Radiation Boundary Conditions for Elastic Wave Propagation

We study a single, motionless three-dimensional droplet growing by adsorption of diffusing monomers on a 2D substrate. The diffusing monomers are adsorbed at the aggregate perimeter of the droplet with different boundary conditions. Models with both an adsorption boundary condition and a radiation boundary condition, as well as a phenomenological model, are considered and solved in a quasistatic approximation. The latter two models allow particle detachment. In the short time limit, the droplet radius grows as a power of the time with exponents of 1/4, 1/2 and 3/4 for the models with adsorption, radiation and phenomenological boundary conditions, respectively. In the long time limit a universal growth rate as $[t/\ln(t)]^{1/3}$ is observed for the radius of the droplet for all models independent of the boundary conditions. This asymptotic behaviour was obtained by Krapivsky \cite{krapquasi} where a similarity variable approach was used to treat the growth of a droplet with an adsorption boundary condition based on a quasistatic approximation. Another boundary condition with a constant flux of monomers at the aggregate perimeter is also examined. The results exhibit a power law growth rate with an exponent of 1/3 for all times.

Constructal entransy optimizations for insulation layer of steel rolling reheating furnace wall with convective and radiative boundary conditions

In this paper is developed a general method of solving Oseen’s linearized equations for a two-dimensional steady flow of a viscous fluid past an arbitrary cylindrical body. The method is based on the fact that the velocity in the neighbourhood of the cylinder can be generally expressed in terms of a pair of analytic functions, the determination of which from the appropriate boundary condition can be effected by successive approximations in powers of the Reynolds number, R. The method enables one to obtain the velocity distribution near the cylinder and the lift and drag acting on it in the form of power series in R, without recourse to manipulation of higher transcendental functions such as Bessel and Mathieu functions for circular and elliptic cylinders, respectively. As an example of the application of the method, the uniform flow past an elliptic cylinder at an arbitrary angle of incidence is considered. Analytical expressions for the lift and drag coefficients are obtained, which are correct to the order of R, the lowest order terms being O(R-1) and numerical calculations are carried out for the thickness ratio t = 0, 0.1, 0.5, 1 and the Reynolds number R = 0.1, 1. It is found that drag increases slightly with increase of either thickness ratio or angle of incidence, and that lift decreases with increase of thickness ratio while, as a function of the angle of incidence, it has a maximum at about 45°.

To investigate the seismic performance of simulated subway tunnels under complex geological conditions of obliquely incident earthquake excitations, a new input method for oblique incidence of SV waves with arbitrary incident angles in 3D space has been proposed. The method relies on explicit finite element method (EFEM) and viscous-spring artificial boundary (VSAB) condition, which simultaneously considers soil-structure interaction, surrounding soil characteristics, and seismic wave incident characteristics. It is applied and verified based on the seismic wave theory. The overall numerical model of subway tunnel and soil foundation is then established, and the dynamic response of subway tunnel in soft and hard interbedded soils under oblique incidence of SV waves is systematically analyzed. The obtained simulated results indicate that the dynamic response of each point in subway tunnel is decreased by increasing incident SV wave angle. It is also found that the dynamic responses of the site interbedded by soft and hard soils are significantly higher than that in other points of the site at the same incident angle, indicating that the mentioned zone has weak seismic resistance. In addition, the dynamic response of each point in tunnel structure tends to become closer when the bulk moduli of soft and hard soils are getting close. The displacement and moment of the site interbedded by soft and hard soils are reduced by increasing the thickness of lining structure, therefore improving the seismic resistance of the tunnel. The results of parametric study show that these factors could have significant influences on the seismic responses of tunnel structures during earthquake and should be considered during earthquake-resistant design of subway tunnels under complex geological conditions.

Boundary value problems in magnetohydrodynamics (and fluid dynamics). I. Radiation boundary condition

Explicit expressions are presented for the radiation fields scattered by rough surfaces. Both electric and magnetic dipole sources are assumed, thus excitations of both vertically and horizontally polarized waves are considered. The solutions are based on a full-wave approach which employs complete field expansions and exact boundary conditions at the irregular boundary. The scattering and depolarization coefficients axe derived for arbitrary incident and scatter angles. When the observation point is at the source these scattering coefficients are related to the backscatter cross section per unit area. Solutions based on the approximate impedance boundary condition are also given, and the suitability of these approximations are examined. The solutions are presented in a form that is suitable for use by engineers who may not be familiar with the analytical techniques and they may be readily compared with earlier solutions to the problem. The full-wave solutions are shown to satisfy the reciprocity relationships in electromagnetic theory, and they can be applied directly to problems of scattering and depolarization by periodic and random rough surfaces.

A coupled three-dimensional fluid-elastic wave propagation mathematical model has been developed to handle environmental interactions in the ocean between a fluid medium and an elastic bottom. The existing model combines three-dimensional fluid and elastic wave propagation models with the incorporation of a set of horizontal fluid-elastic interface conditions. This paper extends the above model to consider an irregular fluid-elastic interface. The theoretical development of the irregular fluid-elastic interface equations is presented. Comparison of the fluid-elastic irregular interface to the horizontal case is one of the major objectives of this paper. Another major objective of this paper is the construction of a complete set of three-dimensional fluid-elastic wave equations, including the irregular interface, in a form suitable for stable numerical solution.

Measurements of the polarization of visible light by transmission through glass plates are discussed as being worthwhile undergraduate laboratory experiments. At the introductory level, the experiments are to becarried out at Brewster's angle of incidence, for which a simple polarization formula is derived without use of Fresnel's equations. At the more advanced level, familiarity with the latter permits the data to be analysed at arbitrary angles of incidence, and allows a single correction to be made for reflections. Examples of data taken with the present apparatus are presented and the results compared with theory.

