The application of finite element analysis to the solution of Stokes wave diffraction problems

A new finite-element based method of calculating non-linear wave loads on offshore structures in extreme seas is presented in this paper. The diffraction wave field is modelled using Stokes wave theory developed to second order. Wave loads and free surface elevations are obtained for fixed surface-piercing structures by solving a boundary value problem for the second-order velocity potential. Special attention has been given to the radiation condition for the second-order diffraction field. Results are presented for three test examples, the vertical cylinders of Kim and Yue and of Chakrabarti, and an elliptic cylinder. These results demonstrate that early problems with the application of second-order theory arising from inadequate radiation conditions have been overcome.     

Application of non-linear strain waves to the study of the growth of long bones

A non-linear dynamic model is developed to account for material inhomogeneities in a growth plate in long bones. The governing equations are obtained to account for non-linear dispersive, viscoelastic and inhomogeneous features of the growth plate. The evolution of non-linear strain waves over the material inhomogeneities is obtained via the asymptotic solutions. It is shown that variations in the amplitude and the width of both the bell-shaped and kink-shaped waves reflect the position and the size of the inhomogeneity. This may be used for a detection of the growing plate features and in the development of the reaction-diffusion equation for the stimulus of the growth of long bones.     

Influence of a rigid skeleton pore structure on wave propagation in a fluid-filling porous medium

This work concerns an analysis of the influence of a rigid skeleton pore structure on wave propagation in a fluid-filling porous medium. The analysis is based on the continuum theory of a deformable porous medium in which the pore structure is described by two macroparameters. Considerations comprise two questions: the influence of the pore structure on wave-propagation velocity analysed for the quasilinear case and the role of structure in the reflection-refraction wave phenomenon in fluid at the contact surface of two porous media. It has been shown that the pore structure reduces the velocity of wave and together with the angle of incidence it defines the reflection-refraction wave phenomenon.     

On the existence of non-polarized azimuthal-symmetric electromagnetic waves in circular dielectric waveguide filled with nonlinear isotropic homogeneous medium

The propagation of monochromatic nonlinear symmetric hybrid waves in a cylindrical nonlinear dielectric waveguide is considered. The physical problem is reduced to solving a transmission eigenvalue problem for a system of ordinary differential equations. Spectral parameters of the problem are propagation constants of the waveguide. The problem is reduced to the new type of nonlinear eigenvalue problem. The analytical method of solving this problem is presented. New propagation regime is found.     

Spectral Analysis of the Balance Equation of Ground Water Hydrology

The spectral analysis of the balance equation of ground water flow, associated to an asymptotic expansion of the conductivity (K) and head (h) fields, permits to show that the high wave number components of the source terms, F, and of the conductivity, K, are attenuated when h is computed as solution to the balance equation. This has important consequences on the inverse mapping (h, F) → K: in fact it is not possible to recover in a reliable way the high wave number components of K, because small errors on the corresponding components of h are amplified so that they can hidden the true signal.     

Some exact expressions for the temporal evolution of long Rossby waves on a beta-plane

Some exact expressions are derived to describe the temporal evolution of forced Rossby waves in a two-dimensional beta-plane configuration where the background flow has constant zonal-mean velocity. The meridional length scale of the problem is assumed to be small relative to the zonal length scale and so the long-wave limit of zero aspect ratio is taken. In the case where the background flow velocity is zero, an exact solution is obtained in terms of generalized hypergeometric functions. A late-time asymptotic approximation is obtained and it shows that the solution oscillates with time and its amplitude goes to zero in the limit of infinite time. In the case of a non-zero background flow velocity, the solution is evaluated using two different procedures which give two equivalent expressions in terms of different generalized hypergeometric functions. The late-time asymptotic behaviour is investigated and it is found that the solution approaches a steady state in the limit of infinite time.We also derive a solution in the form of an asymptotic series expansion for the more general situation where a Rossby wave packet is generated by a zonally-localized boundary condition comprising a continuous spectrum of wavenumbers or Fourier modes. The exact solutions found here can be used as leading-order solutions in weakly-nonlinear analyses and other studies involving more realistic configurations for time-dependent Rossby waves or wave packets.     

Numerical issues related to the modelling of electrochemical impedance data by non-linear least-squares

Non-linear least-squares (NLS) fitting is the typical approach to the modelling of electrochemical impedance spectroscopy (EIS) data. In general the application of NLS to EIS models can give rise to ill-posed problems. On the one side, with ill-posed problems it is not possible to prove a priori that a unique solution exists. On the other side, the relevant numerical approximations cannot ensure that a unique solution exists even a posteriori. It is therefore basically pointless to endeavour to achieve one absolute minimum of any objective function for an EIS model in an NLS problem. A lack of awareness of the above-mentioned factors might render numerical approaches tending to locate the absolute minimum questionable.     

