Piecewise Homogeneous Medium Research Articles

Uniqueness in inverse transmission scattering problems for multilayered obstacles

Assume a time-harmonic electromagnetic wave is scattered by an infinitely long cylindrical conductor surrounded by an unknown piecewise homogenous medium remaining invariant along the cylinder axis. We prove that, in TM mode, the far field patterns for all incident and observation directions at a fixed frequency uniquely determine the unknown surrounding medium as well as the shape of the cylindrical conductor. A similar uniqueness result is obtained for the scattering by multilayered penetrable periodic structures in a piecewise homogeneous medium. The periodic interfaces and refractive indices can be uniquely identified from the near field data measured only above (or below) the structure for all quasi-periodic incident waves with a fixed phase-shift. The proofs are based on the singularity of the Green function to a two dimensional elliptic equation with piecewise constant leading coefficients.

Complex variables boundary element method for elasticity problems with constant body force

The direct formulation of the complex variables boundary element method is generalized to allow for solving problems with constant body forces. The hypersingular integral equation for two-dimensional piecewise homogeneous medium is presented and the numerical solution is described. The technique can be used to solve a wide variety of problems in engineering. Several examples are presented to verify the approach and to demonstrate its key features. The results of calculations performed with the proposed approach are compared with available analytical and numerical benchmark solutions.

The inverse electromagnetic scattering problem in a piecewise homogeneous medium

This paper is concerned with the problem of scattering of time-harmonic electromagnetic waves from an impenetrable obstacle in a piecewise homogeneous medium. The well-posedness of the direct problem is established, employing the integral equation method. In Liu and Zhang (2009 Appl. Anal. 88 1339–55) it was proved, under the condition that the wave numbers in the innermost and outermost homogeneous layers coincide and S0 is known in advance, that the obstacle with its physical property can be uniquely determined from knowledge of the electric far-field pattern for incident plane waves. In this paper, we will remove this restriction by establishing a new mixed reciprocity relation. Furthermore, inspired by Hähner's idea in Hähner (1993 Inverse Problems 9 667–78), we prove that the penetrable interface between layers can also be uniquely determined.

Numerical method for determining the inhomogeneity boundary in the Dirichlet problem for Laplace’s equation in a piecewise homogeneous medium

The Dirichlet problem for Laplace’s equation in a two-dimensional domain filled with a piecewise homogeneous medium is considered. The boundary of the inhomogeneity is assumed to be unknown. The inverse problem of determining the inhomogeneity boundary from additional information on the solution of the Dirichlet problem is considered. A numerical method based on the linearization of the nonlinear operator equation for the unknown boundary is proposed for solving the inverse problem. The results of numerical experiments are presented.

Comparison of different BEM+Born series modeling schemes for wave propagation in complex geologic structures

Complex geologic structures generally consist of irregular subregions with piecewise constant properties. Two different boundary-element method (BEM) plus Born-series schemes have been formulated for wave-propagation simulation in such piecewise homogeneous media by incorporating a Born series and boundary integral equations. Both schemes decompose the resulting boundary integral equation matrix into two parts: (1) the self-interaction operator, handled with a fully implicit BEM, and (2) the extrapolation operator, approximated by a Born series. The first scheme associates the self-interaction operator with each boundary itself and interprets the extrapolation operator as cross-interactions between different boundaries in a subregion. The second scheme relates the self-interaction operator toeach subregion itself and expresses the extrapolation operator as cross-interactions between different subregions in a whole model. In the second scheme, the matrix dimension of the resulting boundary integral equation is reduced by eliminating the traction field. Both numerical schemes have been validated by dimensionless frequency responses to a semicircular alluvial valley, compared with the full-waveform BEM numerical solution. We then extended these schemes to a complex fault model by calculating synthetic seismograms to evaluate approximation accuracies. Numerical experiments indicate that the [Formula: see text] series modeling schemes significantly improve computational efficiency, especially for high frequencies and with multisource seismic surveys. The tests also confirmed that the second modeling scheme has a faster convergence but may cause more noise in higher-order iterations than the first scheme.

The solution of the Cauchy integral equation with a real-definite operator

Boundary-value problems of logarithmic potential for piecewise homogeneous media are classified as problems of linear adjustment of piecewise analytical functions. The simple layer at the interface between contiguous regions satisfies the Cauchy equation with a real definite operator. For a closed K-parametrized analytical contour, this equation is equivalent to the Riemann-Gilbert boundary-value problem for a circle. The subspace of solutions on the functions of symmetry pairs of univalent functions is determined by the Cauchy formula. This paper addresses a new method for solving the boundary-value problem of stationary field adjustment, raised in geophysics applications, and is a natural extension to the earlier publication [Martyshko and Ladovskii, 2005].

