Inverse problem for wave propagation in a perturbed layered half-space

This paper is concerned with the inverse medium scattering problem in a perturbed, layered, half-space, which is a problem related to the seismologial investigation of inclusions inside the earth’s crust. A wave penetrable object is located in a layer where the refraction index is different from the other part of the half-space. Wave propagation in such a layered half-space is different from that in a homogeneous half-space. In a layered half-space, a scattered wave consists of a free wave and a guided wave. In many cases, only the free-wave far-field or only the guided-wave far-field can be measured.We establish mathematical formulas for relations between the object, the incident wave and the scattered wave. In the ideal condition where exact data are given, we prove the uniqueness of the inverse problem. A numerical example is presented for the reconstruction of a penetrable object from simulated noise data.     

Transient response of SH waves in a layered half-space with sub-surface and interface cracks

The transient scattering of SH waves by sub-surface and interface cracks parallel to the free surface in a layered elastic solid is investigated. The problem in frequency domain is solved by using a hybrid method which combines the finite element method of the near field with the boundary integral representation of the far field. The transient responses are then obtained by inverting the spectra via fast Fourier transform with the incident pulse Ricker of wavelet. Numerical results are presented for the surface displacements, dynamic stress intensity factors and wave motion in the layered half-space. Furthermore, the propagations of reflected, diffracted, and direct impact waves at any instant are clearly identified by the present method. To understand the mechanism of elastic wave interaction is very important in the field of ultrasonic non-destructive evaluation (NDE) and fracture mechanics studies.     

Far-field directivity of parametric loudspeaker arrays set on curved surfaces

Parametric loudspeaker arrays comprise arrangements of piezoelectric ultrasonic transducers that emit highly directive audible sound, thanks to the nonlinear parametric array phenomenon. Most parametric loudspeakers consist of planar arrays of transducers though, lately, devices have been developed which involve their distribution on curved surfaces. In this work, we present an extension of the recently proposed convolution model for planar arrays to predict the far field directivity of curved parametric loudspeakers. An expression is first given to compute the audible secondary pressure field generated by a single ultrasonic transducer placed at any point on a general curved surface, and pointing in its normal direction. Assuming weak non-linearity, the total audible pressure produced by all transducers on the surface is then recovered from the superposition principle. As an application, we predict the far-field pressure generated by an omnidirectional parametric loudspeaker consisting of hundreds of ultrasonic transducers set on a sphere. A critical aspect for the performance of the omnidirectional source is that of finding a proper distribution for them on the spherical surface. Two options are analyzed: getting an optimal solution for a Fekete-like problem, and resorting to an equal-area partitioning scheme, which is more feasible for a practical construction of the source. Numerical simulations are carried out for both alternatives.     

Mixed Antiplane Problem of Electroelasticity for a Piezoceramic Half-Space with a Tunnel Hole

Harmonic vibrations of a piezoceramic half-space weakened by a tunnel hole with a system of active surface electrodes are investigated. Based on integral representations for solutions to two types of conditions on the boundary of the half-space, the boundary-value problem of electroelasticity is reduced to a system of singular integrodifferential equations of the second kind. The results of parametric investigations characterizing the behavior of the electroelastic field components on the boundary and in the region of a piecewise-homogeneous half-space are presented.     

Vibration of a stamp partially adhering to an elastic medium : PMM, vol. 42, no. 6, 1978, pp. 1085–1092

There is examined the problem of vibration of a stamp of arbitrary planform occupying a space Ω and vibrating harmonically in an elastic medium with plane boundaries. It is assumed that the elastic medium is a packet of layers with parallel boundaries, at rest in the stiff or elastic half-space. Contact of three kinds is realized under the stamp: rigid adhesion in the domain Ω₁, friction-free contact in domain Ω₂, there are no tangential contact stresses, and “film” contact without normal force in domain Ω₃ (there are no normal contact stresses, only tangential stresses are present.). It is assumed that the boundaries of all the domains have twice continuously differentiable curvature and Ω = Ω₁ Ω₂ Ω₃.

The problem under consideration assumes the presence of a static load pressing the stamp to the layer and hindering the formation of a separation zone. Moreover, a dynamic load, harmonic in time, acts on the stamp causing dynamical stresses which are of the greatest interest since the solution of the static problem is obtained as a particular case of the dynamic problem for ω = 0 (ω is the frequency of vibration). The general solution is constructed in the form of a sum of static and dynamic solutions.

A uniqueness theorem is established for the integral equation of the problem mentioned and for the case of axisymmetric vibration of a circular stamp partially coupled rigidly to the layer, partially making friction-free contact, the problem is reduced to an effectively solvable system of integral equations of the second kind, which reduce easily to a Fredholm system.

These results are an extension of the method elucidated in [1], where by the approach in [1] must be altered qualitatively to obtain them.     

