Geometrically nonlinear flexural vibrations of plates: In-plane boundary conditions and some symmetry properties

This study is devoted to the derivation of some properties of the von Kármán equations for geometrically nonlinear models of plates, with a boundary of arbitrary shape, for applications to nonlinear vibration and buckling. An intrinsic formulation of the local partial differential equations in terms of the transverse displacement and an Airy stress function as unknowns is provided. Classical homogeneous boundary conditions—with vanishing prescribed forces and displacements—are derived in terms of the Airy stress function in the case of a boundary of arbitrary geometry. A special property of this operator, crucial for some energy-conserving numerical schemes and called “triple self-adjointness”, is derived in the case of an edge of arbitrary shape. It is shown that this property takes a simple form for some classical boundary conditions, so that the calculations in some practical cases are also simplified. The applications of this work are either semi-analytical methods of solution, using an expansion of the solution onto an eigenmode basis of the associated linear problem, or special energy-conserving numerical methods.     

Generalization of the rotated parabolic equation to variable slopes

The rotated parabolic equation [J. Acoust. Soc. Am. 87, 1035-1037 (1990)] is generalized to problems involving ocean-sediment interfaces of variable slope. The approach is based on approximating a variable slope in terms of a series of constant slope regions. The original rotated parabolic equation algorithm is used to march the field through each region. An interpolation-extrapolation approach is used to generate a starting field at the beginning of each region beyond the one containing the source. For the elastic case, a series of operators is applied to rotate the dependent variable vector along with the coordinate system. The variable rotated parabolic equation should provide accurate solutions to a large class of range-dependent seismo-acoustics problems. For the fluid case, the accuracy of the approach is confirmed through comparisons with reference solutions. For the elastic case, variable rotated parabolic equation solutions are compared with energy-conserving and mapping solutions.     

A two-way parabolic equation that accounts for multiple scattering

A two-way parabolic equation that accounts for multiple scattering is derived and tested. A range-dependent medium is divided into a sequence of range-independent regions. The field is decomposed into outgoing and incoming fields in each region. The conditions between vertical interfaces are implemented using rational approximations for the square root of an operator. Rational approximations are also used to relate fields between neighboring interfaces. An iteration scheme is used to solve for the outgoing and incoming fields at the vertical interfaces. The approach is useful for solving problems involving scattering from waveguide features and compact objects.     

同轴线-矩形波导结散射特性的数值分析

 提出了一种数值分析同轴线-矩形波导结散射特性的模式匹配方法。采用同轴线和矩形波导的本征模函数表示电磁场分量,通过同轴线-矩形波导结截面横向场分量匹配获得波导结的散射参数,引入电场模式匹配矩阵的解析形式提高了计算效率。给出了基于模式匹配法数值仿真的各种同轴线-矩形波导结散射特性及仿真结果分析,并与3维全电磁波分析软件HFSS的仿真结果进行了比较,二者非常吻合。模式匹配法计算效率高,能广泛应用于微波毫米波元器件及系统结构的设计与优化。     

Polarization analyzers operating in a wide spectral range

Different versions of practical realization of customary optical systems for full measurement of the polarization state of quasi-monochromatic partially polarized light are considered. They can be used to overcome measurement difficulties associated with the finite width of the spectral range being studied. The effect of changes in characteristics of optical and polarization elements over a spectral range on the character of polarization measurements is analyzed. Methods for elimination of chromatism or taking into account its effect on measurements are suggested. Relations for taking into account the effect of chromatism of a phase plate with constant thickness on measurement results are given. Operation of a classical four-channel scheme for studing quasi-monochromatic partially polarized light in the visible and near-IR regions is considered. For this system, expressions are derived which make possible determination of Stokes parameters taking into account the effect of a finite spectral range. A system using phase modulation of radiation by an electrooptical cell and measurement of the maximum and minimum intensities of the modulated signal is considered. Measurements are carried out by using a two-channel system. Expressions are derived which make possible determination of Stokes parameters for the system under consideration in the visible and near-IR spectral regions. A system for measuring the polarization state of radiation in the visible and near-IR regions and based on modulation of radiation by an electrooptical phase cell is described. Measurements are carried out for the fundamental modulation frequency and its harmonics using a two-channel scheme. Relations for determining Stokes parameters of the system are presented.     