Direct and Inverse Obstacle Scattering Problems in a Piecewise Homogeneous Medium

This paper is concerned with the problem of scattering of time-harmonic acoustic waves from an impenetrable obstacle in a piecewise homogeneous medium. The well-posedness of the direct problem is established, employing the integral equation method, and then used, in conjunction with the representation in a combination of layer potentials of the solution, to prove a priori estimates of solutions on some part of the interface between the layered media. The inverse problem is also considered in this paper. A uniqueness result is obtained for the first time in determining both the penetrable interface and the impenetrable obstacle with its physical property from a knowledge of the far field pattern for incident plane waves. In doing so, an important role is played by the a priori estimates of the solution for the direct problem.

Uniqueness in the inverse scattering problem in a piecewise homogeneous medium

The scattering of time-harmonic acoustic plane waves by an impenetrable obstacle in a piecewise homogeneous medium is considered. Having established the well posedness of the direct problem by the variational method, we prove a uniqueness result for the inverse problem, that is, the unique determination of the obstacle and its boundary condition from a knowledge of the far-field pattern for incident plane waves. The proof is based on a generalization of the mixed reciprocity relation.

The method of calculation of the geophysical fields in piecewise homogeneous media

Recurrent algorithms for solving 3D direct problems on the distribution of geophysical fields in the piecewise-homogeneous media are built, based on methods of integral transforms, integral representations, and integral equations. The algorithms allow the recalculation of Green’s functions of one medium into Green’s function of another medium. The computational procedure is applicable to problems with equations of elliptic, parabolic, hyperbolic, and mixed types. The algorithms can be split and implemented on computing clusters and multiprocessor complexes. Inverse problems and algorithms of searching for boundaries of piecewise-homogeneous media as extremals of the Tikhonov regulating functional, approximated by the spline functions, are formulated.

A uniqueness result for the inverse electromagnetic scattering problem in a piecewise homogeneous medium

The scattering of time-harmonic electromagnetic plane waves by an impenetrable obstacle in a piecewise homogeneous medium is considered. The well-posedness of the direct problem is proved by the variational method. Under the condition that the wave numbers in the innermost and outermost homogeneous layers coincide, we then establish a uniqueness result for the inverse problem, that is, the unique determination of the obstacle and its boundary condition from a knowledge of the electric far field pattern for incident plane waves. The proof is based on a generalization of the mixed reciprocity relation.

Method of boundary integral equations as applied to the numerical solution of the three-dimensional Dirichlet problem for the laplace equation in a piecewise homogeneous medium

A Dirichlet problem is considered in a three-dimensional domain filled with a piecewise homogeneous medium. The uniqueness of its solution is proved. A system of Fredholm boundary integral equations of the second kind is constructed using the method of surface potentials, and a system of boundary integral equations of the first kind is derived directly from Green’s identity. A technique for the numerical solution of integral equations is proposed, and results of numerical experiments are presented.

Algorithms and numerical analysis of dc fields in a piecewise-homogeneous medium by the boundary integral equation method

Boundary integral equation methods are considered for computing dc fields in three-dimensional regions filled with a piecewise-homogeneous medium. The problem is formulated and a system of Fredholm boundary integral equations of first kind is constructed, following directly from Green’s formula. The numerical solution stages are considered in detail, including construction and triangulation of the numerical surfaces, evaluation of surface integrals, and solution of a system of block-matrix equations.

Near-surface instability of composites weakly reinforced with short fibers

The paper is concerned with loss of stability near the free flat surface of a weakly short-fiber-reinforced composite material under compression along the fibers. Emphasis is on the effect of the mechanical and geometrical parameters of the composite and the distance between fibers and to the free surface on the critical strains. The piecewise-homogeneous medium model and the rigorous three-dimensional linearized theory of stability of deformable bodies are used

Plane instability problem for a composite reinforced with a periodic row of short parallel fibers

The instability of a composite material reinforced with a periodic row of parallel short fibers is studied considering the interaction of neighboring fibers. Emphasis is on the mutual influence of short fibers in the matrix during loss of stability, depending on the distance between them. A piecewise-homogeneous medium model and the three-dimensional linearized theory of stability of deformable bodies are used

