Discontinuous Media Research Articles

An invariant imbedding analysis of general wave scattering problems

The invariant imbedding technique, via the solution of a Riccati-type equation, is modified to calculate the wave fields inside and scattered from a strongly (laterally and vertically) heterogenous, anisotropic inclusion, which may be large but remains compact. The factorization underlying this approach is carried out with respect to direction of average power flow rather than the more conventional factorization with respect to local direction of propagation. The solution of the operator Riccati equation is related to the Dirichlet-to-Neumann map. The formulation is robust in the sense that it can handle a rather extreme range of modal wave speeds, and allows continuous as well as discontinuous medium variations on different (wave) length scales. It also, inherently, takes care of critical-angle phenomena. The algorithm, based on the invariant imbedding approach, yields the internal fields for a full survey of sources and receivers simultaneously. The wave field solution in the inclusion is coupled to the external field via a boundary element approach.

Effect of structural irregularity on propagation properties of optical waves in discontinuous magneto-optical media with one-dimensional quasirandom aray structures

To elucidate the effect of structural irregularity on propagation properties of optical waves in discontinuous magneto-optical media, theoretical analysis was carried out by employing random matrix technique and a model with one-dimensional quasirandom array structure. The analysis revealed that the effect of structural irregularity was so significant that the dispersion curves of optical waves varied considerably depending on the media structures. However, no matter how the medium structure varies irregularly, dispersion curves of waves always exist within the confines between those of the half (0...01...1) and the alternate (0101...01) structures. This fact suggests that the effect of structural irregularity becomes less significant when the discontinuous media have fine discrete structures, where the dispersion curves of the half and the alternate structures are very close.

Magneto-optical Faraday effect of discontinuous magnetic media with a one-dimensional array structure

To elucidate unique Faraday spectra observed in porous magnetic films, we theoretically analyzed the magneto-optical Faraday effect of discontinuous media with an ideal one-dimensional array structure. The media are constituted by magnetic materials (width DM) and air gaps (width DG) which are aligned interchangingly. Propagation properties of optical waves in the media, such as the dispersion characteristics and the polarization states, were explored circumstantially, and apparent Faraday rotation and ellipticity spectra of the discontinuous media were obtained. The analysis reveals that the apparent Faraday spectra of the media vary to a large extent from those of a continuous medium depending strongly on the media structures. The key factors which govern the magneto-optical effect of the discontinuous media are structural elements of DM and 2(DM+DG): The Faraday spectra are subject to change only when DM<λ (λ: the wavelength of optical wave in a vacuum). Then, the magneto-optical effect of the magnetic materials is related to mode conversion from TE to TM when 2(DM+DG)<λ. If DM<λ<2(DM+DG), the mode conversions of both directions from TM to TE and TE to TM are possible. The mode conversion dominates the apparent Faraday spectra of the media, causing the enhancement in the Faraday rotation and systematic shift of the spectra toward shorter and/or longer wavelength of the optical waves. These results explain very well the unique Faraday spectra of the porous films.

A Saint-Venant type principle for Dirichlet forms on discontinuous media

We consider certain families of Dirichlet forms of diffusion type that describe the variational behaviour of possibly highly nonhomogeneous and nonisotropic bodies and we prove a structural Harnack inequality and Saint Venant type energy decays for their local solution. Estimates for the Green functions are also considered.

Analysis of magneto-optical Faraday effect in discontinuous magnetic medium with periodic air gaps

The magneto-optical Faraday effect in discontinuous magnetic media was analyzed by employing the matrix approach method. The discontinuous media have a one-dimensional array structure composed of magnetic materials (width D/sub M/) and air gaps (width D/sub G/) which are aligned alternately. The analysis reveals that, when the medium meets the condition of D/sub M/</spl lambda/<2(D/sub M/+D/sub G/), the Faraday spectrum varies considerably due to the discontinuous structure.