Using a local radial basis function collocation method to approximate radiation boundary conditions

By means of modal series expansions of electromagnetic fields in terms of prolate spheroidal vector wave functions, an exact solution is obtained for the electromagnetic plane wave scattering by a system of two parallel uniformly lossy dielectric prolate spheroids. The incident excitation is a monochromatic plane electromagnetic wave of arbitrary polarization and angle of incidence. Translational Addition Theorems for spheroidal vector wave functions are employed in order to transform the outgoing wave from one spheroid into the incoming wave at the other spheroid. Application of appropriate boundary conditions gives the column vector of the unknown coefficients of the series expansions of the scattered and transmitted fields expressed in terms of the column vector of the known incident field expansion coefficients and the system matrix which is independent of the direction and polarization of the incident wave. Numerical results in the form of curves for normalized bistatic and monostatic radar cross sections are given for a variety of parallel two-body system of uniformly lossy dielectric prolate spheroids and different values of distances of separation. >

Acoustic radiation force on a cylindrical particle near a planar rigid boundary II. – Viscous fluid cylinder example and inherent radiation torque

In the analysis of antennas and scattering problems by the finite difference time domain (FDTD) method, the dimensions of the problem space are primarily determined by the quality of the radiation boundary condition (RBC) that is used to truncate the computational grid. Higher order boundary conditions give better performance when used in conjunction with pulse type sources. Unfortunately, they are less efficient when applied to scattering problems as shown by Blaschak and Kriegsmann (1988). This suggests that the current one-way wave equation approximations are overlooking an important aspect of the problem. We believe that the problem resides in the inability of the higher order approximations to deal with the evanescent part of the field. Since this evanescent part is much more prominent in scattering type problems, especially at low frequencies, this would explain the apparent failure of the higher order conditions in those cases. We propose a boundary condition that takes into account evanescent as well as non-evanescent plane waves and prove that this new RBC gives excellent results in both scattering and pulse-type problems. We restrict ourselves to two-dimensional problems but our approach can be extended to 3D problems as well.

of SPIE. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited. It seems to be feasible in the near future to exploit the properties of left-handed metamaterials in the telecom or even in the optical regime. Recently, split ring-resonators (SRR’s) have been realized experimentally in the near infrared (NIR) and optical regime. 1,2 In this contribution we numerically investigate light propagation through an array of metallic SRR’s in the NIR and optical regime and compare our results to experimental results. We find numerical solutions to the time-harmonic Maxwell’s equations by using advanced finite-elementmethods (FEM). The geometry of the problem is discretized with unstructured tetrahedral meshes. Higher order, vectorial elements (edge elements) are used as ansatz functions. Transparent boundary conditions (a modified PML method 3) and periodic boundary conditions 4 are implemented, which allow to treat light scattering problems off periodic structures.

We consider the propagation of elastic waves in gas-filled porous media at small but non-zero values of Knudsen numbers $$ {\text{Kn}} $$ , where $$ {\text{Kn}} = \lambda /l $$ , $$ \lambda $$ is the mean free path of gas molecules; $$ l $$ is the characteristic size of inclusion (the so-called slip regime). In this case, it is possible to apply the classic equations of hydrodynamics with modified boundary conditions at solid walls. We have assumed that the gas molecules distribution function is satisfied at the modified Maxwell boundary conditions (Struchtrup 2013; Mohammadzadeh and Struchtrup 2015). We have obtained the expressions for drag and added mass coefficients for the Biot equations of poroelasticity for a system of randomly oriented gas-filled cylindrical capillaries. Our calculations have shown that the drag and added mass coefficients depend considerably on the Knudsen number and the properties of the surface. The influence of the interfacial slip effect on the velocities of the compressional wave of the first kind and shear wave is small, but the velocity and attenuation of the compressional wave of the second kind are considerably influenced by this effect. The results obtained show the fundamental possibility of the determination of the accommodation coefficient by measuring the velocity of the compressional wave of the second kind for different values of the Knudsen number.

In this work, we study seismic wave scattering by cracks in inhomogeneous geological continua with depth-dependent material parameters and under conditions of plane strain. A restricted case of inhomogeneity is considered, with Poisson’s ratio equal to 0.25 and with both shear modulus and density profiles varying proportionally. Also, time-harmonic conditions are assumed to hold. For this type of material, elastic wave speeds remain macroscopically constant and it becomes possible to recover exact Green functions for the crack-free inhomogeneous continuum by using an algebraic transformation proposed in earlier work [1] as a first step. Subsequently, the complete elastodynamic fundamental solution, along with its spatial derivatives and an asymptotic expansion for small argument, are all derived in closed-form using the Radon transform [2]. Next, a non-hypersingular, traction-based boundary element formulation (BEM) is implemented for solving the plane boundary-value problem (BVP) with internal cracks [2]. This formulation is used for computing stress intensity factors (SIF) and scattered wave displacement field amplitudes for the case of an inclined line crack in a continuously inhomogeneous medium swept by either pressure (P) waves or vertically polarized shear (SV) waves at an arbitrary angle of incidence. The numerical results show substantial differences between homogeneous and inhomogeneous materials containing a crack in terms of their dynamic response, with the latter case being a more realistic representation for geological deposits. Finally, these types of results are useful within the context of earthquake engineering, since they can account for the influence of geological cracks in modifying seismically-generated ground motions.