A general numerical method for the solution of gravity wave problems. Part 1: 2D steep gravity waves in shallow water

In this paper, 2D steep gravity waves in shallow water are used to introduce and examine a new kind of numerical method for the solution of non-linear problems called the finite process method (FPM). On the basis of the velocity potential function and the FPM, a numerical method for 2D non-linear gravity waves in shallow water is described which can be applied to solve 3D problems, e.g. the wave resistance of a ship moving in deep or shallow water. The convergence is examined and a comparison with the results of other authors is made. The FPM can successfully avoid the use of iterative methods and therefore can overcome the disadvantages and limitations of such methods. In contrast to iterative methods, the FPM is insensitive to the selection of the initial solution and the number of unknowns. The basic idea of the FPM can be used to solve other non-linear problems. Its disadvantage is that much more CPU time is needed to obtain a sufficiently accurate result.     

Correction of an effect of using a faulty model for the retrieval of a bulk wave slowness of a layered half space

This study concerns the retrieval of a single, real constitutive parameter (the bulk shear wave inverse velocity) of a simple, although representative, geophysical configuration involving two homogeneous, non-dissipative media, from its simulated response to pulsed plane wave probe radiation. This nonlinear inverse problem is solved exactly, at all frequencies, by equating the simulated frequency domain response incorporating the true real inverse velocity to the assumed response incorporating a trial complex inverse velocity. Due to discordance, caused by prior (a parameter of a model that is not retrieved, but rather assumed to be known, although its value may be wrong) uncertainty, between the models associated with the assumed and trial responses (the model giving rise to the latter is therefore qualified as being ‘faulty’), the imaginary part of the retrieved inverse velocity turns out to be non-nil, and, in fact, to be all the greater, the larger is the prior uncertainty. Moreover, the retrieved inverse velocity is found to be dispersive and its real part nearly equal to the true inverse velocity at all frequencies. The reconstructed signals (obtained by inserting the retrieved complex parameter into the faulty response model) are found to coincide exactly with the true signals for three types of probe pulses, even when the prior uncertainty is large.     

Accurate 3-D finite element simulation of elastic wave propagation with the combination of explicit and implicit time-integration methods

One of the big issues in finite element solutions of wave propagation problems is the presence of spurious high-frequency oscillations that may lead to divergent results at mesh refinement. The paper deals with the extension of the new two-stage time-integration technique developed in our previous papers to the solution of wave propagation problems with explicit time-integration methods.The explicit central difference method is used for accurate time-integration of the semi-discrete system of elastodynamics at the stage of basic computations and allows spurious high-frequency oscillations. To filter these oscillations, pre- or/and post-processing (the filtering stage) is applied using a few time increments of the implicit time-continuous Galerkin method with large numerical dissipation.A special calibration procedure is used for the selection of the minimum necessary amount of numerical dissipation (in terms of a time increment) at the filtering stage. In contrast to existing approaches that use a time-integration method with the same dissipation (or artificial viscosity) for all time increments, the new technique yields accurate and non-oscillatory results for wave propagation problems without interaction between user and computer code. The solutions of 3-D wave propagation and impact problems show the effectiveness of the new approach.     

Dynamic homogenization of a complex geophysical medium by inversion of its near-field seismic response

The present study originated in the forward problem of the prediction of the effects of seismic waves (generated by impulsive deep-down sources) in urban areas. The traditional, numerically-intensive approaches to this problem have not, until now, given rise to simple theoretical paradigms which might explain how and why the (often-destructive) response of cities is conditioned by factors such as the city density, building average height, average building composition, site geometry and composition, and characteristics of the solicitation such as incident angles, polarization and frequency. We propose to homogenize the city in order to simplify, and make possible the understanding of, the site-city-solicitation interaction. This homogenization is treated as an inverse problem, i.e., by which we: (1) generate near-field response data for a ‘real’ city, (2) replace (initially by thought) the city by a homogeneous (surrogate) layer above, and in firm contact, with the underlying site, (3) compute the response of the surrogate layer/site response for various trial constitutive properties, (4) search for the global minimum of the discrepancy between the response data and the various trial parameter responses (5) attribute the homogenized properties of the city to the surrogate layer for which the minimum of the discrepancy is attained. We carry out this five-step procedure for a host of ‘real’ city and solicitation parameters, notably the frequency. The result is that: (i) for low frequencies and/or large city densities, the effective constitutive properties are their static equivalents, i.e., the effective shear modulus is the product of a factor related to the city density with the shear modulus of a generic substructure of the city and the effective complex velocity is equal to the complex velocity of the said generic substructure, (2) at higher frequencies and/or smaller city densities, the effective constitutive properties are dispersive and do not take on a simple mathematical form, with this dispersion compensating for the discordance between the ways the inhomogeneous city structure and the homogeneous surrogate layer respond to the seismic wave. For typical seismic solicitation frequencies, the city, represented as a layer with static homogenized properties, is quite adequate to account for the principal features of the response (notably those of the time-domain response). The model of the layer with dispersive homogenized properties is more suitable to account for such features as resonances due to the excitation of surface wave modes.     

A finite amplitude analysis of tunnel deformation by the finite element method with highly viscous fluid

A finite element method for highly viscous fluid is used to calculate the velocity and stress fields in the surrounding soft rock of a tunnel. In order to fit the calculated values with the measured displacement of tunnel wall, we inverted the boundary forces and the mechanical parameters of the surrounding rocks.     