Modelling storage behaviour in a fractured rock mass

A transient model was used to represent single borehole injection tests in fractured rock masses, on the basis of a three-dimensional stochastic description of the fracture network and of the mechanical behaviour of the fractures. The rock matrix remains impervious and storage occurs in the fractures, with a nonlinear compliance effect depending on both fluid pressure and normal closure stress acting on the joints. The aim of the study is to show how the shortest, weakly connected joints can be removed and replaced in the system by an equivalent parameter. These efforts can save computer time or enlarge the modelled rock volume. Moreover, this approach may be helpful in addressing the troublesome up-scaling problem in discontinuous media.

Elastic waves in discontinuous media: Three-dimensional scattering

This report contains an exact study of elastic wave propagation and its scattering in discontinuous media where hard reflectors are onionlike sets of surfaces. In order to reformulate the problem as a finite set of boundary integral equations, the wave motion between reflectors is represented by means of elastic potentials which involve vectorial densities on the surfaces. In the external medium, an outgoing asymptotic condition generalizes the Silver–Müller (and the Sommerfeld) condition to the case of coupled waves (S and P waves) moving with different velocities. The uniqueness of the Green’s function, which guarantees the uniqueness of the direct problem solution, is proven. For any incident wave and arbitrary number of surfaces, the transmission and scattering problems are studied, with and without the simplification obtained by assuming constant Poisson ratios. According to the parameter ranges, the equations which are obtained are well posed, either as second kind Fredholm equations, or because they reduce to the inverse of the sum of the identity operator and a ‘‘small norm’’ bounded operator. The results can be used to describe rigorously the three-dimensional scattering of elastic waves in the frequency domain for any kind of incident wave function (P,S,...) as well as the response to a localized source.

Elastic modeling in discontinuous media

Results from elastic‐wave simulations in a simple model show that for models characterized by a set of layers with sharp boundaries (discontinuous stiffness tensor), traditional finite‐difference methods fail to correctly describe the dynamics of the propagation process. The failure comes from the lack of distinction between model and field variables; the same differential operator is applied to discontinuous (model) and continuous (wavefield) components. This problem is solved with a modified high‐order finite‐difference modeling scheme (dual‐operator method) that uses two distinct operators for evaluating the model and field derivatives, a modified version of Virieux‐staggered grid, and Shoenberg‐Muir averaging relations. The dual‐operator method generates a dynamic response that is more accurate than traditional finite‐difference schemes and comparable to Haskell‐Thomson propagator‐matrix schemes, with the advantage that it can be applied to structurally complex models. The method produces stable, accurate results for both solid and liquid layers.

Development of Mathematical Tools to Determine Optimum Enclosure Designs for Controlling Electromagnetic Fields

Electromagnetic fields are present wherever sources of currents are found. Conventional shielding against these electromagnetic fields is accomplished by using com plete ferromagnetic enclosures. Maxwell's equations govern the phenomenon of electro magnetic fields in discontinuous media, and analytical solutions in the presence of enclo sures of arbitrary shapes are not available. Hence, Maxwell's equations are solved numerically using the finite element method in conjunction with the magnetic scalar potential approach. The test case for the verification of the numerical code is a long cylin drical ferromagnetic shell placed in a transverse uniform magnetic field for which an ana lytical solution is derived here. The comparison of analytical and numerical solutions vali dates the accuracy of the numerical code. As expected the magnetic field inside the complete enclosure varies inversely to the relative permeability of the enclosure. Using this approach, enclosures of different shapes can be designed.

Action functional for diffusions in discontinuous media

The action functional, i.e. the rate function governing the large deviations is obtained for a family of stochastic processes with discontinuous drift and small diffusion. A well-known method of continuous mapping is developed which proves to be efficient in a so called ‘stable case’.

