An asymmetric wave field in a transversely isotropic elastic medium

A transversely isotropic homogeneous elastic medium excited by a point force perpendicular to the anisotropic axis is considered. The wave field in this medium is constructed and investigated. The front sets of the SV and SH waves are in contact with one another at a point. The front sets in the vicinity of this point are investigated additionally. If we consider the SH wave (or the SV wave) separately, then a false plane front set arises in this region. In considering the SH and SV waves in combination, this false front set disappears. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 163–174.     

Wavelength selection in convection near a lateral boundary

^** Corresponding author. This paper considers the way in which a lateral boundary restrictsthe wavelength of convective rolls parallel to the boundary.Unlike previous theories of wave number selection, the presentwork allows for perpendicular cross rolls adjacent to the boundary,the existence of which is indicated by stability considerations.The present theory also incorporates forcing at the boundary,equivalent to a wall that is imperfectly insulated, as in manyexperimental investigations of Rayleigh–Bénardconvection. The results are based on weakly non-linear solutionsof amplitude equations derived from the Swift–Hohenbergmodel and are compared with numerical solutions of the fullnon-linear Swift–Hohenberg equation.     

Wave generation in a computation domain

One of the possible methods to deal with the reflecting waves at the incident boundary in numerical modeling is to generate waves in the computation domain and absorb the outgoing waves at the incident boundary. A source function is introduced into the momentum equation of Boussinesq equations for generating wave in a computation domain in this paper. Typical numerical examples are given for the verification of the proposed method. Numerical examination for the wave diffraction through a breakwater gap shows that the proposed method is especially useful for multidirectional waves.     

Numerical solution of a thermomechanical coupled model governing glacier evolution

In this paper the numerical solution of a highly nonlinear model for the thermomechanical behavior of polythermal glaciers is presented. The modeling follows the shallow ice approximation (SIA) for glaciers introduced in Fowler (1997) [13]. The model has been extended to incorporate additional moving boundaries and other nonlinear features. Moreover, a fixed domain formulation is proposed to avoid the computational drawbacks of a time-dependent domain in the numerical simulation with front tracking methods. In this setting, the coupled problem is decomposed into different nonlinear problems which allow one to obtain sequentially the profile evolution, the velocity field, the glacier surface and atmospheric temperatures, basal magnitudes and the temperature distribution inside the ice mass. A fixed point iteration algorithm converges to the solution of the nonlinear coupled problem. Among different numerical methods involved in the solution of the subproblems, characteristic schemes for time discretization, finite elements for spatial discretization, duality methods for the nonlinearities associated to maximal monotone operators and a Newton scheme for the nonlinear viscous term are proposed. Several numerical simulation examples illustrate the performance of the numerical methods and the behavior of the involved physical magnitudes.     

A simple and robust boundary treatment for the forced Korteweg–de Vries equation

In this paper, we propose a simple and robust numerical method for the forced Korteweg–de Vries (fKdV) equation which models free surface waves of an incompressible and inviscid fluid flow over a bump. The fKdV equation is defined in an infinite domain. However, to solve the equation numerically we must truncate the infinite domain to a bounded domain by introducing an artificial boundary and imposing boundary conditions there. Due to unsuitable artificial boundary conditions, most wave propagation problems have numerical difficulties (e.g., the truncated computational domain must be large enough or the numerical simulation must be terminated before the wave approaches the artificial boundary for the quality of the numerical solution). To solve this boundary problem, we develop an absorbing non-reflecting boundary treatment which uses outward wave velocity. The basic idea of the proposing algorithm is that we first calculate an outward wave velocity from the solutions at the previous and present time steps and then we obtain a solution at the next time step on the artificial boundary by moving the solution at the present time step with the velocity. And then we update solutions at the next time step inside the domain using the calculated solution on the artificial boundary. Numerical experiments with various initial conditions for the KdV and fKdV equations are presented to illustrate the accuracy and efficiency of our method.     

Scattering of the plane wave by a transparent wedge

In the paper it is proved that the problem of scattering of the plane wave by a transparent wedge has a unique solution, provided that the radiation condition should be meant in the following form: if one subtracts from the solution the incident wave and all reflected and refracted waves, then the remainder satisfies the radiation condition in integral form. The problem is scalar, the velocities of the wave inside and outside the wedge are not equal, the wave process is described by the classical Helmholtz equations, and the conjugation boundary condition is satisfied on the sides of the wedge. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 5–18.     

Reflection of Shock Fronts in a van der Waals Fluid

In this paper, the reflection phenomenon of a vapor shock front （both sides of the front are in the vapor phase） in a van der Waals fluid is considered. Both the 1-dimensional case and the multidimensional case are investigated. The authors find that under certain conditions, the reflected wave can be a single shock, or a single subsonic phase boundary, or one weak shock together with one subsonic phase boundary, which depends on the strength of the incident shock. This is different from the known result for the reflection of shock fronts in a gas dynamical system due to Chen in 1989.     

