A simplified two-dimensional boundary element method with arbitrary uniform mean flow

To reduce computational costs, an improved form of the frequency domain boundary element method(BEM) is proposed for two-dimensional radiation and propagation acoustic problems in a subsonic uniform flow with arbitrary orientation. The boundary integral equation(BIE) representation solves the two-dimensional convected Helmholtz equation(CHE) and its fundamental solution, which must satisfy a new Sommerfeld radiation condition(SRC) in the physical space. In order to facilitate conventional formulations, the variables of the advanced form are expressed only in terms of the acoustic pressure as well as its normal and tangential derivatives, and their multiplication operators are based on the convected Green's kernel and its modified derivative. The proposed approach significantly reduces the CPU times of classical computational codes for modeling acoustic domains with arbitrary mean flow. It is validated by a comparison with the analytical solutions for the sound radiation problems of monopole,dipole and quadrupole sources in the presence of a subsonic uniform flow with arbitrary orientation.     

From imaging to material identification: A generalized concept of topological sensitivity

To establish a compact analytical framework for the preliminary stress-wave identification of material defects, the focus of this study is an extension of the concept of topological derivative, rooted in elastostatics and the idea of cavity nucleation, to 3D elastodynamics involving germination of solid obstacles. The main result of the proposed generalization is an expression for topological sensitivity, explicit in terms of the elastodynamic Green's function, obtained by an asymptotic expansion of a misfit-type cost functional with respect to the nucleation of a dissimilar elastic inclusion in a defect-free “reference” solid. The featured formula, consisting of an inertial-contrast monopole term and an elasticity-contrast dipole term, is shown to be applicable to a variety of reference solids (semi-infinite and infinite domains with constant or functionally graded elastic properties) for which the Green's functions are available. To deal with situations when the latter is not the case (e.g. finite reference bodies or those with pre-existing defects), an adjoint field approach is employed to derive an alternative expression for topological sensitivity that involves the contraction of two (numerically computed) elastodynamic states. A set of numerical results is included to demonstrate the potential of generalized topological derivative as an efficient tool for exposing not only the geometry, but also material characteristics of subsurface material defects through a local, point-wise identification of “optimal” inclusion properties that minimize the topological sensitivity at sampling location. Beyond the realm of non-invasive characterization of engineered materials, the proposed developments may be relevant to medical diagnosis and in particular to breast cancer detection where focused ultrasound waves show a promise of superseding manual palpation.     

Elastodynamics and Elastostatics by a Unified Method of Potentials for x 3-Convex Domains

A new general solution in terms of two scalar potential functions for classical elastodynamics of x ₃-convex domains is presented. Through the establishment and usage of a set of basic mathematical lemmas, a demonstration of its connection to Kovalevshi–Iacovache–Somigliana elastodynamic solution, and thus its completeness, is realized with the aid of the theory of repeated wave equations and Boggio’s theorem. With the time dependence of the potentials suppressed, the new decomposition can, unlike Lamé’s, degenerate to a complete solution for elastostatic problems.      

Energy identities in water wave theory for free-surface boundary condition with higher-order derivatives

A modified form of Green's integral theorem is employed to derive the energy identity in any water wave diffraction problem in a single-layer fluid for free-surface boundary condition with higher-order derivatives. For a two-layer fluid with free-surface boundary condition involving higher-order derivatives, two forms of energy identities involving transmission and reflection coefficients for any wave diffraction problem are also derived here by the same method. Based on this modified Green's theorem, hydrodynamic relations such as the energy-conservation principle and modified Haskind-Hanaoka relation are derived for radiation and diffraction problems in a single as well as two-layer fluid.     

Stress distribution in multilayered composites near loaded edges

The edge effect in layered composite material is studied using the piecewise-homogeneous body model and the exact equations of the theory of elasticity. It is assumed that continuously distributed normal forces act at the edges of the reinforcing layers. A plain strain state is considered and the stresses are expressed in terms of the solutions of a system of dual singular integral equations. The singularity of the stresses is determined by the solution procedure. The concentration of the reinforcing layers is assumed low and the interaction between them is not taken into account. A numerical algorithm is developed and numerical results on the stress distribution are presented Published in Prikladnaya Mekhanika, Vol. 44, No. 4, pp. 134–144, April 2008.     