Explosion phenomenology in jointed rocks - new insights : Heuze, F E; Butkovich, T R; Walton, O R; Maddix, D M Proc 7th ISRM International Congress on Rock Mechanics, Aachen, 16–20 September 1991V2, P1043–1048. Publ Rotterdam: A A Balkema, 1991

This article deals with the effects of high-explosive and nuclear explosions in rock masses. We first highlight the strong influence of geological discontinuities, such as joints and faults, on ground motion characteristics. Then, we briefly introduce the Discrete Element method, as a new technique for numerical simulations. It is shown to be far superior to continuum-based approaches, when dealing with the dynamics of discontinuous media such as jointed rocks. Finally, we simulate the ground effects from a generic contained nuclear explosion, which bears some similarity to the SHOAL event in granite. This calculation is intended to emphasize the influence of the near-source geology on the distribution of energy, and on the motion at various azimuths in the medium. 17 refs., 10 figs.

Homogeneous stress hypothesis and actual fault slip: a distinct element analysis

The different inverse methods used in slip data analysis depend on the general validity of the mechanical model adopted by Carey and Brunier, and others. The mechanical scheme is based on the Wallace-Bott relationship assuming a faulted rock mass as a system of rigid blocks interacting mechanically without friction. Until now the validity of this conceptual model was supported by internally consistent results obtained by applying numerical inversion techniques to fault slip data. This paper presents the first results of a numerical modelling investigation of this simple mechanical model by a direct approach to the problem. The numerical method used here is a three-dimensional Distinct Element Method which is suitable to study discontinuous media. First, by modelling the behaviour of one fault, we propose an extrinsic verification of the Wallace-Bott relation in three dimensions and considering rock linear elasticity and friction effects on faults. According to the small discrepancy obtained (i.e. the deviation between the modelling results and the theoretical results), we conclude that the assumptions of the basic model are justified in the range of stress values that we study. Second, using a two-fault model, the effects of overlap in stress perturbations around the faults results in particular fault slip interactions. The existence of such phenomena invalidates, in a generally minor way taking into account practical uncertainties, the assumption of fault slip independence in the basic model. Finally, we discuss limits to inverse methodologies and the necessity for interactions between data collection, interpretation of these data and the ability of the basic model to be used in microtectonic analyses.

Reconstruction of the permittivity profile of an inhomogeneous medium using an equivalent network method

A novel network method used to reconstruct the permittivity profile of a dielectric slab of arbitrary depth or a half-space medium is investigated. The nonlinear Riccati type differential equation for the reflection coefficient of the medium is derived in a new way, from which an approximate closed-form reconstruction formula is obtained, relating the permittivity profile of the medium to the Fourier transform of the reflection coefficient. It is shown that this approximation is valid when it is used to invert the permittivity profile in shallow region. By means of the equivalent network of an inhomogeneous dielectric slab, the approximation can be subsequently used to invert the permittivity profile in an arbitrarily deep inhomogeneous region. Reconstruction examples for both continuous and discontinuous media are given to show the applicability of this scheme. The effect of noise on the inversion process is discussed. >

Soliton Transmission and Reflection in Discontinuous Media

A new approach to nonlinear waves in one-dimensional discontinuous media is presented. As a model a nonlinear lattice which has a discontinuity of the mass distribution is considered. Two kinds of physically interesting waves, slowly varying waves and carrier wave modulations, are investigated. For the former (latter), incident, reflected and transmitted waves are described by Korteweg-de Veries (Nonlinear Schrödinger) equations on different coordinates. The reflected and transmitted waves are constructed from the incident wave analytically. Fission and reflection of a soliton due to the discontinuity are explicitly shown in the lowest order.

A Comparison of Three Mixed Methods for the Time-Dependent Maxwell’s Equations

Three mixed finite-element methods for approximating Maxwell’s equations are compared. A dispersion analysis provides a Courant–Friedrichs–Lewy (CFL) bound that is necessary for convergence when a uniform mesh is used. The dispersion analysis also allows a comparison of the stability properties of the methods. Superconvergence at the interpolation points is proved for uniform grids, and demonstrated by three numerical examples. All three methods are shown to be able to handle discontinuous media without modification of the finite-element spaces. Since all three methods have three-dimensional counterparts, this study suggests that all three methods could be the basis of a successful three-dimensional code.