On the behavior near the caustic of a nonsteady wave field with singularity (A homogeneous generalized function) at the initial front

The Cauchy problem for the wave equation in the case of discontinuity on the initial front is investigated. The discontinuity is described by a homogeneous generalized function of degree λ. The transformation of the initial front while passing the space-time caustic is studied. The structure of the wave front and the space-time rays near the caustic is considered. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 122–133, 1990. Translated by N. Ya. Kirpichnikova.     

Effect of the dynamic pressure on the similarity solution of cylindrical shock waves in a rarefied polyatomic gas

The similarity solution for a strong cylindrical shock wave in a rarefied polyatomic gas is analyzed on the basis of Rational Extended Thermodynamics with six independent fields; the mass density, the velocity, the pressure and the dynamic pressure. A new ODE system for the similarity solution is derived in a systematic way by using the method based on the Lie group theory proposed in the context of the spherical shock wave in a rarefied monoatomic gas in Donato and Ruggeri (J Math Anal Appl 251:395, 2000). The boundary conditions are also specified from the Rankine–Hugoniot conditions for the sub-shock. The derived similarity solution is characterized by only one dimensionless parameter \(\alpha \) related to the relaxation time for the dynamic pressure. The numerical analysis of the similarity solution is also performed. The solution agrees with the well-known Sedov–von Neumann–Taylor (SNT) solution when \(\alpha \) is small. When \(\alpha \) is larger, due to the presence of the dynamic pressure, the deviation from the SNT solution is evident; the strength of a peak near the shock front becomes smaller and the profile becomes broader.

     

Finite difference method in the problem of diffraction of a plane acoustic wave in a half-plane with a cut

In the numerical solution of the diffraction problem for an acoustic plane wave in a half-plane with a cut, boundary conditions that are equivalent to the radiation conditions at infinity are set in a neighborhood of the points of the cut. Joining the physical boundary conditions on the cut, a closing set of equations of order 4N, where N is the number of grid points on the cut, is obtained. The so-called Green’s grid function for the half-plane is used, which makes it possible to pass from one grid layer to another one for the solution satisfying certain conditions at infinity.     

Determining the internal boundary of a region in the two-dimensional initial–boundary-value problem for the heat equation

The article investigates the reconstruction of the internal boundary of a two-dimensional region in the two-dimensional initial–boundary-value problem for the homogeneous heat equation. The initial values for the determination of the internal boundary are provided by a boundary condition of second kind on the external boundary and the solution of the initial–boundary-value problem at finitely many points inside the region. The inverse problem is reduced to solving a system of integral equations nonlinear in the function describing the sought boundary. An iterative numerical procedure is proposed involving linearization of integral equations.     

Scattering of plane harmonic waves on a cylindrical cavity with an elliptical cross section in an orthotropic medium

The diffraction of a plane longitudinal harmonic wave on a cavity with a smooth curvilinear cross section in a rectilinearly orthotropic medium is solved by using small perturbations in the elastic moduli and introducing generalized wave potentials. Results are presented from a numerical analysis of the dynamic stresses in the near diffraction field and at the boundary of an elliptical cavity including variations in the relative incident wavelength, eccentricity of the cavity, and degree of anisotropy of the medium. Donetsk State University. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 29, pp. 102–110, 1999.     

Diffraction of a plane elastic wave by a wedge with a special form of boundary conditions

An exact solution of the antiplane problem of the diffraction of a plane elastic SH-wave with a step profile by a wedge is obtained. The stresses on the wedge sides are assumed to be proportional to a linear combination of the displacements, velocities and higher derivatives with respect to time of the displacements along the wedge axis. A solution of the problem is obtained using integral transformations with subsequent transformation using Cagniard's method. Solutions of the corresponding problems with boundary conditions of the Winkler and inertial types are considered. When a wave with a linear profile is incident on the wedge the stresses suffer a discontinuity of the second kind on the diffraction wave front; the same type of feature is observed in the problem with the inertial condition.     

An effective model of a porous-fluid medium

For a medium containing alternating porous Biot layers and fluid layers, an effective model is established by the method of matrix averaging. An investigation of equations of this effective model shows that the wave field consists of a leading front and two triangular fronts. The velocities of these fronts along the axes are determined. If the thicknesses of the fluid layers are very small, then the second triangular front is converted into a back concave front, and a slow wave arises. This slow wave is of interest for seismology. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 190–211.     