A new look at 2-D time-domain elastodynamic Green''s functions for general anisotropic solids

2-D time-domain elastodynamic displacement Green's functions for general anisotropic solids are obtained by a new method. This method is based on the use of a cosine transform with respect to time and exponential Fourier transforms with respect to both spatial coordinates. By use of a change of variables and the homogeneity and symmetry of the problem, the inverse transforms are reduced to an integral which can be evaluated by a simple use of redidue calculus. The solutions are expressed in terms of three wave fields. The field inside a wavefront corresponds to a complex root of a polynomial of order six with real coefficients. A simple relation between the spatial and time derivatives is found, and is used to reduce the corresponding stresses to a form that is directly applicable to the boundary element method. Numerical implementations are explained in some detail and are demonstrated by three examples.     

Transmission in a rectangular waveguide loaded with an arbitrary number of capacitive screens

Summary Green's theorem is applied to individual domains consisting of the walls of a rectangular waveguide and two neighbouring slotted capacitive screens. In addition this theorem is applied to the two end domains, that is, the one containing the incident field and the other containing the transmitted field only. Using the continuity of the electromagnetic field intensities in the slits, a system of simultaneous integral equations is obtained in terms of field intensity functions within the slits. By a change of variables a system of ² linear algebraic equations is derived, and formulae for the principal mode transmission and reflection coefficients are given to any approximation desired. Conditions for a full transmission are derived, especially to a first order approximation. Some applications with regard to the filter properties of the waveguide system are discussed and the theory developed is compared with measurements.     

Theoretical analysis of generalized loadings and image forces in a planar magnetoelectroelastic layered half-plane

The problem of a planar transversely isotropic magnetoelectroelastic layered half-plane subjected to generalized line forces and edge dislocations is analyzed. The complete solutions consist only of the simplest solutions for an infinite magnetoelectroelastic medium with applied loadings. The physical meaning of this solution is the image method. It is shown that the explicit solutions include Green's function for originally applied singularities in an infinite medium and the other image singularities are induced to satisfy free surface and interface continuity conditions. The mathematical method used in this study provides an automatic determination for the locations and magnitudes of all image singularities. The locations and magnitudes of image singularities are dependent on the roots of the characteristic equation which is related to the material constants of the layered half-plane. With the aid of the generalized Peach-Koehler formula, the explicit expressions of image forces acting on dislocations are easily derived from the full-field solutions of the generalized stresses. Numerical results for the full-field distributions of stresses, electric fields, and magnetic fields in the layered half-plane medium are presented based on the analytical solutions. The image forces and equilibrium positions of one dislocation, two dislocations, and an array of dislocations are presented by numerical calculations and are discussed in detail.     

Fixed Points of Multi-valued Maps and Static Coulomb Friction Problems

The existence of solutions to the Signorini problem with Coulomb friction is a long standing open question. We prove the existence of generalized solutions that satisfy the pointwise Coulomb friction conditions on the entire interface and the normal nonpenetration condition on the complement of a subset with arbitrarily small but possibly positive measure. Furthermore, the penetration itself can also be made arbitrarily small. Although “measure zero” instead of “arbitrarily small measure” would be needed to fully resolve the issue, these generalized solutions seem to be the closest answer available to date. Their existence is proved by a suitable application of Ky Fan's fixed point theorem for multi-valued maps. The same method can be used with a number of variants involving contact of two or more elastic bodies and possible debonding phenomena. This revised version was published online in August 2006 with corrections to the Cover Date.     

Scattering of water waves by thin vertical plate submerged below ice-cover surface

The present paper is concerned with scattering of water waves from a vertical plate, modeled as an elastic plate, submerged in deep water covered with a thin uniform sheet of ice. The problem is formulated in terms of a hypersingular integral equation by a suitable application of Green's integral theorem in terms of difference of potential functions across the barrier. This integral equation is solved by a collocation method using a finite series involving Chebyshev polynomials. Reflection and transmission coefficients are obtained numerically and presented graphically for various values of the wave number and ice-cover parameter.     