Physalis method for heterogeneous mixtures of dielectrics and conductors: Accurately simulating one million particles using a PC

Prosperetti’s seminal Physalis method, an Immersed Boundary/spectral method, had been used extensively to investigate fluid flows with suspended solid particles. Its underlying idea of creating a cage and using a spectral general analytical solution around a discontinuity in a surrounding field as a computational mechanism to enable the accommodation of physical and geometric discontinuities is a general concept, and can be applied to other problems of importance to physics, mechanics, and chemistry. In this paper we provide a foundation for the application of this approach to the determination of the distribution of electric charge in heterogeneous mixtures of dielectrics and conductors. The proposed Physalis method is remarkably accurate and efficient. In the method, a spectral analytical solution is used to tackle the discontinuity and thus the discontinuous boundary conditions at the interface of two media are satisfied exactly. Owing to the hybrid finite difference and spectral schemes, the method is spectrally accurate if the modes are not sufficiently resolved, while higher than second-order accurate if the modes are sufficiently resolved, for the solved potential field. Because of the features of the analytical solutions, the derivative quantities of importance, such as electric field, charge distribution, and force, have the same order of accuracy as the solved potential field during postprocessing. This is an important advantage of the Physalis method over other numerical methods involving interpolation, differentiation, and integration during postprocessing, which may significantly degrade the accuracy of the derivative quantities of importance. The analytical solutions enable the user to use relatively few mesh points to accurately represent the regions of discontinuity. In addition, the spectral convergence and a linear relationship between the cost of computer memory/computation and particle numbers results in a very efficient method. In the present paper, the accuracy of the method is numerically investigated by example computations using one dielectric particle, one isolated conductor particle, one conductor particle connected to an external source with imposed voltage, and two conductor/dielectric particles with strong interactions. The efficiency of the method is demonstrated with one million particles, which suggests that the method can be used for many important engineering applications of broad interest.     

Time-dependent shell-model theory of dissipative heavy-ion collisions

A transport theory is formulated within a time-dependent shell-model approach. Time averaging of the equations for macroscopic quantities lead to irreversibility and justifies weak-coupling limit and Markov approximation for the (energy-conserving) one- and two-body collision terms. Two coupled equations for the occupation probabilities of dynamical single-particle states and for the collective variable are derived and explicit formulas for transition rates, dynamical forces, mass parameters and friction coefficients are given. The applicability of the formulation in terms of characteristic quantities of nuclear systems is considered in detail and some peculiarities due to memory effects in the initial equilibration process of heavy-ion collisions are discussed.     

Spectral solvers for spherical elliptic problems

A new family of direct spectral solvers for the 3D Helmholtz equation in a spherical gap and inside a sphere for nonaxisymmetric problems is presented. A variational formulation (no collocation) is adopted, based on the Fourier expansion and the associated Legendre functions to represent the angular dependence over the sphere and using basis functions generated by Legendre or Jacobi polynomials to represent the radial structure of the solution. In the present method, boundary conditions on the polar axis and at the sphere center are not required and never mentioned, by construction. The spectral solution of the vector Dirichlet problem is also considered, by employing a transformation that uncouples the spherical components of the Fourier modes and that is implemented here for the first time. The condition numbers of the matrices involved in the scalar solvers are computed and the spectral convergence of all the proposed solution algorithms is verified by numerical tests.     

A dynamically adaptive wavelet approach to stochastic computations based on polynomial chaos – capturing all scales of random modes on independent grids

In stochastic computations, or uncertainty quantification methods, the spectral approach based on the polynomial chaos expansion in random space leads to a coupled system of deterministic equations for the coefficients of the expansion. The size of this system increases drastically when the number of independent random variables and/or order of polynomial chaos expansions increases. This is invariably the case for large scale simulations and/or problems involving steep gradients and other multiscale features; such features are variously reflected on each solution component or random/uncertainty mode requiring the development of adaptive methods for their accurate resolution. In this paper we propose a new approach for treating such problems based on a dynamically adaptive wavelet methodology involving space-refinement on physical space that allows all scales of each solution component to be refined independently of the rest. We exemplify this using the convection–diffusion model with random input data and present three numerical examples demonstrating the salient features of the proposed method. Thus we establish a new, elegant and flexible approach for stochastic problems with steep gradients and multiscale features based on polynomial chaos expansions.     

An analytical least squares solution applied to spectral series limit determinations

A method is derived to determine spectral series limits analytically. It is assumed that the graphical quantum defect solution (n?n^* vs ν_lim?ν) is approximately linear (for the correct ν_lim) so that a least-squares fit may be obtained. The problem reduces to a non-linear system of three equations in three unknowns (two least-squares coefficients and the series limit) with a unique solution found numerically. Least-squares errors are also obtained for each variable. Results using this method agree well with published values obtained by the traditional graphical method. This procedure provides an operational method for series-limit determinations, independent of the subjective problems involved with the graphical method.     

Discrete Newtonian gravitation and the three-body problem

Newtonian gravitation is studied from a discrete point of view, in that the dynamical equation is an energy-conserving difference equation. Application is made to planetary-type, nondegenerate three-body problems and several computer examples of perturbed orbits are given.     