On the far-field operator for electromagnetic scattering in chiral media

We consider time-harmonic electromagnetic waves propagating in a homogeneous three-dimensional unbounded chiral medium where a perfect conductor has been immersed. Assuming that the incident electric field is a superposition of plane incident electric waves, the corresponding scattered field and the far-field pattern are expressed as the superposition of the scattered fields and the far-field patterns respectively. It is also proved that the sets of far-field patterns are complete if and only if there does not exist an eigenfunction to the interior perfect conductor problem that vanishes on the boundary of the scatterer which is an electric Herglotz field. The Left-Circularly Polarized and the Right-Circularly Polarized far-field operators are defined and studied and using them the electric far-field operator is defined too. The properties of the above operators and Herglotz functions are related to the solution of the interior perfect conductor boundary value problem.     

Large deformation shape uncertainty quantification in acoustic scattering

We address shape uncertainty quantification for the two-dimensional Helmholtz transmission problem, where the shape of the scatterer is the only source of uncertainty. In the framework of the so-called deterministic approach, we provide a high-dimensional parametrization for the interface. Each domain configuration is mapped to a nominal configuration, obtaining a problem on a fixed domain with stochastic coefficients. To compute surrogate models and statistics of quantities of interest, we apply an adaptive, anisotropic Smolyak algorithm, which allows to attain high convergence rates that are independent of the number of dimensions activated in the parameter space. We also develop a regularity theory with respect to the spatial variable, with norm bounds that are independent of the parametric dimension. The techniques and theory presented in this paper can be easily generalized to any elliptic problem on a stochastic domain.     

A heuristic approach to hard constrained shortest path problems

Constrained shortest path problems have many applications in areas like network routing, investments planning and project evaluation as well as in some classical combinatorial problems with high duality gaps where even obtaining feasible solutions is a difficult task in general.We present in this paper a systematic method for obtaining good feasible solutions to hard (doubly constrained) shortest path problems. The algorithm is based essentially on the concept of efficient solutions which can be obtained via parametric shortest path calculations. The computational results obtained show that the approach proposed here leads to optimal or very good near optimal solutions for all the problems studied.From a theoretical point of view, the most important contribution of the paper is the statement of a pseudopolynomial algorithm for generating the efficient solutions and, more generally, for solving the parametric shortest path problem.     

A modified scaled boundary finite element method for dynamic response of a discontinuous layered half-space

In this paper, a modified scaled boundary finite element method is proposed to deal with the dynamic analysis of a discontinuous layered half-space. In order to describe the geometry of discontinuous layered half-space exactly, splicing lines, rather than a point, are chosen as the scaling center. Based on the modified scaled boundary transformation of the geometry, the Galerkin's weighted residual technique is applied to obtain the corresponding scaled boundary finite element equations in displacement. Then a modified version of dimensionless frequency is defined, and the governing first-order partial differential equations in dynamic stiffness with respect to the excitation frequency are obtained. The global stiffness is obtained by adding the dynamic stiffness of the interior domain calculated by a standard finite element method, and the dynamic stiffness of far field is calculated by the proposed method. The comparison of two existing solutions for a horizontal layered half-space confirms the accuracy and efficiency of the proposed approach. Finally, the dynamic response of a discontinuous layered half-space due to vertical uniform strip loadings is investigated.     

Parametric Vibrations of a Three-Layer Conical Piezoshell

Based on the Kirchhoff-Love hypotheses and adequate supplementary hypotheses for the distribution of electric field quantities, a model for parametric vibrations of composite shells of revolution made of a passive (without a piezoeffect) middle layer and two active (with a piezoeffect) surface layers under the action of harmonic mechanical and electric loads is developed. The dissipative material properties are taken into account by linear viscoelastic models. Since the vibrations on the boundary of the main domain of dynamic instability (MDDI) are harmonic, the investigation of this domain, in a first approximation, is reduced to generalized eigenvalue problems, which are solved by the finite-element method. The problem on parametric vibrations of a three-layer conical shell under harmonic mechanical loading is considered. The influence of the shell thickness, dissipation, and electric boundary conditions on the MDDI is investigated. Two limiting cases of electric boundary conditions are considered, where the electrodes are short-circuited or not. The curves presented show a considerable influence of the electric boundary conditions on the characteristics of the MDDI, namely on its width and position on the frequency axis and on the critical parameter of excitation.     

Diffraction by an Acoustic Grating Perturbed by a Bounded Obstacle

An original approach to solve 2D time harmonic diffraction problems involving locally perturbed gratings is proposed. The propagation medium is composed of a periodically stratified half-space and a homogeneous half-space containing a bounded obstacle. Using Fourier and Floquet transforms and integral representations, the diffraction problem is formulated as a coupled problem of Fredholm type with two unknowns: the trace of the diffracted field on the interface separating the two half-spaces on one hand, and the restriction of the diffracted field to a bounded domain surrounding the obstacle, on the other hand.     

A coupled far-field formulation for time-periodic numerical problems in fluid dynamics

Consider uniform flow past an oscillating body generating a time-periodic motion in an exterior domain, modelled by a numerical fluid dynamics solver in the near field around the body. A far-field formulation, based on the Oseen equations, is presented for coupling onto this domain thereby enabling the whole space to be modelled. In particular, examples for formulations by boundary elements and infinite elements are described.     