Impulsive motion of a flat plate in a Rivlin-Ericksen fluid

The problem of the impulsive motion of a flat plate in a Rivlin-Ericksen fluid is reconsidered. An exact solution for the velocity distribution is found in terms of a definite integral. It is shown that a solution obtained earlier does not satisfy the boundary condition at the plate in the sense of generalized functions.     

Features of the stress field near tunnel inclusions in an inhomogeneous anisotropic space

This paper proposes a method to solve problems for interface tunnel defects in a piecewise-homogeneous elastic material that is under generalized plane strain and has no planes of elastic symmetry. The method is based on integral relations between the discontinuities and sums of the components of the displacement vector and stress tensor at the interface. Closed-form solutions are obtained for a system of interface tunnel inclusions with mixed contact conditions between the space and the inclusions. The dependences of the indices of singularity of the solutions on orthogonal coordinate transformation are established for different combinations of materials of monoclinic and orthorhombic systems. The effect of the antiplane component on the behavior of the solutions is revealed __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 36–45, June 2008.     

A Green's function approach to the deflection of a FGM plate under transient thermal loading

Summary  Green's function approach is adopted for analyzing the deflection and the transient temperature distribution of a plate made of functionally graded materials (FGMs). The governing equations for the deflection and the transient temperature are formulated into eigenvalue problems by using the eigenfunction expansion theory. Green's functions for solving the deflection and the transient temperature are obtained by using the Galerkin method and the laminate theory, respectively. The eigenfunctions of Green's function for the deflection are approximated in terms of a series of admissible functions that satisfy the homogeneous boundary conditions of the plate. The eigenfunctions of Green's function for the temperature are determined from the continuity conditions of the temperature and the heat flux at interfaces. Received 9 October 2000; accepted for publication 3 April 2001     

Recent advances on the fast multipole accelerated boundary element method for 3D time-harmonic elastodynamics

This article is mainly devoted to a review on fast BEMs for elastodynamics, with particular attention on time-harmonic fast multipole methods (FMMs). It also includes original results that complete a very recent study on the FMM for elastodynamic problems in semi-infinite media. The main concepts underlying fast elastodynamic BEMs and the kernel-dependent elastodynamic FM-BEM based on the diagonal-form kernel decomposition are reviewed. An elastodynamic FM-BEM based on the half-space Green’s tensor suitable for semi-infinite media, and in particular on the fast evaluation of the corresponding governing double-layer integral operator involved in the BIE formulation of wave scattering by underground cavities, is then presented. Results on numerical tests for the multipole evaluation of the half-space traction Green’s tensor and the FMM treatment of a sample 3D problem involving wave scattering by an underground cavity demonstrate the accuracy of the proposed approach. The article concludes with a discussion of several topics open to further investigation, with relevant published work surveyed in the process.     

Complete eigenfunction expansion form of the Green's function for elastic layered half-space

Summary  The complete eigenfunction expansion form of the Green's function for a 3-D elastic layered half-space in the frequency domain is derived in this paper. The expression of the Green's function presented here is an extension of that represented by the residue terms and the branch line integrals given by Lamb [1]. The present expression, however, clarifies the mathematical common frame between the residue terms and the branch line integrals with respect to the eigenfunctions and energy integrals. For the derivation, the concept of an energy integral for the improper eigenfunctions is newly developed. The improper eigenfunctions, which can be found in the wavenumbers for the branch cuts, are not in L ₂ space, so the definition of the energy integral requires some treatment. The energy integral is defined as the limit of the inner product of the improper eigenfunction and the definition function of the improper eigenfunction, for which the inner product remains finite. Via the definition of the energy integral, the kernel of the branch line integral is decomposed into the improper eigenfunction, and the complete eigenfunction expansion form of the Green's function is derived. The Green's function can thus be expressed by summation of normal modes, complex modes pointed out in [2], the integral of the improper eigenfunction and the residue at k=0 due to the singularity of the horizontal wavefunction. Received 3 May 2001; accepted for publication 23 August 2001     