The effect of cracks’ interaction for transversely isotropic layered material under compressive loading

Laminar composites due to their internal structure and manufacturing methods contain a number of inter- and intra-component defects whose size, dispersion and mutual interaction alter significantly the critical compression strain level. The current paper investigates crack interaction in transversely isotropic materials compressed in a static manner along inter-laminar defects. For laminated composites compressed along layers and, therefore, along the mentioned interfacial defects, the classical Griffith–Irvin criterion of fracture or its generalisation is inapplicable since all stresses intensity factors and crack opening displacements are equal to zero. The statement of the problem is based on the most accurate approach, the model of piecewise-homogeneous medium. The moment of stability loss in the microstructure of material is treated as the onset of the fracture process. The behaviour of each constituent is described by the three-dimensional equations of solid mechanics, provided certain boundary conditions are satisfied at the interfaces. The complex non-classical fracture mechanics problem is solved by finite elements analysis, using a linear buckling model. The results are obtained for fibre volume fraction V f = 0.1 – 0.73 for the typical dispositions of cracks. The model of classical cracks is utilised. It is found that the fibre volume fraction, the crack length and the mutual position of cracks influence the critical strain of the composite.

The effect of the interaction of cracks in orthotropic layered materials under compressive loading

The non-classical problem of fracture mechanics of composites compressed along the layers with interfacial cracks is analysed. The statement of the problem is based on the model of piecewise homogeneous medium, the most accurate within the framework of the mechanics of deformable bodies as applied to composites. The condition of plane strain state is examined. The layers are modelled by a transversally isotropic material (a matrix reinforced by continuous parallel fibres). The frictionless Hertzian contact of the crack faces is considered. The complex fracture mechanics problem is solved using the finite-element analysis. The shear mode of stability loss is studied. The results are obtained for the typical dispositions of cracks. It was found that the interacting crack faces, the crack length and the mutual position of cracks influence the critical strain in the composite.

Studying edge effects near rectangular mine workings weakened by a crack

Edge effects in a rock mass near a system of periodic mine workings crossed by a crack (fold fault) are considered. The stress state near the mine workings is determined by solving an elastic problem using the piecewise-homogeneous medium model. The original differential problem is reduced to a discrete problem based on the concept of a base scheme. Methods are proposed to solve the discrete problem, and the issue of optimizing the numerical analysis is discussed. The stress distribution around mine workings is analyzed. The stress isolines that represent the boundaries of edge effect zones and indicate their maximum decay length are plotted

Generalized hybrid Mehler-Fock-type integral transformation of the first kind and its applications

We introduce a generalized hybrid integral transformation of the Mehler-Fock type on a segment [0; R] with n conjugate points. We consider examples of application of this transformation to the solution of typical singular boundary-value problems for linear partial differential equations of the second order in piecewise-homogeneous media.

The effect of fibre volume fraction on the onset of fracture in laminar materials with an array of coplanar interface cracks

The onset of fracture process in a layered composite with coplanar interface cracks under compression along layers is studied within the scope of 3-D theory of stability using the model of piecewise-homogeneous medium. Each layer consists of a matrix reinforced by continuous parallel fibres and is modelled by a compressible, linear elastic, transversely isotropic material. The eigenvalue problem with respect to the controlled parameters, namely, the crack length, the crack spacing and the 0°-plies layer volume fraction is solved for the shear and the extensional modes utilizing the finite element analysis by commercially available FE code. The results are obtained for 0°-plies volume fraction V 0° = 0.1 … 0.9 for the typical dispositions of cracks. The effect of the crack length, the crack spacing and the 0°-plies volume fraction on the critical microbuckling strain in the composite was determined.

Plane Problem for Layered Composites with Periodic Array of Interfacial Cracks Under Compressive Static Loading

The current paper addresses the problem of 2-D modelling of the onset of failure process in a layered composite with periodic array of interfacial cracks under static compression along layers. The statement of the problem is based on the most accurate approach, the model of piecewise-homogenous medium. The condition of plane strain state is considered. The shear and the extensional buckling modes are examined. The laminae are modelled by transversally isotropic material (a matrix reinforced by continuous parallel fibres). The complex non-classical failure mechanics problem is solved utilizing finite element analysis. It is found that the 0°-plies volume fraction, the crack length and the mutual position of cracks influence the critical strain in the composite.