Spatial shapes of magnetoelastic shear body waves at the transmission edges in a periodically inhomogeneous magnetostrictive medium

The spatial shapes of magnetoelastic shear body waves at the transmission edges in a periodically inhomogeneous magnetostrictive medium are studied. Numerical results are obtained for a two-component composition of ferrite and nonmagnetic dielectric __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 3, pp. 61–69, March 2006.     

Analysis of worm-like locomotion driven by the sine-squared strain wave in a linear viscous medium

In this paper, a worm-like locomotion in a linear resistive medium is studied to achieve controlled shape changes of the worm-like body by choosing a kind of driving with low energy expended and high-velocity locomotion in certain condition. To this end, we first develop the full dynamic model of the system under consideration to obtain the mean velocity related to friction coefficient, wave speed, linear density, body length and wave width. Correspondingly, a quasi-static model is also given from which the velocity can be expressed analytically. In the case of the shape change driven by the sine-squared strain wave (SSSW), it is seen that these two velocities will tend to uniformity with the friction coefficient or length of the body increasing or the wave speed decreasing when keeping the other parameters unchanged. Thus, the inertia term is ignorable for a large friction, a long body-length but a small wave-speed of the SSSW, which implies that the dynamical model can be reduced to the quasi-static one. The relative criterion is approximately given. As a result, the corresponding quasi-static model is employed to consider two typical drives, namely, the SSSW and the square strain wave (SSW). The result shows the shape change driven by the SSSW has an advantage in both the mean velocity and the average energy expended over that by the SSW when the necessary condition is satisfied. The analytical results are verified by numerical simulation.     

Reconstruction of the three mechanical material constants of a lossy fluid-like cylinder from low-frequency scattered acoustic fieldsReconstruction des trois constants mécaniques matériels d'un cylindre fluide dissipatif à partir de champs acoustiques basses fréquences

The inverse medium problem for a circular cylindrical domain is studied using low-frequency acoustic waves as the probe radiation. To second order in k₀a (k₀ the wavenumber in the host medium, a the radius of the cylinder), only the first three terms (i.e., of orders 0, ?1 and +1) in the partial wave representation of the scattered field are non-vanishing. This enables the scattered field to be expressed algebraically in terms of the unknown material constants, i.e., the density ρ₁, and the real and imaginary parts of complex compressibility κ₁ of the cylinder. It is shown that these relations can be inverted to yield explicit, decoupled expressions for ρ₁ and κ₁ in terms of the totality of the far-zone scattered field. These expressions furnish accurate estimations of the material parameters provided the probe frequency is low and the radius of the cylinder is known very precisely. To cite this article: T. Scotti, A. Wirgin, C. R. Mecanique 332 (2004).     

A boundary element solution to scattering of elastic SH wave by arbitrarily shaped inclusions embedded in an infinite anisotropic medium

Scattering problems for inhomogeneous bodies are investigated by the integral equation method. The boundary integral equation (BIE) for the scattered displacement field associated with finite inhomogeneities in an anisotropic medium are derived with the help of the generalized Green's identity. The discretization of BIE is based upon the constant element, linear element and quadratic element. Several numerical examples for calculating the scattering displacement, stress and scattering cross section from a cylinder, an interface crack, and two elliptic cylinders are given. Results show that the present method can be advantageously applied to a wide range of scattering problems of elastic waves.     

Propagation of nonlinear waves in an inhomogeneous gas-liquid medium. Derivation of wave equations in the Korteweg-de Vries approximation

Equations describing the propagation of waves of small but finite amplitude in a liquid with gas bubbles are derived. The bubble distribution density is a continuous function of bubble size and spatial coordinates. It is found that, for a uniform bubble distribution, the obtained equations become the Korteweg-de Vries, Kadomtsev-Petviashvili and Khokhlov-Zabolotskaya equations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 188–197, March–April, 2009.     

Gas-kinetic description of shock wave structures by solving Boltzmann model equation

On the basis of the mesoscopic theory of Boltzmann-type velocity distribution function, the modified Boltzmann model equation describing the one-dimensional gas flows from various flow regimes is presented by incorporating the molecular interaction models relating to the viscosity and diffusion cross-sections, density, temperature and the dependent exponent of viscosity into the molecular collision frequency. The gas-kinetic numerical method for directly solving the molecular velocity distribution function is studied by introducing the reduced distribution functions and the discrete velocity ordinate method, in which the unsteady time-splitting method and the NND finite difference scheme are applied. To study the inner flows of non-equilibrium shock wave structures, the one-dimensional unsteady shock-tube problems with various Knudsen numbers and the steady shock wave problems at different Mach numbers are numerically simulated. The computed results are found to give good agreement with the theoretical, DSMC and experimental results. The computing practice has confirmed the good precision and reliability of the gas-kinetic numerical algorithm in solving the highly nonequilibrium shock wave disturbances from various flow regimes.