Resolution of linear viscoelastic equations in the frequency domain using real Helmholtz boundary integral equations

Boundary integral equations are well suitable for the analysis of seismic waves propagation in unbounded domains. Formulations in elastodynamics are well developed. In contrast, for the dynamic analysis of viscoelastic media, there are very seldom formulations by boundary integral equations. In this Note, we propose a new and simple formulation of time harmonic viscoelasticity with the Zener model, which reduces to classical elastodynamics if a compatibility condition is satisfied by boundary conditions. Intermediate variables which satisfy the classical elastodynamic equations are introduced. It makes it possible to utilize existing numerical tools of time harmonic elastodynamics. To cite this article: S. Chaillat, H.D. Bui, C. R. Mecanique 335 (2007).     

HALF-SPACE GREENS'S FUNCTION DUE TO A SPATIALLY HARMONIC LINE LOAD AND SCATTERED FIELDS IN THE FAR FIELD REGION

Half-space Green's function due to a spatially harmonic line load has been expressed asa sum of the full-space Green's functions and a 2-D integral representation of the reflected waves bythe free surface of the half-space.By using the obtained half-space Green's function,an integral rep-resentation of the scattered waves by a cylindrical obstacle is then derived.Finally,by analyzing thefar-zone behavior of the integrands of the integral representation.the far-field pattern of the scatteredwaves in a half-space obtained.     

Selection of kernel functions used in a new constitutive equation for viscoelastic fluids

A selection of kernel functions is given to be used in a new integral constitutive equation proposed by Piau whereby the deviatoric stress is calculated from the integral of the history of the past intrinsic rate of rotation and rate of deformation tensors through a representation theorem. Piau has demonstrated the objectivity of a frame moving with a given particle whose axis are directed along the eigenvectors of the rate of deformation tensor. The use of such a framework provides a new approach in the attempt to reduce the computational difficulties associated with conventional constitutive equations written in co-deformational or co-rotational reference frames.The shear and primary normal-stress material functions and the extensional (elongational) stress growth function are defined for the proposed integral constitutive equation. These material functions are used to calculate the kernel functions using steady state, stress relaxation and stress growth data of Attané in simple shear flow for monodisperse polystyrene solutions. The shear and extensional stress growth data of Meissner for a polyethylene melt are also used to show the flexibility of the rheological model.The material functions are first written in terms of five monotonically decreasing functions of the time lag between the past and the present time. Then kernel functions are chosen such that when substituted in the new integral constitutive equation they yield the functions used to describe the data. A further condition imposed on the normalized kernel functions is that they be decreasing functions of time lag.     

Lamb's integral formulas of two-phase saturated medium for soil dynamic with drainage

When dynamic force is applied to a saturated porous soil,drainage is common.In this paper,the saturated porous soil with a two-phase saturated medium is simulated,and Lamb's integral formulas with drainage and stress formulas for a two-phase saturated medium are given based on Biot's equation and Betti's theorem(the reciprocal theorem).According to the basic solution to Biot's equation,Green's function Gij and three terms of Green's function G4i,Gi4,and G44 of a two-phase saturated medium subject to a concentrated force on a spherical coordinate are presented.The displacement field with drainage,the magnitude of drainage,and the pore pressure of the center explosion source are obtained in computation.The results of the classical Sharpe's solutions and the solutions of the two-phase saturated medium that decays to a single-phase medium are compared.Good agreement is observed.     

Steady Free-surface Flow Past an Uneven Channel Bottom

New free-surface flows past a semi-infinite ‘step’ in the bottom of a channel are considered. Surface tension is neglected but gravity is included in the dynamic boundary condition. Fully nonlinear solutions are computed by boundary integral equation methods. Additional weakly nonlinear solutions are derived analytically. A thorough analysis of the weakly nonlinear problem provides a systematic approach to identify all the possible types of solutions and the number of independent parameters.