Discontinuous media and underdetermined scattering problems

The problem of ambiguities in trying to determine a shape by means of scattering experiments, with one or a few illuminating angles and all directions of receivers, is discussed by means of numerical experiments. The model equation gives a good representation of scattering of scalar waves which can taken into account impedance discontinuities inside the scatterer. Physical problems include, for instance, acoustical waves in media where the density and the Lame parameter lambda may vary continuously everywhere except across a finite number of smooth surfaces through which they jump.

Modelling of nonlinear dilatation response of fluids containing columns plastic and shear relaxation considered

The non-linear wave equation governing the propagation and scatter of dilatation waves in discontinuous media is presented and its Eulerian numerical analogue is used to study scatter of acoustic dilatation waves by colums and ducts in elastic and visco-elastic fluids. Plastic and visco-elastic relaxation mechanisms are considered. The spectral form of the wave equation is developed and used to discuss dispersion. Selected applications of the numerical analogue to simulation of scatter, propagation and echo attenuation presented.

個別要素法を用いたサイロ内粒状体の静置時圧力に関する研究

The inside pressure of a silo, especially the dynamic pressure during discharge is a very important factor in its structural design. However, numerical simulations of this problem have seldom been performed because of its complicated behaviors. In this report, the distinct element method (DEM) which is suitable for discontinuous media is used for analysis of the static state of granular materials in silos, and the possibility of its use to analyze the silo problem is indicated. Also, the characteristic distribution of pressures on walls are studied using the DEM simulation, and the influence of friction coefficients and particle arrangement are examined. The result of this study is that the pressure distributions from the DEM simulation are similar to the theoretical or experimental ones. This result of static analysis is used for the initial condition of dynamic analysis, and it is expected to simulate the complicated behaviors of granular materials during discharge in silos.

Hydrogeological approach to investigation in karst for possible modification of groundwater regime and increase of recoverable reserves

An artificial contribution to groundwater reserves in the areas of interest for water supply is a principal methodological target in modern hydrogeology. Investigations directed to this goal are of increasing significance all over the world to meet the growing demand for good water, which groundwater generally can be. Progress has been made in the sphere of practical development in permeable rocks of intergranular porosity, which cannot be said of discontinuous karst media, although it seems to offer greater opportunities. The ingrained notion and fear, even among specialists, of the inherent risk and uncertainty were invariably present wherever a resource was discovered in karst of a geosynclinal area; consequently progress has been limited. The reasons, however, for such a cautious approach are diminishing, because much knowledge has been gained about these aquiferous rocks, especially through investigations in the regions of surface storage reservoirs. Better knowledge of karst features and the results achieved in construction and consolidation of surface reservoirs have indicated that large amounts of groundwater can be recovered. The conventional water investigation and recovery methods have made available only small safe yields equal to the lowest natural discharge on the order of 100 I/sec). A reasonable use of a karst water resource and its better management cannot be considered without artificial control of the groundwater regime, i.e., without adjusting the regime to human demands. Groundwater flow balance in karst is becoming one of the principal problems, and future activities should be directed to the search for a bolder solution. A multidisciplinary team of geologists, geomorphologists, hydrogeologists, hydrologists, hydraulic engineers, etc., is required. In this paper a variety of solutions for water resource utilization in naked geosynclinal karst is suggested and far greater activity in this field is encouraged.

MODIFIED DISTINCT ELEMENT METHOD SIMULATION OF DYNAMIC CLIFF COLLAPSE

Usual Distinct Element Method (DEM), in which soil is represented as a system of numerous discrete particles, does not account for two factors; the continuity of the medium and wave propagation. The modified method, which is proposed in this paper, has another physical structure which presents the effect of the internal material between particles as pore water or clay. We simulated the two dimensional dynamic fracture of a cliff using this method. The fracture process is as follows; many small cracks occur widely then they form fracture lines. Results confirmed that this method can simulate the continuous and discontinuous medium.