloaking is the formation of an anisotropic medium with inhomogeneous components of the density tensor and bulk modulus, which change in a wide range and have limiting values of ∞ or 0 at the cloaking-region boundaries. Some estimates are obtained, according to which a stratified medium composed of a sequence of layers with different densities and compressibilities appears to be more promising, because it (i) is more easy to implement and (ii) opens possibilities for broadband cloaking. Analysis of different versions of the cloaking-region symmetry revealed that, using a layered medium, one can implement efficient acoustic cloaking only in the case of a spherically symmetric region.     

矩形单元声场波函数构造及其应用*

在波叠加法中,结构外部声场是在离散边界上对Green函数进行积分并叠加得到,但数值积分的计算效率较低。而等效源法虽然提高了计算效率,但其面源简化为点源的过程中存在较大的积分近似误差。针对上述两种方法的缺陷,构造了一种波函数以替代离散单元关于Green函数积分的声场。首先,利用球坐标系下Helmholtz方程的解,推导了替代矩形单元积分的一般形式波函数及效率更高的内推波函数。其次,当离散单元为正方形时,将其近似成圆形域,进一步简化了内推波函数的表达式。最后,将所构造的波函数应用于声场计算。数值结果表明,在计算单个矩形单元外部辐射声场时,构造的波函数不仅保证了计算精度,而且相比于直接积分大幅度提高了计算效率。其中,矩形域一般形式和内推形式的波函数计算效率是直接积分的5～6倍,圆形域内推波函数计算效率达到了直接积分的12～13倍。在简支板声源和立方箱体辐射声源数值算例中,圆形域内推波函数在整个计算频段的声场计算精度均高于等效源法。     

Application of a diagram technique for Hamiltonians with Coulomb interactions to the Hubbard model

We give a general statistical Wick theorem for Hamiltonians containing the Coulomb interaction. The diagram technique developed for some class of interactions is applied to the calculation of the transverse Green function and effective interaction within the Hubbard model.     

The effect of wave reflection and refraction at soft tissue interfaces during ultrasound hyperthermia treatments.

An improved numerical model for calculating the ultrasonic power deposition in layered medium was developed and experimentally tested. The new model takes into account the ultrasound wave reflection and refraction at the tissue interfaces thereby providing improved accuracy in ultrasound hyperthermia treatment planning. The model was compared with a simplified model to evaluate when the tissue interfaces could be ignored in the hyperthermia treatment planning and evaluation. The effect of variations in water and tissue temperatures, the fat layer thicknesses, and the beam entrance angle were also investigated to establish guidelines for treatment execution. It was found that in most cases the effects of the soft tissue interfaces can be ignored. However, in some instances the acoustic focus may be shifted several millimeters off axis in layered medium. This is important when sharply focused transducers are used for ultrasound surgery or under the condition of pulsed, high-temperature hyperthermia treatments.