Un modèle paraxial de propagation de la lumière : problème aux limites pour l'équation d'advection Schrödinger en coordonnées obliques

We study the Schrödinger equation which comes from the paraxial approximation of the Helmholtz equation in the case where the direction of propagation is tilted with respect to the boundary of the domain. Our primary interest is in the boundary conditions successively in a half-space, then in a quadrant of $\text{R}{}^2$. We also sketch a numerical method for this problem. To cite this article: M. Doumic et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Uniqueness and stability result for Cauchy’s equation of motion for a certain class of hyperelastic materials

We consider Cauchy’s equation of motion for hyperelastic materials. The solution of this nonlinear initial-boundary value problem is the vector field which discribes the displacement which a particle of this material perceives when exposed to stress and external forces. This equation is of greatest relevance when investigating the behavior of elastic, anisotropic composites and for the detection of defects in such materials from boundary measurements. This is why results on unique solvability and continuous dependence from the initial values are of large interest in materials’ research and structural health monitoring. In this article we present such a result, provided that reasonable smoothness assumptions for the displacement field and the boundary of the domain are satisfied for a certain class of hyperelastic materials where the first Piola–Kirchhoff tensor is written as a conic combination of finitely many, given tensors.     

Possibilities and Challenges in Kinematic Modeling of Highly Redundant,Non-Holonomic and Omnidirectional Mobile Robots

Industry as well as the private domain show an increasing interest to the field of mobile robotics. With a higher number of applications, the complexity of the required tasks rises, and therefore the community tends to use robots with many degrees of freedom (DOF). The present paper takes this topic up and focuses on challenges of the kinematical modeling of redundant, non-holonomic mobile robots by considering a 12 DOF platform. To reach an omnidirectional behaviour, the actuated wheels are diagonally mounted on the chassis. Well known problems resulting from this set-up are parametric singularities which unnecessarily restrict the motion of the mathematical model. As it turns out, this problem can be avoided by the use of a non-minimal parametrized model. An additional challenge results from the inverse kinematic (IK) problem on velocity level. The corresponding equations are highly underdetermined and, due to the non-holonomic wheels, not directly affected by the steering angle velocities. This problem is solved by performing a local optimization on acceleration level. Finally, simulation results are presented. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comportamiento dinámico de viaductos cortos considerando la interacción vehículo-vía-estructura-suelo

The dynamic vehicle-track-bridge-soil interaction is studied on high speed lines. The analysis is carried out using a general and fully three dimensional multi-body-finite element-boundary element model, formulated in the time domain to predict vibrations due to the train passage over the bridge. The vehicle is modelled as a multi-body system, the track and the bridge are modelled using finite elements and the soil is considered as a homogeneous half-space by the boundary element method. Usually, moving force model and moving mass model are employed to study the dynamic response of bridges. In this work, the multi-body system allows one to consider the quasi-static and dynamic excitation mechanisms. Soil-structure interaction is taken into account on the dynamic structure behaviour on simply-supported short span bridges. The influence of soil-structure interaction is analysed in both resonant and non-resonant regimes.     

3D dynamic Green’s functions in a multilayered poroelastic half-space

The complete 3D dynamic Green’s functions in the multilayered poroelastic media are presented in this study. A method of potentials in cylindrical coordinate system is applied first to decouple the Biot’s wave equations into four scalar Helmholtz equations, and then, general solutions to 3D wave propagation problems are obtained. After that, a three vector base and the propagator matrix method are introduced to treat 3D wave propagation problems in the stratified poroelastic half-space disturbed by buried sources. It is known that the original propagator algorithm has the loss-of-precision problem when the waves become evanescent. At present, an orthogonalization procedure is inserted into the matrix propagation loop to avoid the numerical difficulty of the original propagator algorithm. At last, the validity of the present approach for accurate and efficient calculating 3D dynamic Green’s functions of a multilayered poroelastic half-space is confirmed by comparing the numerical results with the known exact analytical solutions of a uniform poroelastic half-space.     

Domain sensitivity analysis of the elastic far-field patterns in scattering from nonsmooth obstacles

We consider elastic scattering problems described by the Dirichlet or the Neumann boundary value problem for the elastodynamic equation in the exterior of a 2D bounded domain or in the exterior of a crack. The boundary of the domain is assumed to have a finite set of corner points where the scattered wave may have singular behaviour. The paper is concerned with the sensitivity of the far scattered field with respect to small perturbations of the shape of the scatterer. Using a modification of the method of adjoint problems (K. Dems, Z. Mróz, Internat. J. Solids Structures 20 (1984) 527-552) we obtain a representation for the shape derivative which is well suited for a numerical realization with boundary element methods and which shows in some cases directly the influence of the singularities of the solution on the sensitivity of the far-field patterns.     

The mathematical modeling of the electric field in the media with anisotropic objects

We present a numerical scheme for modeling the electric field in the media with tensor conductivity. This scheme is based on vector finite element method in frequency domain. The numerical computations of the electric field in the anisotropic medium are done. The conductivity of the anisotropic medium is positive defined dense tensor in general case. We consider the electric field from anisotropic layer, inclined anisotropic layer and some anisotropic objects in isotropic half-space.