N-soliton solution of a coupled integrable dispersionless equation

<正>A new coupled integrable dispersionless equation is presented by considering a spectral problem.A Darboux transformation for the resulting coupled integrable dispersionless equation is constructed with the help of spectral problems.As an application,the N-soliton solution of the coupled integrable dispersionless equation is explicitly given.     

Non-Markovian relaxation of atomic systems and a calculation of the shape of spectral lines

A non-Markovian kinetic equation for a system of two identical interacting two-level atoms has been derived. The solution to this equation has been used for calculating the shape of spectral lines of this system.     

A theory of Doppler and binary collision broadening by foreign gases—II.: Isolated spectral line structure

The kinetic equation, derived in the preceeding paper I and describing combined Doppler and binary collision broadening is examined. A series solution is developed for the case of an isolated spectral line which, to first order in the density, is Lorentzian under appropriate conditions. The general solution retains the effects of non-instantaneous collisions and binary collision correlations. An approximation for the collision terms of the kinetic equation is also proposed.     

Approximate analytical solution to normal emittance of semi-transparent layer of an absorbing, scattering, and refracting medium

A two-step approximate analytical solution for the normal emittance of a plane layer of an absorbing, scattering and refracting medium is derived analytically. The analysis is based on the transport approximation and the two-step solution method for radiative transfer. The high accuracy of the approximate solution, examined by comparing its results to those obtained independently by the discrete ordinates and Monte Carlo methods, makes it suitable for application in combined experimental-analytical studies to identify selected spectral radiative properties of dispersed media in the range of semi-transparency.     

Spectral broadening and kinetic properties of a Dapoxyl fluorescent probe

The steady-state spectra of Dapoxyl in polar solutions under selective excitation demonstrate considerable spectral inhomogeneity (about 600 cm^?1) in the case of a high viscosity solution. The time characteristics of the luminescence decay in different spectral regions and a long-wavelength shift of the instantaneous emission spectra indicate the occurrence of relaxation processes leading to a decrease in the excited-state energy. A correlation between the luminescence spectra and the processes of intermolecular orientational relaxation of the solution is established.     

Estimation of the plasma temperature by using the resolved spectrum of N<Subscript>2</Subscript><Superscript>+</Superscript>

This publication analyzes the problems of estimating the gas temperatures in the nitrogen-containing plasma in the temperature regions between 4 and 9 kK. Problems of low and high resolution molecule bands spectroscopy and temperature derivation is discussed and the proposition of the quick procedure is presented. Its validity is tested by comparison of the temperatures derived from different molecular bands by the help of the Boltzmann rotational structure diagrams. The perturbation in the molecular spectrum is exploited to choose the most easily resolved spectral lines.     

An approximate block Newton method for coupled iterations of nonlinear solvers: Theory and conjugate heat transfer applications

A new, approximate block Newton (ABN) method is derived and tested for the coupled solution of nonlinear models, each of which is treated as a modular, black box. Such an approach is motivated by a desire to maintain software flexibility without sacrificing solution efficiency or robustness. Though block Newton methods of similar type have been proposed and studied, we present a unique derivation and use it to sort out some of the more confusing points in the literature. In particular, we show that our ABN method behaves like a Newton iteration preconditioned by an inexact Newton solver derived from subproblem Jacobians. The method is demonstrated on several conjugate heat transfer problems modeled after melt crystal growth processes. These problems are represented by partitioned spatial regions, each modeled by independent heat transfer codes and linked by temperature and flux matching conditions at the boundaries common to the partitions. Whereas a typical block Gauss–Seidel iteration fails about half the time for the model problem, quadratic convergence is achieved by the ABN method under all conditions studied here. Additional performance advantages over existing methods are demonstrated and discussed.     

Sensitivity of atmospheric spectral transmittance to line and meteorological parameters

A study is made of the variance of the monochromatic molecular absorption coefficient and of the corresponding slant-path transmittance due to variations in the tabulated Lorentzian line-parameter data and atmospheric meteorological variables. General expressions are derived from five basic assumptions, which are useful for calculations involving specific atmospheric profiles and spectral line-parameter data. The results of calculations for 8 representative frequencies, assuming typical variations in the parameters, indicate that transmittance errors greater than 0.01 should be expected for transmittance calculations in the range from about 0.1 to 0.9.     

Sylvester theorem and the multichannel transfer matrix method for arbitrary transverse potential profile inside a wave guide

Based on the Sylvester and Frobenius theorems, we drastically enhance the feasibility of the transfer-matrix approach to deal with problems involving a large number of propagating and interfering modes, which require the solution of coupled differential equations and the evaluation of functions of matrix variables. We report closed formulas for the spectral decomposition of this type of functions. As specific example, besides the calculation of simple and well-known 1D one channel transfer matrices, we derive the multi-channel transfer matrix for an electron gas in the presence of a transverse electric field.