The discrete singular convolution for analyses of elastic wave propagations in one-dimensional structures

In this paper, the problem of wave propagations in one-dimensional (1D) structures is investigated for the first time by using the discrete singular convolution (DSC), a relatively new and promising numerical approach. For simplicity, the non-regularized Lagrange’s delta sequence kernel is adopted in the DSC for most cases. For comparisons, the Regularized Shannon’s delta kernel is also adopted in the DSC for two cases. Methods for applying the free boundary conditions, concentrated loads and concentrated masses are proposed and validated. Detailed formulations and solution procedures are given. Travelling waves in an isotropic aluminum bar and Timoshenko beam are studied. A high degree of accuracy in simulations by the DSC is observed. Numerical results are compared to those obtained from the spectral finite element (SFE) approach, proved a very efficient method in modeling elastic wave propagations. The comparison highlights the efficiency of the DSC in modeling elastic wave propagations. The present research extends the application range of the discrete singular convolution to problems of elastic wave propagations.     

Compressible gas flow inside a cylindrical channel with a source on the channel axis

The article solves the problem of compressible nonviscous gas flow inside a cylindrical channel in the presence of a source on the channel axis. The cases considered include a supersonic spherically symmetrical source and a hypersonic source modeling jet flow. The numerical solution is obtained by two methods: one method treats the flow singularities (shocks, etc.) in explicit form, while the other uses the “through calculation” procedure. The solution is applied to analyze the physical flow pattern and the distribution of gasdynamic parameters in the shock layer, in particular on the channel walls. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 169–180.     

Scattering of guided SH-wave by a partly debonded circular cylinder in a traction free plate

A solution of the scattering problem of guided SH-wave by a partly debonded circular cylinder centered in a traction free plate has been set up. The plate is divided up into three regions with two imaginary planes perpendicular to the plate walls. In the central region where the partly debonded cylindrical obstacle is posted, the wave field is expanded into the cylindrical wave modes and Chebyshev polynomials. In the other two exterior regions the fields are expanded into the plate wave modes. A system of fundamental equations to solve the problem is obtained according to the traction free boundary condition on the plate walls and the continuity condition of the traction and the displacement across the imaginary planes. The approximate numerical method termed mode-matching technique is used to construct a matrix equation to obtain curves showing the coefficient of reflection and transmission versus the ratio of the cylinder’s radius to the plate’s half-thickness and the angular width of the debonded region. A comparison of the numerical results between the welded interface condition and the debonded interface condition is made, and the results are discussed.     

Simulation of wave processes in a vapor-liquid medium

Numerical methods for the simulation of nonlinear wave processes in a vapor-liquid medium with a model two-phase spherical symmetric cell, with a pressure jump at its external boundary are considered. The viscosity and compressibility of the liquid, as well as the space variation of pressure in the vapor, are neglected. The problem is described by the heat equations in the vapor and liquid, and by a system of ODEs for the velocity, pressure, and radius at the bubble boundary. The equations are discretized in space by an implicit finite-volume scheme on a dynamic adaptive grid with grid refinement near the bubble boundary. The total time derivative is approximated by a method of backward characteristics. “Nonlinear” iterations are implemented at each time step to provide a specified high accuracy. The results of numerical experiments are presented and discussed for the critical thermodynamic parameters of water, for some initial values of the bubble radius and pressure jump.     

管道内激波的不稳定性

本文将无限大激波阵面的激波不稳定性理论[1]推广到矩形截面管道内的激波不稳定性问题.首先,给出这个问题的数学提法,包括扰动方程与三类边界条件.其次,给出扰动方程的普遍解.上游和下游的普遍解分别含有5个待定常数.再次,在一类边界条件和一个假定下,证明了激波前扰动为0,激波后两个声扰动之一为0.边界条件是,X→±∞处扰动物理量为0.假定只讨论激波不稳定性问题,从而可先设ω=iγ,γ是不稳定性增长率,为正实数.另一类边界条件是管壁上法向速度扰动为0,它使波数只能取一组离散值.最后,用扰动激波上的5个守恒方程这一边界条件来决定激波后4个待定常数和扰动激波振幅这个未知量时,导出了色散关系.结果表明,正实数γ确是存在.不稳定激波有两种模式,一种模式为γ=-W·k(W<0)它代表激波的绝对不稳定性,是新得到的模式.另一种模式与过去工作中给出的[2,3]大体相同.本文则进一步给出了这种模式的激波不稳定性增长率,并指出j2((?V/?P)_H=1+2M为最不稳定点(即无量纲化的不稳定性增长率Г=∞).如果不假定ω是纯虚数,而是复数,其虚部为正实数Im(ω)≥0.本文也严格证明了其不稳定性判据仍有两种模式,ω仍为纯虚数.     

Modeling binary alloy solidification by a random projection method

This paper addresses the numerical modeling of the solidification of a binary alloy that obeys a liquidus–solidus phase diagram. In order to capture the moving melting front, we introduce a Lagrange projection scheme based on a random sampling projection. Using a finite volume formulation, we define accurate numerical fluxes for the temperature and concentration fields which guarantee the sharp treatment of the boundary conditions at the moving front, especially the jump of the concentration according to the liquidus–solidus diagram. We provide some numerical illustrations which assess the good behavior of the method: maximum principle, stability under CFL condition, numerical convergence toward self‐similar solutions, ability to handle two melting fronts.